The Great Tree-List Recursion Problem (2000) | Hacker News Reader

The Great Tree-List Recursion Problem (2000)

cslibrary.stanford.edu

Isn't this just in-order traversal...

    class Linked:
        def __init__(self, data, left=None, right=None):
            self.data = data
            self.left = left
            self.right = right
    
        def tree_to_list(self):
            if self.left is not None:
                first, left = self.left.tree_to_list()
                left.right = self
                self.left = left
            else:
                first = self
    
            if self.right is not None:
                right, last = self.right.tree_to_list()
                right.left = self
                self.right = right
            else:
                last = self
    
            first.left = last
            last.right = first
    
            return first, last

This is one of the problems in 'Elements of Programming Interviews' book.

Convert a sorted doubly linked list into a BST

C++: http://elementsofprogramminginterviews.com/solutions/cpp/sor...

Java: http://elementsofprogramminginterviews.com/solutions/java/So...

Python: http://elementsofprogramminginterviews.com/solutions/python/...

Convert a BST to a sorted doubly linked list

C++: http://elementsofprogramminginterviews.com/solutions/cpp/bst...

Java: http://elementsofprogramminginterviews.com/solutions/java/BS...

Python: http://elementsofprogramminginterviews.com/solutions/python/...

The reversed problem is more interesting: given a doubly-linked sorted list (and its length), convert it to a balanced binary search tree.

https://mitpress.mit.edu/sicp/full-text/book/book-Z-H-16.htm...

That's the DSW algorithm.

https://en.wikipedia.org/wiki/Day-Stout-Warren_algorithm

Pretty trivial, since you can easily find the middle of a doubly-linked list and step back. Then you simply do "find middle, use it as root, recursively build two subtrees".

With a singly-linked list, it's much harder; I don't know (without googling) if a more efficient than "build and rebalance as you go" algorithm exists.

If you know the length of the initial list (and that's obviously possible in O(N)), it's all easy.

You could convert a singly-linked list to a doubly-linked list in O(n), and then apply your favorite doubly-linked solution.

You could even do this in place without increasing the size of each node at the cost of doubling the size of variables that point to nodes and slightly lowing down stepping through the list. You do this by storing the next and previous pointers for each node in the same location, xor'ed together.

That doubles the size of variables that point to nodes and slows down stepping because you have to turn them into structs that store both a pointer to the node and a pointer to an adjacent node and update both of these pointers as you move through the list.

If you're willing to burn O(n) memory on it you could just create an array of pointers to the nodes in the list and then use that to build the balanced tree. It's probably faster than iterating over the sublist every time to find the center.

This assumes of course that the list is sorted. If not, you should probably sort the list first to avoid driving yourself insane.

If memory is too tight to make temporary storage and the list is too big to sort and you want a perfectly balanced tree then you've got an interesting problem on your hands.

You can sort a linked list in O(nlog(n)) time with O(log(n)) space, and then convert it to a balanced binary tree in O(nlog(n)) time and O(log(n)) space.

EDIT: demonstration:

  sortList [] = []
  sortList [x] = [x]
  sortList xs = merge (sortList as) (sortList bs)
    where
      (as, bs) = split xs

  split [] = ([], [])
  split [x] = ([x], [])
  split (x:y:rest) = let (xs, ys) = split rest in (x:xs, y:ys)

  merge xs [] = xs
  merge [] ys = ys
  merge (x:xs') ys@(y:_) | x <= y = x : merge xs' ys
  merge xs (y:ys) = y : merge xs ys

  data Tree a = Empty | Node a (Tree a) (Tree a) deriving(Show)

  balancedTree xs = mktree (length xs) xs
    where
      mktree 0 [] = Empty
      mktree n xs = Node x (mktree n' less) (mktree (n - 1 - n') more)
        where
          n' = (n - 1) `div` 2
          (less, (x:more)) = splitAt n' xs

  listToBalancedTree :: Ord a => [a] -> Tree a
  listToBalancedTree = balancedTree . sortList

Lol, I like how no one is talking about DSW in the replies to the comment about DSW.

> [F]ind middle, use it as root, recursively build two subtrees.

That's not DSW. DSW should be way more efficient.

> With a singly-linked list, it's much harder; I don't know [...] if a more efficient [...] algorithm exists.

This is DSW. DSW starts with a sorted linked list--called a "vine"--and builds the tree from that, in place. The vine is a singly linked list in the sense that you only need the successor pointer (though each node obviously needs space for a predecessor pointer since it's being turned into a tree).

The algorithm is O(n) time and O(1) space and is actually very elegant. Essentially, you treat the list as a completely unbalanced tree skewed to the right. Then you repeatedly apply a `compress` routine that scans down the far right path in the tree to balance it.

From Wikipedia

    routine vine-to-tree(root, size)
        leaves ← size + 1 − 2^⌊log2(size + 1))⌋
        compress(root, leaves)
        size ← size − leaves
        while size > 1
            compress(root, ⌊size / 2⌋)
            size ← ⌊size / 2⌋

    routine compress(root, count)
        scanner ← root
        for i ← 1 to count
            child ← scanner.right
            scanner.right ← child.right
            scanner ← scanner.right
            child.right ← scanner.left
            scanner.left ← child

Both the naive algorithm and DSW are O(n), but the naive algorithm has a time complexity of about 2n while DSW has a time complexity of about 0.63n, assuming we start with a vine.

Here is a C library I wrote in 1999 which can convert, in-place, a container between Red-Black, AVL and linked list:

http://www.kylheku.com/~kaz/austin.html

The declarative programming language Prolog is very suitable for reasoning about trees, lists and recursive structures in general.

For example, here is a Prolog definition of the tree from the article:

    tree(node(4,
              node(2,
                   node(1,nil,nil),
                   node(3,nil,nil)),
              node(5,nil,nil))).

I am using terms of the form nil/0 and node/3 to represent the tree.

Let us start by converting the tree to a list. As requested, we shall consider the in-order traversal of the tree elements, described as a list. For such situations, Prolog DCG notation is very convenient, since it easily allows us to describe a list in relation to other terms (in this case, a tree):

    tree_to_list(nil) --> [].
    tree_to_list(node(E,Left,Right)) -->
            tree_to_list(Left),
            [E],
            tree_to_list(Right).

Here is a sample query and its result:

    ?- tree(T), phrase(tree_to_list(T), Ls).
    T = node(4, ...),
    Ls = [1, 2, 3, 4, 5].

OK, so we now have the in-order traversal. Of course, lists are already implicitly "linked" in Prolog in one direction: Given a list, we can readily reason about the first element and the remaining elements. The task requires us to do more, i.e., reason about the reverse direction as well.

One way to express this is to use rational trees, also known as cyclic terms in Prolog. For example, here is a case of cyclic term:

    ?- append([a,b,c], Ls, Ls).
    Ls = [a, b, c|Ls].

Here, Ls represents the circular list [a,b,c,a,b,c,a,...].

We can use this feature to our advantage as follows:

    list_with_backlinks([], []).
    list_with_backlinks([L|Ls0], [L-Prev|Ls]) :-
            append([L|Ls], Cs, Cs),
            foldl(with_backlink, Ls0, Ls, Cs, Prev).

    with_backlink(E, E-Cs0, Cs0, Cs) :- Cs0 = [_|Cs].

Sample query and answer:

    ?- list_with_backlinks([a,b,c], Cs).
    Cs = [a-[_S2|_S1], b-_S1, _S2], % where
        _S1 = [a, b-_S1, _S2|_S1],
        _S2 = c-[b-_S1, _S2|_S1] .

Thus, as desired, we can generate a doubly-linked list from the original tree:

    ?- tree(T), phrase(tree_to_list(T), Ls0),
       list_with_backlinks(Ls0, Ls).
    T = node(4, ...),
    Ls0 = [1, 2, 3, 4, 5],
    Ls = [1-[_S4|_S1], 2-_S1, _S2, _S3, _S4], % where
        _S1 = [1, 2-_S1, _S2, _S3, _S4|_S1],
        _S2 = 3-[2-_S1, _S2, _S3, _S4|_S1],
        _S3 = 4-[_S2, _S3, _S4|_S1],
        _S4 = 5-[_S3, _S4|_S1] .

Of course, this does not completely solve the task, which requires in-place modifications of the original tree. Pure Prolog lacks destructive modifications, so we cannot fully translate the code to Prolog. On the plus side, no pointers at all were necessary, and the recursions are pretty straight-forward.

Neat!

Also the C answer is missing a brace at the end of treeToList and that makes me a little crazy

Not really a useful problem, considering that it will be impossible to solve if the tree would be properly balanced.

What do you mean?

I think it is irrelevant whether the binary search tree is balanced or not. Why do you think it matters?

I don't agree. The tree being balanced of not has not bearing on the solvability of the problem.

Even a naive approach would work balanced or unbalanced and that would be simply to pop the lowest value in the tree and construct your list until you run out of nodes.

The trick is to do it in O(n) time and use recursion.

In a time where we're finally realizing the benefits of functional programming (i.e. no side effects), a problem like this really seems like a callback to a more barbaric era. Like writing a state machine using gotos.

EDIT: Never expected this to devolve into an argument about the merits of gotos! I thought we were all past that, but apparently not.

Asking honestly because I would like to know: What is wrong with writing a state machine using gotos? Assuming that one is solving a problem that definitely needs a state machine for some reason.

Gotos are generally considered a bad idea in high-level programming.

http://www.u.arizona.edu/~rubinson/copyright_violations/Go_T...

This is lazy thinking, the gotos described in that essay can jump anywhere in the program while modern gotos are limited to the function scope and have various warnings about initialization order and whatnot.

All the arguments it makes no longer apply but people still quote the snappy headline.

That essay does make an excellent argument against programming with many tiny functions though, as those do cause the execution cursor to jump all over the source code document.

It seems to me that a substantial fraction of the people who quote that paper, assuming they have even read it, have failed to think critically about it.

https://maniagnosis.crsr.net/2011/11/link-o-day-donald-knuth...

> What is wrong with writing a state machine using gotos?

Nothing. But the thing they want to do -- altering the datastructure on the fly -- is not the best way to solve the problem.

I would solve this by writing two functions: one that creates an array from a binary tree, and one that creates a doubly-linked list from an array.

Would this use up more RAM and take longer to run? Yes

Would the code be easier to understand (and therefore more bug-free)? Yes

Nothing wrong with goto, as long as it's clear.

Many programmers treat 'goto' the same way that we treat radioactive waste, but they don't know why. Perhaps it is a holdover from courses with teachers who automatically graded 'fail' on programs that used a goto. Perhaps the programmers have only heard that gotos were bad.

Sometimes 'goto' is the correct answer. For state machines it can be an pretty efficient technique. For "ordinary" code it is sometimes the best way out of a situation (e.g., busting out of a deeply nested set of loops).

The "Goto considered harmful" movement came out of the 1970s, when GOTO was used all the time and the normal state for a big project was abject spaghetti. I think we can all agree that was a very bad thing. But the utter avoidance of goto is an overreaction that seems to be based on unreasoning fear, or toxic peer pressure.

>busting out of a deeply nested set of loops

so, goto is efficient at escaping inefficiently structured code? That doesn't make it really efficient. I mean there's a reasonable rule of thumb: don't indent the code too deep. Because what you are talking about is one kind of spaghetti, the kind cut into digestable chunks. The interleaving of active and inactive blocks leads to unreadable code sooner or later.

It's "barbaric", kinda like eating with your hands.

Of course this type of dogmatism gives us FP gems akin to [eating] a piece of Sushi with a steak knife.

Gotos are a natural choice for state machines, we basically emulate goto when we use while+switch or while+function pointer. The "goto considered harmful" paper is about structured programming, saying that you should use structured programming instead of goto, but it doesn't apply here because goto is the right structure to begin with.

I suggest reading the actual "goto" paper sometime… it's a classic.

I just want to add that the "goto" criticized on that paper does not exist anymore in any modern language (that includes C). Goto has been nerfed into something that can not destroy structured programming.

But I do think the while+switch representation is superior.

The while+switch, to me, looks like goto with extra typing.

    state = START;
    break;

Translates directly to:

    goto start;

Except while+switch is nicer if you have transition tables. YMMV

Yes, but it forces you to keep your state as data, and makes it much more clear how the data relates to the code.

But it does make for the same exact behavior.

The state is data either way... either an explicit variable or the program counter. With goto, the compiler is more likely to be able to see across state change boundaries (e.g. phi() will work correctly without doing anything fancy). So you can, e.g., have the compiler help you out by giving you warnings for uninitialized variables, when the variables are initialized in one state and used in another.

> I just want to add that the "goto" criticized on that paper does not exist anymore in any modern language (that includes C).

It might not be called goto, but isn't setjmp/longjmp essentially the same?

No... you can only longjmp if the jmp_buf corresponds to a setjmp created within a function call which hasn't returned yet and in the same thread. I would call this a primitive form of structured programming with some serious limitations--it's kind of like throwing an exception, but more primitive, and it's kind of like the Common Lisp block/return-from, except less convenient, and it's kind of like call/cc but with a bunch of limitations added.

You can do some dangerous things with longjmp to implement coroutines, but it's horribly non-portable.

Pattern matching is so much sexier than switch statements.

Writing state machines is one of the moments when the 1d top-down method of programming really hurts. I feel like I can easily understand an FSM drawing, but splay it out in code and it's difficult to track.

Pssst! Lambda, the ultimate goto.

http://library.readscheme.org/page1.html

You could always clone the tree into a new linked-list if you wanted to keep things purely functional, but that would add needless complexity and still not make it easier to create a correct implementation.

Functional programming as a paradigm makes sense when you want to limit the context a programmer has to be aware of when working on a problem, but in this case recursion and imperative programming accomplish the same goal.

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