Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fast Deterministic Distributed Degree Splitting

Yannic Maus, Alexandre Nolin, Florian Schager

Abstract

We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log n)$ for computing a balanced orientation with discrepancy at most $\varepsilon \cdot \mathrm{deg}(v)$ for every vertex $v \in V$. This improves upon a previous result by [GHKMSU, Distrib. Comput. 2020] of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log \varepsilon^{-1} \cdot (\log \log \varepsilon^{-1})^{1.71} \cdot \log n)$. Further, we show that this result can also be extended to compute undirected degree splits with the same discrepancy and in the same runtime. As as application we show that $(3 / 2 + \varepsilon)Δ$-edge coloring can now be solved in $\mathcal{O}(\varepsilon^{-1} \cdot \log^2 Δ\cdot \log n + \varepsilon^{-2} \cdot \log n)$ rounds in LOCAL. Note that for constant $\varepsilon$ and $Δ= \mathcal{O}(2^{\log^{1/3} n})$ this runtime matches the current state-of-the-art for $(2Δ- 1)$-edge coloring in [Ghaffari & Kuhn, FOCS'21].

Fast Deterministic Distributed Degree Splitting

Abstract

We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity for computing a balanced orientation with discrepancy at most for every vertex . This improves upon a previous result by [GHKMSU, Distrib. Comput. 2020] of complexity . Further, we show that this result can also be extended to compute undirected degree splits with the same discrepancy and in the same runtime. As as application we show that -edge coloring can now be solved in rounds in LOCAL. Note that for constant and this runtime matches the current state-of-the-art for -edge coloring in [Ghaffari & Kuhn, FOCS'21].

Paper Structure

This paper contains 33 sections, 28 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The LOCAL model linial92.
  3. Degree splitting.
  4. Which discrepancy is possible?
  5. Degree splittings in the bigger scope of distributed complexity theory.
  6. Our Contributions
  7. Limitations for Degree Splitting Algorithms.
  8. Open Problem I: Optimality of the runtime.
  9. Open Problem II: Multi-way splitting.
  10. Related Work
  11. Degree splitting
  12. (Hypergraph) sinkless orientation.
  13. Related work on edge coloring.
  14. Our Algorithm in a Nutshell
  15. Split and orient.
  16. ...and 18 more sections

Key Result

Theorem 1

For every $\varepsilon > 0$, there is a deterministic Local algorithm with round complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log n)$ that computes an (un)directed degree splitting of any $n$-vertex graph such that the discrepancy at each vertex $v$ of degree $\mathrm{deg}(v)$ is at most $\vareps

Figures (1)

  • Figure 1: Doubling the length of a single short cycle in $G_\mathrm{split}$.

Theorems & Definitions (58)

  • Theorem 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Definition 4: Directed Degree Splitting
  • Definition 5: Undirected Degree Splitting
  • Definition 6: Sinkless Orientation
  • Definition 7: min-degree-$\delta$ Sinkless Orientation
  • Lemma 8: Sink- and Sourceless Orientation GHKMSU_dc20
  • Definition 9: Hypergraph sinkless orientation (HSO) BMNSU_soda25
  • ...and 48 more