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Girth Approximations in the CONGEST Model

Shiri Chechik, Gur Lifshitz, Doron Mukhtar

Abstract

This paper advances the state of the art in girth approximation within the CONGEST model. Manoharan and Ramachandran [PODC '24] provided the first significant improvement in girth approximation in over a decade. We build on this momentum and make progress on all fronts: we provide a unified family of algorithms yielding girth approximation-round tradeoffs for undirected networks; we obtain improved bounds for directed networks; and we establish better lower bounds for directed and undirected weighted networks. Together, these results substantially narrow the remaining complexity gaps across all settings. Specifically, for networks with $n$ nodes and hop-diameter $D$, we show that one can compute, with high probability: $(1)$ An $f$-approximation for unweighted undirected girth in $\tilde{O}(n^{1/f}+D)$ rounds, for every constant integer $f>2$, $(2)$ A $(2k-1+o(1))$-approximation for weighted undirected girth in $\tilde{O}(n^{(k+1)/(2k+1)}+D)$ rounds, for every constant integer $k>1$, and $(3)$ A $2$-approximation for directed unweighted girth, and a $(2+\varepsilon)$-approximation for directed weighted girth, both in $\tilde{O}(n^{2/3}+D)$ rounds. We also prove new lower bounds for directed networks and for undirected weighted networks: for every integer $k > 2$ and $\varepsilon>0$, assuming the Erdős-Simonovits' even cycle conjecture (and unconditionally for $k\in\{3,4,6\}$), any $(k-\varepsilon)$-approximation for the girth requires $\tildeΩ(n^{k/(2k-1)})$ rounds, even when $D = O(\log n)$.

Girth Approximations in the CONGEST Model

Abstract

This paper advances the state of the art in girth approximation within the CONGEST model. Manoharan and Ramachandran [PODC '24] provided the first significant improvement in girth approximation in over a decade. We build on this momentum and make progress on all fronts: we provide a unified family of algorithms yielding girth approximation-round tradeoffs for undirected networks; we obtain improved bounds for directed networks; and we establish better lower bounds for directed and undirected weighted networks. Together, these results substantially narrow the remaining complexity gaps across all settings. Specifically, for networks with nodes and hop-diameter , we show that one can compute, with high probability: An -approximation for unweighted undirected girth in rounds, for every constant integer , A -approximation for weighted undirected girth in rounds, for every constant integer , and A -approximation for directed unweighted girth, and a -approximation for directed weighted girth, both in rounds. We also prove new lower bounds for directed networks and for undirected weighted networks: for every integer and , assuming the Erdős-Simonovits' even cycle conjecture (and unconditionally for ), any -approximation for the girth requires rounds, even when .

Paper Structure

This paper contains 56 sections, 17 theorems, 48 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Unweighted Networks
  3. Weighted Networks
  4. Related Work
  5. Unweighted Undirected Networks
  6. Weighted Undirected Networks
  7. Directed Networks
  8. Lower Bounds
  9. Preliminaries
  10. Unweighted Undirected Networks
  11. Overview
  12. The Estimation Procedure
  13. The Approximation Algorithm
  14. Correctness.
  15. Complexity.
  16. ...and 41 more sections

Key Result

Lemma 3.1

For an unweighted and undirected network $G = (V,E)$ with diameter $D$, a set of sources $S \subseteq V$ and a natural number $k$, partial BFS trees rooted at the sources can be deterministically constructed in $O(k+D)$ rounds, such that each vertex participates only in the trees of its $\min(k,|S|)

Figures (1)

  • Figure 1: The graph construction $G$ (Shortcut Tree is omitted). Here $r,r',r^*,r' \in R$ and $l,l',l^*,l' \in L$. The dashed gray directed edges correspond to the set of all potential edges (i.e., if $E_A = E_B = E(H)$), while the pink edges correspond to the actual edge sets given to Alice and Bob.

Theorems & Definitions (39)

  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Claim 3.1
  • proof
  • Claim 3.2
  • proof
  • Claim 3.3
  • ...and 29 more