Multi-Way Co-Ranking: Index-Space Partitioning of Sorted Sequences Without Merge

TL;DR

This paper introduces a merge-free, index-space binary search algorithm for multi-way co-ranking of sorted sequences, enabling efficient partitioning without merging or value-space search, applicable to various distributed and parallel data processing tasks.

Contribution

It extends two-list co-ranking to multiple sequences, providing a merge-free, index-space binary search method with proven correctness and broad application potential.

Findings

Runs in $O(\log( ext{sum of sequence lengths}) \\ \log m)$ time

Uses $O(m)$ space, independent of $K$

Applicable to distributed fractional knapsack, parallel merge partitioning, and multi-stream joins

Abstract

We present a merge-free algorithm for multi-way co-ranking, the problem of computing cut indices $i_1,\dots,i_m$ that partition each of the $m$ sorted sequences such that all prefix segments together contain exactly $K$ elements. Our method extends two-list co-ranking to arbitrary $m$, maintaining per-sequence bounds that converge to a consistent global frontier without performing any multi-way merge or value-space search. Rather, we apply binary search to \emph{index-space}. The algorithm runs in $O(\log(\sum_t n_t)\,\log m)$ time and $O(m)$ space, independent of $K$. We prove correctness via an exchange argument and discuss applications to distributed fractional knapsack, parallel merge partitioning, and multi-stream joins. Keywords: Co-ranking \sep partitioning \sep Merge-free algorithms \sep Index-space optimization \sep Selection and merging \sep Data structures