Lower bounds for sorting of sums

Abstract

This paper addresses the following question: How much can sorting (by comparisons) be speeded up if some information about the possible ordertypes the input sequence (x₁,...,x_n) might have is given in advance? We extend a lower bound due to Fredman [3] concerning sorting of sums of the form (y_i+z_j | 1≤i, j≤m) to the problem of sorting all sums of up to d out of m numbers:

Let d ≥ 2, n=Σ_0≤s≤d( ^m_s . Then every decision tree for inputs from Rⁿ that sorts all sums of up to d out of m numbers w₁,...,w_m, i. e., that determines the ordertype of sequences of the form $(\mathop \Sigma \limits_{r \in S} w_r |S \subset \{ 1,...m\} , |S| \leqslant d)$, for $\overline w \in R^m$, has depth Θ(m^d)=Θ(n).

This is an optimal lower bound. Furthermore, the case of sorting all subset sums of a vector is considered:

Let n=2^m. Then every decision tree for inputs from Rⁿ that determines the ordertype of sequences of the form $(\mathop \Sigma \limits_{r \in S} w_r |S \subset \{ 1,...m\} ),\overline w \in R^m$, has depth ≥2^⌊m/3⌋=Θ(n^1/3).

This lower bound is exponentially larger than those previously known for this problem. It may be viewed as another step in an attempt to analyze how hard the Rⁿ-version of the NP-complete Knapsack problem is on structured computational models like the linear decision tree.

Written under partial support by NSF-grant DCR-8504247.

This work is based on a part of the author's Ph.D.-thesis at the University of Illinois at Chicago.