Solving k-SUM Using Few Linear Queries

The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P , and it is in particular open whether it admits an algorithm of complexity O(n) with c < dk2 e. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n3 log2 n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in Õ(nd k2 e+8) time, and performs O(n3 log2 n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n3 log2 n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-à-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P . We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth Õ(n3) for the k-SUM problem. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems