Lecture 5. Figure 1: The colored edges indicate the member edges of several maximal matchings of the given graph.

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1 5.859-Z Algorithmic Superpower Randomization September 25, 204 Lecture 5 Lecturer: Bernhard Haeupler Scribe: Neil Shah Overview The next 2 lectures are on the topic of distributed computing there are lots of interesting problems in this field where randomization can be applied for performance gain. Specifically, the topics covered will include maximal matching, clustering, coloring and computing the maximal independent set of a graph. The setting we consider for our distributed system is a connected graph G with n nodes, where the nodes know their neighbors but do not know the global topology of the network. This assumption is realistic given that the structure of a distributed network may change over time, and nodes may die or be added. Communication between nodes is done only in synchronous rounds, where each node can send any one message to each of its neighbors. The typical message size we expect to send is roughly O(log n) bits long. Maximal Matching The maximal matching problem involves selecting an edge set E in G such that no node is adjacent to more than one edge, with the condition that if another edge is added to E, we no longer have a matching. Figure : The colored edges indicate the member edges of several maximal matchings of the given graph. Note that a maximal matching is different from a maximum matching a maximum matching is a matching that contains the largest possible number of edges (there may be many of these for a given graph G). Furthermore, any maximum matching is maximal, but not every maximal matching is a maximum matching. It can be shown that any maximal matching is a 2-approximation of a maximum matching. It is easy to check if a given matching is maximal by looking locally at whether another edge can be added and whether it violates 5-

2 the property that a single node touches more than edge this process involves each node looking only at its immediate neighbors. d-clustering The d-clustering problem involves selecting a node set S of cluster centers, and assigning all nodes in G to some center c S such that no two centers are closer than d distance (in terms of shortest-path length), with the property that every node in G is close to one of these centers. Graph Coloring The graph coloring problem involves assigning a single color to each node in G such that no node is of the same color as any of its neighboring nodes. Typically, the goal in graph coloring problems is to use the minimal number of colors to color the entire graph (that is, assign a color to each node in a fashion that satisfies the aforementioned property). Maximal Independent Set The maximal independent set problem involves selecting a node set S such that no two nodes in S are directly connected by an edge, with the property that we can not add any additional nodes to S without violating this rule. Figure 2: The colored nodes indicate the member nodes of one maximal independent set of the given graph. Claim: Computing the maximal independent set (MIS) of a graph G also enables one to solve the maximal matching, d-clustering and graph coloring problems. w.r.t maximal matching: We can compute the maximal matching of G by computing the MIS of the line graph of G. The line graph of G is another graph, denoted L(G) which represents the adjacencies between edges of G. Specifically, each node in L(G) represents an edge of G, and two nodes are adjacent in L(G) if and only if their corresponding edges share a common node in G. A MIS 5-2

3 in L(G) identifies a maximal set of nodes which are not adjacent, which corresponds to a maximal set of edges in G which are not adjacent. Figure 3: An example depicting the construction of the line graph L(G) from the original graph G. w.r.t d-clustering: We can compute a d-clustering on G by computing the MIS of G d, which connects all the nodes which are within distance d of each other. An MIS on G d guarantees that the constituent nodes (which are now cluster centers) are at least d distance away from each other in G. If a node in G does not have a center within d distance away from it, it can itself become a center. w.r.t graph coloring: The reduction is non-trivial and is not covered in this lecture. We now discuss one algorithm which can be implemented in a distributed setting for computing MIS, called Luby s algorithm. 5-3

4 Luby s Maximal Independent Set Algorithm Algorithm : Luby s algorithm to compute MIS Data: input graph G(V, E) with V = n Result: maximal independent set S Let S = ; 2 while G is non-empty do 3 Randomly assign each node v in G a unique priority π v from [, n 5 ]; 4 Let W be the set of nodes in G with higher priority than all of their neighbors (nodes v such that u N(v), π v > π u ); 5 S = S W ; 6 G = G \ (W N(W )); 7 end Note that the range [, n 5 ] is simply used here in order to ensure that each node will be assigned a unique priority w.h.p. In practice, any such range can be used. Luby s algorithm works by choosing an independent set in G at every iteration. By removing the nodes in the independent set as well as their neighbors from G before choosing the next independent set, S must always contains an independent set, since no two nodes in S can be neighbors. Furthermore, the independent set the algorithm produces must be a MIS. If the resulting independent set S was not maximal, it would imply that one or more nodes from G \ S could be added to S to make it maximal. But, each node in G \ S was removed from G because it neighbored some node in S, so adding any nodes from G \ S to S would violate the property of S being an independent set. Thus, S must be a MIS. We can also make the following claim about the performance of Luby s algorithm. Claim log n iterations will suffice to produce an MIS using Luby s algorithm. Proof We will prove this claim by showing that a constant fraction of edges are removed from G in every iteration of the algorithm in expectation. Consider an edge (u, v) in G. Let X u denote the indicator variable for the event that node u s priority is greater than all of its neighbors and all of v s neighbors (excluding u). If X u =, u will be included in the MIS and all edges touching u and v will be removed from G let us say that if in this case, u preemptively removes v and its incident edges. We can identify the probability of this event occurring as P (X u = ) d(u) + d(v) where d(n) denotes the degree of node n. The statement is an inequality because u and v may share some neighbors. () 5-4

5 Note that any vertex can be preemptively removed at most once and any edge (u, w) can be preemptively removed twice (once when u is preemptively removed and once when w is preemptively removed). Then, we can write the expected number of edges removed as 2 (u,v) E (d(u)p (X v = ) + d(v)p (X u = )) (2) where the 2 factor is used because the summation double counts removed edges. Then, the quantity is at least 2 (u,v) E ( d(u) d(u) + d(v) + d(v) ) = E d(u) + d(v) 2 (3) Thus, half of the edge set is removed in expectation in every iteration of Luby s algorithm. It follows that log n iterations are needed for all edges to be removed from G. References [] Line graph. [2] Matching. [3] Costas Busch. Maximal independent set. courses/distributed/fall20/slides/mis.ppt, 20. [4] Nancy A Lynch. Distributed Algorithms. Morgan Kaufmann,