Approximate Maintenance of Maximum Subarray Sum in the Sliding Window Model

TL;DR

This paper develops space-efficient algorithms for approximately maintaining the maximum subarray sum in a sliding window data stream, achieving near-optimal space complexity with adjustable accuracy.

Contribution

It introduces a refined Smooth Histogram framework that provides a tunable approximation of the maximum subarray sum using sublinear space.

Findings

Achieves a constant-factor approximation with $O((\log n)^2)$ bits of space.

Provides a $(1 \pm \epsilon)$-approximation with $O(rac{1}{\epsilon} (\log n)^2)$ bits of space.

Space complexity is asymptotically optimal.

Abstract

In the sliding window model, we are required to maintain the target statistics over the most recent $n$ elements of a data stream, which is captured by a window of size $n$ sliding over the data stream. Exact computation usually requires space linear in $n$, and the central goal is approximate maintenance using sublinear space. In this paper, we study the problem of maintaining the maximum subarray sum in the sliding window model. While the classical Kadane's algorithm computes the exact answer using constant space in the static setting, it does not extend directly, because a new element makes the oldest one expire, which may invalidate the optimal subarray so far. Our first observation is that the so-called Smooth Histogram framework can lead to a constant-factor approximation (in the sense of relative error) using $O((\log n)^2)$ bits of space. We then refine this framework accordingly, which enables for any $\epsilon > 0$ to maintain a $(1 \pm \epsilon)$-approximation using $O(\epsilon^{-1}(\log n)^2)$ bits of space and $O(\epsilon^{-1}\log n)$ operations per update. The space complexity is asymptotically optimal.