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 [1], 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.


[1]: Chris Okasaki. Purely Function Data Structures.
     http://www.cs.cmu.edu/~rwh/theses/okasaki.pdf
     (This is the thesis on which the book is based.)


Regards,
apfelmus

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