Distributed SymmetryBreaking Algorithms for Congested Cliques Leonid Barenboim
Distributed Symmetry-Breaking Algorithms for Congested Cliques Leonid Barenboim and Victor Khazanov
Distributed LOCAL model • • • The communication network is represented by an n-vertex graph Vertices have unique ID’s of size O(log n) each A message passes over an edge with one round Running time is the number of rounds of distributed communication Local computation comes for free
Congested Clique Model • Set of n machines communicating over complete graph • In every round, send to each vertex only O(log n) bits • In every round, every pair of vertices can exchange O(log n) bits.
Previous results • Routing scheme: each node needs to send at most O(n log n) bits and receive at most O(n log n) bit. Then O(1) rounds are sufficient. • C. Lenzen • MST in O(log n) rounds • Z. Lotker, E. Pavlov, B. Patt-Shamir, D. Peleg. (deterministic algorithm) • MST(Minimum Spanning Tree) in O(log* n) rounds • M. Ghaffari and M. Parter (Randomized algorithm) • MIS(Maximal Independent Set) in O(log ∆ log n) • K. Censor-Hillel, M. Parter, G. Schwartzman. (deterministic algorithm) • O(∆) – Coloring in O(log n) rounds • J. Hegeman, and S. Pemmaraju. (Randomized algorithm)
Our results
Main idea • Lenzen’s Routing Problem in Congested Clique – constant number of rounds
Main idea • Lenzen’s Routing Problem in Congested Clique – constant number of rounds
Main idea • Lenzen’s Routing Problem in Congested Clique – constant number of rounds
Main idea • Arboricity - a(G) • Any graph G can be expressed as a sum of forests • Determine the minimum number of edge-disjoint forests into which G can be decomposed. • This number is arboricity of G: a(G)
Main idea • Arboricity - a(G) • Any graph G can be expressed as a sum of forests • Determine the minimum number of edge-disjoint forests into which G can be decomposed. • This number is arboricity of G: a(G)
Main idea • H-partition • computes an H-partition with degree at most O(a) and size k = O(log n) within O(log a) rounds O(1) rounds
Main idea • Forest decomposition
Main idea • Forest decomposition • H-Partition
Main idea • Forest decomposition • H-Partition
Main idea • Forest decomposition • Orientation
Main idea • Forest decomposition • Partitioning the edge set into forests by assigning a distinct label to each outgoing edge of vertex
Better than O(a)-time algorithms •
Main idea • Defective-coloring • In defective coloring, on the other hand, vertices are allowed to have neighbors of the same color to a certain extent (in example d=0, 1, 2)
Main idea • Arbdefective-coloring • An r-arbdefective k-coloring is a coloring with k colors, such that all the vertices colored by the same color i, 1 ≤ i ≤ k, induce a subgraph of arboricity at most r. (Barenboim and Elkin 2011)
Main idea • Arbdefective-coloring • Compute a forests-decomposition
Main idea • Arbdefective-coloring • Compute a forests-decomposition • A vertex selects a new color, once all its parents have selected their colors.
Main idea • Arbdefective-coloring • Compute a forests-decomposition • A vertex selects a new color, once all its parents have selected their colors. • Vertex selects the color used by minimal number of parents
Main idea • Arbdefective-coloring • A partition of k subgraphs with arboricity O(a/k) each
Better than O(a)-time algorithms • O(1) rounds
Better than O(a)-time algorithms • Proper-Coloring-CC(G, a, p) • Arbdefective-Coloring-CC with arboricity a
Better than O(a)-time algorithms • Proper-Coloring-CC (G, a, p) • Arbdefective-Coloring-CC with arboricity a=a/p
Better than O(a)-time algorithms • Proper-Coloring-CC(G, a, p)
Better than O(a)-time algorithms •
Better than O(a)-time algorithms •
Conclusion •
Open questions • MM (maximal matching ) in Congested Clique • Deterministic algorithm for MST better than in O(log n) rounds • Edge-coloring algorithms
Thanks
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