“Dynamic programming” is not referring to “computer programming” | Hacker News Reader

“Dynamic programming” is not referring to “computer programming”

vidarholen.net

422 pointsr4um211 comments

Sri Lankan here. I used to compete in the IOI (International Olympiad in Informatics), back in the '90s.

I aced the competition in Sri Lanka, but I failed to win a medal the first time I competed internationally. I solved several problems with recursion, but these failed to complete within the strict time limits allowed.

One of the contestants from another country told me: "You needed dynamic programming to solve problems 1 and 5." I then spent the next year trying to figure out what dynamic programming was. The folks over here weren't familiar with the term. In fact, I was often asked "did you mean dynamic memory allocation?"

After a while, I managed to find a book that talked about dynamic programming, and I was like "Oh, it's recursion + storing results" and "Ah, you can also do this iteratively in certain circumstances"

Bottom line, armed with this new knowledge, I ended up winning a gold medal at IOI 2001. Fun times.

I had the same experience: no one knew of my mentors what "dynamic programming" was and our country-level competition (that had created problems inspired by IOI) required dynamic programming for 2 out of 5 problems. And of course I failed the first time (2004). Then I learned what it was about and aced the next time (2005).

I was part of the hosting team at IOI 1997 in Cape Town and had similar conversations with the participants that you described and also learned about Dynamic Programming there. Your description of it as "recursion + storing results which you can sometimes do iteratively" is the best summary. Good example is iterative Fibonacci algorithm.

Another fun anecdote from that time is that we had to take special precautions with the scoring program which connected to the participants machine over the serial port. The previous year one participant had hacked the serial port interface to manipulate the results. :-)

Nice! I guess you must know Bruce Merry. Once, when practicing, I kept on hitting my head on a problem. On the spur of the moment, I decided to fire off an email to Bruce (given his track record, I was like "if anyone can figure this out, he can"). I was shocked (and very pleased) to get a reply a few days later outlining his thought process and how he went around solving the problem.

I used to know the ZA team leaders as well (after competing, I was a team leader for several more years).

Yes, I do indeed know Bruce. He won the national Maths and Computer olympiads in my final year of school even though he was 4 grades below me and had just started high school. :-D

The term always made me insufficient as I always assumed it meant much more and I just didn’t understand how.

In the Fibonacci example, it seems more like common sense to avoid recomputing expensive things. Instead we should give derogatory names to the inefficient implementations, not an opaque name to the only sane implementation.

Has there been an "arms race" of sorts with programming contests, as knowledge percolates and harder categories of problems need to be chosen? Since these days dynamic programming is basically table stakes for ICPC-type problems.

Yes, absolutely. I did programming competitions back in high-school (around 10 years ago) and common folklore was that back in the days knowing dynamic programming could win you a medal, but today it was just basic expected knowledge.

There's a lot of variety in DP. Knowing about DP doesn't help much with solving DP problems. I'm sure you've seen all the fun problems on codeforces or leetcode hards. The one quote I remember from Erik Demaine's lecture on DP is where he comments, "How do we solve this with DP ? - With difficulty."

That's my impression as well.

I think it's because access to knowledge has become a lot easier, so contestants know more; plus, the competitions themselves have become more organized.

For example, last I checked, the IOI had a syllabus outlining the various algorithms contestants needed to know. During my time, it was more of a wild west.

That's how Competition Makes All Great Again. Looks like it's by God's design. ^_^

Loved your wonderful story—here’s my plain one.

I came to programming after I graduated, and I never really understood Dynamic Programming until I watched one of Stanford’s algorithm courses (kudos to MOOCs).

But honestly, I’ve never done anything truly useful with it—except that it gave me a new perspective on ordinary things. For example, I found that there may be great wisdom in the Taiji symbol (image here: https://commons.wikimedia.org/wiki/File:Yin_yang.svg). We can divide the current problem and conquer it at the lower levels (like one of the dots in yin and yang). And of course, those lower levels still contain their own divisions and computations—until there are none left.

At the same time, we can also start from the base and build upward (bottom-up), moving toward higher-level computations—where possible divisions await, until eventually one big unified answer emerges.

Pretty useless for its actual usage here, isn’t it? ^^

Dynamic programming is for me one of the things that really scratched an itch and made me start liking programming. The invariant/inductions of composing smaller things into a bigger solution is so elegant.

I do feel dynamic programming influences how I solve problems in my work. How, if you compose the problem correctly, it can never reach a fault state in a sense. Not sure if I'm able to explain what I mean, but for instance I recently refactored an old application at work that had lots of bugs, was hard to grasp and had loads of error checking code / asserts. However, by composing it a bit differently, modeling it properly, I could remove swaths of code / defensive programming because I with the help of the compilator now can guarantee the base cases holds etc.

Edit: one a bit contrived example might be an Order. If you have one big model tracking the lifetime of an order, some fields will have to be null until they have a value. For instance a field sent_at_time. Then you have a function that generates an email to the customers, which uses that field. How do you handle that field possibly being null? Do you say "I know that at this point in time, this field is always set since the email is generated after sending" and tell the compiler to ignore the possibly null value? Or do you handle the possible null with some error handling / fallback value, that in practice is dead and cluttered code and will just confuse a future developer seeing it with questions like "how can this happen"?

With the lessons from dynamic programming in mind, I think both are bad solutions. You want to be able to go from state n to n+1 with safety (order purchased, ordered sent, email sent). So model it in a way that the email sending code only ever can get an order in the correct previous state as input. Then if the assumption were to break in the future (maybe some orders are digital and the email is sent without the order having been physically sent), that breaking assumption will immediately be obvious in how the state/models are composed, and not in runtime when either the not-null-assert fails or the emails start having the fallback value.

Maybe "Making illegal state unrepresentable" is the term you're looking for?

https://fsharpforfunandprofit.com/posts/designing-with-types... https://inside.java/2024/06/03/dop-v1-1-illegal-states/

IMO it's doesn't have much relation with dynamic programming though.

It was mainly just one example of the thought pattern that I feel translates. In my head it has a relation. In dynamic programming, you compose a solution of smaller, but still valid solutions. When calculating fib(8), you can trust that combining fib(7) and fib(6) is valid, and so on down to the base case. If you apply that concept to other parts of programming, it's much easier to reason about.

I wonder if you can clear up a memory of mine from IOI 2001. The first day results were delayed, and I seem to remember this is because a contestant encoded the search in one of the problems into a compile time c++ template metaprogram. On the second day, there was then also a compile time limit for solutions, from what I remember. Do you remember any more details of this?

That sounds very smart :D

But aren't the programs given access to the problem data _after_ the program has been compiled?

There are plenty of opportunities to precompute _everything_ for a certain class of problems.

Sure, but the input might be bounded/finite, or the operations needed similarly constrained (e.g. trigonometry operations). Then you can offload lots of the computation to the compilation, sometimes all of it.

Yeah, but they didn't precompute the solution for that specific program data, they precomputed the solution for all possible ones, and then selected the correct one when the program data was provided.

No, sorry. I vaguely remember compile time limits, but they were high enough (30 seconds, I think?) that I didn't bother worrying about them (at least, that's my memory).

I think I know the culprit. The story I was told is that the person precomputed some of the results and submitted a huge source file (in tens of megabytes) of hard coded data. This probably led to subsequent amendment of the rules regarding source file sizes.

I'll report back once I find the persons involved.

Wasn't this for one of the questions on the second day? Where you had to submit a series of outputs, but these were trivial to generate once you cracked the problem.

I remember getting an "aha" moment, writing a program, and then submitting (scored 100% too!). Then, I met a guy who also cracked the problem and realized that he just needed to paste the test data into a spreadsheet, do some basic sorting there, and then paste the output into a text file; no coding necessary.

I felt pretty stupid afterwards.

I think you're referring to "double crypt" of the day 2 problem set. For this problem you submit the result/answer instead of the code, so it shouldn't trigger a compilation/source code limit.

His comment when I showed him this thread: "That was my ah-ha moment of recognizing the beauty of codegen"

One thing I always wondered is: what makes dynamic programming more "dynamic" than regular programming? Why is recursion "static" but it becomes "dynamic" if you store results?

According the article, the term “dynamic” was chosen because it has positive connotations.

[flagged]

Two hard problems in computer science…

1. Cache invalidation

2. Naming things

3. Off-by-one errors

0. Race conditions

4: Threconcadiparaurrenllecy, lism.ng,

If you have to impress the bureaucrats to get funding, you go and impress the bureaucrats.

Oh, then you will really love and want to fuck the name of the "Wave Function Collapse" algorithm! (Maybe Chuck Tingle will write a book about it.) Also "Extreme Learning Machines", "Reservoir Computing", "Liquid State Machines", and "Crab Computing" (I shit you not)! Don't even get me started on "Moveable Feast Machines".

[Welcome to Coffee Talk, with Linda Richman. Todays topic: Wave Function Collapse is neither quantum physics, particles, waves, nor collapsing functions. Discuss amongst yourselves, while I am constrained to solve all these Sudoko puzzles. Oh, I am so verklempt!]

https://news.ycombinator.com/item?id=42749675

DonHopkins 6 months ago | parent | context | favorite | on: Generating an infinite world with the Wave Functio...

Ha ha, great comment! You're right, WFC's name just refers to quantum mechanics because it sounds cool, not because the algorithm has anything to do with physics, waves, or collapsing functions (it's more just a constraint based Sudoko solver). But I perceive it more as a playful buzzword for a useful algorithm, and not as maliciously deceptive like Deepak Chopra, who I was just commenting on recently (Quantum was cool until Deepak Chopra ruined it for everybody):

https://news.ycombinator.com/item?id=42714477

I do think it's useful combined with other techniques like procedural programming, cellular automata, Voronoi diagrams / jump flooding, noise, etc, to steer the higher and lower level structures. Check out Maxim Gumin's work on Markov Junior!

https://twitter.com/ExUtumno/status/1531992699997507586

Maxim Gumin @ExUtumno: I published a new project about combining rewrite rules and solving constraint problems made of rewrite rules

https://github.com/mxgmn/MarkovJunior

I think the really nice property of WFC is that it's very naturally and easily artist-driven. It doesn't require programming skills to use, you just supply it with a bunch of concrete before and after examples. (Like "Programming by Example".) That makes it much easier for a wide range of people to use, which is an important feature for democratizing custom world generation. Especially when you can construct or discover the examples in-world, and have it operate kind of like a 2-d tile or 3-d cube auto-complete, that looks at the rest of the stuff you've built or seen, like Canvas does with 1-d code.

https://news.ycombinator.com/item?id=42751176

DonHopkins 6 months ago | parent | context | favorite | on: Generating an infinite world with the Wave Functio...

The difference between Deepak Chopra's abuse of Quantum Physics terminology and WFC's is that WFC actually works and is useful for something, and its coiner publishes his results for free as open source software and papers, so he deserves more poetic license than a pretentious new-age shill hawking books and promises of immortality for cash like Deepak.

Here are some notes I wrote and links I found when researching WFC (which is admittedly a catchier name than "Variable State Independent Decaying Sum (VSIDS) branching heuristics in conflict-driven clause-learning (CDCL) Boolean satisfiability (SAT) solvers"):

https://donhopkins.com/home/wfc-notes.txt

    Here are some notes I wrote and links I found when researching Wave
    Function Collapse (WFC). -Don Hopkins

    Wave Function Collapse

    Maxim Gumin

    Paul Merrell

    https://paulmerrell.org/research/

    https://paulmerrell.org/model-synthesis/

    Liang et al

    Jia Hui Liang, Vijay Ganesh, Ed Zulkoski, Atulan Zaman, and
    Krzysztof Czarnecki. 2015. Understanding VSIDS branching heuristics
    in conflict-driven clauselearning SAT solvers. In Haifa Verification
    Conference. Springer, 225–241.

    WaveFunctionCollapse is constraint solving in the wild

    https://escholarship.org/content/qt1f29235t/qt1f29235t.pdf?t=qwp94i

    Constraint Satisfaction Problem (CSP)
    Machine Learning (ML)

[...lots more stuff...]

Even more fun stuff about WFC, AgentSheets, and defining cellular automata rules by example:

https://news.ycombinator.com/item?id=42749578

And now about "Extreme Learning Machines" and "Liquid State Machines":

https://news.ycombinator.com/item?id=42750857

DonHopkins 6 months ago | parent | context | favorite | on: Generating an infinite world with the Wave Functio...

Like calling random vector functional link networks and single-layer feed-forward networks with random hidden weight an "Extreme Learning Machine".

https://en.wikipedia.org/wiki/Extreme_learning_machine#Contr...

>Controversy

>There are two main complaints from academic community concerning this work, the first one is about "reinventing and ignoring previous ideas", the second one is about "improper naming and popularizing", as shown in some debates in 2008 and 2015.[33] In particular, it was pointed out in a letter[34] to the editor of IEEE Transactions on Neural Networks that the idea of using a hidden layer connected to the inputs by random untrained weights was already suggested in the original papers on RBF networks in the late 1980s; Guang-Bin Huang replied by pointing out subtle differences.[35] In a 2015 paper,[1] Huang responded to complaints about his invention of the name ELM for already-existing methods, complaining of "very negative and unhelpful comments on ELM in neither academic nor professional manner due to various reasons and intentions" and an "irresponsible anonymous attack which intends to destroy harmony research environment", arguing that his work "provides a unifying learning platform" for various types of neural nets,[1] including hierarchical structured ELM.[28] In 2015, Huang also gave a formal rebuttal to what he considered as "malign and attack."[36] Recent research replaces the random weights with constrained random weights.[6][37]

But at least it's easier to say, rolls off the tongue smoothly, and makes better click bait for awesome blog postings!

I also love how the cool buzzwords "Reservoir Computing" and "Liquid State Machines" sounds like such deep stuff.

https://news.ycombinator.com/item?id=40903302

>"I'll tell you why it's not a scam, in my opinion: Tide goes in, tide goes out, never a miscommunication." -Bill O'Reilly

How about rebranding WFC as "Extreme Liquid Quantum Sudoko Machines"? ;)

Then there's "Crab Computing"!

https://news.ycombinator.com/item?id=42701560

[...] If billiard balls aren't creepy enough for you, live soldier crabs of the species Mictyris guinotae can be used in place of the billiard balls.

https://web.archive.org/web/20160303091712/https://www.newsc...

https://www.wired.com/2012/04/soldier-crabs/

http://www.complex-systems.com/abstracts/v20_i02_a02.html

Robust Soldier Crab Ball Gate

Yukio-Pegio Gunji, Yuta Nishiyama. Department of Earth and Planetary Sciences, Kobe University, Kobe 657-8501, Japan.

Andrew Adamatzky. Unconventional Computing Centre. University of the West of England, Bristol, United Kingdom.

Abstract

Soldier crabs Mictyris guinotae exhibit pronounced swarming behavior. Swarms of the crabs are tolerant of perturbations. In computer models and laboratory experiments we demonstrate that swarms of soldier crabs can implement logical gates when placed in a geometrically constrained environment.

https://www.futilitycloset.com/2017/02/26/crab-computing/

And of course "Moveable Feast Machines":

https://news.ycombinator.com/item?id=15560845

https://news.ycombinator.com/item?id=24157104

https://news.ycombinator.com/item?id=34561910

The most amazing mind blowing MFM demo is Robust-first Computing: Distributed City Generation:

https://www.youtube.com/watch?v=XkSXERxucPc

Not quite sure what this all is, but it is an interesting way to wind up in the a.m. fully reminded that I am not and have never been, nor will be, as smart as I would like to be, but it’s an enjoyable ride. Thanks

See https://github.com/fcampelo/EC-Bestiary

Marketing matters almost too much in programming.

You'd hardly see people rallying behind concepts like pure functions or clean code if they were called brown functions or moist code.

Hydrated pages seems to work though, so you could maybe swing wet or dry code?

As I always say, some people think DRY stands for Do Repeat Yourself. You can say that again!

Hydration has good connotations though. Hydration is healthy and good for your skin. If you'd used an adjective like "waterboarded" it would have less marketing appeal.

If you use too much moist code, is there a risk of getting a soggy bottom? Considering how many names seem to be derived from memes these days (all the -strikes and so on), a language that guarantees " safety" as a metaphor for correctness would probably do well.

I seem to hear a lot about "red" and "blue" functions these days https://journal.stuffwithstuff.com/2015/02/01/what-color-is-...

Meanwhile Linux users be killing children in the command line using bash.

If I ever invent anything I’m calling it “dynamic entropy,” that is clearly the coolest name.

Quantum chromodynamics is still in the lead.

Nobody wants to buy a next-generation spam generator but everyone is lining up to bid on "AI".

When I learned dynamic programming, I was told this was in contrast to simple memoization in that memoization is a cache of computation, dynamic programming involves a "choice" in each step.

For example let's look at the fibonacci sequence, F(n) = F(n-1) + F(n-2), naively you just fill in a (1 dimensional) table with the results [1, 1, 2, 3, 5, 8, 13...] and it feels static because there's nothing to choose or decide.

In contrast for example the longest common subsequence looks like this:

  lcs[m, n] = 0                 if m = 0 or n = 0
  lcs[m, n] = lcs[m-1, n-1] + 1  if X[m] = Y[n]
  lcs[m, n] = max(lcs[m-1, n], lcs[m, n-1])  if X[m] != Y[n]

At every step there's a "choice" to be made depending on the situation. The functions like "max()" are quite typical for dynamic programming. At least this is how I interpret the "dynamic" part.

I also discovered the term "dynamic programming" 20 years in my career, when doing leetcode training for a job interview.

I still have no idea why it's called "dynamic".

> I still have no idea why it's called "dynamic".

Neither did I – until I read TFA :)

Ditto, it took me years to figure out "dynamic programming". It sounded hard and complicated! When I finally figured out wtf it meant I felt anger instead of relief - what an absolutely asinine name for what is otherwise a simple (but valuable) concept.

Our industry has a bad habit of this, where we name things in such a way that make them harder to explain.

See for example one of my other pet peeves: "lambdas". Sounds fancy and difficult! Sounds like a concept you may read about in a research paper! The reality is that it's really quite simple, and we do ourselves a disservice when we name things like this.

As others have mentioned the term was chosen to be distinctive and attractive.

Marketing, Richard Bellman was trying to get funding and needed his work to sound high tech and fancy.

According to wikipedia: https://en.wikipedia.org/wiki/Dynamic_programming#History_of...

The word dynamic was chosen by Bellman to capture the time-varying aspect of the problems, and because it sounded impressive.The word programming referred to the use of the method to find an optimal program, in the sense of a military schedule for training or logistics. This usage is the same as that in the phrases linear programming and mathematical programming, a synonym for mathematical optimization.

Also, Harold J. Kushner stated in a speech that, "On the other hand, when I asked [Bellman] the same question, he replied that he was trying to upstage Dantzig's linear programming by adding dynamic.

With dynamic programming the order of computation isn't necessarily fixed, even though you can sometimes order it to reduce or eliminate recursion. You are just reusing past results when they exist, else computing and storing if they don't.

I know this is trite and arguably against the guidelines and I'm sorry. But this is simply a crazy thing to comment on an article whose whole point is to explain why it's called that.

Oh that's the year I went. Trip down memory lane. I was woefully unprepared. The mobile phone cell summation problem underlined how much I didn't know, and later figuring it out, how much I had to learn; it cemented my love for the field. I just loved the experience.

As a US born native English speaker, I always struggled with the phrase. Eventually I just mentally called it a misnomer as the phrase provided no value to the actual problem set they described.

For my first programming competition (the national ICPC prequalifier, in 2002), I asked my teammates (who I didn't know very well, we had just cobbled together a team) a couple of minutes before the contest what this dynamic programming thing was. One of them was “oh, it's only caching the output of a recursive function” (technically, that's memoization, but who cares).

I ended up solving a problem using DP a couple of minutes before the deadline, which was enough to get us to 3rd and to ICPC. Fun times.

In the university, we had a team for ACM ICPC. We helped our professor organize a local competition, to encourage more people to practice. We participated in that competition, of course. One question would give you a few numbers no greater than 13, and you would output the number of ways to put N queens on the cheeseboard of those sizes. Not a difficult problem, but our first implementation, needed 2 minutes while time limit was 1 minute. We could optimize, but we decided to run it for all possible numbers, write a new program with just a switch-case. This was q perfectly legal move. We could see his face while judging our submission, his huge surprise when code ended in milliseconds, his look of admiration at us, the curiosity about the solution, and finally his reaction when he checked out our code. Priceless.

Fun enough, in commercial software engineering your solution won’t raise eyebrows as something routine and expected, but an efficient algorithm could do.

Its called https://en.wikipedia.org/wiki/Memoization

More like "precomputed table" and "table lookup"...

Memoization is the kind of dynamic programming the first post on the thread is talking about.

Had a similar problem in a highschool comp. sci. class with a maze-following algorithm where you got points off for back-tracking. Turns out your code could pre-fetch the entire maze before making a move, which I think goes against the spirit of the exercise.

I had to laugh hard the first time I heard about it, I mean it's a really obvious technique if you've ever done low-level programming on limited machines as it's very close to something we used to do all the time - calculate result tables and use lookups which we built in to our code.

Trades memory for CPU when it's worth it. Dynamic programming takes the same idea but does it at runtime.

To be fair, DP generally also involves recursion in some way. You could describe it as a lookup table where, to fill in later entries of the table, you need to use the earlier entries.

I found this video very helpful:

https://m.youtube.com/watch?v=oBt53YbR9Kk

Yes, it is 5 hours long. At least 4 of those are worth it.

Many people confuse caching with DP. I’ve had that conversation too often. I think it’s down to the memoization examples people choose being too toy and just looking like caching. Caching is global shared state, whereas memoization can be scoped to the current calculation and still be useful.

But they always skip over tabulation, which is where I believe DP distinguishes itself.

But dp is a form of caching, no? It sacrifices space by caching intermediate results to compute successive results. The only reason it is called dp is because the "inventor" (Iirc Bell) needed a cool name.

No, memoization is not caching. That’s a reductive understanding that causes no end of trouble.

Caching is global shared state. It brings nearly every problem that global shared state brings with it. If you conflate the two you miss all of the advantages of DP and “just use caches”.

Memoization in the scope of DP (some languages misuse the word in their APIs) is acknowledging that a problem is self recursive and eliminating the duplicate work by either inverting the problem or storing previously calculated values. These are usually call graph scoped. Nobody else sees the value. For instance in the case of solving Fibonacci with iteration, you need only remember the previous two values, which you store in registers. There is no table. There is no O(n) storage space.

Again, tabulation is when the contrast becomes stark. You’re creating an array where the relationship between the values contains information that makes the calculations cheap.

No definition of "caching" I know imply global shared state. Local private caches are commonplace.

I don’t know what rock you’re living under but Redis is absolutely GSS, and any lookup table accessible between different tasks is shared state. In a language with threads anything the entire app can see is considered global shared state.

I think you misunderstand. DP is not equal to "caching", but in the set of all things that are "caching" DP is contained. I cannot think of any DP algorithm that does not explicitly or implicitly use a memory cache. Can you? Even in the case of DP Fibonacci the two registers of previous values constitute a cache.

DP is not a form of caching. DP is a way of breaking down a problem into subproblems. Caching (memoization) is one way to use this subproblem structure to reduce the amount of computation that ends up being performed in the end.

DP is a resolution technique that uses progressive (i.e. "dynamic") memoization (i.e. "programming" as in "tabulation", or resulting data stored in and retrieved from a table) as a way to curtail the cost of overlapping (i.e. intermediary and repetitive) subproblems. Memoization is a form of deterministic caching.

I was given a programming interview once where the interviewer gave me a DP problem (which I didn't know at the time), and he failed me because I used a cache instead of memoization. Asides from being inelegant, what are the downsides to caching?

Global shared state. High memory overhead. Evictions mean that the beginning of your call can see potentially different values than the end, and most of us write code like this as if it is transactional. We cannot handle a user’s account being closed or upgraded to an admin in the middle of a transaction. So using a cache to keep refetching the user data instead of passing it with the call graph is fraught (and makes unit testing terrible).

As a concrete example, my coworker was trying to walk a DAG with many leaf nodes that were repeated (think dependency tree). He used a cache to do this. It had bugs (2 I knew, but 6 I found). It was one of three large calculations we made at startup, and it should have been the least of these. But it ate memory like candy and added a lot to our start time which was slowing down deployments.

So I replaced it with a DP solution that just looked like normal code, reducing startup memory by ~80 MB and start time by half. By walking the tree and making a list of all the nodes in dependency order and then processing them once, after the list was compiled. And for the microservices and dev tools where we ran this same code the startup time became negligible.

Part of the problem is that the library we used did not like being called recursively, so you had to dance around reentrancy bugs and that cost time and space. But a linearized run has no recursion.

Edit to add: I realize now that I’ve left the conclusion ambiguous.

His two main sins were one, that the cache lived beyond the useful lifetime, and two, that even with the cache some amount of post-processing was required. By unrolling the tree and running each calculation once I reduced the post processing by about 3/4 and climbing as the tree grew in width and height.

Memoization is caching so that's a strange reason to fail someone. You cache based on the function call and its arguments.

  SOME_FUN = dict(base_cases)
  def some_fun(a, b, c):
    if (a,b,c) in SOME_FUN: return SOME_FUN[(a,b,c)]
    result = some_fun(a - 1, b - 1, c) + some_fun(a, b, c-1) # or whatever the recursive calls would be
    SOME_FUN[(a,b,c)] = result
    return result

That's it. Some languages (Python, Lisp) make it easy to take the naive version and wrap it (Python decorators are useful here) so that you don't have to write that code above. That's what the @cache decorator in functools does for you. You can memoize the naive version without needing to actually write a specific memoized version.

No, this is what forty years of semantic dilution gets you.

https://en.wikipedia.org/wiki/Memoization

> In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls to pure functions and returning the cached result when the same inputs occur again. Memoization has also been used in other contexts (and for purposes other than speed gains), such as in simple mutually recursive descent parsing.[1] It is a type of caching, distinct from other forms of caching such as buffering and page replacement. In the context of some logic programming languages, memoization is also known as tabling.[2]

It includes citations so I won't bother repeating them. Memoization is caching, there is no confusion about this in the literature.

Because Wikipedia has never been wrong and doesn’t suffer from dilution.

The distinction I am making is that caching is not memoization.

> It is a type of caching, distinct from other forms of caching

Distinct. As in not to be confused with.

If I wrote "a square is a rectangle" would you also waste time shouting that rectangles aren't squares? I didn't write that all rectangles are squares. Nor did I write that all caching is memoization.

Even your quote doesn't support your quibbling. It says that memoization is distinct from other forms of caching. Which does not mean that it is not a form of caching.

EDIT: Removed a particularly rude bit.

All dogs are mammals, but not all mammals are dogs.

If everyone starts talking about dogs every time you mention mammals, eventually you will get touchy about it.

Yeah, dogs are great. But so are cats, and llamas.

Memoization is deterministic caching, meaning that the structure is a perfect stand-in for a deterministic function. The "determinism" of the function itself is relative to some agreed upon lifespan, during which if you feed it some input, the corresponding output is guaranteed to always be the same (e.g. until the next reboot, the runtime of the main program, the span of a function call, etc). Within that timespan, the result of computations can be reliably stored and retrieved from the cache. When it's exceeded, the function's corresponding cache also goes stale and should be considered unreliable and to be discarded.

Note that in the case of memoization, the guarantee of determinism (and associated lifespan) is attached to the function, not the caching structure. Unlike some other caching approaches that rely on a replacement policy.

The key difference between caching and memoization is not global shared state, it's determinism. That's why the latter mostly applies locally. In that sense, they're only semantically different. Data in a cache can be expected to go stale; it must not if you rely on the structure as a memoized equivalent to calling the function with the given inputs. In practice, you can have global cache whose data is deterministic and you can also have local cache that goes stale. The latter is not memoization.

I think you’re quibbling. Sharing state between independent tasks leads to nondeterminism. That’s the whole problem with shared state. Spooky action at a distance creates nondeterminism and even undefined behavior.

> I think you’re quibbling.

I can assure you I'm not.

There are absolute claims to make, they just aren't the ones you're making. Determinism is an absolute claim of memoization. "Local rather than shared state" is simply not, no matter how many time you repeat it is.

Excluding shared state from memoization implies that you can't rely on the technique in parallel scenarios, which is just not true. I can think of more than a handful of DP problems where you can further optimize with parallelism if the function is modified to rely on shared state memoization. For example, the slew of problems that rely on the tabulation to simply fetch one value in the "previous row" and the value at the "previous column" of the current row. In practice, instead of iterating through rows and waiting to completely process them one at a time, you could simply start processing another row in parallel once you have just enough of the requisite "previous" data. This is absolutely valid DP. Some DP solutions even receive their tabulation. How does that not qualify as global?

If you thought "memoization is never shared", then in practice, not only your possibly very deterministic functions would end up solving the same problems over and over each time they're called, you would also deny yourself the opportunity of tremendous optimization techniques given by a guarantee of determinism.

Determinism is key. Not shared state.

> Many people confuse caching with DP. I’ve had that conversation too often.

This is the beginning of my thesis. Without this no other conversation is productive.

The problem with the DP space in specific and coding in general is that a substantial fraction of 'everyone' hears 'memoization' and immediately substitutes caching into their thought processes, "oh that's just caching, got it," and then believes they are productively contributing to the followup conversation when really all they are doing is confusing everyone else and themselves.

That's my beef, frustration, and frankly, institutional trauma. Equating them in your brain makes everything you say from them on practically useless.

You have a fair point about parallelism, but even here the sharing you illustrate is between parallel sub-problems, the 'sub' is IMO doing all of the work in that sentence. Versus me caching a header because you and I have the same account settings, only there's a bug because it's now showing you my unread message status due to incorrect sharing.

Technically, "dynamic programming" (god, I hate that term) is focused on memoization/tabulation. It doesn't necessarily require recursion or an iterative approach, although they almost always go hand-in-hand in common leetcode/hackerrank/etc exercises.

Programming terminology is in the same vein as "Linear Programming..."

As per Wikipedia, that origin story is somewhat disputed:

"According to Russell and Norvig, the above story "cannot be strictly true, because his first paper using the term (Bellman, 1952) appeared before Wilson became Secretary of Defense in 1953." Also, Harold J. Kushner stated in a speech that, "On the other hand, when I asked [Bellman] the same question, he replied that he was trying to upstage Dantzig's linear programming by adding dynamic. Perhaps both motivations were true."

I can't find the resource, but, according to Bellman he needed a name that wouldn't sound like something the army would scrape— effectively protecting his research. He had a few options and "dynamic" sounded "cool" enough, in his opinion, for army.

This said, the "programming" part is pretty accurate, but IMO the author misses the point with what "computer programming" meaning is, afterall.

Linear programming gets confused with programming, but also with linear algebra and linear modelling.

Programming in this sense means just optimization

I've seen this many times before and it always presents Charles Wilson's aversion to "research" in a "haha what a luddite" manner.

You have to understand that Eisenhower believed that the military industrial complex was out of control. He brought Wilson in to help rein it in and reorganize the defense department.

The military and military contractors were conducting "research" in the form of doing chemical and biological weapons experimentation on soldiers and civilians without their knowledge or consent. Wilson was trying to stop that.

And of course many of the think tanks were trying to lead the United States into an offensive war against the Soviet Union, which Wilson and Eisenhower did not want or think was necessary. It's entirely probable that the only reason Bellman got funding to do this work at RAND was because they intended to use it for running their desired war.

Dynamic programming is a subfield of mathematical programming. The field of studying optimization problems.

It's called programming because programming means scheduling. And most of the problems the field initially concerned itself with were logistics scheduling problems.

Computer programming is called programming because it involves scheduling instructions.

I did not know that. Fascinating.

> Try thinking of some combination that will possibly give [the word, dynamic] a pejorative meaning. It’s impossible*.

> *This Haskell fan’s contender is “dynamic typing” :P

Nothing is impossible!

It’s certainly possible to use it as a euphemism, if not an outright pejorative. “Dynamic truthfulness”, “dynamic sense of morality”, etc.

There's also dynamic scoping; as opposed to lexical scoping a.k.a. static scoping.

You can find defenders of dynamic typing, but dynamic scope is now widely seen as a mistake. Or at least dynamic scope by default; it has specialised uses--and in Haskell the Reader Monad is basically isomorphic to dynamic scoping and no one complains about it.

> dynamic scoping

Right you are. Even more horrible. The tcl language still has it!

I think EmacsLisp also does dynamic scoping. Or at least they used to about ten years ago. Not sure, if they fixed it?

Well, Tcl is kind of a mixture of scope rules. In some respects, e.g., within a proc, it's mainly a lexical environment, but of course dynamic scoping is introduced by commands like upvar/uplevel. FWIW Tcl programmers don't concern themselves very much with sorting out dynamic vs. lexical. In any case, Tcl programmers are careful to maximize static scoping. No doubt that's necessary to create reliable larger programs many of which have been written in Tcl.

Dynamic scope/binding is super useful, but I do not agree tcl does it or does it well, because it's not just shadowing the closure environment (ha), or changing globals in a dynamic-wind.

Perl's the only non-lisp I'm aware of that does dynamic binding well-enough to get the idea, but too many things just don't work:

    local *int = sub { return 69 }; print int("42");

is a particularly annoying one: You just can't do it. Other names might have better chances, e.g.

    package foo; sub int { 42 };
    package main; sub a{local *foo::int = sub { return 69 };&b(@_);} sub b{goto \&foo::int};
    print b("x"), a("x"), b("x"), a("x");

but this is a little unsatisfying.

> Perl's the only non-lisp I'm aware of that does dynamic binding well-enough to get the idea, [...]

Well, Haskell does dynamic binding in the form of the Reader Monad. It's well received there.

Erm no. The point is I want to have the ability to make any variable dynamic, not just work with a global variable that is really a getter.

The reader monad doesn't need to be global.

But yeah, by that standard, Haskell implements both dynamic scoping and mutable state via getters and setters. (Though, of course, you can go all out on these via lenses.)

So does emacs lisp, by default, although you can set it to lexical scope per-file (and it’s recommended to always do so).

Isnt dynamic scoping a bit similar to Dependency Injection? Maybe there's a form of it that could be useful. Like explicit dynamic scopes?

Well, isn't dependency injection just a more cumbersome way to say 'function arguments'? Dynamic scoping is exactly the same, it's basically equivalent to extra implicit function arguments that are passed through everywhere.

And yes, dynamic scopes can be useful in specific cases. What's wrong is having your variables scoped dynamically by default.

Those are not pejorative names. They are descriptive names for (arguably) bad things.

And they are all much more enticing than the things deserve.

Compare and contrast https://en.wiktionary.org/wiki/euphemism_treadmill

'Idiot' and 'retard' didn't start out as pejorative either.

It seems dynamic programming (some predetermined sequence of steps) as envisioned by Bellman is strictly less dynamic than ordinary programming today, hence the confusion?

Elevating "memoization+recursion" to "dynamic programming" seems like a development that Bellman didn't anticipate

https://cs.stackexchange.com/questions/99513/dynamic-program...

There is another mid century term which is oddly specific for something oddly general (or the other way?) "Operations Research"

Guess Bellman should have called it "Reactive Optimization", then we would have the totally appropriate "Recursive Memoization" or "Memoized Recursion"

You should think of the “programming” in dynamic programming the same way you think of in linear programming, integer programming, and constraint programming.

Indeed even in layman’s terms, thinking of it as in television programming is more accurate than thinking it is related to computer programming (as is mentioned in TFA)

> linear programming, integer programming, and constraint programming

Can't think of a universe in which you'd learn about these things before learning "computer programming".

There was a universe before digital computers were popular or a thing at all.

Computing predates computers.

Before digital computers existed, there were still computers. They were people. A baker is a person who bakes. A computer was a person who computes (computing mathematical results).

Sure, that's why I had computers italicized to indicate the specific modern definition, not everything that computes.

In the UK at A-level (age 16-18) you may still be taught linear and dynamic programming before ever touching a line of code! (Indeed, that was the same for me!)

Did your school used the specific terms "linear programming" and "dynamic programming" to refer to those topics? My original comment didn't phrase this clearly, but I was thinking less about techniques themselves, and more about encountering them under those specific labels.

As an analogy, it's not unusual to learn a good chunk of calculus before learning that "calculus" is a thing they're part of - for example, by having an over-eager physics teacher who teaches you some of it so we can understand physics material deeper, but without ever mentioning the branch of math we're now using.

I know two people who are experts in linear programming who have never written a single line of code.

I meant that in particular you should _not_ think of the “programming” part of “dynamic programming” in the sense of computer programming

I always find funny how many meanings "programming" has.

Television programming isn't a separate meaning from computer programming. Programming means creating a program. The program, whether you're talking about a computer, a television channel, or a live staged performance, is a list of what's going to happen in what order.

That's what I was hinting to. Pre mechanical programs were 'prewritten' sequences. Mostly linear, like a (literature) script. Musical score would fit that definition too. Then you can have slightly more complicated structures like dags and data flow, still linear in a way. When you add state,loops, feedback it gets interesting though.

I think of "programming" in "dynamic programming" the exact same way I think of it in "linear programming", "integer programming" and "constraint programming": it's probably some kind of software development that some computer scientists came up once and that I don't need to think about, because my normal programming has worked out pretty well so far

(Except, well, I guess I understand what "dynamic programming" is more than I understand what the other forms of programming you mention is; "dynamic programming" is solving certain classes of recursive problems by using arrays, sometimes nested arrays, to avoid re-computation, and somehow that's supposed to be more "dynamic" than not doing that)

The sequence of steps is the result of dynamic programming. Not every sequence of steps is a dynamic programme. And (what I would argue are) the core results are fairly mathematical, memoization and recursion don't feature, but partial differential equations do. Check out the Hamilton Jacobi Bellman equation to get a flavour.

> Elevating "memoization+recursion" to "dynamic programming" seems like a development that Bellman didn't anticipate

Well, it also isn't a development that happened. You can (always) implement dynamic programming as recursion with memoization. But an algorithm that is recursive and memoized isn't necessarily an example of dynamic programming.

The point of dynamic programming is that you set up the recursion so that recursive calls do hit the cache, not that there is a cache and if you're lucky they might hit it.

Usually though when using recursion to solve an algorithmic problem, people memoize results they know will be needed later again. So in most cases that should automatically become the same thing in practice.

My favorite dynamic programming problem is computing the percent of relatedness between two people, given a set of people and a partial graph of parenthood among them (with the assumption that any relatedness not implied by the graph is zero). If seems very confusing at first, but then you realize that if A is not a descendant of B, rel(A,B)=(rel(A,father(B))+rel(A,mother(B)))/2, which allows you to compute all relatedness values from top to bottom of the graph as fast as possible.

DP(Dynamic Programming) is such a beautiful technique .It took a while for me to grasp it . There are many applications in real world systems as well. Eg Duck DB's join order optimizer is built on https://15721.courses.cs.cmu.edu/spring2020/papers/20-optimi... .

https://github.com/duckdb/duckdb/blob/61f07c3221e01674d8fe40...

If anyone's interested, I published a short, ad-free YouTube series on Dynamic Programming about 4-5 years ago [1]. I haven't had a chance to continue it, but many people have found the course eye-opening.

[1] https://www.youtube.com/@andreygrehov

I first learned about Dynamic Programming in the algorithms and datastructures class in the first semester at uni. But the lecturer very quickly moved on to only ever refer to it as "DP". This makes me think he may have done that as to not confuse students.

Even though I knew it wasn't referring to programming I did wonder why it was called that, interesting!

Also, the term "dynamic programming" has slightly different meanings in the context of leetcode problems and in the context of reinforcement learning.

In the first it's for optimizing a relatively small deterministic system, in the latter it is for optimizing a stochastic process.

Pretty confusing.

The idea is way older than both leetcode and reinforcement learning and is used everywhere (for example when planning SQL queries). If reinforcement learning invented a new way to use the word then that is all their fault because leetcode is true to the original meaning.

Both meanings derive from Bellman. "Reinforcement learning" did not invent a new way, and the field of reinforcement learning predates leetcode by a large margin.

So what is the confusion then? Leetcode got it from real computer science and engineering and retained the original meaning in both steps without any changes. Every bigger piece of software you use probably uses dynamic programming somewhere. Or do you think that e.g. PostgreSQL is leetcode?

There is nothing derived here. It is exactly the same meaning.

Perhaps we misunderstand each other. I'm saying that the mathematical optimization method differs from the algorithmic paradigm, as explained on this Wikipedia page for example: https://en.wikipedia.org/wiki/Dynamic_programming. That is to say, they serve different purposes and are distinct concepts. They sure have the same fundamental basis laid out by Bellman, but the applications have diverged quite a bit.

Do you mean to say that these two concepts also have exactly the same meaning?

Fun facts the venerable Border Gateway Protocol or BGP is using distant-vector algorithm for global Internet routing calculation based on Bellman-Ford dynamic programming approach due to its scability in updating the routing table [1].

For intranet routing however shortest path algorithm based on Dijkstra approach underlies the OSPF and IS-IS protocols.

In the old days distant-vector protocol like Routing Information Protocol protocol or RIP is also used but it's now becoming obsolete and humorously known as Rest In Peace protocol.

Distant-vector now is making a comeback for intranet in the form of mobile ad hoc and mesh networks (guerilla networks for routing without infrastructure) with Ad hoc On-Demand Distance Vector Routing protovol or AODV [2]. This setup can be very beneficial for disaster and also for mobile group or platoon communication.

[1] Distance-vector routing protocol:

https://en.wikipedia.org/wiki/Distance-vector_routing_protoc...

[2] Ad hoc On-Demand Distance Vector Routing:

https://en.wikipedia.org/wiki/Ad_hoc_On-Demand_Distance_Vect...

For all the people looking for different terminology, the word you are looking for is optimization.

Dynamic optimization makes just as much sense. Same as Mathematical optimization does for the wider field of study.

Richard Bellman on the Birth of Dynamic Programming (2002) [pdf] https://news.ycombinator.com/item?id=42482289

Just call it caching. The whole concept is a bit too simple to be a word.

That thought comes from a programmer perspective.

What’s interesting in terms of the theoretical properties and classification of a problem is the recursive nature with overlapping recursion.

It doesn't require caching, though. So that's a wrong description of the concept.

You're thinking of memoization, which is related to but distinct from DP. Memoization is "top down", DP is "bottom up". DP generally has better space complexity (memory requirements). For example, for a function that computers the Fibonacci sequence, space complexity with memoization is O(n), vs O(1) with DP.

https://en.wikipedia.org/wiki/Dynamic_programming#Fibonacci_...

Both top down and bottom up approaches can be used with dynamic programming. DP is about how you approach solving a problem by breaking it into subproblems (see the term used, "optimal substructure"). Whether you use that information for a top-down (often memoized) solution or a bottom-up solution is a choice for the implementor (and which depends on how you're using your solver and solutions).

If you need to solve one problem, bottom-up is often most efficient. But if you need to solve many related problems (say trialing many different initial conditions) then you may want the memoized solution.

Fibonacci is a bad example because it relies on the linearity of `prev + current` to be O(1) memory complexity. A lot of divide-and-conquer schemes exploit symmetry and linearity of their internals but if the operation is nonlinear then it requires memoization/caching.

On that note, I really struggle to understand what's special about "bottom up" versus "top down" DP or what categorizes it as unique from any other algorithms.

Bottom-up solutions to DP problems allow you, if the structure permits, to throw away parts of the full table. This is more memory efficient. Or, you could build the table bottom-up which avoids the repeated verification step in memoized, top-down solutions because when you reach an entry that isn't filled, you know that its related subproblems have already been computed and cached.

Someone posted an Euler problem with finding the maximal sum along any path (binary choice at each step) down a pyramid. The bottom-up solution only ever generates a "cache" of row elements (starting the row count from 1). The 100th row has 100 elements. When you go up to the 99th row you only need 99 elements by the end of calculating it, and so on. This is more efficient in memory than the n*(n+1)/2 (approx.) elements you'd need doing a top-down recursive, memoized version. Though in that case it's not awful on modern computers, it is a consideration for complex and large problems.

The bottom-up version also lends itself to not needing to bring the entire pyramid into memory, only one row + the cache at a time so you only have |row| + |row| + 1 items in memory at any given time (plus some temp variables or index variables). Again, it can be very memory efficient. Technically, you don't even need the entire current row in memory, only one element of it and then read each new element as you step across the row which further reduces the memory overhead.

With regard to building a full table, I realize Fib is a poor example in general but it is illustrative and fits in a comment. Let's say you memoize the naive version, then you have to include this check either before the recursive calls or at the start of each call:

  if n in FIB: # either return if at top of the function, or retrieve rather than make the recursive call

This is unneeded if you build the table bottom-up:

  def fib_bottom_up(n):
    FIB = {1: 0, 2: 1}
    i = 3
    while i <= n:
      FIB[i] = FIB[i - 1] + FIB[i - 2]
    return FIB

Bottom-up construction of the table skips those checks which can add up for larger table constructions, or if you're building up many different tables. Consider a more complex optimization problem where different initial parameters result in different tables being generated (but by the same method). This bottom-up approach will be more efficient than the top-down approach.

Of course, a benefit for top-down is when an input only needs some subset of the table. If the generated table will be sparse, but may not be easily constructed in a sparse manner bottom-up, then top-down can be better.

Dynamic programming has absolutely nothing to do with caching (or programming). You should read the article.

I learned what 'dynamic programming' was from studying the 'BLAST' algorithm for DNA sequence search. BLAST is a heuristic approximation for the otherwise Dynamic Programming algorithm created by 'Smith & Waterman' Before that, I would have misunderstood the term too.

Reminds me of "Extreme Programming" [0], which is referring to project programming rather than computer programming.

[0] https://en.m.wikipedia.org/wiki/Extreme_programming

Compare with "Extreme Ironing": https://www.ebaumsworld.com/pictures/23-insane-examples-of-e...

If it weren't for the fact that the page is 10 years old, I would've thought those were AI-generated images at first glance.

Thanks for sharing. I always viewed DP as finding partials to cache. Something not done a lot in day to day algos. Did not appreciate the name as being related to execution order optimization.

I always found that badly named things did make learning harder / more jarring; especially if an explanation for the incongruous name wasn't provided.

Yup: And stochastic dynamic programming and stochastic optimal control! Suppose a cargo plane is to fly from Kansas City to St. Louis, Chicago, Indianapolis, Dayton, Nashville, and Memphis and at each city pickup cargo. Suppose the cargo weights are known only in probability distribution from history and otherwise are probabilistically independent. Have weather reports for the winds and temperatures, but the actual values can vary from the reports. And suppose the fuel prices vary significantly among the stops but are known.

Then, how much fuel should the pilot buy at each stop to minimize expected cost for the trip? I.e., in simple terms, buy extra fuel where it's cheap and carry (tanker) it to where it's expensive but in so doing burn extra fuel from the extra weight of the extra fuel.

Solution: Stochastic dynamic programming.

Hint: In Canada, the solution is illegal!

Was meeting with Nemhauser (see references below); my cab back to the airport was waiting; and he gave me a 15 second lecture. On the plane thought about his lecture, ..., and stood for my Ph.D. orals and passed (extra credit for guessing my employer).

Some references (with TeX markup):

Stuart E.\ Dreyfus and Averill M.\ Law, {\it The Art and Theory of Dynamic Programming,\/} ISBN 0-12-221860-4, Academic Press, New York.\ \

Dimitri P.\ Bertsekas, {\it Dynamic Programming: Deterministic and Stochastic Models,\/} ISBN 0-13-221581-0, Prentice-Hall.\ \

George L.\ Nemhauser, {\it Dynamic Programming,\/} ISBN 0-471-63150-7, John Wiley and Sons, New York\ \

E.\ B.\ Dynkin and A.\ A.\ Yushkevich, {\it Controlled Markov Processes,\/} ISBN 0-387-90387-9, Springer-Verlag, Berlin.\ \

Wendell H.\ Fleming and Raymond W.\ Rishel, {\it Deterministic and Stochastic Optimal Control,\/} ISBN 0-387-90155-8, Springer-Verlag, Berlin.\ \

Michael Athans and Peter L.\ Falb, {\it Optimal Control:\ \ An Introduction to the Theory and Its Applications,\/} McGraw-Hill Book Company, New York.\ \

Dimitri P.\ Bertsekas and Steven E.\ Shreve, {\it Stochastic Optimal Control: The Discrete Time Case,\/} ISBN 0-12-093260-1, Academic Press, New York.\ \

E.\ B.\ Lee and L.\ Markus, {\it Foundations of Optimal Control Theory,\/} ISBN 0471-52263-5, John Wiley, New York,

Dynamic optimality is a way more interesting concept to me. Data structures like the splay tree can effectively write the "ideal program" for accessing data based on the current use patterns by maintaining the data in a particular way over time. Instructions and data are mostly the same thing when you really think about it.

Dynamic optimality sounds like a very euphemistic description of Levin's Universal Search to me – it is definitely optimal, just in a very, hmm, dynamic, let's say, way.

Possibly the worst named thing in computing? Not only is "programming" not referring to programming, but "dynamic" is meaningless and just there to try to make it sound fancy.

I prefer "cached recursion".

Well, cached recursion is a bit too generic. Almost anything can be done with both caches and recursion.

We had a different characterisation in one of our mathematical optimisation classes: dynamic programming is basically every problem that isomorphic to finding the longest path in a graph, with suitable 'overloading' of 'maximum' (for picking among multiple alternative paths) and 'addition' (for combining multiple sub-paths).

First, to illustrate what I mean by this overloading: matrix multiplication usually has multiplication () and addition (+). However, in the min-plus-algebra you overload (+) with minimum and () with plus, and then multiplying matrices becomes equivalent to hopping two paths in the adjacency matrix of a graph. (Sorry, if this is a bit confusing.) Specifically, taking an adjacency matrix A and calculating A* := 1 + A + AA + AAA + ... is equivalent to taking the transitive closure of edge hopping in your graph, ie the shortest paths between all pairs of vertices.

If you overload (+) with maximum and () with plus instead, you can find longest paths. For that to make sense, you should not have cycles in your graph.

Now let's go back to dynamic programming: the shared optimal sub-structure property of dynamic programming is exactly the same as what you have in finding longest paths.

The structure might sound overly restrictive, but you'll find that it works for all (most?) examples of dynamic programming.

P.S. My recollection was a bit vague, so I asked ChatGPT for some help, and apparently I wasn't completely wrong: https://chatgpt.com/share/687de24c-c674-8009-9984-0fda56d1c1...

P.P.S. Despite everything I said above, I agree that 'cached recursion' is still a better name than dynamic programming.

I actually think overly focusing on the memoized recursion undermines the core conception of dynamic programming. Remember, as noted in the post, the term programming refers to a tabular method of planning, i.e. specifically, the bottom-up calculation to solve a problem. The emphasis here is on a predetermined order on how to tabulate the plan to get to the desired result which will be the solution to the broader problem.

The memoized recursive function is an implementation methodology on a digital computer that specifically side-steps the question of in which order one should do the tabulation. I would argue that recursive memoization is a possible solution technique to dynamic programming problems (defined as the universe of problems that admit a dynamic programming solution,) but strictly speaking, in and of itself, is not dynamic programming.

I'd actually go the other way round: figuring out a suitable recurrence relation is usually the first step in any dynamic programming solution. As a next step, sure, you can explicit construct a table to fill in, but that's not actually a requirement: dynamic programming also works when your parameters aren't things like numbers that you can easily use to enumerate rows in a table.

Memoisation is a more general tactic that hands more of the busy work over to the computer.

I think "cached recursion" is too broad. I tend to go with "Minimal Memory Memoization".

1) the recursion solution is often too slow, but uses little memory.

2) the memoization solution can make the algorithms from 1 a lot faster, but blows up memory use.

3) the dynamic programming solution only keeps the previous partial solutions in memory that will be needed for future partial solutions. Therefore it is the "Minimal Memory Memoized" solution. It often requires a smart ordering of partial solutions that allows for the earliest cache eviction.

Your "cached recursion" sounds like number 2 to me, and the crux about dynamic programming is to figure out when to remove an entry from your cache.

Even just keeping the partial solutions in memory that will later be needed can use much more memory than a bottom up solution. Recursion being slow depends on the implementation of your programming language.

> Even just keeping the partial solutions in memory that will later be needed can use much more memory than a bottom up solution.

Can you give an example of this? I don't think that is correct. But we might be saying the same thing, because in practice by the time you achieve minimal memory, you will have a bottom-up solution.

This problem is an example: https://projecteuler.net/problem=18 (assuming you are not brute-forcing it).

At a quick glance, the minimal memory requirement is the O(N) partial solutions you need to keep, with N being the number of rows.

Is your comment that this is higher than the memory requirement of the recursive solution, because that doesn't seem correct to me? The recursive solution also has O(N) memory, which is the depth of the tree it is searching.

Recursive from top or recursive bottom up?

If you go bottom up, you don't need any memoization. You just keep the paths and compare them, discarding the ones that are guaranteed to be suboptimal. This way initially, you will have N paths of length 1 (the leaves) and each step up you will have -1 path to keep, where the paths grow by 1 element.

If you go recursively from the top, then you don't know anything about the lower levels and thus must "fork" into left and right branch at each step, until you reach the bottom, from which you start filling the cache with the results of subproblems.

Of course one problem of using a global cache is, that parallelizing this becomes ugly and requires a mutex, while in contrast with the bottom up idea you can cleanly separate, because you don't need any global state at all.

Dynamic is definitely not meaningless. I am familiar with dynamic programming from the optimal control theory point of view, and it's squarely focused on dynamical systems. The central equation of dynamic programming is the Hamilton-Jacobi-Bellmann equation, a generalization of the Hamilton-Jacobi equations of classical mechanics. And the terminology programming in this context also exists in linear programming and stochastic programming. It's definitely confusing today, but these terms simply predate the modern use of programming, as well as modern computers. I am honestly a bit puzzled by everyone in this comment section talking about recursion + memoization (as a way to efficiently implement backward induction I presume?). It seems like computer science teaches a particular application and implementation of dynamic programming ideas and people mistake that for the whole thing?

Well, to be fair, you are questioning terminology bastardization in "computer science," a field that is named after "computers" and "science" but has little to do with either. One should temper their expectations :) Informatics, a commonly used term in Europe, would have been a much better name.

Well, just pretend that the whole discussion happened in German, where despite using your preferred term of Informatik, they still talk about Dynamische Programmierung.

Some of these terms sound sooo awkward in German. I vote for calling it memoization.

Well, memoisation ain't exactly the same as dynamic programming. You can use memoisation to do dynamic programming, but you can also use it in other contexts (and you can implement dynamic programming in different ways, too).

However, information science (and in Finnish, informatiikka) is a synonym of library science. Computer science isn't too bad a name for a science about computing on (abstract) machines.

I prefer the Danish term: Datalogi

Naming things is considered one of the hard problems in the field, after all.

I don't understand most of what you wrote. But can you, in simple terms, explain to me why recursion without caching is "static"?

Why do you think that what I wrote implies they are static?

I didn't say that recursion and caching are the opposite of dynamic, I said they are essentially orthogonal concepts to it.

Generally speaking, dynamical systems are systems that have some sort of dependence on time [1]. In dynamic programming we have an optimization/decision problem in which time plays an essential role. The optimality is measured with respect to the whole time evolution. Bellmann showed that you can break this intertemporal optimization down into individual time steps.

So the key observation of dynamic programming is how to turn an "all times at once" problem into a "one time at a time" problem.

The reason that Hamilton's name shows up next to Bellmann here is that physicists and mathematicians (Lagrange, Hamilton, Jacobi) figured out that you can do the inverse: You can write the dynamical equations of physics as an intertemporal (all times at once) optimization problem. This has been one of the most influential ideas in theoretical physics _ever_, especially after Noether's theorems leveraged this structure to connect symmetries and conserved quantities. In many fields of theoretical physics you no longer write down differential equations to model a dynamical system, you write down a "Lagrangian", and mean that your system follows those trajectories that minimize the integral of the Lagrangian over all time. So the central ideas of dynamic programming are extremely tightly related to physics and dynamical systems.

[1] https://en.wikipedia.org/wiki/Dynamical_system

Edit: After reading a bit more I think I understand why people focus on this. The algorithmic examples with shortest paths sort of show the point. To me the core point of Dynamic Programming is before you start thinking about recursion or memoization. It's when you establish that you have optimal substructures:

https://en.wikipedia.org/wiki/Optimal_substructure

what is dynamical is that the subproblems depend on each other, in the way that what happens at time t+1 depends on what happens at time t, or, more to the point, what is the optimal decision at time t depends on what the optimal decision at time t+1 is. If your subproblems are ordered by time, then of course you don't need recursion to solve them, you just iterate over time steps, hence my confusion.

I don't think that what you wrote implies they are static, but the name "dynamic programming" implies that the alternative is in some way static.

The alternative are optimization problems without explicit time dependency. As in linear programming.

I'm not sure that "static" is the appropriate antonym to "dynamic" here.

If we set aside the fact that the term "dynamic" was apparently chosen here basically because it sounded cool, and for no good reason, and are trying to retroactively redefine it in a way where the word has meaning, then:

With recursion the computation order is predefined and deterministic.

With dynamic programming the computation order is more flexible since the core of the technique is just memoizing prior work. Your DP solution could be organized in recursive top-down fashion, or more optimally in bottom-up fashion if the algorithm is strictly recursive by nature, or neither of the above.

i.e. With dynamic programming in general you are just reusing results if/when they exist, else calculating them - it a more "go with the flow" dynamic approach than recursion where the order of calls is strictly prescribed.

Again, if the term linear programming makes sense for linear optimization problems, then the term dynamic makes sense for time dependent ones.

Also: the core of dynamic programming is breaking down a global optimization problem into many interacting local pieces. The Bellmann equation, which is the core mathematical result of dynamic programming, features neither recursion nor memoization.

OK, although that seems to be a different (maybe more thoughtful!) use of the term "dynamic programming".

It reminds also of "blackboard-based" AI systems where a bunch of cooperating agents with different capabilities jointly work on a problem by posing sub-problems and sharing them on a common blackboard where other agents can grab them and post a solution back on the board.

Look at the Wikipedia article on dynamic programming. They are very much related. My impression is that the deep part of the math of dynamic programming, the decomposition into interrelated subproblems, is trivial in the CS text book examples. Then all that's left are implementation aspects.

Yup. I don’t mind it though as this field is riddled with names that convey absolutely nothing about the actual thing.

It's deliberately obscure. Per the article it's for promotion not description.

Same for "linear programming".

The "linear" in linear programming at least is related to the problem it's solving (linear optimization).

> Thus, I thought dynamic programming was a good name. It was something not even a Congressman could object to. So I used it as an umbrella for my activities.

That reminds me of, much later, getting my first basic programming book. In communist Romania. It was titled "Let's learn interactive microelectronics" because "programming" books were considered a waste of resources and did not get approved.

I thought communist nations heavily relied on programs (in the author's sense) for their planned economies?

No, on "plans". Mostly five year plans. At least in Romanian.

The only other use of "program" I remember from back then is "tv program".

According to [1], the optimization done in the Soviet Union (the largest planned economy) was based on linear programming.

[1]: https://www.rtsg.media/p/soviet-planning-demystified

But this subthread is not about what they were using but about localized names for computer programming vs other forms of programming.

Also I don't think Soviet examples help because I don't think the last few soviet dictators before the iron curtain's fall were against computers, while Ceausescu definitely was.

Huh, I've never realised this. It actually makes more sense if I revert to my native British spelling of programme in this sense.

A program can use dynamic programming to dynamically figure out the programme required to perform a computation.

Similar for “linear optimization”, which doesn’t refer to program optimization.

Ok, but how do you call what we are doing:

We have an interpreted language in which you can call "subroutines" by name. But you can build the name in you program "on the fly", for example out of data.

The article would probably be more complete with references to similar terms "linear program" or "integer program", unless this is also a slightly different meaning?

If you haven’t, everyone should read Eye of the Storm

It’s Bellmans autobiography and it’s very good though he doesn’t go as much into depth as I wanted on how he came up with DP and his inequality

I love that origin story.

James Gosling's infamous Emacs screen redisplay algorithm uses dynamic programming.

https://en.wikipedia.org/wiki/Gosling_Emacs

It computed the minimal cost path through a cost matrix of string edit operations (the costs depended i.e. on the number of characters to draw, length of the escape codes to insert/delete lines/characters, padding for slow terminals, etc).

The algorithm used is a dynamic programming one, where

   M[i,j] = MIN (M[i-1,j]+dcost,             # UP,   implies delete
                 M[i,j-1]+icost+redraw cost, # LEFT, implies ins+redraw
                 M[i-1,j-1]+rewrite cost)    # BACK, implies rewrite

Each terminal type would configure the display driver according to its supported escape codes and screen update times (some terminals were REALLY SLOW inserting and deleting lines or even scrolling, and you had to send null padding to wait for it to finish).

It's infamous both for its skull-and-crossbones warning comment [designed by Brian Reed] and correspondingly poisonous complex code, and also RMS's battle with UniPress software, incorporating it into Gnu Emacs, getting threatened by UniPress, then rewriting it from scratch.

https://features.slashdot.org/story/13/01/06/163248/richard-...

>Q: Give me your best hack. [...]

>RMS: I can't remember all the hacks that I was proud of, so I can't pick the best. But here's something I remember fondly. The last piece of Gosmacs code that I replaced was the serial terminal scrolling optimizer, a few pages of Gosling's code which was proceeded by a comment with a skull and crossbones, meaning that it was so hard to understand that it was poison. I had to replace it, but worried that the job would be hard. I found a simpler algorithm and got it to work in a few hours, producing code that was shorter, faster, clearer, and more extensible. Then I made it use the terminal commands to insert or delete multiple lines as a single operation, which made screen updating far more efficient.

There's some more discussion about it here that ended up being pretty accurate once all was said and done:

https://www.reddit.com/r/emacs/comments/bek5b2/til_emacs_was...

Emacs's infamous "Ultra-hot screen management package" with its "Skull and Crossbones" warning was definitely black magic!

The algorithm worked great and was well worth it over a 300 / 1200 / 2400 baud connection to a Vax 780 / 750 / etc, and as modems and computers got faster it was still useful, but at today's network bandwidths and cpu speeds it's thunderous overkill.

https://news.ycombinator.com/item?id=38996713

A redisplay algorithm, by James Gosling (ACM SIGPLAN Notices, April 1981):

https://donhopkins.com/home/documents/EmacsRedisplayAlgorith...

>7. Acknowledgements: The people who did the real work behind this paper are Mike Kazar, Charles Liescrson and Craig Everhart; all from CMU.

>Bibliography: 1. Kevin Q. Brown. Dynamic Programming in Computer Science. CMU, February, 1979.

That code was also used in Maryland Windows (a text based overlapping/tiled window system developed at the University of Maryland by Chris Torek, like Emacs - Text Editor + Window System, kind of like "screen" or "mux" or "mgr").

https://donhopkins.com/home/archive/emacs/mw/display.c

https://en.wikipedia.org/wiki/Dynamic_programming

https://wiki.c2.com/?DynamicProgramming

https://en.wikipedia.org/wiki/ManaGeR

https://news.ycombinator.com/item?id=22849522

>Gosling Emacs was especially noteworthy because of the effective redisplay code, which used a dynamic programming technique to solve the classical string-to-string correction problem. The algorithm was quite sophisticated; that section of the source was headed by a skull-and-crossbones in ASCII art, warning any would-be improver that even if they thought they understood how the display code worked, they probably did not.

https://donhopkins.com/home/archive/emacs/mw/display.c

    /*  1   2   3   4   ....            Each Mij represents the minumum cost of
          +---+---+---+---+-----        rearranging the first i lines to map onto
        1 |   |   |   |   |             the first j lines (the j direction
          +---+---+---+---+-----        represents the desired contents of a line,
        2 |   |  \| ^ |   |             i the current contents).  The algorithm
          +---+---\-|-+---+-----        used is a dynamic programming one, where
        3 |   | <-+Mij|   |             M[i,j] = min( M[i-1,j],
          +---+---+---+---+-----                      M[i,j-1]+redraw cost for j,2
        4 |   |   |   |   |                           M[i-1,j-1]+the cost of
          +---+---+---+---+-----                        converting line i to line j);
        . |   |   |   |   |             Line i can be converted to line j by either
        .                               just drawing j, or if they match, by moving
        .                               line i to line j (with insert/delete line)
     */

Trivia: That "Skull and Crossbones" ASCII art is originally from Brian Reid's Scribe program, and is not copyrighted.

https://donhopkins.com/home/archive/emacs/skull-and-crossbon...

                         /-------------\ 
                        /               \ 
                       /                 \ 
                      /                   \ 
                      |   XXXX     XXXX   | 
                      |   XXXX     XXXX   | 
                      |   XXX       XXX   | 
                      \         X         / 
                       --\     XXX     /-- 
                        | |    XXX    | | 
                        | |           | | 
                        | I I I I I I I | 
                        |  I I I I I I  | 
                         \             / 
                          --         -- 
                            \-------/ 
                    XXX                    XXX 
                   XXXXX                  XXXXX 
                   XXXXXXXXX         XXXXXXXXXX 
                          XXXXX   XXXXX 
                             XXXXXXX 
                          XXXXX   XXXXX 
                   XXXXXXXXX         XXXXXXXXXX 
                   XXXXX                  XXXXX 
                    XXX                    XXX 

                          ************** 
                          *  BEWARE!!  * 
                          ************** 

                        All ye who enter here: 
                    Most of the code in this module 
                       is twisted beyond belief! 

                           Tread carefully. 

                    If you think you understand it, 
                              You Don't, 
                            So Look Again.

But if you're not trying to support old terminals (but might still have a slow "thin wire" network connection), there is an orders of magnitude better approach: The Emacs NeWS display driver (for both UniPress and Gnu Emacs) downloaded PostScript code to define an efficient application specific network protocol with instantaneous local input tracking and feedback (unlike how X-Windows uses a fixed protocol, but like how AJAX uses JavaScript).

The source code to Gosling's UniPress Emacs 2.20 just recently surfaced, and the display code is well commented (still including the skull and crossbones and ascii art diagrams):

https://github.com/SimHacker/NeMACS/blob/main/src/DspVScreen...

And the lower level terminal driver layer:

https://github.com/SimHacker/NeMACS/blob/main/src/DspTrm.c

The NeWS terminal drivers I worked on for UniPress and Gnu Emacs were layered on top of that dynamic programming screen update code. But instead of sending escape codes to a dumb terminal, it downloaded PostScript code into the NeWS server to implement drawing, mouse tracking, text selection feedback, pie menus, tabbed windows, etc, and sent binary PostScript tokens back and forth (a practice now called "AJAX" for X = XML or JSON text instead of binary PostScript data and code):

TrmPS.c: https://github.com/SimHacker/NeMACS/blob/main/src/D.term/Trm...

TrmPS.cps: https://github.com/SimHacker/NeMACS/blob/main/src/D.term/Trm...

PostScript stuff for Emacs: https://github.com/SimHacker/NeMACS/tree/main/ps

Emacs 2.20 Demo (NeWS, multiple frames, tabbed windows, pie menus, hypermedia authoring):

https://www.youtube.com/watch?v=hhmU2B79EDU

Here's a brochure from February 1988 about UniPress Emacs 2.20 and "SoftWire" (NeWS without graphics, kind of like Node with PostScript instead of JavaScript).

What is Emacs: https://www.donhopkins.com/home/ties/scans/WhatIsEmacs.pdf

I also worked on the Gnu Emacs 18 NeWS display driver (supporting a single tabbed windows and pie menus in The NeWS Toolkit 2.0):

tnt.ps: https://www.donhopkins.com/home/code/emacs18/src/tnt.ps

tnt.cps: https://www.donhopkins.com/home/code/emacs18/src/tnt_cps.cps

tnt.c: https://www.donhopkins.com/home/code/emacs18/src/tnt.c

tnt-win.el: https://www.donhopkins.com/home/code/emacs18/lisp/term/tnt-w...

More on the NeWS versions of Emacs with links to code here:

https://news.ycombinator.com/item?id=26113192

Makes sense as the most widely known algorithm for finding the minimal edit distance between two strings also relies on DP. It was invented around 1968.

https://en.wikipedia.org/wiki/Wagner%E2%80%93Fischer_algorit...

In strength training 'programming' means the details of your workout plan. Like how many sets, reps and exercises.

That's a really interesting origin story. I had no idea, but I also totally believe it. I've seen things, man...

Top 3 worst name ever (along with "toxic masculinity" and "oxygen").

What would we call it if it were invented today?

Probably AI something.

The name "dynamic programming" was chosen to impress bureaucrats, and AI is where it is at today.

"Subtask training" maybe, as the idea is to train the algorithm on smaller tasks to to make it more efficient at solving larger tasks. Ok, it is just storing intermediate results, but it sounds more AI-ish like that.

Partial/Intermediate Solution Storage

Tabularized memoization?

Bottom-up memoization?

Iterative memoization?

I thought both recursive top-down memoization and iterative bottom-up tabulation are considered instances of "dynamic programming".

I read the post and all these comments and I still have no idea what dynamic programming is. I’m also slightly infuriated by the rationalization for the name mentioned in the post.

Many past related threads, but these seem to be the interesting ones. Other interesting ones?

Richard Bellman on the Birth of Dynamic Programming (2002) [pdf] - https://news.ycombinator.com/item?id=42482289 - Dec 2024 (9 comments)

Dynamic programming is not black magic - https://news.ycombinator.com/item?id=38988948 - Jan 2024 (202 comments)

Dynamic Programming vs. Divide-and-Conquer (2018) - https://news.ycombinator.com/item?id=26930667 - April 2021 (115 comments)

Real-world dynamic programming: seam carving - https://news.ycombinator.com/item?id=20285242 - June 2019 (66 comments)

Graphical Introduction to Dynamic Programming - https://news.ycombinator.com/item?id=19682258 - April 2019 (11 comments)

Dynamic Programming for Technical Interviews - https://news.ycombinator.com/item?id=19395578 - March 2019 (218 comments)

Solving dynamic programming interview problems - https://news.ycombinator.com/item?id=17102604 - May 2018 (164 comments)

Dynamic Progamming: First Principles - https://news.ycombinator.com/item?id=15545686 - Oct 2017 (98 comments)

What is difference between memoization and dynamic programming? - https://news.ycombinator.com/item?id=13281884 - Dec 2016 (1 comment)

Dynamic Programming: The Name (1984) - https://news.ycombinator.com/item?id=12476597 - Sept 2016 (61 comments)

Fibonacci series and Dynamic programming - https://news.ycombinator.com/item?id=5893237 - June 2013 (28 comments)

Dynamic Programming versus Memoization - https://news.ycombinator.com/item?id=4434671 - Aug 2012 (26 comments)

Visualizing dynamic programming - https://news.ycombinator.com/item?id=2243297 - Feb 2011 (11 comments)

Ask HN: Have you ever used Dynamic programming in your dev job? - https://news.ycombinator.com/item?id=1796861 - Oct 2010 (11 comments)

Why the name "dynamic programming"? - https://news.ycombinator.com/item?id=1290397 - April 2010 (25 comments)

The Edit Distance Problem - https://news.ycombinator.com/item?id=571570 - April 2009 (13 comments)

Introduction to Dynamic Programming - https://news.ycombinator.com/item?id=329953 - Oct 2008 (12 comments)

Looks like someone discovered this reddit post [1] and wrote a whole blog around the top answer. Since the example is copied verbatim, it might even be an LLM that was hooked up to web search.

[1] https://www.reddit.com/r/learnprogramming/comments/1ac9zbl/d...

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