Solution


Approach 1: Dynamic Programming

Intuition

We don't care about the profit of the scheme if it is , because it surely will be over the threshold of profitability required. Similarly, we don't care about the number of people required in the scheme if it is , since we know the scheme will be too big for the gang to execute.

As a result, the bounds are small enough to use dynamic programming. Let's keep track of cur[p][g], the number of schemes with profitability and requiring gang members: except we'll say (without changing the answer) that all schemes that profit at least P dollars will instead profit exactly P dollars.

Algorithm

Keeping track of cur[p][g] as defined above, let's understand how it changes as we consider 1 extra crime that will profit p0 and require g0 gang members. We will put the updated counts into cur2.

For each possible scheme with profit p1 and group size g1, that scheme plus the extra crime (p0, g0) being considered, has a profit of p2 = min(p1 + p0, P), and uses a group size of g2 = g1 + g0.

Complexity Analysis

  • Time Complexity: , where is the number of crimes available to the gang.

  • Space Complexity: .


Analysis written by: @awice.