NOTE: To change the image on this slide, select the picture and delete it. Then click the Pictures icon in the placeholder to insert your own image. Fast Distributed Algorithms for (Weakly) Connected Dominating Sets and Linear- Size Skeletons Devdatt Dubhashi a Alessandro Mei, Alessandro Panconesi, Jaikumar Radhakrishnan,Aravind Srinivasan Dolev Felman -Adas

Motivation- ad hoc wireless networks The network doesn’t rely on a pre existing infrastructure Each node participates in routing by forwarding data for other nodes Hence it is necessary to set up a so-called backbone: a set of vertices and links among them that is in charge of routing. general consensus that the backbone should be a dominating set 2

Dominating Set each vertex is either in the dominating set (backbone) or next to some vertex in it 3

Dominating Set each vertex is either in the dominating set (backbone) or next to some vertex in it 4

Dominating Set each vertex is either in the dominating set (backbone) or next to some vertex in it 5

Dominating Set - MCDS dominating set (backbone) should be : small connected or weakly connected minimum connected dominating set (MCDS) = NP complete 6

Connected Dominating Set DS is connected if the subgraph induced by it is connected DS is weakly connected: connect every two vertices of DS that are at distance 1 or 2. If the resulting graph (with vertex set DS) is connected, then DS is weakly connected. 7

Dominating Set -low stretch Since the vertices in the dominating set take care of routing, it is desirable that dominators of “nearby” vertices are “relatively nearby” low stretch : for any pair of nodes u and v in G, the distance (in the subgraph induced by DS) between their respective dominators is at most O(log n) times the distance between u and v in G 8 d(u`,v`)<=O(log n)* d(u,v) u v v` u`

Model wireless net-works can be modeled as message-passing vertices are processors edges are communication links network is synchronous processors are required to run in polynomial-time all processors know the size of the graph, or at least a reasonable upper bound on it. 9

Algorithms polylogarithmic-time distributed algorithms compute CDS whose size is at most construct dominating sets that have the above properties, as well as the “low stretch” property. being the maximum degree of the input network 10

Algorithms - compute CDS 11

Let's try to think together how to connect the graph!

connect it up try 1 Let u and v be two vertices in the dominating set at distance at most 3,let Puv be a shortest path connecting them. A connected dominating set can be obtained by inserting all vertices lying on all such paths this method will not preserve the approximation guarantee, for the number of vertices inserted can be quadratic in the size of the original dominating set 13

connect it up try 2 14

Algorithms instead, we will be satisfied as long as the final connected graph has a linear number of edges in the size of DS. We will see 2 algorithms : 1. Randomized: It may (very rarely) fail to have a linear number of edges, but the resulting graph is always connected. 2.deterministic version of the randomized algorithm and never fails Before that we will see some definitions to help us understand the algorithms. 15

Definition- Powers of Graphs 16 3

Proposition 1 17

Proposition 2 v w G’ G w v d(v,w) ≥ 4 18

Continue Proof Let v = v0, v1, v2, v3,..., vd = w be a shortest path from v to w in G. Let u be the vertex of S that is closest in G to v2. since S is a dominating set, we have d(u, v2) ≤ 1. So, d(u, v) ≤ 3, hence u and v are in the same component of G′. But then, u and w must be in different components, d(u,w) ≤ d(v,w) − 1, contradicting the choice of (v,w). Hence, G′ must be connected. G v d =w v=v 0 v1v1 v2v2 u v w G’ u 19

Linear size of CDS - Lemma 20

Linear size of CDS The subgraph H we obtain will have at most 3|S| edges( by removing cycles of length less than 1+2 log( n ) ) each edge corresponding to a path of length at most 3 in G. By adding the intermediate vertices on each such path (at most 2) to S, we get a connected dominating set of size at most 6|S| + |S| = 7|S| vertices, giving us a solution within a factor O(log) of the optimum. 21

For notational convenience, we present these two algorithms as removing all cycles of length less than log n from an n-vertex network G = (V,E) in our application, these algorithms will be run on G′. In view of Proposition 1, there is a blow-up of a constant factor (three) in running the algorithms on G′. 22

Algorithm 1 23

Algorithm 1 - Correctness 24

Algorithm 1- Claim 25

Algorithm 1- Claim 26 g/3

Algorithm 1- Analysis 27

Algorithm 2 We assumed that each edge has a unique ID if not a simple distributed way of achieving this is to make each edge choose a random real in [0, 1] as its ID. Algorithm 2 : Each edge, in parallel, drops out if it is the edge with the smallest ID in a cycle of length less than log n. Running time :O(log n). 28

Algorithm 2 - preserves connectivity sequential process: Order the edges according to their ID number. go over the edges one-by-one from the list and delete them if they are currently in a cycle of length less than log n. when an edge is deleted it is in a cycle, so its deletion cannot disconnect the graph. Thus, in the end we will be left with a connected graph of girth at least log n. 29

low stretch Connected Dominating Sets

Dominating sets with low stretch 31

How to make sequential algorithm distributed? Is there an efficient distributed algorithm for generating a connected dominating set whose induced subgraph H has the “low stretch” property guaranteed by the proposition? Note: Algorithms 1 and 2 are not guaranteed to produce such a solution. Yes! And to see it we will need some more definitions ! 32

Theorem of Linial and Saks 33

The Algorithm 34

The Algorithm 35

The Algorithm 36

Correctness 37

H has a linear number of edges 38

Running time: 39

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For low stretch : If (u, v) is an edge in G, then it is easy to check that the dominators u′ ∈ S and v′ ∈ S of u and v respectively, are adjacent in G′. we know that there is a path of length O(log n) connecting u′ and v′ in H. Finally, since each edge in H corresponds to a path of length at most 3 in G the distance of u` - v` is at most O(log n). 41 u v v` u` d(u`,v`)<=O(log n)* d(u,v)

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