ICALP 1987: Automata, Languages and Programming pp 457-466

# Lower bounds for sorting of sums

• Martin Dietzfelbinger
Algorithms And Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)

## 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,...,xn) might have is given in advance? We extend a lower bound due to Fredman [3] concerning sorting of sums of the form (yi+zj | 1≤i, j≤m) to the problem of sorting all sums of up to d out of m numbers:

Let d ≥ 2, n=Σ0≤s≤d( s m . Then every decision tree for inputs from Rn that sorts all sums of up to d out of m numbers w1,...,wm, 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(md)=Θ(n).

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

Let n=2m. 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(n1/3).

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.

## Keywords

Decision Tree Knapsack Problem Sorting Problem Major Open Problem General Decision Tree

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