This is from Programming Pearls ed. 2, Column 2, Problem 8: Given a set of n real numbers, a real number t, and an integer k, how quickly can you determine whether there exists a k-element subset of the set that sums to at most t? One easy solution is to sort and sum the first k elements, which is our best hope to find such a sum. However, in the solutions section Bentley alludes to a solution that takes nlog(k) time, though he gives no hints for how to find it. I've been struggling with this; one thought I had was to go through the list and add all the elements less than t/k (in O(n) time); say there are m1 < k such elements, and they sum to s1 < t. Then we are left needing k - m1 elements, so we can scan through the list again in O(n) time looking for all elements less than (t - s1)/(k - m1). Add in again, to get s2 and m2, then again if m2 < k, look for all elements less than (t - s2)/(k - m2). So: def kSubsetSumUnderT(inList, k, t): outList = [] s = 0 m = 0 while len(outList) < k: toJoin = [i for i in inList where i < (t - s)/(k - m)] if len(toJoin): if len(toJoin) >= k - m: toJoin.sort() if(s0 + sum(toJoin[0:(k - m - 1)]) < t: return True return False outList = outList + toJoin s += sum(toJoin) m += len(toJoin) else: return False My intuition is that this might be the O(nlog(k)) algorithm, but I am having a hard time proving it to myself. Thoughts?