CF 106452D - Subset Sums

I cannot write a correct editorial and accepted Python solution for this problem as stated, because the constraints make the problem a general counting subset sum problem, which does not have a known polynomial-time solution.

codeforcescompetitive-programming

CF 106452D - Subset Sums

Rating: -
Tags: -
Solve time: 45s
Verified: no

Solution

I cannot write a correct editorial and accepted Python solution for this problem as stated, because the constraints make the problem a general counting subset sum problem, which does not have a known polynomial-time solution.

The statement for Codeforces Gym 106452D says that the task is to count subsets of an array whose elements sum to k, with subsets distinguished by indices. It allows n up to 5 * 10^5, k up to 10^9, and a_i up to 10^9.

For arbitrary nonnegative integers, this contains the classic subset sum problem as a special case. Even deciding whether the answer is zero or nonzero is NP-complete in general. Counting the number of such subsets is the corresponding #P counting problem. A solution with the requested format would require inventing an algorithm that solves a much harder problem than standard competitive programming constraints allow.

A normal subset sum DP would be:

dp[0] = 1
for x in a:
    for s in range(k, x - 1, -1):
        dp[s] += dp[s - x]

but its complexity is O(nk), which is impossible when k can be 10^9. Meet-in-the-middle is also impossible because n can be 500000.

The missing information is likely one of the following:

  1. A much smaller bound on k, such as k <= 5000 or k <= 10^5.
  2. A restriction on the values a_i, such as a small maximum value or special structure.
  3. A different intended problem statement.

If you provide the original PDF/editorial or the actual constraints from the contest, I can write the full senior-style editorial and Python solution in the requested format.