Approximate Maintenance of Maximum Subarray Sum in the Sliding Window Model

2026-04-25

AI Summary

This work addresses the problem of efficiently maintaining the maximum subarray sum in the sliding window model. It presents the first adaptation of the Smooth Histogram framework to this setting, yielding an approximation algorithm that supports dynamic updates within sublinear space. The proposed method achieves a $(1 \pm \varepsilon)$-approximation for any accuracy parameter $\varepsilon > 0$, with an update time of $O(\varepsilon^{-1} \log n)$ and a space complexity of $O(\varepsilon^{-1} (\log n)^2)$ bits, which is asymptotically optimal. By circumventing the high space overhead inherent in exact algorithms, this approach provides a theoretically optimal solution for real-time approximate computation of the maximum subarray sum over streaming data.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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 $ε> 0$ to maintain a $(1 \pm ε)$-approximation using $O(ε^{-1}(\log n)^2)$ bits of space and $O(ε^{-1}\log n)$ operations per update. The space complexity is asymptotically optimal.

Problem

• sliding window
• maximum subarray sum
• approximate maintenance
• sublinear space
• data stream

Innovation

• sliding window
• maximum subarray sum
• smooth histogram
• sublinear space
• approximation algorithm