Longest equally-spaced subsequence

Question

I have a million integers in sorted order and I would like to find the longest subsequence where the difference between consecutive pairs is equal. For example

Answer 1

Here is another answer, working in time O(n^2) and without any notable memory requirements beyond that of turning the list into a set.

The idea is quite naive: like the original poster, it is greedy and just checks how far you can extend a subsequence from each pair of points --- however, checking first that we're at the start of a subsequence. In other words, from points a and b you check how far you can extend to b + (b-a), b + 2*(b-a), ... but only if a - (b-a) is not already in the set of all points. If it is, then you already saw the same subsequence.

The trick is to convince ourselves that this simple optimization is enough to lower the complexity to O(n^2) from the original O(n^3). That's left as an exercice to the reader :-) The time is competitive with other O(n^2) solutions here.

A = [1, 4, 5, 7, 8, 12]    # in sorted order
Aset = set(A)

lmax = 2
for j, b in enumerate(A):
    for i in range(j):
        a = A[i]
        step = b - a
        if b + step in Aset and a - step not in Aset:
            c = b + step
            count = 3
            while c + step in Aset:
                c += step
                count += 1
            #print "found %d items in %d .. %d" % (count, a, c)
            if count > lmax:
                lmax = count

print lmax