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

Provided that C(m) is so magical that you cannot use any better technique to find your solution directly and thus you really need to traverse all a*b in decreasing order, this is what I would do:

Initialize a max-heap with all pairs (a, b) such that a = b. This means that the heap contains (0, 0), (1, 1), ... , (1.000.000, 1.000.000). The heap should be based on the a * b value.

Now continuously:

1. Get the max pair (a, b) from the heap.
2. Verify if (a, b) satisfies C(a * b). If so, you are done.
3. Otherwise, add (a, b-1) to the heap (provided b > 0, otherwise do nothing).

This is a very simple O(n log n) time and O(n) space algorithm, provided that you find the answer quickly (in a few iterations). This of course depends on C.

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If you run into space problems you can of course easily decrease the space complexity by splitting up the problem in a number of subproblems, for instance 2:

1. Add only (500.000, 500.000), (500.001, 500.001), ... , (1.000.000, 1.000.000) to the heap and find your best pair (a, b).
2. Do the same for (0, 0), (1, 1), ... (499.999, 499.999).
3. Take the best of the two solutions.