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Distributed way’s to create the Connected Dominating Set Arun Kumar Gupta 201111049
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Jie Wu and Hailan Li’ Algo(Node Marking Algorithm)• Approach• Given a Connected and unweighted graph(Provided Graph is not Complete) and every node have a Marker called m(v) where v is a node for all v ∈ V .• Every node v have N(v) open neighbor set of v .• The marking process is following:• Initially assign marker F to every v in V.• Every v exchanges its open neighbor set N(v) with all its neighbors.• Every v assigns its marker m(v) to T if there exist two unconnected neighbors.
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• they propose two rules to reduce the size of a connected dominating set generated from the marking process.• We first assign a distinct id, id(v), to each vertex v in G’.• Rule 1: Consider two vertices v and u in G’. If N[v] ⊆ N[u] in G and id(v) < id(u), change the marker of v to F. if node v is marked, i.e., G’ is changed to G’ - {v}. Condition N[v] ⊆ N[u] is showing that they are connected .• Rule 2: Assume u and w are two marked neighbors of marked vertex v in G’ . If N(v) ⊆ { N(u) ∪ N(w)} in G and id(v) = min{id(v), id(u), id(w)}, then change the marker of v to F.• The condition N(v) ⊆ {N(u) ∪ N(w) } in Rule 2 implies that u and w are connected. ?
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Here all node are black after Marking Process , nodes 2 , 5 ,6 will become the gray (F) nodes because there id is less than theirneighbor’s id .Rule 2 is creating problem , because every node is taking decision by own (Pure Local).
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Performance• In this approach, the time complexity of the marking process at each vertex is O(Δ2), where Δ(G) = max degree of the graph• The total amount of message exchanges is O(Δv), where v = |V|
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Jie Wu (Extended Concept)• Here there was only one change. they didn’t consider the UDG ,so not all links was the Bidirectional Link.• Two Nodes• If (u,v) is an edge in D, we say that u dominates v and v is an absorbent of u.• Dominator set : A set V’ ⊂ V is a dominating set of D if every vertex v ∈ V – V’ is dominated by at least one vertex u ∈ V’• Absorbent set : A set V’ ⊂ V is a absorbent set of D if every vertex u ∈ V – V’ there exist a vertex v ∈ V’ which is an absorbent of u.
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• Marking Process• Initially assign F to each u ∈ V.• U changes it’s marker m(u) to T if there exist vertices v and w such that (w,u) ∈ A and (u,v) ∈ A but (w,v) ⊄ A.• As before each node v ∈ V’ will assign a Random Id.• v ∈ V’ have two sets Nd (v) , Na (v) .• N(v) = Nd (v) ∪ Na (v)• Rule 1. Assume that u is a marked vertex in V’ and v is a vertex in V . Unmark u if both conditions hold• 1) Nd (u) – {v} ⊆ Nd (v) and Na (u) – {v} ⊆ Na (v)• 2) id(u) < id(v)• U and v are be bidirctionally connected .• Note that u and v may or may not be (bi directionally or uni-directionally) connected . ?• If I believe that statement is true then this not a distributed algorithm. Provided assumption is that every node have 2 hop neighbor information only .
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• Rule 2. Assume that u is a marked vertex in V’ and v and w are vertices in V . Unmark u if all conditions hold• 1. Nd (u) – {v ,w} ⊆ { Nd (v) ∪ Nd (w)} , Na (u) – {v ,w} ⊆ { Na (v) ∪ Na (w)} and• 2. id(u) = min{ id(u) , id(v) , id(w)}• 3. v and w are bidirectionally connected .• Isn’t it it’s for specific condition ?
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Jie Wu’s next Extension• Marking Process was same as last .• DOMINANT PRUNING COVERAGE THROUGH k-NEIGHBOR Rule• G’ = (V’ , E’) is the induced subgraph of a given directed graph G = (V, E).• In this they represents Na (V’k ) , Nd (V’k ) to represent the absorbent and dominating set of vertex set V’k Where V’k is the Strongly connected set .• If Nd (u) – V’k ⊆ Nd (V’k) , and Na (u) – V’k⊆ Na (V’k) in G and id(u) < {id(1) , id(2) , ….. , id(k)} change the marker to the F.• Thing to be noted that so this is distributed algorithm so V’ k information is available to each node u ∈ V’• Then they talked that two rules• restricted rule where each node wants only 2 hop V’ 2 .• Non- restricted where each node have whole nodes information .• At last author suggested to use the restricted rule .• But this not increasing the performance .
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At start all nodes are made themselves black .• if we are applying rule 1 only no node will change to F•If we are applying rule 2 only1.Restricted (with 2 hop Knowledge ) so , no node will become F also2.Only the way is if we have whole Graph Knowledge Non Restricted . Which isdifficult to implement .
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AlZOUBI’s Algo(CDS Via Clustering)• Main Steps 1. Creating MIS 2. Connecting Them• To create the MIS first Create a arbitrary rooted Spanning Tree T , which can be created• Phase 1 :
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