Let $S(X) = \{\sum_{i\in Y} i | Y\subset X \}$, the set of subset sums of $X$. $S_n(X) = S(X)\cap \{1,\ldots,n\}$. Consider the following variant of subset sum.

INPUT: positive integer $n$ and set $X\subset \{1,\ldots,n\}$.
OUTPUT: $S_n(X)$

There is the common dynamic programming algorithm taught in algorithm classes, which takes $O(n |X|)$ time. It takes $O(n^2)$ time for large input sets. It's not hard to devise and output sensitive version that takes $O(|X||S_n(X)|+n)$ time.

It's possible to solve it in $O((n \log n)^\frac{3}{2})$ time by decompose the problem to $\sqrt{n\log n}$ ALL-SUBSET-SUMS with small output size, and we can combine the solutions through FFT.

How fast can we solve this problem? There are subset sum algorithms using analytical number theory[1], but they have many technical conditions on the input. It can't be applied directly on ALL-SUBSET-SUMS.


[1] M. Chaimovich , G. Freiman , Z. Galil, Solving dense subset-sum problems by using analytical number theory, Journal of Complexity, v.5 n.3, p.271-282, Sept. 1989

  • $\begingroup$ Why pseudo-polynomial? $\endgroup$
    – Saeed
    Commented Mar 14, 2014 at 10:04
  • $\begingroup$ Because there is no polynomial time algorithm for this problem. We can find $|X|=O(\log n)$ and $|S_n(X)| = \Omega(n)$. The output alone takes exponential time with respect to the input size. $\endgroup$
    – Chao Xu
    Commented Mar 14, 2014 at 19:53
  • $\begingroup$ I see, I thought may be |X| is in $\Omega(n)$. $\endgroup$
    – Saeed
    Commented Mar 14, 2014 at 22:54


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.