Prompted by this question on Stack Overflow, 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 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.

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?

from random import randrange
from itertools import islice

def randomSeq(max):
  while True: yield randrange(max)

def randomList(N,max):
  return list(islice(randomSeq(max),N))

## Returns the longest subsequence (non-contiguous) of X that is strictly increasing.
def subsequence(X):
    L = 1     ## length of longest subsequence (initially: just first element)
    M = [0]   ## M[j] = index in X of the final member of the lowest subsequence of length 'j' yet found
    P = [-1]
    for i in range(1,X.__len__()):
        ## Find largest j <= L such that: X[M[j]] < X[i].
        ## X[M[j]] is increasing, so use binary search over j.
        j = -1
        start = 0
        end = L - 1
        going = True
        while going:
            if (start == end):
                if (X[M[start]] < X[i]):
                    j = start
                going = False
            else:
                partition = 1 + ((end - start - 1) / 2)
                if (X[M[start + partition]] < X[i]):
                    start += partition
                    j = start
                else:
                    end = start + partition - 1

        if (j >= 0):
            P.append(M[j])
        else:
            P.append(-1)
        
        j += 1
        if (j == L):
            M.append(i)
            L += 1
        if (X[i] < X[M[j]]):
            M[j] = i

    ## trace subsequence back to output
    result = []
    trace_idx = M[L-1]
    while (trace_idx >= 0):
        result.append(X[trace_idx])
        trace_idx = P[trace_idx]
    
    return list(result.__reversed__())


l1 = randomList(15,100)

Edit:

from random import randrange
from itertools import islice

def randomSeq(max):
  while True: yield randrange(max)

def randomList(N,max):
  return list(islice(randomSeq(max),N))

## Returns the longest subsequence (non-contiguous) of X that is strictly increasing.
def subsequence(X):
    L = 1     ## length of longest subsequence (initially: just first element)
    M = [0]   ## M[j] = index in X of the final member of the lowest subsequence of length 'j' yet found
    P = [-1]
    for i in range(1,X.__len__()):
        ## Find largest j <= L such that: X[M[j]] < X[i].
        ## X[M[j]] is increasing, so use binary search over j.
        j = -1
        start = 0
        end = L - 1
        going = True
        while going:
            if (start == end):
                if (X[M[start]] < X[i]):
                    j = start
                going = False
            else:
                partition = 1 + ((end - start - 1) / 2)
                if (X[M[start + partition]] < X[i]):
                    start += partition
                    j = start
                else:
                    end = start + partition - 1

        if (j >= 0):
            P.append(M[j])
        else:
            P.append(-1)
        
        j += 1
        if (j == L):
            M.append(i)
            L += 1
        if (X[i] < X[M[j]]):
            M[j] = i

    ## trace subsequence back to output
    result = []
    trace_idx = M[L-1]
    while (trace_idx >= 0):
        result.append(X[trace_idx])
        trace_idx = P[trace_idx]
    
    return list(result.__reversed__())


l1 = randomList(15,100)