LOCAL Distributed Algorithms Mohsen Ghaffari ETH Zurich Based

LOCAL Distributed Algorithms Mohsen Ghaffari (ETH Zurich) Based on joint works with Manuela Fischer (ETH), Fabian Kuhn (U. Freibrug), Yannic Maus (U Freiburg), & Hsin-Hao Su (MIT)

LOCAL Distributed Graph Algorithms •

LOCAL Graph Algorithms • Maximal Independent Set Maximal Matching simple local reductions

Randomized vs. Deterministic LOCAL Algorithms • Maximal Independent Set Maximal Matching

Part I: Randomized Improved Randomized Algorithm for Maximal Independent Set

Maximal Independent Set (MIS) A central problem in distributed graph algorithms Many other problems reduce to MIS: Identify a set S of nodes such that: 1. Set S is an independent set: no two nodes are adjacent 2. Set S is maximal in independence: no node can be added LOCAL Algo. for Lovasz Local Lemma Maximal Matching Ruling Sets Maximal Independent Set

Distributed MIS, Prior Results BEPS 12 Luby 85 round complexity • KMW 06 Linial 92

round complexity Distributed MIS, Old & New Luby 85 New Alg • BEPS 12 KMW 06 Linial 92

The Algorithm’s Outline: • Graph Shattering in the style of Beck’s 1991 LLL algorithm

The Pre-Shattering Randomized Algorithm 0. 5 • 0. 5 0. 25 0. 5

The Post-Shattering Deterministic Algorithm •

The Randomized State of the Art Maximal Independent Set Maximal Matching

Part II: Deterministic Improved Deterministic Algorithm for Edge Coloring

The Deterministic State of the Art Maximal Independent Set Holy Grail: finding polylog n-round deterministic distributed algorithms. Maximal Matching

Our Deterministic Algorithm for Edge Coloring • Maximal Independent Set Maximal Matching

Edge Coloring via Degree Splitting •

A Few Words about the Splitting Algorithm 1. The algorithm for graphs with max degree polylog n 2. The extension to graphs of higher degrees

• t=4 Before Augmentation After Augmentation

For a deterministic distributed algorithm, there are three crucial points: 1. Augmenting paths must be short, e. g. , polylog n hops. 2. There must be “many” short augmenting paths, to fix many nodes simultaneously. 3. Should distributedly & deterministically compute many “disjoint” augmenting paths. Item 3 is where most of the technical issues and novel ideas are. In a nutshell: we simultaneously construct “disjoint” exponentially-growing “trees” from “all ” sources until each reveals an augmenting path for its source. See the paper http: //arxiv. org/abs/1608. 03220

Degree Splitting for Higher Degrees •

Other Applications of the Splitting Techniques •

Part III: Randomized vs Deterministic Rounding is “complete”!

Hypergraph Degree Splitting • Nodes or rounding constraints Hyperedges or rounding variables

Relaxed Hypergraph Degree Splitting • Nodes or rounding constraints Hyperedges or rounding variables

Relaxed Hypergraph Degree Splitting is “complete” • Hypergraph Degree Splitting “All local problems” Network Decomposition (à la Linial-Saks) Awerbuch. Peleg’s Ball Carving Hypergraph Conflict-free Coloring Defective Vertex Coloring

Part IV: Concluding Remarks

Further Improvements for Maximal Matching & Edge Coloring Maximal Independent Set Maximal Matching

! s k n a Th Open Questions BEPS 2012 Luby 1986 G. , 2016 round complexity • Maximal Independent Set KMW 2006 Maximal Matching Linial 1992 Polylog n Fischer, G. , & Kuhn ‘ 17 Randomized LOCAL MIS