Linear Time Computation of the Maximal Linear and Circular Sums of Multiple Independent Insertions into a Sequence

TL;DR

This paper presents a method to compute the maximal sum of sequences with multiple insertions efficiently in linear time, improving over naive approaches and applicable to network buffer optimization.

Contribution

It introduces algorithms for fast computation of maximal sums after multiple insertions, extending Kadane's algorithm to handle multiple updates in linear time.

Findings

Maximal sums after multiple insertions can be computed in O(n+m) time.

The approach extends to circular and contiguous subsequences.

Algorithms are practical and motivated by wireless network buffer minimization.

Abstract

The maximal sum of a sequence "A" of "n" real numbers is the greatest sum of all elements of any strictly contiguous and possibly empty subsequence of "A", and it can be computed in "O(n)" time by means of Kadane's algorithm. Letting "A^(x -> p)" denote the sequence which results from inserting a real number "x" between elements "A[p-1]" and "A[p]", we show how the maximal sum of "A^(x -> p)" can be computed in "O(1)" worst-case time for any given "x" and "p", provided that an "O(n)" time preprocessing step has already been executed on "A". In particular, this implies that, given "m" pairs "(x_0, p_0), ..., (x_{m-1}, p_{m-1})", we can compute the maximal sums of sequences "A^(x_0 -> p_0), ..., A^(x_{m-1} -> p_{m-1})" in "O(n+m)" time, which matches the lower bound imposed by the problem input size, and also improves on the straightforward strategy of applying Kadane's algorithm to each sequence "A^(x_i -> p_i)", which takes a total of "Theta(n.m)" time. Our main contribution, however, is to obtain the same time bound for the more complicated problem of computing the greatest sum of all elements of any strictly or circularly contiguous and possibly empty subsequence of "A^(x -> p)". Our algorithms are easy to implement in practice, and they were motivated by and find application in a buffer minimization problem on wireless mesh networks.