Minimum Spanning Trees in Temporal Graphs

Transcription

1 Minimum Spanning Trees in Tempora Graphs Siu Huang Chinese University of Hong Kong Ada Wai-Chee Fu Chinese University of Hong Kong Ruifeng Liu Chinese University of Hong Kong ABSTRACT The computation of Minimum Spanning Trees (MSTs) is a fundamenta graph probem with important appications. However, there has been itte study of MSTs for tempora graphs, which is becoming common as time information is coected for many existing networks. We define two types of MSTs for tempora graphs, MST a and MST w, based on the optimization of time and cost, respectivey. We propose efficient inear time agorithms for computing MST a. We show that computing MST w is much harder. We design efficient approximation agorithms based on a transformation to the Directed Steiner Tree probem (DST). Our soution aso soves the cassica DST probem with a better time compexity and the same approximation factor compared to the state-of-the-art agorithm. Our experiments on rea tempora networks further verify the effectiveness of our agorithms. For MST w, our soution is capabe of shortening the runtime from 1 hours to seconds. 1. INTRODUCTION The increasing popuarity of socia networks and the emergence of other arge scae networks have attracted much reated research interest in recent years. Whie most works so far focus on static networks, many networks of interest carry time information, which can be highy usefu and important for network anaysis. Such networks are termed tempora networks [15]. An important category of tempora networks is that of communication, incuding phone cas, emais, messaging, and socia contact networks, where each edge represents a correspondence between two parties or two individuas. Another exampe is a network of airports, with edges abeed by the times of fight departures and arrivas. Other tempora networks incude neura and brain networks, ecoogica networks, etc. A comprehensive survey of tempora networks is given in [15]. Consider the ca detai records (CDR) generated by teecom companies as a tempora graph exampe. CDR records the caing party, the receiving party, the start time, the ca duration and biing information. We thus assume that we are given a directed graph G = (V, E), where each edge e in E is of the form (u, v, t u, ˆt v, w), e inks source vertex u to destination vertex v with start time t u and end time ˆt v, and carries a weight (or cost) of w. A path in G is a Permission to make digita or hard copies of a or part of this work for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. SIGMOD 15, May 1 June, 215, Mebourne, Victoria, Austraia. Copyright c 215 ACM /15/5...$ sequence of edges e 1, e 2,..., e k for some k V, where the destination vertex of e i is the source vertex of e i+1. The path is timerespecting if the start time of e i+1 is not earier than the end time of e i [2]. Information fows aong a path P in the network ony if P is time respecting. The spread of information, rumor, or disease (epidemics) can foow ony time respecting paths. Therefore, in a tempora graph, we need to consider time-respecting paths. This necessity often makes tempora graphs much harder to hande. For exampe, computing strongy connected components takes inear time in a static graph, but is NP-compete in a tempora graph []. The computation of minimum spanning trees (MST) is a cassica probem in graph theory. One major appication of MST is network broadcasting. In a socia network, this corresponds to information dissemination, which is important in socia campaigns, vira marketing, study of rumor spreading, etc. In view of this appication, we define two different optimization probems for spanning trees in a tempora graph. The first is a time based optimization, which is for the fastest spread from a source vertex to each reachabe vertex. We refer to such an optima spanning tree as a MST a. The second is a cost based optimization, assuming each edge in the graph carries some weight (cost), a spanning tree with a minimum tota weight is a MST w. Given a non-tempora (static) directed graph G = (V, E), the MST probem can be soved in O( E + V og V ) time [12]. However, the MST probems for tempora graphs behave very differenty. Our study shows that a MST a can be computed by efficient agorithms with inear time. However, the probem of computing a MST w is MAX-SNP hard. Hence, there is no poynomia time approximation scheme for computing MST w uness P = NP. We transform the MST w probem on tempora graphs to the cassica Directed Steiner Tree (DST) probem. The state-ofthe-art approximation agorithm for DST requires O(n i k 2i ) computation time, where n is the number of vertices, k is the number of terminas, and i is the number of iterations in the agorithm [8]. We derive an improved agorithm that runs in O(n i k i ) time with the same approximation ratio. Our main contributions are summarized as foows. (1) We consider MSTs for tempora graphs. Based on the appication of information dissemination, we define two types of MST, namey, MST a for optimizing time, and MST w for optimizing cost. (2) For computing a MST a, we propose inear time agorithms which are optima and improve on the resuts in []. () We show that computing a MST w is MAX-SNP hard. We transform the MST w probem to the minimum Directed Steiner Tree probem (DST) with a conversion of the tempora graph to a static graph. () DST is a cassica probem of independent interest. Our resut improves on the best-known approximation agorithm in [8], achieving a better time compexity whie having the same approximation guarantee.

2 (5) We evauate our approaches with a comprehensive set of experiments on rea tempora networks. We show that our DST agorithm is up to 5 orders of magnitude faster than the state-of-the-art agorithm. This paper is organized as foows. Section 2 contains our probem definitions. Section describes the probem of MST a and our agorithms for this probem. Section is about the MST w probem and its soutions. Our empirica study is reported in Section 5, reated work is discussed in Section 6. We concude in Section PROBLEM DEFINITION In this section we introduce some definitions and notations, and two probems of minimum spanning trees in a tempora graph. 2.1 Notations for Tempora Graphs Let G = (V, E) be a tempora graph, where V is the vertex set and E is the tempora edge set. Let V = n and E = M. We aso denote V by V (G) and E by E(G). Each tempora edge e E is abeed with a starting point u, an end point v, a starting time t u from the starting point, an arriva time ˆt v at the end point, and a weight (or cost) w, denoted as e = (u, v, t u, ˆt v, w) where u, v V, and t u, ˆt v, w are non-negative rea numbers. The starting point, end point, starting time, arriva time, and weight of e are aso denoted by s(e), a(e), t s(e), t a(e), and w(e), respectivey, i.e., s(e) = u, a(e) = v, t s(e) = t u, t a(e) = ˆt v, and w(e) = w. We assume that t a(e) t s(e). Aso, et d(e) be the duration on tempora edge e, i.e., d(e) = ˆt v t u, d(e). We say that e is an in tempora edge incident to v, and it is an out tempora edge incident to u. Let N o(u) and N i(u) denote the set of out tempora edges and the set of in tempora edges incident to vertex u, respectivey, i.e., N o(u) = {e s(e) = u, e E}, N i(u) = {e a(e) = u, e E}. Note that G can be a cycic graph. <1,> <,5> <1,5> <,6> [2] [1] [] [] <8,9> <,6> [2] [1] <5,8> <7,9> <6,8> [] [2] [2] Figure 1: Tempora Graph G EXAMPLE 1. Figure 1 is a tempora graph. Each tempora edge is abeed with t u, ˆt v and [w]. In this exampe, we set the weight on each tempora edge as the duration,i.e., ˆt v t u. However, note that in genera, w can be set as any vaue. The graph can represent phone cas. Each vertex is an individua and an edge from u to v is a phone ca from u to v. The tempora edge in red (bod) refers to e = (, 1, 1,, 2), meaning that there is a phone ca from individua to individua 1 starting from time 1, ending (arriving) at time and the weight is 2. The graph may aso represent fights, an edge from u to v is a fight from u to v. A tempora graph can be converted into a static graph by discarding the time information. Given a tempora graph G = (V, E), et G S = (V S, E S) be the corresponding static graph. Then, V S = V, E S = {(u, v, w) (u, v, t u, ˆt v, w) E}. V = V S = n, et E S = m. P = {e 1, e 2,..., e k } is caed a path in a tempora graph if and ony if a(e i) = s(e i+1) and t a(e i) t s(e i+1), where 1 i < k. We say that P is a path from vertex s(e 1) to a(e k ). We ca t a(e i) t s(e i+1) a time constraint. We ca t a(e k ) the arriva time of P, and denote it by A(P ). Given a time interva [t α, t ω]. 5 <1,> <1,5> <,6> <5,8> <6,8> 5 <1,> <,6> [2] [] <,6> [2] <7,9> <6,8> [2] [2] (a) MST a (b) MST w Figure 2: Minimum Spanning Tree of Figure 1 We say that a path P in G is within [t α, t ω] if for each edge e i in P, t α t s(e i) and t a(e i) t ω. A vertex v is reachabe from u in G in [t α, t ω] if v = u or there is a path from u to v within [t α, t ω]. 2.2 Two Types of Minimum Spanning Trees Given a tempora graph G = (V, E), a root r V, and a time interva [t α, t ω]. Let P(r, v) be the set of paths within [t α, t ω] from r to v, if P P(r, v) has the eariest arriva time among a paths in P(r, v), we ca the arriva time A(P ) the eariest arriva time for v, and denote it by Ã(v). Let V r be the set of vertices reachabe from r in G in [t α, t ω]. A spanning tree ST (r) = (V ST, E ST ) rooted at r is a subgraph of G with the foowing properties: (1) V ST = V r. (2) v V where v r, there is exacty one in tempora edge incident to v; r has no in tempora edge, E ST = V r 1. () v V where v r, there exists a path from root r to v in ST (r) within [t α, t ω]. We define the foowing two types of minimum spanning trees in G and our probems are the computation of such MSTs. MST a (Minimum spanning tree with eariest arriva times). A spanning tree ST (r) = (V ST, E ST ) rooted at r is a MST a if and ony if v V ST, v r, the path P from r to v in ST (r) has the eariest arriva time for v, i.e. A(P ) = Ã(v); MST w (Minimum spanning tree with smaest tota weight): A spanning tree ST (r) = (V ST, E ST ) rooted at r is a MST w if and ony if e E ST w(e) is minimum among a spanning trees rooted at r. We refer to e E ST w(e) as the weight or cost of ST (r) and aso denote it by ζ(st (r)). EXAMPLE 2 (MINIMUM SPANNING TREES). Let us set the root r as vertex in tempora graph G in Figure 1, we obtain V r = {1, 2,,, 5}. Figure 2(a) is a MST a for G, whie Figure 2(b) is a MST w for G, rooted at vertex. In Figure 2(a), the arriva times for vertices 1, 2,,, 5 are, 5, 6, 8, 8, respectivey, which are the corresponding eariest arriva times in G. If the graph represents phone cas, these are the eariest times for a message to be transmitted from to the other individuas. In Figure 2(b), the tota weight of a edges is = 11, which is the smaest among a possibe spanning trees of G rooted at. If the graph represents phone cas, then the minimum cost to transmit a message from to a other individuas is 11. If the graph represents fights, then 11 is the minimum cost to distribute some goods from airport to a other airports. 2. Motivation for MST a and MST w Some discussion about our probem definitions is in order. For non-tempora graphs, a branching is a forest in which each edge is directed towards a different vertex. Given a directed graph G = (V, E), a branching G = (V, E ) in G is a spanning arborescence, or a directed spanning tree (DSPT), of G iff E = V 1. 5

3 A minimum DSPT (MDSPT), has the smaest tota edge weight among a DSPTs [1, 11, 25]. Ceary, G contains a DSPT iff a vertices in G are reachabe from some vertex r in G. Hence, previous studies on MDSPT assume that a vertices in V are reachabe from some root vertex r in V. Then, either the root vertex is prescribed or it is not prescribed. In the eary work by Edmonds, appication of a east costy way to have a message communicated from the root to a other nodes is described, where the root is prescribed [11]. To our knowedge, no appication is known for the case where the root is not prescribed. Thus, as in the best known agorithm for MDSPT in [12], we dea with a prescribed root r. In rea tempora graphs, different vertices may reach different sets of vertices, and a given vertex r typicay cannot reach a vertices. Thus, we ensure the reachabiity condition by considering the subgraph induced by a vertices reachabe from r. Our work is motivated by a number of appications. Information fows foow time respecting paths in a network, and a MST a from a source vertex s tes us the eariest time that a piece of information can be dispatched to other sites. MST a is aso usefu for the study of epidemioogy, the spread of infectious diseases (or computer virus), when the network is about individua contacts (or a computer network) [, 2]. Another appication is the schedue of transportation. MST a gives a schedue of transports for distribution of goods from a source ocation with the eariest arriva time for each destination. Information dissemination is the major motivating appication for the study of tempora networks in [2]. An important appication of MST w is to minimize the cost of message communication or information propagation, where each edge represents a ink of direct communication from one party to another, with a weight that indicates the cost of the communication, provided that the cost is additive. E.g. phone communication invoves some cost for each ca. Given a time window [t α, t ω], MST w gives us the minimum information dissemination cost from a person. As the time window sides forward, we can predict the minimum cost for the future. The transportation probem cited in [2] is another appication. An exampe is a network of airports with edges corresponding to fights, and each fight invoves some cost. A MST w minimizes the tota cost to transport some given resource from a given ocation r to a destinations. MST w can aso be usefu for custering [2, ], which is reated to community search in socia networks. MST a bears some resembance to a shortest path tree in a nontempora muti-graph in that the path from the root to each vertex is optimized. The difference is that for a shortest path tree, the tota ength of edges in the path from r to each vertex v is minimized, whie in a MST a, the arriva time at v is minimized. MST a aso soves the probem of minimizing the maximum arriva time since the arriva time for each vertex is minimized.. TIME-MINIMUM SPANNING TREE We consider the computation of a MST a in a tempora graph. Given tempora graph G, an interva [t α, t ω], and root r, et Ã(v) be the eariest arriva time from root r to reachabe vertex v in V r. We can show that there exists a MST a rooted at r which is made up of paths from r to a reachabe vertices with the eariest arriva times. LEMMA 1. Given a tempora graph G = (V, E) and the root r, there must exist a minimum spanning tree (MST a) rooted at r. In the foowing we propose two efficient agorithms for MST a with inear compexity. Let us first present the data format for the tempora graph. We assume that a tempora edge is of the Agorithm 1: MST a(r, G, t) Input : A tempora graph G (chronoogica edge ist), root r, time interva [t α, t ω]; Output : MST a (given by P()) 1 Initiaize: A(u) =,P(u) = r u r, u V ;A(r) = t α; 2 for e = (u, v, t u, ˆt v, w) G do if (t u A(u))&(ˆt v < A(v))&(ˆt v t ω) then A(v) ˆt v; 5 P(v) u; form e = (u, v, t u, ˆt v, w). We are given a ist of M tempora edges, {e 1, e 2,..., e i,..., e M }, as the input raw data. Assume the edges are ordered by non-decreasing starting time t s(e i). We ca this raw data a chronoogica edge ist input, denoted by G = [e 1, e 2,..., e M ]. The first agorithm is given in Agorithm 1. EXAMPLE. Consider the graph in Figure 1. We set the time interva of interest as t α =, t ω =. The first edge in the input is (, 1, 1,, 2), we update A(1) to, and P(1) to. The next input is (, 2, 1, 5, ), we set A(2) to 5, and P(2) to, according to Lines -5 in Agorithm 1. For the next inputs of (, 2,, 6, ), (, 1,, 5, 1), the if condition at Line is not met, no update is triggered. Simiary, we process the remaining inputs and the MST a in Figure 2(a) is generated. THEOREM 1. Agorithm 1 returns a MST a in O(M) time given t s(e) t a(e), e E. (M = E ) PROOF. Let S T be the set of eariest arriva time for vertices in V r. Let S T = {t 1, t 2,...t i,...t ST }, where t j < t j+1, 1 j S T 1. First we prove that A(u) in Agorithm 1 correcty records the eariest arriva time for each vertex u V r whose Ã(u) = t1. Vertex u must be one hop from root r. Suppose {e} is the eariest arriva path from r to u, where t a(e) = t 1, then A(u) must be set to t 1 when processing e in ine 2 since t s(e) t α and A(r) is set to be t α initiay, and note that t α is a ower bound of starting times. We prove by induction. Assume that the fina A(v) in Agorithm 1 correcty records the eariest arriva time for any vertex v with Ã(v) < ti. We need to prove that A(v ) correcty record the eariest arriva time for any vertex v with Ã(v ) = t i. Let P = {e 1, e 2,..., e k } be a path from r to v with eariest arriva time Ã(v ) = t i; et v j = a(e j), where 1 j < k. From inequaity t a(e k 1 ) t s(e k ) < t a(e k ) = t i, we have t a(e k 1 ) < t i. Aso, Ã(v k 1 ) t a(e k 1 ), since Ã(v k 1) is the eariest arriva time for vertex v k 1 = a(e k 1 ). Thus, Ã(v k 1 ) t i. By the induction hypothesis, the fina A(v k 1 ) = Ã(v k 1). Note that when processing e k at Line 2, A(v k 1 ) has aready been set to Ã(v k 1) since t s(e k ) t a(e k 1 ) > t s(e k 1 ) and tempora edges are processed in ascending order of t s(e), e E. According to Line, A(v ) wi be set to t i when edge e k is processed. Hence, Agorithm 1 correcty returns a MST a as recorded in {P(v) v V r}. Finay, Agorithm 1 needs ony one scan of the tempora edges, thus, the time compexity is O(M). Agorithm 1 aso appies when the input tempora graph consists of edges sorted in ascending order of the arriva time instead of the starting time. This can be proved by induction, simiar to the proof of Theorem 1, as foows. Foowing the notations in the induction in the proof of Theorem 1, et P = {e 1, e 2,..e k 1, e k } be a path from r to v with the eariest arriva time Ã(v ) = t i = t a(e k ); et v j = a(e j), where 1 j < k. According to the induction hypothesis, A(v k 1 ) is set to Ã(v k 1) correcty since

4 t a(e k 1 ) < t i. Note that A(v k 1 ) must be set to Ã(v k 1) before the occurrence of e k since t a(e k ) > t a(e k 1 ) Ã(v k 1) and tempora edges are sorted in non-descending order of arriva time t a(e). Thus, after processing e k in Line 2, A(v k ) equas t a(e k ), i.e., A(v ) = Ã(v ). Hence the agorithm is correct. However, in Theorem 1, we have the constraint that t s(e) t a(e), e E. That is, Agorithm 1 requires that the duration at each edges is non-zero. In some rea tempora graphs, such as the graphs for pubications, facebook wa posting, or emai communication, the edge durations are given as zero, in such a case the agorithm may fai, as shown in the foowing exampe. EXAMPLE. The tempora graph G in Figure is an exampe where Agorithm 1 cannot correcty return a MST a. Suppose start time is, the edges of the input graph are sorted as foows: (, 1, 1, 1, ), (2,, 2, 2, ), (, 1, 2, 2, ), (1,,,, ), (, 2,,, ), (,,,, ). In this case, when (, 2,,, ) is visited, A() =, the condition at Line wi fai and edge (, 2) wi not be added to the MST. Hence, vertex 2 wi be missing in the resuting tree. Hence, we need another agorithm to dea with the genera case where an edge duration can be zero, i.e., t s(e) t a(e), e E. For this probem, we first transform the input graph G from the raw stream format into the foowing sorted adjacency edge ist format. Each input ine corresponds to the out tempora edges incident to one vertex u V, in the foowing form: u, < v 1, t 1, ˆt 1, w 1 >,..., < v i, t i, ˆt i, w i >,... In each ine, edges e N o(u) are sorted by the start times, t s(e), in non-increasing order, u V. That is, t i t i+1, where 1 i < N o(u). We imit the edges so that the start time is eariest t α and end time is atest t ω, we ca the resuting edge ist input G[t α, t ω]. The transformation takes O(M) time. <1,1> <2,2> <,> <2,2> <,> <,> <1,1> <,> <,> <,> Figure : G with zero durations and its MST a We propose another efficient agorithm, Agorithm 2, with inear time compexity to sove the genera MST a probem. The input to the agorithm adopts the sorted adjacency edge ist format above. In Agorithm 2, we assume that for each vertex u V, tempora edges incident to u are sorted in non-ascending order of t s(e) and stored in array N o(u). We ony need to scan N o(u) once and pos(u) is kept to indicate the current position in N o(u). Ony smaer arriva times are recorded as shown in Lines and 5, and the time constraint for each path is maintained by Line 9. The compexity for Agorithm 2 is bounded by O(M). Our empirica study aso verifies the efficiency of this proposed agorithm. THEOREM 2. Agorithm 2 returns a MST a in O(M) time, where M = E. PROOF. We first prove the correctness. That is, given tempora graph G and root r, the path from r to v in MST a, v V, v r, resuts in eariest arriva time of v in G, where MST a is the resuting minimum spanning tree from Agorithm 2. Let the path from r to v in MST a be P = {e 1, e 2,...e k }, where v = a(e k ). Note that t a(e k ) is the output A(v) of Agorithm 2. We say that vertex v is at eve k of MST a. We prove by induction on k. Agorithm 2: MST a(r, G) Input : A tempora graph G[t α, t ω] (in sorted adjacency edge ist format), root r, time interva [t α, t ω]; Output : MST a (given by P()) 1 Initiaize: A(u) =, pos(u) = 1, u V ; push < r, r, t α > onto stack S; 2 whie S do < u, v, ˆt v > top(s), pop(s); if ˆt v < A(v) then 5 A(v) ˆt v; 6 P(v) u; 7 if pos(v) N o(v) then 8 < v, v, t v, ˆt v, w > N o(v)[pos(v)]; 9 whie pos(v) N o(v) and A(v) t v do 1 push < v, v, ˆt v > onto stack S; 11 pos(u) + +; 12 < v, v, t v, ˆt v, w > N o(v)[pos(v)]; When v is at eve 1, i.e., k = 1 and P = {e 1}, then t a(e 1) must be the eariest arriva time for a(e 1), i.e., t a(e 1) = Ã(a(e1)). This is because a tempora edges meeting the time constraint wi be visited as described in Line 9 of Agorithm 2 and we aways keep the smaest arriva time for each vertex as described in Line of Agorithm 2. Assume that when k < i, t a(e k ) is the eariest arriva time for vertex v, i.e., t a(e k ) = Ã(v), where v is at eve k of MST a. We want to prove that t a(e i) is the eariest arriva time for vertex v = a(e i) when v is at eve i, i.e., when k = i. Suppose there exists a path from r to v in G, P = {e 1, e 2,..., e }, where a(e ) = v and t a(e ) < t a(e i). There must exist one vertex ũ = s(e j) such that ũ is at eve h of MST a, where h < i. Then according to the hypothesis, A(ũ) in Agorithm 2 is equa to Ã(ũ). Thus, according to Line 9 in Agorithm 2, e j must be visited since t s(e j) t a(e j 1) Ã(ũ), and A(a(e j)) must be no more than t a(e j) according to Lines -5. Repeat the above anaysis, we can concude that A(a(e )), i.e., A(v ) (or t a(e i)), must be no more than t a(e ), which contradicts with t a(e ) < t a(e i). Hence, t a(e i) is the eariest arriva time for vertex v = a(e i) when v is at eve i, i.e., k = i. Next we anayze the time compexity for Agorithm 2. There are at most M outer whie oops (Line 2), hence, the time required for Lines -8 is bounded by O(M). The inner (Line 9) and outer whie oops (Line 2) together take up one scan of the tempora graph. Hence, the tota compexity is bounded by O(M).. WEIGHT-MINIMUM SPANNING TREES In this section, we study the probem of computing a MST w, which turns out to be far more difficut than computing a MST a. In the foowing we prove the hardness of the MST w probem even for approximation soutions. Then we show how we can transform the probem to the cassica directed Steiner Tree probem (DST) on non-tempora graphs. Whie the transformed probem can be soved by a state-of-the-art approximation agorithm with some post-processing steps, scaabiity is a major probem, we propose an agorithm with better time compexity whie preserving the approximation guarantee..1 Hardness of the MST w Probem In this section, we discuss the hardness of computing a MST w. The probem of MST w is: given a tempora graph G = (V, E),

5 a root vertex r and an interva [t α, t ω], compute a MST w. Our main resuts are that the probem is NP-hard and aso MAX-SNP hard. A proof of the foowing is given in the Appendix. THEOREM. Computing a MST w is NP-hard. A poynomia time approximation scheme (PTAS) is an agorithm which takes an instance of an optimization probem and a parameter ɛ > and, in poynomia time, produces a soution that is within a factor 1+ɛ of being optima for a minimization probem (or 1 ɛ for maximization probems). A MAX-SNP hard probem does not have a PTAS uness P = NP [1]. THEOREM. Computing a MST w is MAX-SNP hard. Given a (f, g, α, β) L-reduction from probem P to probem Q 1, and a (1 ± ε)-approximation agorithm for Q, we obtain a poynomia-time (1 ± δ)-approximation agorithm for P where δ = αβε [2, 18]. The proof of Theorem first converts the transformation in the proof of Theorem into an L-reduction X from the maximum eaf spanning tree probem to the MST w probem. Next we adopt the L-reduction Y in [1], which reduces the MAX- SNP compete probem of Min Dominating set-b to the probem of maximum eaf spanning tree. We form an L-reduction by a composition of the functions in Y and the corresponding functions in X. Thus, we obtain an L-reduction from the minimum dominating set probem to the MST w probem. From Theorem, there is no poynomia time approximation scheme for computing a MST w in a tempora graph, uness P = NP..2 Probem Transformation We have shown that computing MST w is a hard probem and remains hard for approximation agorithms. Here, we aim for an approximation agorithm with good scaabiity. In this section, we show that this probem can be transformed to the minimum directed Steiner tree probem (DST) on a static graph. Hence, we can appy the best known DST approximation agorithm for soving the MST w probem. We first introduce the graph transformation. The directed Steiner tree probem is defined as foows. Minimum Directed Steiner Tree (DST) : Let G = (V, E) be a directed static graph, where V is the vertex set and E is the edge set with weights (or costs) on edges. Given a set of terminas X V and a root vertex r, the minimum directed Steiner tree probem (DST) asks for a minimum-cost tree T = (V T, E T ) which contains a path from root r to each termina x X, such that e E T w(e) is minimum among a such trees. We refer to e E T w(e) as the cost of T and aso denote it by ζ(t ). In the foowing, we present a graph transformation agorithm. For carity, in the remaining discussions about MST w, we assume that [t α, t ω] = [, ] and V r = V. It is straightforward to extend the discussions to the genera case. Graph Transformation: Given a tempora graph G = (V, E), et E in(v) be v s incoming edge set for each vertex v V. If v r, et T (v) be the set of v s arriva time instances, i.e., T (v) = {ˆt v e = (u, v, t u, ˆt v, w) E in(v)} = {ˆt 1 v, ˆt 2 v,..}. For root r, et T (r) = {}, and ˆt 1 r =. The foowing steps transform the tempora graph G = (V, E) into a directed weighted graph G = (V G, E G ), where an edge e from vertex u to v with a weight of w is given by e = (u, v, w). 1 f, g are functions such that if p is an instance of P, then f(p) is an instance of Q, and if q is a soution to f(p), then g(q) is a soution to P. 1 [2] [1] [] [] [] [] [] [] [2] [] [2] [2] [] [] [] [] Figure : The Transformed Graph G of Figure 1 Step 1 Construct V G in G. For each vertex v V, (a) Create T (v) virtua vertices in G, i.e., {v 1, v 2,...v T (v) } with non-decreasing time instances: ˆt 1 v ˆt 2 T (v) v... ˆt v. T (v) +1 (b) If v r, create a dummy vertex v in G. Set ˆt v = Step 2 Construct E G in G. (a) For each vertex v V, create T (v) virtua edges (v 1, v 2, ), (v 2, v, )... (v T (v) 1, v T (v), ) and (v T (v), v, ) in G. (b) For each edge e = (u, v, t u, ˆt v, w) E find virtua vertex u i V G with time instance ˆt i u t u and ˆt i+1 u > t u; find virtua vertex v j V G with time instance ˆt j v = ˆt v. Create soid edge (u i, v j, w) in G. EXAMPLE 5 (GRAPH TRANSFORMATION). According to the 2 steps above, tempora graph G in Figure 1 is transformed into static graph G in Figure. [ Step 1 ]: take vertex 1 in G as an exampe, vertices 1 1 and are created in Figure corresponding to arriva times of and 5, respectivey, according to Step 1(a); at the same time dummy vertex 1 is created by Step 1(b). [ Step 2 ]: take vertex 1 in G as an exampe, virtua edges (1 1,, ) and (, 1, ) are created in Figure, as described in Step 2(a); meanwhie take tempora edge (1,,, 6, 2) in G as an exampe, according to Step 2(b) we find the starting point 1 1 in G (since 1 1 corresponds to time, corresponds to time 5 and < < 5); then find the end point 1 (since 1 corresponds to time 6); we connect 1 1 to 1 by a soid edge with weight 2. LEMMA 2. Given a tempora graph G = (V, E), the transformed graph G contains O( E ) vertices and O( E ) edges. The foowing theorem says that after the graph transformation, we can appy a soution for the minimum DST probem and then we can obtain the MST w resut from the DST resut. THEOREM 5 (PROBLEM TRANSFORMATION). A MST w with root r in a tempora graph G can be derived in inear time from a minimum Directed Steiner Tree (DST) rooted at r in G s corresponding transformed graph G, where the termina set is the dummy vertex set S created in G, i.e., X = S. In the next subsection, we describe the best-known approximation agorithm for DST and how to use it for the MST w probem.. Approximation Agorithm on the Transformed Graph The best known approximation agorithm for the minimum directed Steiner tree probem in static graph is shown in Agorithm [8], which greediy picks the owest density subtrees in the computation. Note that the input graph for Agorithm is a transitive cosure. That is, given a directed static graph G = (V, E), we first need preprocessing to get the transitive cosure G t so that there is

6 Agorithm : A i (k, r, X) Input : Termina set X, number of terminas k to be covered, root r, eve number i, transitive cosure G t; Output : A tree T with height i rooted at r covering k terminas in X 1 Initiaize: T ; 2 if i = 1 then whie k > do (r, v) arg (r,v) min cost(r, v), v X; 5 T T (r, v); k k 1; X X {v}; 6 ese 7 whie k > do 8 T BEST ; den(t BEST ) ; 9 for each vertex v V and each k, 1 k k do 1 T A i 1 (k, v, X) (r, v); 11 if den(t BEST ) > den(t ) then 12 T BEST T ; 1 T T T BEST ; k k X V (T BEST ); X X V (T BEST ); 1 return T ; an edge (u, v) whenever v is reachabe from u, and the weight of (u, v) is the weight of the shortest path from u to v. Agorithm is a recursive function on i, where i is the eve number, r is the root, X refers to the termina set and k denotes how many terminas need to be covered in this function. The approximate resut for DST is derived from A i ( X, r, X). When i = 1, k edges from root r to terminas are seected with smaest weights. Since the input graph is a transitive cosure G t, the seected edges correspond to k shortest paths from the root to k terminas in G. When i 1, Agorithm recursivey cas A i 1 (k, v, X) for different v and k as shown in Lines 9-1, picking the v and k whose resuting T is of the best density. The density is given by den(t ) = cost(t )/k(t ), where cost(t ) is the tota weight of tree T and k(t ) is the number of covered terminas in T. As proved in [8], for i > 1 iterations, the approximation ratio of A i (k, r, X) is bounded by i 2 (i 1)k 1/i and the time compexity is O( V i k 2i ), which is sti very costy. 2 In the next 2 subsections, we introduce strategies that ead to a much faster agorithm. After getting the approximation spanning tree from Agorithm, we need the foowing two postprocessing steps to get the corresponding spanning tree in the origina tempora graph G to approximate MST w. We shoud point out that the first step is aso required for the minimum DST probem in G. Step 1 Get corresponding spanning tree T in transformed graph G: (a) Repace edges in the resuting spanning tree of Agorithm with shortest paths in G. (b) Each vertex keeps ony one incoming edge. If there are mutipe incoming edges, choose one with the smaest weight. Step 2 Get corresponding spanning tree in tempora graph G: (a) For T from Step 1, deete virtua edges by merging a virtua and dummy vertices for each vertex in G into one vertex. According to the transformation scheme in Section.2, restore the corresponding tempora edges and vertices in G. (b) Each vertex keeps ony one incoming tempora edge, the one with the smaest arriva time among such edges. 2 In [8], when i = og k, an approximation ratio of O(og 2 k) is derived based on an erroneous resut in [2], with a corrected resut in [1], the ratio is given by O(og k). 1 [1] [] [] [] [5] 5 1 [1] 1 [] [] [] [2] [2] 2 5 (a) i = 1 (b) i = 2 [] 1 [] Figure 5: Resuting Approximation Tree in Agorithm 1 [2] [1] [] [] [] [2] [] [2] [] [] [] [] [2] [1] [] [] [] [2] [2] [2] [] [] [] (a) i = 1 (b) i = 2 <,6> [2] Figure 6: Postprocess Step 1 <1,> <,6> [2] [] <5,8> <6,8> [] [2] 5 <,6> [2] <1,> <,6> [2] [] <7,9> <6,8> [2] [2] (a) i = 1 (b) i = 2 Figure 7: Postprocess Step 2 EXAMPLE 6 (ALGORITHM ). Take the graph in Figure as the given static graph G before preprocessing for Agorithm with X = {1, 2,,, 5}, k = 5 and r =. Figure 5(a) is the resuting spanning tree for i = 1 with tota weight 17. That is, pick the shortest path from root to each termina vertex in G. Figure 5(b) is the resuting spanning tree for i = 2 with tota weight 12. For each 1 k 5 and each vertex in Figure, we cacuate the density for T in Lines 9-12, and pick v =, k = 1 whose resuting T has the smaest density den(t ) = 1 = 1. Then, update T, 1 k and X according to Line 1 and go to the next whie iteration, where v = 2 2, k = are chosen with density den(t ) = Finay v =, k = 1 are chosen with density den(t ) = =. 1 Postprocess: Figure 6 is the resuting spanning tree after Step 1 in G (Figure ). Figure 7 is the resuting tempora spanning tree after Step 2 in G (Figure 1). Take vertex 1 in G as an exampe, there are two incoming tempora edges after Step2(a) and tempora edge (, 1, 1,, 2) is chosen according to Step 2(b). Though we have a guarantee on the approximation for the DST probem, it does not transate to the MST w probem directy since the postprocessing changes the cost or weight of the fina spanning tree. The foowing theorem guarantees that the approximation ratio sti hods. THEOREM 6. Given a tempora graph G = (V, E), the DSTbased method with Agorithm provides an i 2 (i 1)k 1/i approximation to the MST w probem in O( E i k i ) time, where k < V. Ceary the postprocessing steps can be executed in inear time. With the graph transformation steps in Section.2, the DST agorithm and the postprocessing steps above, we can sove the MST w probem. The time compexity is dominated by Agorithm, which is sti high and severey imits the number of eves it can process. We are interested in reducing the runtime whie preserving the answer and hence the approximation guarantee. This probem wi be studied in the next subsection. 5

7 . An Improved DST Agorithm We begin with some observations about Agorithm. Consider A i (k, r, X), in order to find T BEST in Lines 7 1, we need to process each A i 1 (k, v, X) (r, v), v V, 1 k k. Some further anaysis can hep to greaty reduce this compexity. First we need some definitions. DEFINITION 1. The process tree of Agorithm is defined as foows. For each node v in the tree, v is the root of the tree returned by a recursive ca with r = v, the chidren are the roots of the returned trees for the recursive cas at the next eve, arranged in the order of the function cas. For each subtree T v rooted at v, we ca an edge from v to a chid w a branch of T v. We aso refer to the subtree rooted at the chid as a branch of T v. In Agorithm, et us ca an execution of the whie oop at Lines -5 or Lines 7-1 a w-iteration. In the foowing, et T i 1 (k ) be the tree returned by A i 1 (k, v, X), for simpicity, we may aso overoad the term A i 1 (k, v, X) to refer to T i 1 (k ). LEMMA (PROPERTY 1). Let be the smaer number of w- iterations in A i 1 (k, v, X) and A i 1 (j, v, X). A i 1 (j, v, X) and A i 1 (k, v, X) must share the same outcomes for the first ( 1) w-iterations. That is, T i 1 (k ) and T i 1 (j ) have the same first ( 1) branches in the process tree. DEFINITION 2 (L, p i 1 ). Denote by L the tota number of w- iterations in A i 1 (k, v, X). Let p i 1 be the number of covered terminas (terminas in T ) after w-iterations in A i 1 (k, v, X), where L. p i 1 =, p i 1 L = k. LEMMA (PROPERTY 2). For v V, A i 1 (p i 1, v, X) (r, v) has the smaest density among a A i 1 (k, v, X) (r, v), k (p i 1 1, pi 1 ], where 1 L. PROOF. Let T i 1 (k ) be the returned tree of A i 1 (k, v, X) for a given k and vertex v. Note that when k p i 1 1, there are at east subtrees in T i 1 (k ) and from Lemma, the first 1 branches in T i 1 (k ) are the same for a such k. For k p i 1 i 1 1, et T (k ) be the th branch in T i 1 (k ). T i 1 (k ) is obtained in the th w- iteration (in the function ca at Line 1) in A i 1 (k, v, X). First we show that den(t i 1 (k )) den(t i 1 (p i 1 )) k (p i 1 1, pi 1 ]. If there exists a k such that den(t i 1 (k )) < den(t i 1 (p i 1 )), then k shoud be the number of covered terminas after w-iterations in A i 1 (k, v, X), instead of p i 1. From Lemma, the first 1 branches in T i 1 (k ) are exacty the same k (p i 1 1, pi 1 ]. Note that T i 1 (k ), where p i 1 1 < k < p i 1, may have more than branches whie T i 1 (p i 1 ) has exact branches. According to the greedy seection in Agorithm, the density of the th branch shoud be greater than that of the th branch in T i 1 (k ) when >. Hence den(t i 1 (p i 1 )) den(t i 1 (k )). Let α be the cost of (r, v). The density of A i 1 (k, v, X) (r, v) is α/k + den(t i 1 (k )), k (p i 1 1, pi 1 ]. Since α/p i 1 α/k and den(t i 1 (p i 1 )) den(t i 1 (k )), we can concude that the density of A i 1 (p i 1, v, X) (r, v) is the smaest among a A i 1 (k, v, X) (r, v), k (p i 1 1, pi 1 ] and a specific v. Property 1 indicates repeated computations in trying different k, 1 k k, for each v V. Property 2 further confirms the possibe saving by skipping certain k vaues for a specific v. It reveas that one ony need to consider A i 1 (p i 1, v, X), where 1 L. What is more, according to Property 1, we ony need to try A i 1 (k, v, X) since each of A i 1 (p i 1, v, X) (1 < L) is aready covered in the process of A i 1 (k, v, X). It remains to work out the design of an improved agorithm that can make use of these two properties. For our design, another crucia observation is the foowing. Let α be the cost of (r, v). As increases, p i 1 α/p i 1 density α/p i 1 increases, and the covered terminas increases, )) increases. Hence the + den(t i 1 (p i 1 )) can increase or decrease as decreases, whie den(t i 1 (p i 1 increases. This means that we need to keep track of the smaest density with increasing. Agorithm : Ãi (k, r, X) Input : Termina set X, covered number of terminas k, root r, eve number i, transitive cosure G t; Output : A tree T with height i rooted at r covering k terminas in X 1 Initiaize: T ; 2 if i = 1 then whie k > do (r, v) arg (r,v) min cost(r, v), v X; 5 T T (r, v); k k 1; X X {v}; 6 ese 7 whie k > do 8 T BEST ; den(t BEST ) ; 9 for each vertex v V do 1 T B i 1 (k, v, X, (r, v)) (r, v); 11 if den(t BEST ) > den(t ) then 12 T BEST T ; 1 T T T BEST ; k k X V (T BEST ); X X V (T BEST ); 1 return T ; Agorithm is our improved agorithm. Within Agorithm, Agorithm 5 is caed at Line 1. The main improvement in Agorithm is that for each v, recursive ca (Agorithm 5) is triggered ony once instead of k times when compared to Line 9 in Agorithm. Agorithm 5 repaces the checking of different k in Line 9 in Agorithm according to the anaysis above and we wi show that it returns the T i 1 (p i 1 ) where the density (denoted by den()) of T i 1 (p i 1 ) (r, v) is the smaest (the best) for 1 L. Note that the number of covered terminas in the returned tree of B i (k, r, X, e) is not necessariy k, instead it shoud be p i 1 for some so that the density of T i 1 (p i 1 ) (r, v) is the smaest. From Lines 16-17, the best density T i 1 (p i 1 ) (r, v) is aways tracked in Agorithm 5. Before we prove the correctness, et us first compare our agorithm with Agorithm with an exampe. EXAMPLE 7 (ALGORITHM ). Compared to Agorithm, Agorithm ony cas function B i 1 (k, v, X, (r, v)) once for each v whie Agorithm cas function A i 1 (k, v, X) k times for each v. Given the graph in Figure. Compare Ã2 (5,, X) with A 2 (5,, X) where X = {1, 2,,, 5}, taking vertex as r. B 1 (5,, X, (, )) is caed in Ã2 (5,, X) and T = {(, )} is found, where T {(, )} is of the best density. Whie in A 2 (5,, X), A 2 (5, k, X) is caed for different k vaues, so that the resuting T {(, )} is of the best density. Here i = 2, the cases of i work in a simiar way according to Lines 9-17 instead of Lines -7 in Agorithm 5. For our proof of correctness, et us aso ca an execution of the whie iteration at Lines -5 or Lines 7-1 of Agorithm ; or at Lines -5, or Lines 9-17 of Agorithm 5 a w-iteration. THEOREM 7. [Correctness] Given k, r and X, Agorithm (Ãi (k, r, X)) and Agorithm (A i (k, r, X)) return the same tree.

8 Agorithm 5: B i (k, r, X, e) Input : Termina set X, maximum number of avaiabe terminas k, root r, eve number i, the incoming edge e of r, G t; Output : A tree T with height i rooted at r covering at most k terminas in X so that the density of T e is the smaest 1 Initiaize: T ; T c, where den(t e) = den(t c e) = ; 2 if i = 1 then whie k > do (r, v) arg (r,v) min cost(r, v), v X; 5 T c T c (r, v); k k 1; X X {v}; 6 if den(t e) > den(t c e) then 7 T = T c; 8 ese 9 whie k > do 1 T BEST ; den(t BEST ) ; 11 for each vertex v V do 12 T B i 1 (k, v, X, (r, v)) (r, v); 1 if den(t BEST ) > den(t ) then 1 T BEST T ; 15 T c T c T BEST ; k k X V (T BEST ); X X V (T BEST ); 16 if den(t e) > den(t c e) then 17 T = T c; 18 return T ; PROOF. Agorithm differs from Agorithm ony at Lines 9-1. Hence, to prove that Ãi (k, r, X) equas A i (k, r, X), it suffices to prove that T returned by B i 1 (k, v, X, (r, v)) (r, v) (Line 1) in Agorithm has the smaest density among a T returned by A i 1 (k, v, X) (r, v) for different 1 k k (Lines 9-1) in Agorithm. According to Lemma, we ony need to consider the density of A i 1 (k, v, X) (r, v), where k = p i 1, 1 L. Thus, the probem is reduced to proving that the tree T returned by B i 1 (k, v, X, (r, v)) (r, v) in Agorithm has the best density among a T returned by A i 1 (k, v, X) (r, v) for different k = p i 1, 1 L in Agorithm. Hence, it remains to prove the foowing: For a given k and a vertex u V, T c in the th w-iteration of B i 1 (k, u, X, (r, u)) is equa to T returned by A i 1 (p i 1, u, X), where 1 L. Reca that p i 1 refers to the number of covered terminas (in T ) after w-iterations in A i 1 (k, v, X). We prove by induction on i. For the base case where i = 2, consider Lines -5 in Agorithm and Agorithm 5, where one termina is chosen in each whie iteration. Hence p i 1 = and L = k. In the th w-iteration of B i 1 (k, u, X, (r, u)), T c consists of the shortest paths to cosest terminas. Whie in A i 1 (p i 1, u, X), the resuting T aso consists of the shortest paths to cosest terminas. The induction hypothesis states that given k and v V, when i j, T c in the th w-iteration of B i 1 (k, v, X, (r, v)) is equivaent to T returned by A i 1 (p i 1, v, X), where 1 L. Consider i = j + 1. We first note that the tree T returned by B j 1 (k, v, X, (r, v)) (r, v) (Line 12) v V in B j (k, r, X, e) has the smaest density among a trees T that are returned by A j 1 (p j 1, v, X) (r, v) (Line 1), v V and 1 L in A j (k, r, X), due to the induction hypothesis and Lines in Agorithm 5. According to Lemma, for each v V, the tree T returned by B j 1 (k, v, X, (r, v)) (r, v) in B j (k, r, X, e) has the best density among a T returned by A j 1 (k, v, X) (r, v) in A j (k, r, X), where 1 k k. Hence, T BEST in the first w- iteration of B j (k, r, X, e) is the same as that in the first w-iteration of A j (k, r, X). That is, T c in the first w-iteration of B j (k, r, X, e) is the same as T in the first w-iteration of A j (k, r, X). In addition, the tree returned by A j (p j 1, r, X) is the T in the first w-iteration of A j (k, r, X). Thus, T c in the first iteration of B j (k, r, X, e) is the same as T in A j (p j 1, r, X). In the first w-iteration of A j (k, r, X) and B j (k, r, X, e), k and X are updated according to Line 1 in Agorithm and Line 15 Agorithm 5, respectivey. The updates are the same since the T BEST s are the same. Let k and X be the updated k and X vaues, respectivey, at the end of the th w-iteration in A j (k, r, X) and B j (k, r, X, e). In the second w-iteration of A j (k, r, X) and B j (k, r, X, e), the above anaysis on B i 1 (k 1, v, X 1, (r, v)) and A i 1 (k 1, v, X 1) aso appies. That is, the T BEST trees in the second w-iteration of B j (k, r, X, e) and that of A j (k, r, X) are the same. Hence, T c in this w-iteration of B j (k, r, X, e) is the same as T returned by A j (p j 2, r, X). Simiary, the argument appies to the remaining w-iterations. Thus, the tree T c in the th w-iteration of B j (k, r, X, e) is equivaent to the tree T returned by A j (p j, r, X). This competes the proof by induction. With the above correspondence of T c and T, and according to Lines 16-17, B i 1 (k, v, X, (r, v)) returns T such that T (r, v) has the best density among a w-iterations, we concude that the density of B i 1 (k, v, X, (r, v)) (r, v) (Line 1) in Agorithm is the best among a A i 1 (k, v, X) (r, v), where k = p i 1, 1 L (Lines 9-1) in Agorithm. Thus, given k, r and X, Agorithm (Ãi (k, r, X)) and Agorithm (A i (k, r, X)) return the same tree. THEOREM 8. Given a static graph G = (V, E), Agorithm provides an i 2 (i 1)k 1/i approximation to the optima DST in time O( V i k i ). PROOF. For the approximation bound, the anaysis is the same as Theorem.1 in [8], except that the corrected resut in [1] repaces that of Lemma 2.2 in [8]. Next consider the time compexity anaysis. First, Ãi (k, r, X, e) invokes at most nk B i 1 (k, r, X, e) and each B i 1 (k, r, X, e) invokes at most nk B i 2 (k, r, X, e). Recursivey, we have that the time compexity for Ãi (k, r, X, e) is bounded by O( V i k i ). From Theorem 8, if we choose i = og n, then Agorithm attains an O(og n) approximation for the optima directed Steiner Tree probem in O(n og n k og n ) time. Agorithm achieves the same approximation guarantee but requires O(n og n k 2 og n ) time. From our experimenta study, we show that the improvement in time compexity transates into significanty faster runtime..5 Enhancement by Density Based Ordering In Section., we discuss the pruning of different k (1 k < k), but sti we need to process a different v V at Line 9 of Agorithm. In this section, we introduce a pruning strategy so that we may expore ony a subset of V instead. This pruning strategy is based on vertex ordering. Consider the improved A i (k, r, X) in Agorithm, the function B i 1 (k, v, X, e) is caed for each vertex v V in the first whie iteration. Let τ(v) be the density of B i 1 (k, v, X, (r, v)) (r, v), v V. That is, τ(v) records the best density of a possibe T i 1 (k ), 1 k k, for v V. Let us sort the vertices in non-descending order of τ(). In the next whie iteration, we ca B i 1 (k r, v, X r, e) according to this vertex order. We keep updating τ(v) and record the best τ best so far in this whie iteration. We can substantiay reduce the computation of B i 1 (k r, v, X r, e) for some v, whose τ(v) is bigger than the current best τ best as shown in Line * of Agorithm 6. This pruning strategy is aso appied to B i (k, r, X, e).

9 Agorithm 6: F inaa i (k, r, X) 1* Same as Agorithm except for repacing Lines 9-12 by the foowing: 2* for each vertex v V (in sorted order) do * if τ(v) < τ best then * T F inab i 1 (k, v, X, (r, v)) (r, v); 5* if den(t BEST ) > den(t ) then 6* T BEST T ; 7* τ(v) den(t ); 8* if τ(v) < τ best then 9* τ best τ(v); 1* ese 11* sort V according to τ(v), v V; 12* break; 1* /* F inab i 1 (k, v, X, (r, v)) is the same as Agorithm 5 except for repacing Lines 11-1 by the same for-oop repacement (Lines 2*-12*) as in the above. */ THEOREM 9. Given k, r and X, Agorithm 6 (F inaa i (k, r, X)) and Agorithm (A i (k, r, X)) return the same tree. PROOF. (Sketch) Line * in Agorithm 6 is where pruning occurs. Assume we are now starting the for oop execution for a specific vertex v in the th w-iteration (Line 2* in Agorithm 6), τ(v) in Line * records the best density for v in the ( 1) th w-iteration. The best density for v in the th iteration cannot exceed that in the ( 1) th w-iteration, due to ess avaiabe terminas in X. Hence we can use τ(v) in the previous iteration for possibe pruning of vertices in the for oop. Though Agorithm 6 has the same time compexity as Agorithm, since in the worst case there may be no pruning, in our experiments, pruning by density ordering is shown to be highy effective, and the runtime is improved by more than an order of magnitude in most cases. 5. EXPERIMENTAL RESULTS In this section, we evauate the performance of our proposed agorithms for computing MST a and MST w. We aso evauate the quaity of the resuts of our approximation agorithm on DST for datasets with known soutions. We run a experiments on a machine with a.ghz Inte Core i7-77 CPU and 16 GB RAM, running Ubuntu 12. LTS Linux OS. A agorithms were impemented in C++ and compied with GNU c++ compier. 5.1 Datasets We tested on datasets downoaded from the Kobenz Large Network Coection ( ). Sashdot is a network of user repies coected from technoogy website Sashdot. Epinions records onine product ratings, where edges are trust or distrust inks between users. Facebook is a graph representing posts to other users was on Facebook. Enron records emais sent among empoyees of Enron between 1999 and 2. arxiv HepPh is a coaboration network of authors of scientific papers from High Energy Physics, in which vertices are authors and edges are common pubications between two authors. DBLP is a graph of coauthors from DBLP computer science bibiography. Phone is based on anonymized records of Orange phone cas and SMS exchanges in Ivory Coast for five months from Dec, 211 [5]. A datasets are tempora graphs, each time stamp is either a physica time (for Phone) or a Unix time. For the Phone dataset, each edge has a weight given by the duration_voice_cas attribute. V E E s deg deg s π Γ G Sashdot 51K 1K 1K Epinions 11K 717K 717K Facebook 6K 855K 26K Enron 87K 115K 2K HepPh 28K 919K 6291K DBLP 111K 11957K 851K Phone K 512K Tabe 1: Datasets: sizes of tempora graph G = (V, E) and static graph G s = (V s, E s), distinct time instances (Γ G), and π The other networks are unweighted, we assign weights to the edges based on the weight cascade mode in [19], in which the propagation probabiity for an edge (u, v) is given by pp(u, v) = 1/d(v), where d(v) is the in-degree of v. We use the out-degree instead since it represents better the graph properties. As noted in [9], if we convert this to a weight of og pp(u, v) on (u, v), then we have a maximum infuence path when the tota edge weight is minimum. Hence we adopt og pp(u, v) as the weight (or cost) for each edge (u, v). Next, we describe how to set the window of [t α, t ω], the time span of interest. For each of the six datasets, we compute the tota time range [t A, t Ω] from the time units attached to a edges. Then we set the range [t α, t ω] so that it covers the midde one tenth of the tota range, i.e. (t ω t α).1(t Ω t A). We extract a subgraph G of the given graph G by imiting edges to within [t α, t ω]. For the root vertex for the MST, we require that it can reach at east one tenth of the set of vertices in G. We simpy scan the vertices unti one such vertex is found. We show some statistics of the datasets in Tabe 1. In Tabe 1, π is the maximum number of tempora edges from u to v, for any u and v in V (G). Note that for π = 1, G can sti be very different from G s since paths must be time respecting. 5.2 Efficiency of the Agorithms for MST a Here we evauate the performance of our two MST a agorithms, Agorithm 1, denoted by Ag1, and Agorithm 2, denoted by Ag2. We compare with the existing agorithm in [], denoted by Bhadra, which utiizes a modified Prim-Dijkstra agorithm. The time compexity of Bhadra given in [] is O(n 2 D og T ), where n is the number of vertices, D is the maximum out degree of a vertex and T is the number of time points in G. With more carefu anaysis, the time can be bounded by O(m og n + m og π), where m is the number of edges in the corresponding static graph G S = (V S, E S), and π is the maximum number of time instances incident to each static edge e s E S, which is bounded by T. For the window of [t α, t ω], since the computation time is very short for MST a, other than the defaut range giving rise to G as described above, we aso consider the range of [, ], where MST a spans V r(g). Our first set of experiment ensures that the durations in the edges are non-zero in the given tempora graphs. As in [27], the durations are set to 1. The resuts are shown in Tabe 2. Ag1 outperforms Bhadra by a arge margin in a cases. Our next set of experiment is to compare Ag2 with Bhadra, for networks that may contain zero edge durations. Since Ag1 cannot hande zero duration, it is not incuded. Foowing the given datasets, the edge durations are set as zero. The resuts are shown in Tabe. Ag2 outperforms Bhadra in amost a cases. The root in Tabe 2 and is set as the first vertex in the input order, which can reach more than one tenth of the tota vertex size. Compared with Tabe 2, the numbers of reachabe vertices are simiar, except for DBLP. DBLP is about coaboration of coauthors, there exist paths from an author to other authors in the same year, and not across dif-