Lower bounds for sorting of sums

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


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.


Decision Tree Knapsack Problem Sorting Problem Major Open Problem General Decision Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Martin Dietzfelbinger
    • 1
  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicago

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