Time-space tradeoffs for all-nearest-larger-neighbors problems

Abstract

This paper addresses two versions of a fundamental problem, referred to as the All-Nearest-Larger-Neighbors (ANLN) problem, defined as follows: given a one-dimensional array A of n real-valued keys, find, for each array element A[i], the index of a nearest array element, if one exists, whose key is strictly larger than A[i]. We develop algorithms for one- and two-sided versions of the ANLN problem that run in O(n logb n) time, using Θ(b) work-space, for all b = O(n), exhibiting a full time-space tradeoff that subsumes all known (memory-restricted) special cases. In addition, a non-trivial lower bound is developed for the time complexity of solving both versions on a pointer machine with limited work-space. This lower bound matches the time complexity of our algorithms, when restricted to constant space. The fundamental nature of ANLN problems make them intrinsically interesting to study. They also capture the essence of a variety of other familiar problems, such as determining the forest structure associated with a given string of nested parentheses, and triangulating monotone polygons. For both of these, we describe reductions to versions of the ANLN problem, achieving the same time-space tradeoffs. © 2013 Springer-Verlag.