B-Trees Require Fewer Comparisons Than Balanced Binary Search Trees
databasearchitects.blogspot.com
databasearchitects.blogspot.com
And that I do remember. I suspect he had been doing lecture planning over the weekend (this was in the "we're redoing the class, teach it as it needs to be taught" type mandate) and he had us skip that section of the text. The fist lecture he was going off of overhead transparencies that he was writing as he lectured. The second lecture he gave us photocopies of hand written notes and diagrams.
Instead of AVL and red black trees, we instead learned about 2-3 trees and B trees.
For what it's worth, I still know how to do 2-3 trees from scratch.
Later I looked at AVL trees... and they still give me a headache.
1) that there are many programs that implement the same algorithm, and many algorithms that implement the same function.
2) that there are a few basic tradeoffs we can make when implementing a function/customising an algorithm: space vs time, read vs write, deterministic vs indeterministic, etc.
I'd guess trees have traditionally been used, because there's a whole zoo of them, which clearly presents (1) and allows (2) to be learned (in "cookbook" style) even if they're never explicitly presented (in "textbook" style).
I agree they suck tho.
When we first were taught about B trees in my databases course in college, he professor quipped something like "these are the reason you'll never actually use a red-black tree in practice". I've actually found that in practice I rarely ever up actually needing to use any sort of data structure that maintains sort order; for whatever reason, I've run into cases far more often where I need to maintain insertion order than sort order, but I'm not confident about whether this is due more to the type of stuff I work on or if it's just a more common requirement in general.
Yes really, that’s why the vast majority of languages default to a hashmap (if they even have tree-based maps at all). Turns out the average application is a lot more interested in a low-constant-factor O(1) best case than in a high-constant-factor O(log n) worst case.
> Hashmaps don't have guaranteed performance characteristics
Of course they do.
> require the "correct" hash function to work properly.
And tree maps require correct comparators to work correctly.
> In particular, their performance degrades as they become more "full" requiring a re-allocation to a table of generally n-times the size.
Yes? Also “n-times” seems like pretty fuddy fud. And it seems weird to be more worried by the reallocation of a hashmap when a tree requires an allocation per node, so your average balanced tree requires a heap allocation per entry. And two pointers, for up to 16x overhead not including the allocator’s own metadata.
> Balanced trees could be regarded a more generic solution to the mapping problem and are more suitable in many applications.
And the amphibious cycle could be regarded as the more generic solution to the transport problem.
>Of course they do
No, they do not, at least not worst case O(1). If you write a very poor (but functional!) hash function or an adversary is choosing your items, then congratulations, your hashmap is now a linked list with extra steps.
Sure, you could use a robust cryptographic hash function, but you didn't because (a) it's not the default in your language and (b) it's slow and you are "a lot more interested in a low-constant-factor O(1) best case".
Otherwise, you're more or less correct.
In my own experience, most uses of hash maps are typically O(logn). A good rule of thumb is hash maps are preferable unless you need predictable worst case performance or your hash maps are directly exposed to the public.
Also, obviously, features only one of them provides e.g. if you need range queries then a hash table is quite useless. Meanwhile if you need insertion ordering, that’s reasonably easy to do with a hashmap.
Yes, you can throw the nodes on an array in the heap to allow tail recursion, but at that point you're going to be losing to a map with all but the biggest datasets, and even then the copies really aren't desirable
Yes, it requires allocation for every entry whereas a hash map requires it for some entries. You don't know which ones. So real time constraints are out the window. But no it's not something your average web programmer (including me) needs to worry about. Some people do, though.
For additional context, I work almost entirely in Rust nowadays, and BTreeMap and HashMap are the two map types in the standard library (https://doc.rust-lang.org/std/collections/index.html). I've ended up needing insertion-ordered maps quite often in my work though, which is not something I've seen taught very often or in standard libraries, although there are plenty of third-party implementations out there, and in most languages it's not too difficult to hand-roll an implementation using an unordered map and a linked list by keeping both the value and a reference to the node containing the element in the map, which provides constant time insertion, lookup and deletion. (Rust is somewhat notorious for _not_ being a language where implementing something like this by hand is super easy though, since tracking mutable references to individual nodes of a linked list across separate hashmap values is not something the borrow checker is enthusiastic about)
I've had the exact opposite experience, so I'm guessing it has a lot to do with the type of stuff you've (and I've) worked on.
I think I've had to implement my own traversable data structure literally only in interviews
I'll note that the instructor for this was Professor David DeWitt - https://en.wikipedia.org/wiki/David_DeWitt
He normally taught databases and one of my great regrets was dropping his 500 level database class one semester when things were rough. (Another regret was not taking networking from Professor Landweber https://en.wikipedia.org/wiki/Lawrence_Landweber .)
It could be that my professors were simply not that good, so that I don't really understand this. I mostly groked the subjects by working through the books and exercises in them. The professors' lectures almost always went over some detail too quickly, so in a 3 hour lecture, I usually ended up following only the first hour or so.
My question is maybe better put as: what quality or approach makes them great lecturers?
I had his class before (the aforementioned sophomore data structures and algorithms) and I knew he was a good lecturer. That wasn't a question. It was a "too much going on that semester and I wasn't able to do the assignments in a timely manner".
Professor Landweber's class had a waiting list that I scoffed at as a college student and looked for something that I could get into and fill the requirement without realizing who he was. It wasn't until later while reading the UNIX and Linux System Administration Handbook and seeing an image in there that was attributed to him that the name clicked but he wasn't teaching that next semester and I had filled that requirement elseclass.
Another regret was dropping numerical methods with Professor de Boor ( https://en.wikipedia.org/wiki/Carl_R._de_Boor ) when I couldn't get my head around splines rather than going to his office hours (he was also the undergraduate advisor and very helpful).
Those classes could have changed the trajectory of my career. I was looking at being a sysadmin instead and writing perl and keeping some big iron running. My career pivoted from sysadmin to webmaster to web developer (perl) to web developer (Java). I have no regrets from that path - I do have regret for not learning what I could from the experts in their field when I was in a place and time to learn from them.
I'm not sure what was going through his head at the time, whether he'd just learned how to write them at college, or if he just thought they were the pinnacle of engineering.
None of the cases they were trying to shoe-horn them in to made sense. Places where you might debate that even a straight array would be sufficient, due to the limited number of items. The worst case of searching to the end of the array wouldn't matter. Doubly so given they weren't working on anything anywhere near a hot path, or a time sensitive operation. Standard library components would have been perfectly fine, and not carried the maintenance burden.
They did, on at least a couple of occasions, smugly explain what a black-red tree was to their mentor, who had likely implemented their first red-black tree at college before that Intern had ever even touched a computer for the first time. He showed remarkable patience.
That's understandable and rather common with new fresh grads. They have the energy and motivation but may not quite have the experience. We were all there once, including me.
My history professor from 8th grade used say: "There is a difference between knowing what you're doing and doing what you know".
I think it's easy to forget that a lot of interns might never have actually worked on a "real" codebase before; I know I certainly hadn't before my first internship! For someone who had only ever worked on homework code in classes before and maybe some side projects for fun, it's very easy to imagine that they honestly didn't understand that implementing the hardest data structure they knew by hand wouldn't actually be an impressive display of engineering to the company they were hoping to get a job offer from.
Then you work on real software and realize that computers are pretty fast and basic hashmaps & arrays will likely suffice for the size of the datasets you're working with, and beyond that, performance is more about tool/database choice than rolling your own algorithms.
This particular intern was dead set on this red-black tree, despite it being rejected constructively, repeatedly, every time they tried to introduce it, and despite mentoring on the subject from the experienced developer who was his mentor, and guidance from other engineers in the group.
The only intern I every had to deal with who didn't seem to recognise that production code is different from academic code, where the balance on clever vs maintainable is vastly different :)
Left Leaning Red Black Trees Considerer Harmful: https://www.read.seas.harvard.edu/~kohler/notes/llrb.html
I believe it was the 1992 article by Bruce Schneier:
https://jacobfilipp.com/DrDobbs/articles/DDJ/1992/9204/9204c...
Why? From what I remember AVL trees were much easier to implement than B-trees. Especially if using recursion.
Recently implemented them for an advanced hobby project and having a solid fuzzing tester in place really alleviated a lot of headaches in the long term since it was able to produce quite an exhaustive set of conditions.
In practice, I find that a B-tree implementation is 2 to 3× the length of an AVL tree impl.
I also fuzz-tested my implementations thoroughly. https://www.nayuki.io/page/avl-tree-list , https://www.nayuki.io/page/btree-set , etc.
B-trees require a much longer implementation than AVL trees. https://www.nayuki.io/page/btree-set
if you inline the red nodes into the black nodes, you get a 2-4 tree node. except that the 3-nodes have chirality. but then, you can just use a 2-4 tree
AVL trees seem not worth learning. they have slightly different performance characteristics than red-black trees. but, if i care about performance beyond big O, i'm probably using a hash table or a cache-aware tree of some kind instead of a CS101 data structure
A B-tree of any branching factor (not only 2) is equivalent to a binary tree.
We can take any B-tree node n:
n
|
---------+-----------
| | | | | |
a b c d e f
n
|
/ \
a /\
b/\
c/\
d/\
e/\
f nil
What the unbalanced tree cannot do is binary search through a to f to find which one to descend into; it doesn't have those nodes in a neat array for that.
That binary search makes B-tree more or less equivalent to a very balanced binary tree.
I think the b-tree is using the start/end sequence knowledge at each to do fewer node evaluations. That is, a b-tree is "shorter" so of course it'll take fewer node evaluations to reach the leaf.
"b-tree is equivalent to balanced binary tree", and "unbalanced tree", don't make a ton of sense when ostensibly the article compares b-trees to balanced BST.
[1] https://en.algorithmica.org/hpc/data-structures/binary-searc...
The fact that we can search a B-tree node that has, say, 256 children in 8 steps means that the node effectively contains the equivalent of a small balanced binary tree.
The takeaway was that the break-even point was suprising small - just tens of elements, which is crazy considering a CPU can do like 2x 8-wide AVX2 integer comparisons in a single cycle. Modern CPUs are just that good at executing branchy code.
But with B-trees we can have the best of both worlds - trees with a relatively high branch factor, like 16, that can still be tested in one go using SIMD.
Another advantage is cache locality - which is one of the reasons B-trees are favored when implementing filesystems - much of the tree doesn't need to be fetched into memory while the parts that do are contiguous in memory.
If the whole tree is in CPU cache then likely the difference is noise, but if the tree is large enough to have some data in RAM then the cost to copy that into cache will be larger based not on the size, but on the amount of queries from the CPU to the RAM (unless the size it way out of proportion). RAM has latency that is generally an order of magnitude slower than CPU Cache and generally an order of magnitude slower than fast contemporary storage.
There are many places you can find "Latency numbers every programmer needs to know" https://fullstackengineer.pro/blog/latency-numbers-programme...
But this trend has held for about 30 years and seem fundamental to the economics of building hardware. CPUs will be fastest, and cache slower, and RAM slower still , and fast storage still slower etc.
From your example, if it takes 100ns to get the node containing a~f then the whole of the B-Tree search for f might take 150ns (1 lookup from RAM + a few CPU intructions), but with the strictly binary tree it will need 6 lookups to RAM, and we can see that we can ignore the CPU time because it is so small. This could take 600ns (or 1300ns if each branch is a node with only pointers and the data needs to be dereferenced too). A smart cache prefetcher might guess that these were pointers and precache a and b once n was referenced, and might even cache things pointed to by loaded pointers but even in that highly optimistic scenario it could still take 300ns to load stuff.
Consider filesystems on storage and how they have had this problem but worse for a long time. Tree structures with many pointers are faster than binary trees in proportion to latency between the thing processing the tree and the place the tree it stored.
EDIT - And don't get me started on clever use of SIMD instruction. Imagine if node in a tree had binary layout matching an AVX register, then comparisons looking for the searched item could check a whole node in a single instruction! I am curious if there are any implementations that do this?
EDIT 2 - It does exist and I am late to the show, people have been doing this for like 15 years: https://stackoverflow.com/questions/20616605/using-simd-avx-...
It's simply because the nodes themselves contain the equivalent of a balanced binary tree, in the form of a binary-searched array.
In the ultimate case, we set the number of children to be large enough to contain all the data, so then there is only one B-tree node, which is binary searched to find the element. Then we have the same number of comparisons as well balanced binary search tree.
So for practical sizes of M, as in a few cache lines, and sufficiently large datasets of size N a binary tree will eliminate 1/2 of data each node checked and a B tree with M set to 8 would skip 7/8ths of the data each node. When not descending nodes they are both in Log2(N) time, but when jumping nodes this hypothetical B-Tree is using Log8(N) time.
So yeah if the N and M are close then they are similar, but those aren't interesting cases. Datasets that small are fast to iterate even if unsorted, and large unwieldy values of M are silly and impractical. Once you have thousands or millions of items in N then the savings of eliminating downtree data from checked at all becomes real. Or put another way binary trees are always averaging Log2(N) amount of checks but B trees have some mix of Log2(N)+LogM(N) amount of checks which should be lower in most cases and so close in edge cases as to not matter.
But again, I state that counting comparisons is silly. On any modern hardware (past 20 years) load time dominates the comparison time and B-Trees will just have fewer loads, memory complexity is the "interesting" part of this problem unless you are a C64 or some zany future company with hyper fast RAM.
Keep on thinking. You never know what else you will come up with.
A lot of string comparisons use SIMD now. In C++ if you use std:string on GCCit has done it for years. I bet Rust and Python (in the compiled runtime) ate doing it too.
No sense leaving easy performance gains on the table.
That is true on-disk, where you would not use BSTs anyway. In-memory, mid to high single digit values are practical. For instance iirc Rust’s btree (map and set) uses something like k=6.
... for absl::btree_set<int32_t>, nodes currently have 62 children ...
[1] https://github.com/rust-lang/rust/blob/master/library/alloc/...
[2] https://github.com/abseil/abseil-cpp/blob/74f8c1eae915f90724...
[3] https://github.com/abseil/abseil-cpp/blob/74f8c1eae915f90724...
The AMD64 cache line is only 64 bytes, that would make for very low branching factors given interior nodes need a pointer per child, plus key, plus record pointer if b-tree (as opposed to b+tree).
Binary tree is a good default choice for sorted structure.
Underappreciated structure: sorted arrays (but come with big caveats).
B-tree also enjoys a *massive* advantage when your access block size can hold many keys. With big data this will always win.
For instance, fewer values per node means less information to copy when copying the node. Locking is more granular with fewer values per node because fewer values are locked at a time.
The binary structure also makes it possible to support randomized balancing and self-adjusting strategies like splay trees.
I am disappointed to still see frequent mention of red-black trees – I see no reason for red-black trees to ever be the best choice in practice, or in theory, or in school. LBST's (logarithmic binary search trees) [1] are generally simpler, more intuitive, and more efficient [2] than red-black trees. They also support positional access inherently, so the balancing information is useful (subtree size).
[1] https://www.semanticscholar.org/paper/A-New-Method-for-Balan... [2] https://rtheunissen.github.io/bst
But this also assumes single-threaded computation. Increasingly, high-performance computers are SIMD-compute (aka: GPUs, and if not, at a minimum you're using AVX512 or ARM SVE).
If you have 1024-threads per thread group (common maximum in GPUs), that will commonly be used to create 1024-element B-Tree nodes, as sorting can be accomplished in a single pass network. (Bitonic network: https://en.wikipedia.org/wiki/Bitonic_sorter)
A singular lock that covers the whole 1024-node would better match these very wide GPUs that perform very wide SIMD-execution in practice. Physically, GPUs are 32-wide for NVidia and 32-wide or 64-wide for AMD. But various software tricks increase the width by either ganging-up neighboring compute-units in parallel and synchronizing them... or by running the same code across sequential passes like in AMD GCN architecture.
However, the programming interface of a 1024-wide (looking) piece of hardware/equipment naturally maps to 1024-wide B-Tree nodes, a 1024-wide Bitonic sort (or MergePath sort), and other such parallel sorting networks.
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B-Trees are "some number larger than 2", which I think for modern high-performance SIMD-based GPUs, the number absolutely should be "more than 2".
Where I'm wondering, is if the optimal algorithm is 32-wide (aka: one NVidia wave, the smallest unit of computation on an NVidia GPU), or if the optimal algorithm is 1024-wide (the maximum gang of waves that can work as a team). Or if maybe using the said maximum-gang across an even larger node (ex: 4096-wide) has any benefits.
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We are reaching the point where compute is so cheap that even a 1024-wide parallel sort is a trifling thing.
After n grows large enough, I think you’d want to benchmark your -exact- dataset on your -exact- prod hardware. Some datasets may cause certain trees to perform badly. Some hardware is much faster at memory lookups and cache misses are not as painful.
For instance, developing locally on Intel and deploying to Arm could yield pretty wild performance differences!