# Some simple distributed algorithms for sparse networks

@article{Panconesi2001SomeSD, title={Some simple distributed algorithms for sparse networks}, author={Alessandro Panconesi and Romeo Rizzi}, journal={Distributed Computing}, year={2001}, volume={14}, pages={97-100} }

Summary. We give simple, deterministic, distributed algorithms for computing maximal matchings, maximal independent sets and colourings. We show that edge colourings with at most $2\Delta-1$ colours, and maximal matchings can be computed within ${\cal O}(\log^* n + \Delta)$ deterministic rounds, where $\Delta$ is the maximum degree of the network. We also show how to find maximal independent sets and $(\Delta+1)$-vertex colourings within ${\cal O}(\log^* n + \Delta^2)$ deterministic rounds. All…

## Topics from this paper

## 170 Citations

Faster Deterministic Distributed Coloring Through Recursive List Coloring

- Computer Science, MathematicsSODA
- 2020

An improved deterministic $2^{O(\sqrt{\log\Delta})}\cdot\log^3 n$-round algorithm for $\Delta$-coloring non-complete graphs with maximum degree $\Delta\geq 3$.

Simple and Near-Optimal Distributed Coloring for Sparse Graphs

- Mathematics, Computer ScienceDISC
- 2017

Simple randomized distributed algorithms that, with high probability, color any $n$-node $\alpha$-arboricity graph, and are nearly-optimal, as it is known by results of Linial and Barenboim and Elkin that coloring with $\Theta(\alpha)$ colors, or even poly$(\ alpha) $ colors, requires $\Omega(\log_{\alpha} n)$ rounds.

Improved deterministic distributed matching via rounding

- Mathematics, Computer ScienceDistributed Computing
- 2018

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts.…

Distributed Maximal Independent Set on Scale-Free Networks

- Computer ScienceArXiv
- 2018

The problem of distributed maximal independent set (MIS) is investigated on inhomogeneous random graphs with power-law weights by which the scale-free networks can be produced by state-of-the-art algorithms with the time complexity of $O(\log{n})$.

Exact Bounds for Distributed Graph Colouring

- Computer Science, MathematicsSIROCCO
- 2015

It is shown that for infinitely many n the time complexity is precisely $\frac{1}{2} \log^{*} n$ communication rounds.

Improved Distributed Approximation to Maximum Independent Set

- Computer Science, MathematicsDISC
- 2020

Improved results for approximating Maximum Independent Set ($\MaxIS$) in the standard LOCAL and CONGEST models of distributed computing and implies that finding an $O(\Delta)-approximation to $\MaxIS$.

Distributed approximation algorithms for maximum matching in graphs and hypergraphs

- Computer Science, MathematicsArXiv
- 2018

A randomized and deterministic approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in hypergraphs and an algorithm for hypergraph maximal matching, which is significantly faster than the algorithm of Ghaffari, Harris & Kuhn (2017).

Improved Deterministic Distributed Matching via Rounding

- Mathematics, Computer ScienceDISC
- 2017

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts.…

Distributed deterministic edge coloring using bounded neighborhood independence

- Computer Science, MathematicsPODC '11
- 2011

The main technical contribution is a subroutine that computes an O(Δ/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p2) + log* n time, for a parameter p, 1 ≤ p ≤ Δ.

Fast Distributed Approximation Algorithm for the Maximum Matching Problem in Bounded Arboricity Graphs

- Mathematics, Computer ScienceISAAC
- 2009

The result implies that the known $\Omega(\sqrt{\log n/\log\log n})$ lower bound on the time complexity for a constant or polylogarithmic approximation does not hold for graphs of bounded arboricity.

## References

SHOWING 1-10 OF 18 REFERENCES

Network decomposition and locality in distributed computation

- Computer Science30th Annual Symposium on Foundations of Computer Science
- 1989

The authors introduce a concept of network decomposition, a partitioning of an arbitrary graph into small-diameter connected components, such that the graph created by contracting each component into…

Locality in Distributed Graph Algorithms

- Mathematics, Computer ScienceSIAM J. Comput.
- 1992

This model focuses on the issue of locality in distributed processing, namely, to what extent a global solution to a computational problem can be obtained from locally available data.

Parallel Symmetry-Breaking in Sparse Graphs

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 1988

Efficient deterministic techniques for breaking symmetry in parallel are used to construct fast linear processor algorithms for several problems, including the problem of $(Delta + 1)$-coloring constant-degree graphs and 5-colors planar graphs.

Parallel symmetry-breaking in sparse graphs

- Computer Science, MathematicsSTOC
- 1987

Efficient deterministic techniques for breaking symmetry in parallel are described and applied to construct fast linear processor algorithms for several problems, including (&Dgr; + 1)-coloring constant-degree graphs, 5-coloring planar graphs, and finding depth-first-search trees inPlanar graphs.

Analysis of approximate algorithms for constrained and unconstrained edge-coloring of bipartite graphs

- Mathematics
- 1995

The problem of edge-coloring a bipartite graph is to color the edges so that adjacent edges receive different colors. An optimal algorithm uses the minimum number of colors to color the edges. We…

Low diameter graph decompositions

- Mathematics, Computer ScienceComb.
- 1993

It is shown that this trade-off between the number of blocks and the diameter is nearly best possible, for two families of graphs: the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals.

Parallel ((Greek D)D+1)-Coloring of Constant-Degree Graphs

- Mathematics, Computer ScienceInf. Process. Lett.
- 1987

Abstract This paper presents parallel algorithms for coloring a constant-degree graph with a maximum degree of Δ using Δ + 1 colors and for finding a maximal independent set in a constant-degree…

Analysis of Approximate Algorithms for Edge-Coloring Bipartite Graphs

- Mathematics, Computer ScienceInf. Process. Lett.
- 1995

This work considers several approximation algorithms for edge-coloring bipartite graphs and shows tight bounds on the number of colors they use in the worst case, and presents results on the constrained edge- Coloring problem.

A faster distributed algorithm for computing maximal matchings deterministically

- Computer SciencePODC '99
- 1999

We show that maximal matchings can be computed deterministically in O(log4 n) rounds in the synchronous, message-passing model of computation. This improves on an earlier result by three log-factors.

A Graph-Theoretic Model for the Scheduling Problem and its Application to simultaneous Resource Scheduling

- Computer ScienceComputer Science and Operations Research
- 1992

A graph-theoretic model for formally specifying scheduling problems is presented, used for recognition of the similarity between scheduling problems, the transformation of one scheduling problem to a different scheduling problem, and the decomposition of a problem into subproblems, and an optimal algorithm for parallel I/O in multiple bus architectures is obtained.