Distributed Approximate Matching DAM

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Distributed Approximate Matching (DAM)

• Zvi Lotker, Boaz Patt-Shamir, Adi Rosen
• Presentation Deniz Çokuslu
• May 2008

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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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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

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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

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Distributed Approximate Matching (DAM)

• Works on general weighted graphs
• Static Graph Algorithm
• Finds Maximum weighted matching within a factor
  of 4e
• Time complexity O(e-1 log e-1 log n), for e gt 0
• Dynamic Graph Algorithm
• Nodes are inserted or deleted one at a time
• Unweighted matching (1 e)-approximate, T(1/ e)
  time per delete/insert
• Weighted matching Constant approximate, constant
  running time

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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)

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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

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DAM in Static Graphs

• Aim is to define an approximation algorithm whose
  approximation factor is close to 4
• Let e is a positive constant
• Aim Find a (4 5e)-approximate algorithm
• For simplicity let e e/5, then approximate
  factor is (4 e)
• a 1 1/ e
• ß a / a-1 e 1

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DAM in Static Graphs

• Assume 1/n e 1/2
• Otherwise
• If e gt ½ then run algorithm with e ½
• If e lt 1/n run Hoepman algorithm
• Each class i include edges weighted wai , ai1)
• Each class is divided into k logß a
  subclasses
• Subclass (i, j) contains edges in class i whose
  weights are in ai ßj , ai ßj1)

J.-H. Hoepman. Simple Distributed Weighted
Matchings CoRR cs.DC/0410047, 2004
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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
randomized parallel algorithm for maximal
matching. Info. Proc. Lett., 22(2)7780, 1986
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DAM in Static Graphs

• Running the UWM on the subclasses sequentially,
• From heaviest to the lightest
• Deleting matched nodes from consideration
• Approximation factor 2ß
• 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

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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 ith 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...

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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)

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DAM in Static Graphs

• Analysis
• The number of phases in the first stage
• k logß a
• Each phase takes TUWM
• Since 0 lt e ½
• logß a ln a / ln ß ln ( 11/ e) / ln (1 e)
  (2 log 1/ e) / e
• Total runtime of the first stage O(1/ e log 1/
  e TUWM)
• The number of iterations in the second stage
• 3 loga n O( log n / log 1/ e )
• Total runtime of the complete algorithm
• O( 1/ e log 1/ e TUWM log n / log 1/ e )

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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/
  e)
• Whose output is at least 1/(1e) times the size
  of the maximum matching

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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.

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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
  (k1)/k M where M is the set of
  non-intersecting edges
• The output of the algorithm never contains
  augmenting paths shorter than 2/e

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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/e)1 from itself to findout if
  there is an augmenting path of size at most 2/e
• 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)

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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

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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, 4i1)
• 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

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DAM in Weighted Dynamic Graphs

• The runtime of the algorithm is constant
• Each of the class weight algorithms works only to
  distance O(1/e)
• Since only one hop neighborhood is affected by
  the change, we use e 1, therefore O(1/e) O(1)
  
• Only this neighborhood change the output

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Questions ...