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All-to-all Routing on Kautz Graphs: Regular Routing Beats Shortest Paths

Vance Faber, Noah Streib

TL;DR

It is proved that K(d,D) contains an edge whose shortest-path congestion strictly exceeds tau(d,D) when D is sufficiently large, which means that, for every fixed outdegree d at least 2 and all sufficiently large diameters D, no shortest-path routing scheme can match this makespan.

Abstract

We study packet routing in the Kautz digraph K(d,D), where every ordered pair of distinct vertices is connected by a unique shortest directed path. The regular routing introduced in earlier work schedules all ordered pairs in tau(d,D) = (D-1)d^(D-2) + D d^(D-1) steps. We show that, for every fixed outdegree d at least 2 and all sufficiently large diameters D, no shortest-path routing scheme can match this makespan. More precisely, we prove that K(d,D) contains an edge whose shortest-path congestion strictly exceeds tau(d,D) when D is sufficiently large. Our construction uses edge-words drawn from a subset of ternary unbordered square-free words, together with a trimming inequality that propagates large congestion at distance D down to shorter distances. Computations for d=2 and small D show that for all D at least 4 there is an edge in K(2,D) with congestion greater than tau(2,D).

All-to-all Routing on Kautz Graphs: Regular Routing Beats Shortest Paths

TL;DR

It is proved that K(d,D) contains an edge whose shortest-path congestion strictly exceeds tau(d,D) when D is sufficiently large, which means that, for every fixed outdegree d at least 2 and all sufficiently large diameters D, no shortest-path routing scheme can match this makespan.

Abstract

We study packet routing in the Kautz digraph K(d,D), where every ordered pair of distinct vertices is connected by a unique shortest directed path. The regular routing introduced in earlier work schedules all ordered pairs in tau(d,D) = (D-1)d^(D-2) + D d^(D-1) steps. We show that, for every fixed outdegree d at least 2 and all sufficiently large diameters D, no shortest-path routing scheme can match this makespan. More precisely, we prove that K(d,D) contains an edge whose shortest-path congestion strictly exceeds tau(d,D) when D is sufficiently large. Our construction uses edge-words drawn from a subset of ternary unbordered square-free words, together with a trimming inequality that propagates large congestion at distance D down to shorter distances. Computations for d=2 and small D show that for all D at least 4 there is an edge in K(2,D) with congestion greater than tau(2,D).
Paper Structure (27 sections, 18 theorems, 78 equations, 2 tables)

This paper contains 27 sections, 18 theorems, 78 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Kautz digraphs and unique shortest paths
  3. Edge-words and subword repetition properties
  4. A trimming inequality for geodesic layers
  5. Deficits at distance D and witness templates
  6. General position $t$ and overlap $r$
  7. Witness templates and cost
  8. Example
  9. Bounding Lemmas
  10. High congestion for general outdegree d
  11. Edge-words and overlaps in $K(d,D)$
  12. A universal cost bound from 7/4+-freeness
  13. Main theorem for all d>=2 from $7/4^{+}$-free words
  14. Corollary for d>=3: circular square-free words
  15. Computational evidence
  16. ...and 12 more sections

Key Result

Theorem 1

For every fixed outdegree $d\ge 2$ there exists $D_0(d)$ such that for all $D\ge D_0(d)$ the Kautz digraph $K(d,D)$ contains an edge $e$ whose shortest–path congestion satisfies

Theorems & Definitions (29)

  • Theorem
  • Definition 2.1
  • Lemma 2.2
  • Definition 3.1: Border and unbordered word
  • Definition 3.2: $\alpha$-power, $\alpha$-free, and $\alpha^{+}$-free
  • Definition 3.3: Circular $\alpha$-power-free
  • Remark 3.4
  • Definition 4.1
  • Lemma 4.2: Trimming inequality
  • Lemma 4.3
  • ...and 19 more