Global Predecessor Indexing: Avoiding Binary Search in Weighted Job Scheduling

TL;DR

This paper introduces Global Predecessor Indexing (GPI), a novel preprocessing method that replaces binary search in weighted job scheduling, achieving linear or near-linear time complexity and improving practical performance.

Contribution

The paper presents GPI, a new multi-phase preprocessing technique that computes job predecessors efficiently, enabling faster dynamic programming solutions for weighted job scheduling.

Findings

GPI reduces the time complexity to O(n) with linear-time sorting.

GPI outperforms classical binary search-based methods in practice.

The approach is cache-efficient and simplifies the DP recurrence.

Abstract

We present an improved solution to the Weighted Job Scheduling (WJS) problem. While the classical dynamic programming (DP) solution for $n$ jobs runs in $O(n \log(n))$ time due to comparison-based sorting and per-job binary search, we eliminate the binary search bottleneck. In its place, we introduce a novel multi-phase preprocessing technique called \emph{Global Predecessor Indexing (GPI)}, which computes the latest non-overlapping job (i.e., the predecessor) for all jobs via a two-pointer linear-time pass after sorting. This yields a time complexity of $O(S(n) + n)$ where $S(n)$ is the time to sort all jobs. GPI enables direct use in the classical DP recurrence. When combined with linear-time sorting, GPI yields a complete $O(n)$ solution. Even with comparison-based sorting, GPI significantly outperforms the classical solution in practice by avoiding repeated binary searches in favor of the more cache-efficient extra sort and two-pointer pass.