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Distributed Approximate Matching (DAM). Zvi Lotker, Boaz Patt-Shamir, Adi Rosen Presentation: Deniz Çokuslu May 2008. Motivation. Matching A matching M in a graph G is a set of nonloop edges with no shared endpoints.

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distributed approximate matching dam

Distributed Approximate Matching (DAM)

Zvi Lotker, Boaz Patt-Shamir, Adi Rosen

Presentation: Deniz Çokuslu

May 2008

motivation
Motivation
  • Matching
    • A matching M in a graph G is a set of nonloop edges with no shared endpoints.
    • The vertices incident to M are saturated (matched) by M and the others are unsaturated (unmatched).

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motivation1
Motivation
  • A maximal matching in a graph G is a matching that cannot be enlarged by adding more edges
  • A maximum matching in a graph G is a matching of maximum size among all matchings
  • A perfect matching covers all vertices of the graph (all are saturated)
  • A maximal weighted matching in a weighted graph, is a matching that maximizes the weight of the selected edges
motivation2
Motivation
  • Matching Algorithms
    • A. Israeli and A. Itai, A fast and simple randomized parallel algorithm for maximal matching (1986)
      • Time complexity: O(log n)
    • M. Wattenhofer and R. Wattenhofer. Distributed weighted matching (2004)
      • For trees: 4-approx. Algorithm
        • Time Complexity: Constant
      • For general graphs: 5-approx. Algorithm
        • Time Complexity: O(log2 n)
    • F. Kuhn, T. Moscibroda and R. Wattenhofer. The price of being near-sighted (2005)
        • Lowerbound of any distributed algorithm that approximates the maximum weighted matching:
distributed approximate matching dam1
Distributed Approximate Matching (DAM)
  • Works on general weighted graphs
    • Static Graph Algorithm
      • Finds Maximum weighted matching within a factor of 4+ε
      • Time complexity: O(ε-1 log ε-1 log n), for ε > 0
    • Dynamic Graph Algorithm
      • Nodes are inserted or deleted one at a time
      • Unweighted matching: (1 + ε)-approximate, Θ(1/ ε) time per delete/insert
      • Weighted matching: Constant approximate, constant running time
distributed approximate matching dam2
Distributed Approximate Matching (DAM)
  • The system is modeled as a unidirected graph G(V,E)
  • Time progress in synchronous rounds
  • In each round each processor may send messages to any subset of its neighbors
  • All messages that are sent, are received and processed at the same round
  • Edges may have weights (min weight = 1)
dam general idea
DAM General Idea
  • Sort edges in descendent order
  • Divide the list into classes
  • Divide the classes into subclasses
  • Run a maximal unweighted matching algorithm on the subclasses concurrently
  • Refine resulting edge set
dam in static graphs
DAM in Static Graphs
  • Aim is to define an approximation algorithm whose approximation factor is close to 4
  • Let ε is a positive constant
  • Aim: Find a (4 + 5ε’)-approximate algorithm
  • For simplicity let ε = ε’/5, then approximate factor is (4 + ε)
  • α = 1 + 1/ ε
  • β = α / α-1 = ε + 1
dam in static graphs1
DAM in Static Graphs
  • Assume 1/n ≤ ε ≤ 1/2
  • Otherwise:
    • If ε > ½ then run algorithm with ε = ½
    • If ε < 1/n run Hoepman* algorithm
  • Each class i include edges weighted: w[αi , αi+1)
  • Each class is divided into k = [logβα] subclasses
  • Subclass (i, j) contains edges in class i whose weights are in [αi * βj , αi * βj+1)

*J.-H. Hoepman. Simple Distributed WeightedMatchings CoRR cs.DC/0410047, 2004

dam in static graphs2
DAM in Static Graphs
  • Approach is to reduce the weighted case to multiple instances of unweighted cases
  • Let UWM* is a black-box model for a maximal unweighted matching algorithm
  • TUWM is the runtime of the UWM
  • Run UWM for each subclasses concurrently

* A. Israeli and A. Itai. A fast and simple randomizedparallel algorithm for maximal matching. Info. Proc.Lett., 22(2):77–80, 1986

dam in static graphs3
DAM in Static Graphs
  • Running the UWM on the subclasses sequentially,
    • From heaviest to the lightest
    • Deleting matched nodes from consideration
    • Approximation factor: 2β
    • Running time: # of subclasses * TUWM
  • At the end of concurrent operations, the result may not be a matching
    • First Phase: Run UWM on each subclass of each classes synchronously, this finds matchings in each class
    • Second Phase: Resolve conflicts between different classes
dam in static graphs4
DAM in Static Graphs
  • Second Phase: Resolve conflicts
    • Resulting edges at the end of the first phase is denoted by A
    • Partition edges in A according to weight classes
    • Edges in i’th class is denoted by Ai
    • Note that a node may have at most one incident edge in each Ai
    • If a node has two incident edges in A, the edges are in different classes
    • In such a case, we should select the heaviest edge, BUT...
dam in static graphs5
DAM in Static Graphs
  • The heaviest edge dominates other incident edges of the node
  • However, an edge may dominate others in one endpoint, and be dominated in other endpoint
  • So: Select edges which are dominating in both endpoints (Combine Procedure)
dam in static graphs6
DAM in Static Graphs
  • Analysis
    • The number of phases in the first stage:
      • k = [logβα]
      • Each phase takes TUWM
      • Since 0 < ε ≤ ½
        • logβα = ln α / ln β = ln ( 1+1/ ε) / ln (1 + ε) ≤ (2 log 1/ ε) / ε
      • Total runtime of the first stage : O(1/ ε log 1/ ε TUWM)
    • The number of iterations in the second stage:
      • 3 logα n = O( log n / log 1/ ε )
    • Total runtime of the complete algorithm:
      • O( 1/ ε log 1/ ε TUWM + log n / log 1/ ε )
dam in unweighted dynamic graphs
DAM in Unweighted Dynamic Graphs
  • Each topological change is insertion or deletion of a single node
  • Aim is to develop an algorithm:
    • Whose running time per topological change is O(1/ ε)
    • Whose output is at least 1/(1+ε) times the size of the maximum matching
dam in unweighted dynamic graphs1
DAM in Unweighted Dynamic Graphs
  • AUGMENTING PATH
    • Let G = (V,E) be a graph, let M E be a set of non-intersecting edges in E, and let k ≥ 1.
    • A path v0, v1, . . . , v2(k−1), v2k−1 is an augmenting path of length 2k − 1 with respect to M if for all 1 ≤ i ≤ k − 1, (v2i−1, v2i) M, for all 1 ≤ i ≤ k (v2(i−1), v2i−1) M, and both v0 and v2k−1 are not endpoints of any edge in M.
dam in unweighted dynamic graphs2

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DAM in Unweighted Dynamic Graphs
  • AUGMENTING PATH
    • A node is free if none of its incident edges is in a matching
    • Augmenting path is a path of alternating sequence of matched and unmatched edges with free end nodes
  • Theorem
    • If there is no augmenting path of length 2k-1 then the size of the largest matching is at most {(k+1)/k } * |M| where M is the set of non-intersecting edges
  • The output of the algorithm never contains augmenting paths shorter than 2/ε
dam in unweighted dynamic graphs3
DAM in Unweighted Dynamic Graphs
  • Insertion
    • Algorithm searches for all augmenting paths that starts with the new node v’
    • V’ starts an exploration of the topology of the graph up to (2/ε)+1 from itself to findout if there is an augmenting path of size at most 2/ε
    • If no path is found terminate
    • Otherwise the shortest augmenting path is chosen and roles of the edges are flipped (matching  non-matching and vice versa)
dam in unweighted dynamic graphs4
DAM in Unweighted Dynamic Graphs
  • Deletion
    • If deleted node v was not matched, then terminate
    • Otherwise
      • Find a neighbor on the otherside of the matched edge
      • Re-insert that neighbor using the insertion method
dam in weighted dynamic graphs
DAM in Weighted Dynamic Graphs
  • Basic idea is to reduce the weighted case to the unweighted case
    • Partition the edges into disjoint classes where all edges in class i have weights in [4i, 4i+1)
    • When a node is inserted, it initiates the unweighted algorithm for each weight class according to the weights of its incident edges
    • After O(1) times all algorithms terminate in each classes
    • Each node then picks the matched incident edge in the highest weight class
    • An edge is added iff both its two endpoints choose it
dam in weighted dynamic graphs1
DAM in Weighted Dynamic Graphs
  • The runtime of the algorithm is constant
    • Each of the class weight algorithms works only to distance O(1/ε)
    • Since only one hop neighborhood is affected by the change, we use ε = 1, therefore O(1/ε) = O(1)
    • Only this neighborhood change the output