Prompted by this question on Stackoverflow, I wrote an implementation in Python of the longest increasing subsequence problem. In a nutshell, the problem is: given a sequence of numbers, remove the fewest possible to obtain an increasing subsequence (the answer is not unique). Perhaps it is best illustrated by example.

>>> elems
[25, 72, 31, 32, 8, 20, 38, 43, 85, 39, 33, 40, 98, 37, 14]
>>> subsequence(elems)
[25, 31, 32, 38, 39, 40, 98]

The code below works, but I am sure it could be made shorter and / or more readable. Can any more experienced Python coders offer some suggestions?

edited to add a description: The algorithm iterates over the input array, X, while keeping track of the length longest increasing subsequence found so far (L). It also maintains an array M of length L where M[j] = "the index in X of the final element of the best subsequence of length j found so far" where best means the one that ends on the lowest element. It also maintains an array P which constitutes a linked list of indices in X of the best possible subsequences (e.g. P[j], P[P[j]], P[P[P[j]]] ... is the best subsequence ending with X[j], in reverse order). P is not needed if only the length of the longest increasing subsequence is needed.

from random import randrange
from bisect import bisect_left


def randomList(N, max):
    return [randrange(max) for x in xrange(N)]


def subsequence(seq):
    """Returns the longest subsequence (non-contiguous) of seq that is
    strictly increasing.

    """
    # head[j] = index in 'seq' of the final member of the best subsequence
    # of length 'j + 1' yet found
    head = [0]
    # predecessor[j] = linked list of indices of best subsequence ending
    # at seq[j], in reverse order
    predecessor = [-1]
    for i in xrange(1, len(seq)):
        ## Find j such that:  seq[head[j - 1]] < seq[i] <= seq[head[j]]
        ## seq[head[j]] is increasing, so use binary search.
        j = bisect_left([seq[head[idx]] for idx in xrange(len(head))], seq[i])

        if j == len(head):
            head.append(i)
        if seq[i] < seq[head[j]]:
            head[j] = i

        predecessor.append(head[j - 1] if j > 0 else -1)

    ## trace subsequence back to output
    result = []
    trace_idx = head[-1]
    while (trace_idx >= 0):
        result.append(seq[trace_idx])
        trace_idx = predecessor[trace_idx]

    return result[::-1]


l1 = randomList(15, 100)