O Algorithm for the Miimum Spaig Trees Problem with Diameter Bouded Below Edward Kh. Gimadi 1,2, Alexey M. Istomi 1, ad Ekateria Yu. Shi 2 1 Sobolev Istitute of Mathematics, 4 Acad. Koptyug aveue, 630090 Novosibirsk, Russia 2 Novosibirsk State Uiversity, 2 Pirogova Str. 630090, Novosibirsk, Russia gimadi@math.sc.ru, alexeyistomi@gmail.com, Shius90@yadex.ru Abstract. The miimum spaig trees problem is to fid k edge-disjoit spaig trees i a give udirected weighted graph. It ca be solved i polyomial time. I the k miimum spaig trees problem with diameter bouded below (k-mstbb there is a additioal requiremet: a diameter of every spaig tree must be ot less tha some predefied value d. The k-mstbb is NP - hard. We propose a asymptotically optimal polyomial time algorithm to solve this problem. Keywords: miimum spaig tree, miimum spaig trees problem, asymptotically optimal algorithm, probabilistic aalysis, bouded below, performace guaratees, radom iputs Itroductio The Miimum Spaig Tree (MST problem is oe of the classic discrete optimizatio problems. Give G = (V, E udirected weighted graph, the MST is to fid a spaig tree of a miimal weight. The MST is polyomially solvable, there are classic algorithms by Boruvka (1926, Kruskal (1956 ad Prim (1957. These algorithms have complexity O( 2 ad O(M log where M = E ad = V. The Miimum Spaig Tree with diameter Bouded Below (MSTBB problem is a atural geeralizatio of the MST problem. I the MSTBB aim is to fid a spaig tree of a miimal weight such that its diameter is ot less tha some predefied value d. Diameter of a tree is defied as the umber of edges o the logest path betwee two leaves i the tree. The MSTBB is studied i [6]. It is NP-hard sice with d = 1 it is equivalet to the problem of fidig Hamiltoia Path of a miimal weight i a give graph. Moreover if d is costat the the MSTBB ca be solved i polyomial time by iteratig through all possible simple paths cosistig exactly of d edges. Thus MSTBB is actual with d growig. By F A (I ad OP T (I we deote respectively the approximate (obtaied by some approximatio algorithm A ad the optimum value of the objective fuctio of the Copyright c by the paper s authors. Copyig permitted for private ad academic purposes. I: A. Kooov et al. (eds.: DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
12 Edward Kh. Gimadi, Alexey M. Istomi, Ekateria Yu. Shi problem o the iput I. Accordig to [5] a algorithm A is said to have performace guaratees ( ε A (, δ A ( o the set of radom iputs of the problem of the size, if { Pr F A (I > ( 1 + ε A ( } OP T (I δ A (, (1 where ε A ( is a assessmet of the relative error of the solutio obtaied by algorithm A, δ A ( is a estimatio of the failure probability of the algorithm, which is equal to the proportio of cases whe the algorithm does ot hold the relative error ε A ( or does ot produce ay aswer at all. A algorithm A is called asymptotically optimal o the class of istaces of the problem, if there are such performace guaratees that ε A ( 0 ad δ A ( 0 as. Let us deote by UNI(a, b a class of complete graphs with vertices where weights of edges are idepedet idetically distributed radom variables with uiform distributio o a segmet [a, b ]. By EXP(a, α we will deote a class of complete graphs with vertices where weights of edges are idepedet idetically distributed radom variables with expoetial distributio with a probability fuctio p(x = { 1 α exp ( x a α, if a x <, 0, otherwise. Ad by N(a, σ we will deote a class of complete graphs with vertices where weights of edges are idepedet idetically distributed radom variables with cuttedormal distributio with a probability fuctio p(x = { 2 2πσ 2 exp ( (x a 2 2σ 2, if a x <, 0, otherwise. I [6] preseted a asymptotically optimal algorithm à with complexity O(2 for graphs which belog to UNI(a, b class. O the first stage algorithm à build a path P = {(i 0, i 1, (i 1, i 2,..., (i d 1, i d } of d edges startig from a arbitrary vertex i 0 ad elargig it by addig successive edges of a miimal weight. O the secod stage a spaig tree of a miimal weight D which cotais path P is built usig Prim algorithm. Built tree is take as a solutio. The algorithm à o graphs which belog to UNI(a, b class has the followig performace guaratees: So, à is asymptotically optimal if ( b /a ε = O / l 1, δ = e 0.25( d. d b /a l, d = o(. The same result but for graphs with ulimited edges weights was preseted i [7], where preseted aalysis of algorithm à for graphs from classes EXP(a, α ad
O Algorithm for the k-mstbb 13 N(a, σ. It is show that à fids a solutio with performace guaratees: ( β /a ε = O, δ = O(1/. /l Thus the algorithm is asymptotically optimal i the case the β ( = o, a l where β = α for the graphs from EXP(a, α ad σ i the case of N(a, σ. Aother geeralizatio of the MST problem is the problem of fidig k edge-disjoit spaig trees of a miimal total weight (k-msts. The k-msts is also polyomially solvable, i [4] by Roskid ad Tarja was preseted a algorithm with complexity O( 2 log( + 2 k 2. I curret work a modificatio of the k-msts is cosidered where the aim is to fid k edge-disjoit spaig trees of a miimal total weight such that every tree has a diameter ot less tha some predefied value d. We will call this problem as k miimum spaig trees problem with diameter bouded below (k-mstbb. It is NP-hard sice with d = 1 ad k = 1 it is equivalet to a problem of fidig Hamiltoia Path of a miimal weight i a give graph. We propose a polyomial time algorithm to solve the k-mstbb problem. The algorithm builds k edge-disjoit Hamiltoia chais with d edges each o the preselected set of d + 1 vertices. The by the algorithm by Roskid ad Tarja [4] edge-disjoit spaig trees i the umber of k cotaiig these chais are foud. Also we preset performace guaratees of the algorithm ad sufficiet coditios of asymptotic optimality for the graphs from UNI(a, b. 1 New algorithm A k Let us cosider G = (V, E, a udirected complete -vertex graph belogig to UNI(a, b. Our ew algorithm is based o ideas of from [6], uses the algorithm by Roskid ad Tarja [4] ad algorithm A AV [1] which builds a Hamiltoia path i a give arbitrary graph G p with probability 1 δ AV. Algorithm A k : 1. Build k paths with a umber of edges of d o the set of d + 1 vertices. (a Radomly remove all but d + 1 vertices from graph G, get G[d] iducted by the set of vertices left. (b Build subgraphs G 1 [d],..., G k [d] by a radom procedure: every edge from G[d] radomly gets oe color betwee 1 ad k. (c Remove all edges from G 1 [d],..., G k [d] which weight more tha some threshold w, so we get graphs G 1 w[d],..., G k w[d]. (d I G 1 w[d],..., G k w[d] fid Hamiltoia paths by the algorithm A AV.
14 Edward Kh. Gimadi, Alexey M. Istomi, Ekateria Yu. Shi 2. By the Roskid-Tarja algorithm fid k edges-disjoit spaig trees cotaiig paths built o the previous step. Threshold value w must be chose i a way that G i w[d] cotais a Hamiltoia path. I [2] it was established that o a d + 1 vertex graph where edge exists with a probability p if the value of p is so that graph cotais more tha c(d + 1 log(d + 1 edges, the graph will have a Hamiltoia path. I the paper [1] preseted a algorithm A AV to fid such Hamiltoia path with high probability (whp i O(d l 2 d time. Theorem 1. [1] For all α > 0 there exists a K(α such that if the umber of edges i a radom graph N > (α + K(α(d + 1 l(d + 1, where K(α is sufficietly large, the probability that algorithm A AV fids a Hamiltoia path is 1 O(d α. We will deote a threshold value of algorithm A AV as c AV = (α + K(α. Now let us formulate a Lemma to provide a estimatio of probability that G i w[d] cotais eough edges for A AV to succeed. Lemma 1. The probability that G i w[d] cotais less tha N edges is less tha e d if p 4(N+d ( 1. Proof. δ N = P r{g i w[d] cotais less tha N edges } = Now by the iequality (1 βp k=0 N 1 k=0 ( p k (1 p k e β2 p 2. k ( d(d 1 2 p k (1 p d(d 1 2 k. k which is correct for all, p, β, where is iteger, p [0, 1] ad β [0, 1], we get if p 4(N+d d(d 1. δ N e d(d 1 4 p+n e d, Let s deote δ AV a probability of a case whe the algorithm A AV was uable to give us a solutio: G i w[d] had ot eough edges or algorithm A AV retured failure. By Theorem 1 ad Lemma 1 δ AV = O(d α + e d. Let D be a solutio of the k-mstbb foud by algorithm A k o a -vertex graph. By w 0 we will deote a weight of solutio of the k-msts. Now, let us formulate the followig Lemma. Lemma 2. If all C i, i {1, 2,... k}, were successfully built by the algorithm A k, the w( D w 0 + w(c 1 +... + w(c k kda.
O Algorithm for the k-mstbb 15 Proof. It is obvious that D is a solutio of the k-msts problem with a modified weight fuctio, if w ij is a origial weight of a edge (i, j the modified oe will be w ij, if (i, j / k C t, w ij = t=1 a, if k C t. D is a group of k edge-disjoit spaig trees with a miimum total weight amog a set D of all possible groups of k edge-disjoit spaig trees, where edges weights are w ij. O the oe had we have: mi w ij D D = w ij = (i,j D (i,j D w ij +... + w ij + w( D w(c 1... w(c k = (i,j C 1 (i,j C k t=1 kda + w( D w(c 1... w(c k. O the other had, sice w ij w ij: mi w ij D D mi w ij D D = (i,j D (i,j D mi {w(d D D } = w 0. Combiatio of these two iequalities gives us the result of the Lemma. Lemma 3. If C i was successfully built by the algorithm A k, the w(c i da + 5kc AV (b a log d, i = 1,..., k, where c AV is a threshold value defied for algorithm A AV by Theorem 1 [1]. Proof. Let i {1,..., k}. Let w be (b a p + a, sice edges weights have uiform distributio o a segmet [a, b ] (we are cosiderig graphs from class UNI(a, b, the probability that a edge have weight less tha w is equal to p. To grat to algorithm A AV a possibility to fid a Hamiltoia path graph G i w[d] must have more tha c AV (d+ 1 log(d + 1 edges. Together with Lemma 1 it meas p 5kc AV log d d. It give us that every edge i G i w[d] has weight less tha C i cotais d edges, so 5kc AV (b a log d d + a. w(c i 5kc AV (b a log d + da.
16 Edward Kh. Gimadi, Alexey M. Istomi, Ekateria Yu. Shi Theorem 2. Algorithm A k has the followig performace guaratees: ε = 5kc AV log d b a a, ( ( 1 δ = O k d α + 1 e d, where c AV = (α + K(α is a costat, α > 0. Proof. Let us first assume that we are uder the coditio that every C i, i = 1,..., k, was successfully foud by the algorithm A AV. Usig iequality ka w 0 w( D ad Lemma 2 we have: { P r w( D > (1 + ε w } P r {w 0 + w(c 1 +... + w(c k kda > (1 + ε w } P r {w(c 1 +... + w(c k kda > ε w } P r {w(c 1 +... + w(c k kda > ε ka }. The last probability equals to zero sice the iequality iside is always false due to the defiitio of ε ad Lemma 3 uder the assumptio that every C i, i = 1,..., k, was successfully foud by the algorithm A AV. A probability that C i was successfully foud is 1 δ AV, it give us the followig expressio for the value of the failure probability of the algorithm A k : δ = 1 k (1 δ AV i=1 k i=1 ( ( 1 δ AV = kδ AV = O k d α + 1. d Theorem 3. The algorithm A k is asymptotically optimal if d as, d = o( ad b /a = o(/ log. Proof. It follows from the Theorem 2. 2 Coclusio The problem of fidig k miimum spaig trees with diameter bouded below was studied. The polyomial algorithm was preseted, its performace guaratees ad sufficiet coditios of beig asymptotically optimal were foud for the case where edges weights have idepedet uiform distributio o a segmet [a, b ], b > a > 0. Further aalysis of this problem o graphs with other distributios of edges weights, for example a case whe a = 0, is of a big iterest. Ackowledgmets. This research was supported by the Russia Foudatio for Basic Research (grats 15-01-00976 ad 16-31-00389.
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