Order (a,b) pairs by result of a*b

Question

I would like to find the highest value m = a*b that satisfies some condition C(m), where

1 <= a <= b <= 1,000,000.

In order to do

Answer 1

Here's a not particularly efficient way to do this with a heap in Python. This is probably the same thing as the BFS you mentioned, but it's fairly clean. (If someone comes up with a direct algorithm, that would of course be better.)

import heapq  # <-this module's API is gross. why no PriorityQueue class?

def pairs_by_reverse_prod(n):
    # put n things in heap, since of course i*j > i*(j-1); only do i <= j
    # first entry is negative of product, since this is a min heap
    to_do = [(-i * n, i, n) for i in xrange(1, n+1)]
    heapq.heapify(to_do)

    while to_do:
        # first elt of heap has the highest product
        _, i, j = to_do[0]
        yield i, j

        # remove it from the heap, replacing if we want to replace
        if j > i:
            heapq.heapreplace(to_do, (-i * (j-1), i, j-1))
        else:
            heapq.heappop(to_do)