The Debit Method

An explanation of the debit method for analyzing the time complexity of purely functional data structures in Haskell.

Table of contents

Introduction
Message 1 - Concatenating two lists
Message 2 - Debit passing

This article is a bunch of mailing list posts for now.

Introduction

To analyze the runtime of purely functional data structures in a lazy language, Chris Okasaki developed the debit method in his book.

The following two messages explain it.

Message 1 - Concatenating two lists

Abhay Parvate wrote:

I somehow thought it would be easy to talk about complexity of calculating individual elements in an infinite list should be sufficient, but that seems to be involved, and my over-generalization doesn’t seem to work. Thanks for the link; particularly it has reference to Wadler’s papers exactly on this problem.

Note however that Wadler’s and similar formalisms are still a unsatisfactory in that they are quite clumsy to work with, it’s quite tedious/impossible to analyze examples with a lot of lazy evaluation. But they are a good guideline.

In his book about purely functional data structures, Okasaki takes a different approach; each node of a data structure is given a debit, a cost to evaluate it. For instance, consider

xs = x1 : x2 : x3 : ... : xn : []
        1    1    1 ... 1    1  0

ys = y1 : y2 : y3 : ... : ym : []
        1    1    1 ... 1    1  0

The numbers below indicate the time it takes to evaluate the node to weak head normal form. For demonstration purposes, I arbitrarily chose 1 for each (:) here.

The combined list will then have debits like

xs ++ ys = x1 : x2 : x3 : ... : xn : y1 : y2 : y3 : ... : ym : []
              2    2    2 ... 2    2    1    1    1 ... 1    1  0

In other words, the ys list is copied verbatim but each element of xs incurs an additional cost of 1, corresponding to one step in the evaluation of the concatenation with (++).

In order to force/inspect a constructor/node, you have to pay off its debits first. In the above example, head (xs ++ ys) would have to pay 2 units of time (one unit for head xs and one for the (++)). Now, the thing about debits is that we can relocate them to the top and only overestimate the total running time if we do that. For instance, we could push all debits to the top

xs ++ ys = x1   : x2 : x3 : ... : xn : y1 : y2 : y3 : ... : ym : []
              2n+m   0    0 ... 0    0    0    0    0 ... 0    0  0

so that evaluating head (xs ++ ys) is now estimated to cost (2n+m) units of time while the rest is free/fully evaluated.

The above example is rather useless, but consider the case n == m and

xs = x1 : x2 : x3 : ... : xn : []
        0    0    0 ... 0    0  0

ys = y1 : y2 : y3 : ... : yn : []
        0    0    0 ... 0    0  0

i.e. two fully evaluated lists of the same length. Then, we have

xs ++ reverse ys =
   x1 : x2 : x3 : ... : xn : yn : y{n-1} : ... : y1 : []
      1    1    1 ... 1    1    n        0 ... 0    0  0

because reversing the list ys is “monolithic”, i.e. looking at its head already forces the tail of the list. But now, we can distribute the debits upwards

xs ++ reverse ys =
   x1 : x2 : x3 : ... : xn : yn : y{n-1} : ... : y1 : []
      2    2    2 ... 2    2    0        0 ... 0    0  0

and thereby amortize the cost of reversing the second lists over the n elements of the first list. This is used in the implementation of purely functional queues, see also Okasaki’s book.

Regards,

apfelmus

Message 2 - Debit passing

Ronald Guida wrote:

Thank you, apfelmus. That was a wonderful explanation; the debit method in [1] finally makes sense.

A diagram says more than a thousand words :)

My explanation is not entirely faithful to Okasaki, let me elaborate.

In his book, Okasaki calls the process of transferring the debits from the input

xs = x1 : x2 : x3 : ... : xn : []
        1    1    1 ... 1    1  0

ys = y1 : y2 : y3 : ... : ym : []
        1    1    1 ... 1    1  0

to the output

xs ++ ys = x1 : x2 : x3 : ... : xn : y1 : y2 : y3 : ... : ym : []
              2    2    2 ... 2    2    1    1    1 ... 1    1  0

“debit inheritance”. In other words, the debits of xs and ys (here 1 at each node) are carried over to xs ++ ys (in addition to the debits created by ++ itself). In the thesis, he doesn’t give it an explicit name, but discusses this phenomenon in the very last paragraphs of chapter 3.4 .

The act of relocating debits from child to parent nodes as exemplified with

xs ++ reverse ys =
   x1 : x2 : x3 : ... : xn : yn : y{n-1} : ... : y1 : []
      1    1    1 ... 1    1    n        0 ... 0    0  0 

xs ++ reverse ys =
   x1 : x2 : x3 : ... : xn : yn : y{n-1} : ... : y1 : []
      2    2    2 ... 2    2    0        0 ... 0    0  0

is called “debit passing”, but Okasaki doesn’t use it earlier than in the chapter “Implicit recursive slowdown”. But the example I gave here is useful for understand the scheduled implementation of real time queues. The trick there is to not create a “big” suspension with n debits but to really “physically” distribute them across the data structure

 x1 : x2 : x3 : ... : xn : yn : y{n-1} : ... : y1 : []
    2    2    2 ... 2    2    2        2 ... 2    2  2

and discharge them by forcing a node with every call to snoc . I say “physically” because this forcing performs actual work, it does not simply “mentally” discharge a debit to amortize work that will be done later. Note that the 2 debits added to each yi are an overestimation here, but the real time queue implementation pays for them nonetheless.

My focus on debit passing in the original explanation might suggest that debits can only be discharged when actually evaluating the node to which the debit was assigned. This is not the case, an operation may discharge any debits, even in parts of the data structure that it doesn’t touch. Of course, it must discharge debits of nodes it does touch.

For instance, in the proof of theorem 3.1 (thesis) for queues, Okasaki writes “We can restore the invariant by discharging the first (two) debit(s) in the queue” without bothering to analyze which node this will be. So, the front queue might look like

 f1 : f2 : f3 : ... : fn : f{n+1} : f{n+2} : ... : fm : []
    0    0    1 ... 1    1        n        0 ... 0    0  0

and it’s one of the nodes that carries one debit, or it could look like

      f2 : f3 : ... : fn : f{n+1} : f{n+2} : ... : fm : []
         0    0 ... 0    0       n-3       0 ... 0    0  0

and it’s the node with the large amount of debits. In fact, it’s not even immediate that these two are the only possibilities.

However, with the debit passing from my previous post, it’s easier to say which node will be discharged. But even then, only tail discharges exactly the debits of nodes it inspects while the snoc operation discharges debits in the untouched front list. Of course, as soon as identifying the nodes becomes tractable, chances are that you can turn it into a real-time data structure.

Another good example are skew heaps from

Chris Okasaki. Fun with binary heap trees. In J. Gibbons, O. de Moor. The Fun of Programming.

Here, the “good” nodes are annotated with one debit. Every join operation discharges O(log n) of them and allocates new ones while walking down the tree, but the “time” to actually walk down the tree is not counted immediately. This is just like (++) walks down the first list and allocates debits without immediately using O(n) time to do that.