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

Amit Joshi

TL;DR

The paper addresses the problem of computing a global rank split across $m$ sorted sequences without performing any merge or value-space search. It introduces a merge-free, index-space multi-way co-ranking algorithm that generalizes two-list co-ranking through a donor/receiver greedy strategy with explicit per-list bounds, achieving $O(\log(\sum_t n_t) \ \log m)$ time and $O(m)$ space. Correctness is established via an exchange argument on an imbalance potential, and the approach is shown to readily apply to distributed fractional knapsack, parallel multi-way partitioning, and multi-stream joins. The method provides a deterministic, $K$-independent primitive for fast, scalable partitioning in distributed and database settings, with an accessible implementation path.

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

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

TL;DR

The paper addresses the problem of computing a global rank split across sorted sequences without performing any merge or value-space search. It introduces a merge-free, index-space multi-way co-ranking algorithm that generalizes two-list co-ranking through a donor/receiver greedy strategy with explicit per-list bounds, achieving time and space. Correctness is established via an exchange argument on an imbalance potential, and the approach is shown to readily apply to distributed fractional knapsack, parallel multi-way partitioning, and multi-stream joins. The method provides a deterministic, -independent primitive for fast, scalable partitioning in distributed and database settings, with an accessible implementation path.

Abstract

We present a merge-free algorithm for multi-way co-ranking, the problem of computing cut indices that partition each of the sorted sequences such that all prefix segments together contain exactly elements. Our method extends two-list co-ranking to arbitrary , 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 time and space, independent of . 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
Paper Structure (21 sections, 3 theorems, 5 equations, 2 algorithms)

This paper contains 21 sections, 3 theorems, 5 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem.
  3. Contributions.
  4. Why index space?
  5. Background and Related Work
  6. Two-list co-ranking (partitioners for merge).
  7. Splitter-based parallel sorting (Siebert & Wolf).
  8. Selection in sorted partitions (Frederickson--Johnson).
  9. This paper’s positioning.
  10. Preliminaries and Notation
  11. Data Structure: Indexed (Addressable) Heaps
  12. Algorithm
  13. High-level idea.
  14. Correctness
  15. Complexity Analysis
  16. ...and 6 more sections

Key Result

Lemma 1

If $\Phi(i) > 0$, let $p = \arg\max_t \ell_t$ and $q = \arg\min_t r_t$. Among all feasible infinitesimal transfers of index mass that preserve $\sum_t i_t = K$, the pair $(p,q)$ achieves the greatest non-increasing change in $\Phi$. Any other pair $(u,v)$ can produce at most the same or a weaker dec

Theorems & Definitions (6)

  • Lemma 1: Local extremal optimality
  • proof : Proof sketch
  • Lemma 2: Exchange of transfer order
  • proof : Proof sketch
  • Theorem 1: Convergence and validity
  • proof : Proof sketch