Brief Announcement: Distributed 3/2-Approximation of the Diameter

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1 Brief Announcement: Ditributed /2-Approximation of the Diameter Preliminary verion of a brief announcement to appear at DISC 14 Stephan Holzer MIT holzer@mit.edu David Peleg Weizmann Intitute david.peleg@weizmann.ac.il Roger Wattenhofer ETH Zurich wattenhofer@ethz.ch Liam Roditty Bar-Ilan Univerity liamr@mac.biu.ac.il Contact: Stephan Holzer, , Abtract We preent an algorithm that compute a /2-approximation of the diameter of a graph. Thi algorithm take time O( n log n + D) in the CONGEST model, where in each ynchronou round, every node can tranmit a different (but hort) meage to each of it neighbor. Due to a lower bound tated for graph of mall diameter in [2] thi algorithm i optimal. We extend thi algorithm to compute a (/2 + ε)-approximation of the diameter in time O( n Dε log n + D). 1 Introduction The diameter i one of the mot fundamental propertie of a graph. Thi i true epecially in ditributed computing, where it i ued to define which problem are local (a their runtime i independent of the diameter) and problem that are global (a their runtime i lower bounded by the diameter). Independently of each other the author of [] and [7] preented algorithm that compute the diameter in time O(n). Thi runtime matche a lower bound of [2] tated for network of mall diameter. Beide thi algorithm, the author of [] and [7] alo tate algorithm to approximate the diameter. E.g. [] provide a /2-approximation in time O(n /4 + D) that wa later improved by [5] to O( n log n+d). Thi matche a lower bound of Ω( n+d) derived in [2] for any (/2 ε)-approximation in mall diameter network (and contant ε). However, when the (/2 ε)-approximation lower bound of [2] i analyzed in more detail and generalized to network of arbitrary diameter (and arbitrary ε) it turn into a Ω( (n/d)ε + D) lower bound. In thi brief announcement we preent an algorithm to /2-approximate the diameter in time O( n log n + D) that we obtain by combining reult of [, 7] with idea from [8]. Thi olution i a factor log n fater than the one achieved in [5] and ue a different approach. Our different approach i of interet a we how how to extend it to compute a (/2 + ε)-approximation to the diameter in time O( n/(dε) log n + D). Thu we eentially match the Ω( (n/d)ε + D) lower bound (when the approximation factor i allowed to differ by a mall value). Part of thi work ha been done at ETH Zurich. At MIT the author wa upported by the following grant: AFOSR Contract Number FA , NSF Award CCF, NSF Award CCF , NSF Award number CCF-AF Supported in part by grant from the Irael Science Foundation, the United-State - Irael Binational Science Foundation and the Irael Minitry of Science. Work upported by the Irael Science Foundation (grant no. 822/10) 1

2 2 Model and Baic Definition Model: The CONGEST model [6] i a meage paing model with limited bandwidth. In thi model a network i repreented by an undirected unweighted graph G = (V, E) where node V correpond to proceor (computer or router). Two node are connected by an edge from et E if they can communicate directly with each other. Each node in V ha a unique identifier (ID) in the range of {1,..., 2 O(log V ) }. Node initially know the ID of node in their immediate neighborhood. Communication over edge in E i ynchronou. Every node can end B = O(log V ) bit of information over all it edge in one round of communication. A node can end different meage of ize B to each of it neighbor and receive different meage from each of it neighbor in every round. We are intereted in time complexity, i.e., the number of communication round required by a ditributed algorithm to olve a problem. Subequently, internal computation are neglected. We denote the number V of node of a graph by n, and the number E of it edge by m. For implicity, for u V, we ometime ue u alo to refer to u ID, when thi i clear from the context. Let u denote by d(u, v) the (hop-)ditance of node u and v in G, which i the length of a hortet u-v path in G. A k-dominating et for a graph G i a ubet DOM of vertice with the property that for every v V there i ome u DOM at ditance of at mot k to v. Definition 1 (Eccentricity, diameter). The eccentricity of a node u V i ecc(u) := max v V d(u, v), namely, the maximum ditance to any other node in the graph. The diameter D := max u V ecc(u) = max u,v V d(u, v) of a graph G i the maximum ditance between any two node of the graph. Definition 2 (Approximation). Given an optimization problem P, denote by OP T the value of the optimal olution for P and by ol A the value of the olution of an algorithm A for P. Let ρ 1. We ay A compute a ρ-approximation (multiplicative approximation) for P if OP T ol A ρ OP T for any input. Definition (APSP, S-SP). Let G = (V, E) be a graph. The all pair hortet path (APSP) problem i to compute the hortet path between any pair of vertice in V V. In the S-Shortet Path (S-SP) problem, we are given a et S V and need to compute the hortet path between any pair of vertice in S V. A /2-Approximation to the Diameter We decribe an O( n log n + D)-time algorithm that compute a /2-approximation to the diameter. Thi algorithm i baed on a equential algorithm that wa recently preented in [8] which in turn extend an algorithm of Aingworth et al. [1]. Let C k (w) denote the et of k cloet vertice to w viited by a (partial) BFS tarting in w that top after viiting k node (tie are broken arbitrarily, e.g. by lexicographical order in the tree topology). Thi et C k (w) i computed only for a ingle vertex w (e.g. with mallet ID). Algorithm 1 preented below i a ditributed verion of the non-ditributed algorithm of [8]. The author of [8] provide more intuition behind Algorithm 1 and a proof of correctne. Theorem 1. Algorithm 1 compute a /2-approximation of the diameter w.h.p. in O( n log n+ D) time. Proof. In [8], Theorem 1, it i tated that Algorithm 1 of [8] compute the deired approximation. Our Algorithm 1 i Algorithm 1 of [8] adapted to the ditributed etting. We analyze the runtime of our algorithm: The firt tep can be done locally by every node and w.h.p. create a et S of ize Θ((n/) log n). In tep two, we compute S-SP in time O( S + D) = O((n/) log n + D) by uing the S-SP algorithm from Section 6.1. in []. The reult of thi can be ued by each 2

3 Algorithm 1 Ditributed verion of [8] a executed by each node v G. Output: /2-approximation to the diameter of G 1: each node v join et S with probability log n ; 2: compute a BFS from each node in S; : for every v V, compute p S (v) := the cloet node in S to v; 4: w := arg max v V d(v, p S (v)); 5: compute a BFS tree from w a well a C (w); 6: for every v C (w), compute a BFS tree from v; 7: return the maximum depth of any BFS tree that wa computed; node internally to olve tep without communication. The node w in Line 4 can be found by max-aggregation in time O(D). Computing BFS w and C (w) in Line 5 can be done in O(D) a well. To compute C (w) node w eentially aggregate information on how many node are in each level of the BFS w a e.g. done in Algorithm Diam DOM in [4]. From thi information w compute an i uch that N i (w) < N i+1 (w). Next each node at level i tell it parent how many node at level i + 1 are in it ubtree. Accordingly the node in level i 1 proceed in the ame way and o on. Baed on thi information, exactly N i (w) node can be elected in level i + 1 in time O(D). The next line can be realized by computing C (w)-sp in time O( + D). The return value can be found by a max-aggregation in time O(D). Thu the total time complexity i O( + (n/) log n + D). By chooing := n log n we obtain the deired runtime of O( n log n + D). 4 A (/2 + ε)-approximation to the Diameter Theorem 2. ( For any 0 < ε ) 1/, a (/2 + ε)-approximation of the diameter can be computed w.h.p. in O n D ε 1 log n + D time. To how thi reult, we extend and modify Algorithm 1 lightly by uing idea of the algorithm to (1 + ε)-approximate the diameter preented in [], Section 6.2. Firt we compute a 2-approximation D of D by executing a BFS from the node with mallet ID. Next we et ε /2 ε ε ε := and compute a 8 D -dominating et DOM of ize at mot 8n/(ε D ) in Line. To compute thi dominating et we can ue Algorithm Diam DOM preented in [4]. Now we execute Algorithm 1 retricted to the node in DOM in the ene that other node only implicitly participate (mainly by forwarding meage) a decribed below. In more detail, we firt compute a et S DOM by aking each node in DOM to join S with probability log n, where i choen ( ) later. Therefore S i of ize Θ DOM log n = Θ( n log n ε D ) w.h.p.. Next, for each v S, we compute BF S(v) in graph G. Baed on thi information each node v DOM can compute p S (v), a cloet node in S to v. By a max-aggregation convergecat tarted in the node with mallet ID, we can identify a node w DOM of larget ditance to S, that i a node w DOM uch that d(w, p S (w)) d(u, p S (u)) for all node u DOM. Next we compute a et C DOM (w), which i defined to be a et that conit of cloet node in DOM to w. Thi computation i done in a imilar way to the (partial) BFS in previou ection. Then we perform a BF S(u) for each u C DOM (w). The algorithm return 2(1+ε ) time the maximal depth of any BFS that wa computed during the execution. The proof of Theorem 2 can be found in the appendix and will be included in a full verion of thi brief announcement.

4 Reference [1] D. Aingworth, C. Chekuri, P. Indyk, and R. Motwani. Fat etimation of diameter and hortet path (without matrix multiplication). SIAM Journal on Computing, 28(4): , [2] S. Frichknecht, S. Holzer, and R. Wattenhofer. Network Cannot Compute Their Diameter in Sublinear Time. In Y. Rabani, editor, Proceeding of the 2rd annual ACM-SIAM Sympoium on Dicrete Algorithm, SODA 2012, Kyoto, Japan, January 17-19, 2012, page , [] S. Holzer and R. Wattenhofer. Optimal ditributed all pair hortet path and application. In D. Kowalki and A. Panconei, editor, Proceeding of the 1t annual ACM SIGACT-SIGOPS Sympoium on Principle of Ditributed Computing, PODC 2012, Funchal, Madeira, Portugal, July 16-18, 2012, page 55 64, [4] S. Kutten and D. Peleg. Fat ditributed contruction of mall k-dominating et and application. Journal of Algorithm, 28(1):40 66, [5] C. Lenzen and D. Peleg. Efficient ditributed ource detection with limited bandwidth. In P. Fatourou and G. Taubenfeld, editor, Proceeding of the 2nd annual ACM SIGACT- SIGOPS Sympoium on Principle of Ditributed Computing, PODC 201, Montreal, Quebec, Canada, July 22-24, 201, page 75 82, 201. [6] D. Peleg. Ditributed computing: a locality-enitive approach. Society for Indutrial and Applied Mathematic, Philadelphia, Pennylvania, USA, [7] D. Peleg, L. Roditty, and E. Tal. Ditributed algorithm for network diameter and girth. In A. Czumaj, K. Mehlhorn, A. M. Pitt, and R. Wattenhofer, editor, Automata, Language, and Programming - 9th International Colloquium, ICALP 2012, Warwick, UK, July 9-1, 2012, Proceeding, Part II, volume 792 of Lecture Note in Computer Science, page Springer, Berlin & Heidelberg, Germany, [8] L. Roditty and V. V. William. Fat approximation algorithm for the diameter and radiu of pare graph. In D. Boneh, T. Roughgarden, and J. Feigenbaum, editor, Proceeding of the 45th annual ACM Sympoium on Theory of Computing, STOC 201, Palo Alto, California, USA, June 1-4, 201, page ,

5 Appendix A Proof of Theorem 2 Algorithm 2 Ditributed verion of [8] a executed by each node v G. Input: accuracy parameter ε Output: (/2 + ε)-approximation to the diameter of G 1: compute and broadcat D := 2 ecc(id min ); 2: ε := ε /2 ε ; k := ε D /8 ; : compute DOM := k-dominating et of ize at mot max{1, n/(k + 1) }; 4: v join et S with probability log n ; 5: compute a BFS from each node in S; 6: For every v V, compute p S (v) := the cloet node in S to v; 7: w := arg max v V d(v, p S (v)); 8: compute a BFS tree from w a well a C DOM (w); 9: For every v C DOM (w), compute a BFS tree from v; 10: return 2(1+ε ) time the maximum depth of any BFS tree that wa computed; We already know from Theorem 5 in [8] that Algorithm 1 produce a /2-approximation when executed on G. We follow along the line of their proof and adopt it to our modified algorithm to how that it compute a (/2 + ε)-approximation to the diameter. Lemma 1. Let G = (V, E) be a graph with diameter D = h + z, where h 0 and z {0, 1, 2}. The maximal depth of any BFS tree that wa computed in Algorithm 2 i at leat 2h(1 ε ) w.h.p.. Proof. Let a, b V uch that d(a, b) = D. Then there are node a, b DOM uch that d(a, b ) D 2k = D(1 ε 2 ). Let w DOM be a vertex that atifie d(w, p S(w)) d(u, p S (u)) for all node u DOM. Cae 1, (d(w, p S (w)) h): Then d(a, p S (a )) h. A the algorithm compute BF S(v) for every v S, it follow that BF S(p S (a )) i computed a well. Since ecc(a ) i at leat D k = D(1 ε /4), it follow that ecc(p S (a )) ecc(a ) h = 2h + z ε 4 (h + z) 2h(1 ε ) a required. Cae 2, (d(w, p S (w)) > h): We can alo aume that ecc(w) < 2h(1 ε ) ince the algorithm compute BF S(w) and if ecc(w) 2h(1 ε ) then it compute a BFS tree of depth at leat 2h(1 ε ) a required. Since ecc(w) < 2h(1 ε ) it follow that d(w, b ) < 2h(1 ε ). Moreover, ince d(w, p S (w)) > h we can conclude that S hit C DOM (w) w.h.p., that i S C DOM (w). Therefore it mut be the cae that C DOM (w) contain a node at ditance greater h from w, and hence N h (w) C DOM (w). Thi implie that there i a vertex w C DOM (w) on the path from w to b uch that d(w, w ) = h and hence d(w, b ) < 2h(1 ε ) h = h 2ε. Since d(a, b ) D(1 ε ε 2 ) = (h + z)(1 2 ), we alo have that d(a, w ) d(a, b ) d(w, b ) > 2h(1 ε ). The algorithm compute BF S(u) for every u C DOM (w), and in particular, it compute BF S(w ), which ha depth at leat d(a, w ) 2h(1 ε ). Proof. (of Theorem 2). Correctne follow from Lemma 1 combined with the choice of ε and the requirement that ε 1/: By multiplying the depth of the deepet BFS performed with 2(1 ε ), we obtain an etimate ˆD uch that D ˆD ( 2 + ε)d. 5

6 Analyzing the runtime i almot the ame a in Theorem 1. We only need to add O(D) for n computing the k-dominating et in Line 1. The total runtime i O( S ++D) = O( ε D ++D). Chooing := n log n ε D yield the deired runtime. 6