Efficiently Monitoring Link Bandwidth in IP Networks Zhiping Cai, Jianping Yin, Fang Liu, Xianghui Liu, Shaohe Lv School of Computer, National University of Defense Technology 410073 Changsha, Hunan, P.R.China Email:caizhiping
[email protected],
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[email protected] Abstract— Link bandwidth utilization is obviously critical for numerous network management tasks. Using the flowconservation law, we could reduce the number of activated monitor agents. The problem of efficiently monitoring linkbandwidth based on flow-conservation law could be reduced to Weak Vertex Cover problem, which is NP-hard. In this paper, we demonstrate an approximation preserving reduction from the vertex cover problem to Weak Vertex Cover problem. Due to this reduction, it follows that it is very difficult to get an approximation algorithm with approximation ratio lower than 2 for Weak Vertex Cover problem. Using the primal-dual method, we give an approximation algorithm with approximation ratio 2 to solve the problem. The effectiveness of our monitoring algorithm is validated by simulations evaluation over a wide range of network topologies. We also demonstrate the problem of weak vertex cover with blackout vertices could be reduce to Weak Vertex Cover problem. Hence we could use the approximation algorithms for Weak Vertex Cover problem to solve the problem of weak vertex cover with blackout vertices.
I. I NTRODUCTION Knowledge of the up-to-date link bandwidth utilizations is critical for numerous important network management tasks, including identifying and relieving congestion points, proactive and reactive resource management and traffic engineering, as well as providing and verifying QoS guarantees for end-user applications. Some novel tools and infrastructures for measuring network bandwidth have been developed and proposed by researchers and industries, like as SNMP and RMON measurement probes [1], Cisco’s NetFlow tools [2], packet-pair algorithms for measuring link bandwidth [3], [4] and Pathneck tool for locating network bottlenecks [5], [6]. These measurement tools periodically query and collect detailed traffic data on packet flows for monitoring and measuring network flows and bandwidth usage. Unfortunately, processing queries can adversely impact routers performance and monitoring data transfers can result in significant volumes of additional network traffic [7]. In particular, as the network monitoring process requires more data to be collected and at much higher frequencies, the overhead that a polled monitoring agent imposes on the underlying router can be significant and can adversely impact the router’s throughput. The number of placed monitors of a monitoring system should be kept as small as possible in order to reduce the deployment cost and the actual monitoring operating cost [8]. Several measurements over backbone routers show each IP router satisfies a flow-conservation law that, the sum of the
traffic flowing into router is approximately the same as those of the traffic flowing out [7]. The flow-conservation law could be applied to reduce the number of activated monitor agents used to monitor link bandwidth usage. Using the flow-conservation law, we no longer need to ensure that all edges of a router are covered by a monitor: if a router has k links incident on it and the bandwidth utilization of k−1 of the links is known, then the bandwidth utilization of the remaining link can be derived from the flow-conservation equation for that router [7]. Thereby the application results in a substantial reduction in the monitoring method impacting on the underlying router’s throughput and performance. Breitbart Y. et al. [7] first address the challenging problem of efficiently monitoring bandwidth utilization. And they reduce the problem to a novel NP-hard problem, termed Weak Vertex Cover problem. For solving Weak Vertex Cover problem, Xianghui Liu et al. [9] brought forward a greedy approximation algorithm which gives an approximation ratio 2(1 + ln d), where d = maxv∈V {deg(v)}. And Xianghui Liu et al. [10] proved that the weak vertex cover problem is NP-complete. Yong Zhang et al. [11] gave an approximation algorithm with approximation ratio 1 + ln d. In this paper, we demonstrate an approximation preserving reduction from the vertex cover problem to Weak Vertex Cover problem. Due to this reduction, it implied that it is difficult to get an approximation algorithm with approximation ratio lower than 2. Using the primal-dual method, we give an approximation algorithm with approximation ratio 2 to solve the problem. In some scenarios, the monitoring agent could not deploy in some network nodes. We designate these network nodes in which could not be activated monitor agent and call it blackout vertices. In this paper, we also demonstrate the weak vertex cover with blackout vertices problem could be reduce to the weak vertex cover problem. Hence we could use the approximation algorithms for the weak vertex cover problem to solve the weak vertex cover with blackout vertices problem. The rest of the paper is organized as follows. In section II, we describe the problem formulation and demonstrate an approximation preserving reduction from the vertex cover problem to the weak vertex cover problem. In next section, we give an approximation algorithm to solve the weak vertex cover problem with approximation ratio 2. We discuss the problem of weak vertex cover with blackout vertices in section
0-7803-9415-1/05/$20.00 (C) 2005 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2005 proceedings.
IV. The effectiveness of our monitoring algorithm is validated by simulations evaluation over a wide range of network topologies in section V. Finally, conclusions are presented in the last section. II. W EAK V ERTEX C OVER P ROBLEM A. Problem Formulation The problem of efficiently monitoring the link bandwidth utilization based on the flow-conservation law is to find the minimum Link Bandwidth Monitoring Set of a graph [7], [9]. Definition 1 (Link Bandwidth Monitoring Set): Given an undirected graph G = (V, E), where V denotes the set of nodes, E represents the edges between two nodes, we say S ⊆ V is a link bandwidth monitoring set of G, if monitoring the link bandwidth utilization of those edges that are incident on nodes in S is sufficient to infer the bandwidth utilization of every edge in E. And the following two constraints must be satisfied: (1)∀v ∈ V , d(v) ≥ 2, where d(v) denotes the degree of node v; (2)∀v ∈ V , u∈V f (u, v) = 0, where f (u, v) denotes the flow from node u to node v. The minimum link bandwidth monitoring set is a link bandwidth monitoring set that contains minimum number of nodes of the graph. The problem of finding the minimum link bandwidth monitoring set from an underlying network could be abstracted to the problem of finding the minimum Weak Vertex Cover set (WVC) of a graph [7]. Hence we could solve the problem of finding the minimum link bandwidth monitoring set by solving the problem of finding the minimum weak vertex cover set for a given graph. Definition 2 ( Weak Vertex Cover): Given an undirected graph G = (V, E), where ∀v ∈ V , d(v) ≥ 2 holds, we say S ⊆ V is a Weak Vertex Cover Set of G, if and only if every edge in G can be marked by performing the following three steps: (1) Mark all edges that are incident on vertices in S; (2) Mark the edge if it is the only unmarked edge among all of the edges that are incident on the same vertex; (3) Repeat step (1) until no new edge can be marked. B. Approximation Preserving Reduction from Vertex Cover to Weak Vertex Cover A vertex cover of an undirected graph is a subset of the vertex set which is incident with every edge in the graph. The vertex cover problem is to find a minimum weight vertex cover. As all edge could be marked by step (1) of Definition 2, the vertex cover of a graph is also a weak vertex cover. We demonstrate an approximation preserving reduction from the vertex cover problem to the weak vertex cover problem. Due to this reduction, it follows that it is very difficult to get an approximation algorithm with approximation ratio lower than 2 for the weak vertex cover problem. Given a undirected graph G = (V, E), we extend G to a undirected graph H by adding a vertex ve for each edge e, and connecting ve with the vertices which is incident to e in G.
We will show that there always exists a minimum weak vertex cover S ∗ of the extended graph H = (V , E ) while S ∗ is also a minimum vertex cover of graph G, where S ∗ ⊆ V . We give a lemma as follows at first. Lemma 1: Let the undirected graph G = (V, E) have all vertices of degree 2 or more. S is a vertex cover of graph G if and only if S is a weak vertex cover of the extended graph H = (V , E ), where S ⊆ V and V = V ∪ {ve |e ∈ E}, E = E ∪ {(ve , ve1 ), (ve , ve2 )|e = (ve1 , ve2 ) ∈ E}. Proof: Supposed that S is a vertex cover of G. For two vertices ve1 and ve2 incident with edge e, there must exist one vertex belonging to S. Let ve1 ∈ S, the edge (ve , ve1 ) could be marked according to step (1) of Definition 2. And the edge (ve , ve2 ) could be marked according to step (2) of Definition 2. So all edges in E could be marked. Therefore S is also a weak vertex cover of H. Conversely, supposed that S is a weak vertex cover of the graph H. As three edges (ve , ve1 ),(ve , ve2 ) and (ve1 , ve2 ) / S, there must exist one vertex of compose a cycle and ve ∈ ve1 and ve2 belonging to S. Thus S is also a vertex cover of graph G. Theorem 1: Let the undirected graph G = (V, E) have all vertices of degree 2 or more. There always exists a minimum weak vertex cover S ∗ of the extended graph H = (V , E ), where S ∗ ⊆ V and V = V ∪ {ve |e ∈ E}, E = E ∪ {(ve , ve1 ), (ve , ve2 )|e = (ve1 , ve2 ) ∈ E}. And this weak vertex cover S ∗ is also a minimum vertex cover of graph G. Proof: Supposed that S ∗ is a minimum weak cover of / V , we graph H. If there exists vertex ve ∈ S ∗ and ve ∈ discuss three cases of ve1 and ve2 as follows, where ve1 and ve2 are incident with edge e. 1) If both ve1 and ve2 belong to S ∗ , edge (ve , ve1 ) and (ve , ve2 ) could be marked according to step (1) of Definition 2. So S ∗ − {ve } is a weak vertex cover. It conflicts with the fact that S ∗ is a minimum weak vertex cover. 2) If there is only one vertex of ve1 and ve2 belonging to S ∗ , let ve1 be in S ∗ . Replace ve by ve2 to be added into S ∗ , edge (ve , ve1 ) and all edges incident with ve2 could be marked according to step (1) of Definition 2. So S ∗ − {ve } + {ve2 } is a minimum weak vertex cover. 3) If neither of ve1 and ve2 belongs to S ∗ , except two edges (ve , ve1 ) and (ve1 , ve2 ), there exists at most one unmarked edge incident with ve1 , and ve2 likewise. We could replace the ve by either ve1 or ve2 to be added into S ∗ . Let ve1 be added into S ∗ , all edges incident with ve1 could be marked according to step (1) of Definition 2. As (ve , ve1 ) being marked, edge (ve , ve2 ) could be marked by using step (2) of Definition 2. Therefore all edges incident with ve2 could be marked. So both S ∗ − {ve } + {ve1 } and S ∗ − {ve } + {ve2 } are the minimum weak vertex cover. Therefore, for every minimal weak vertex cover S ∗ of graph H, if S ∗ contains some vertices being not belong to V , we could get a minimal weak vertex cover with all vertices from V by replacing some vertices. Hence, there exists a corresponding minimum weak vertex cover whose vertices are all in V for any minimum weak vertex cover.
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Due to lemma 1, the minimal weak vertex cover S ∗ is a vertex cover of graph G. If there exists a vertex cover Sc of G whose size is smaller than that of S ∗ , Sc is also a weak vertex cover of H. It conflicts with the fact that S ∗ is a minimum weak vertex cover of graph H. Thus, a minimum weak vertex cover whose vertices are all in V of graph H is a minimum vertex cover of graph G by Lemma 1. Due to theorem 1, it follows that the performance ratio obtainable for the weak vertex cover problem cannot be better than the one obtainable for the vertex cover problem. There have been many attempts to achieve an approximation ratio small than 2 for the vertex cover problem [12], and so far these attempts were unsuccessful. And it is difficult to get an approximation algorithm with approximation ratio lower than 2 for Weak Vertex Cover problem. III. A PPROXIMATION A LGORITHMS
A. Some Properties We have some properties of the Weak Vertex Cover as follows. Proposition 1: Given an undirected graph G = (V, E), where ∀v ∈ V , d(v) ≥ 2, the set S is a weak vertex cover, if and only if G = (V , E ) is acyclic, where S ⊆ V ,V = V −S and E = {(u, v)|(u, v) ∈ E ∧ u ∈ V ∧ v ∈ V }. For a given graph G = (V, E), we let τ (V ) denote the cardinality of the smallest weak vertex cover for G. Let d(v) denote the degree of vertex v in G. Given a subset S of vertices, let E[S] denote the subset of edges that have both endpoints in S. Let G[S] denote the subgraph (S, E[S]) induced by G, and let ds (v) denote the degree of v in G[S]. We let b(S) = |E[S]| − |S| + 1 and b(V ) = |E| − |V | + 1. We say that a weak vertex cover F is minimal if for any v ∈ F ,F − v is not a weak vertex cover. Then we can state some theorems. The details of this proof are omitted due to space limitations [15]. Theorem 2: Let F denote any weak vertex cover of a graph G = (V, E), where ∀v ∈ V, d(v) ≥ 2 holds. Then [d(v) − 1] ≥ b(V ). (1) v∈F
d(v) ≥ b(V ) + τ (V ).
(2)
v∈F
Let FM is any minimal weak vertex cover, then d(v) ≤ 2(b(V ) + τ (V )) − 2. v∈Fm
ds (v) ≥ |E[S]| − |S| + 1 + τ (S) = b(S) + τ (S) (4)
v∈F ∩S
B. A Primal-Dual Algorithm By Corollary 1, the integer programming formulation of Weak Vertex Cover problem is the following: Min wv xv v∈V
Subject to: ds (v)xv ≥ b(S) + τ (S) S ⊆ V, E[S] = 0 (IP) v∈S
We give a 2-approximation algorithm for the weak vertex cover by using the primal-dual method for approximation algorithms, which has been used to derive approximation algorithms for network design problems [12-14]. At the beginning, we give some inequalities that will be needed in proving the performance guarantees of the algorithms and in giving the integer programming formulation. In fact, we could get different primal-dual algorithms by using different integer programming formulation [15].
Observe that if F is a weak vertex cover for G, then F ∩ S is clearly a weak vertex cover for G[S]. Hence we have the following corollary of inequality (2). Corollary 1: Let F be any weak vertex set. Then for any S ⊆ V , E[S] = 0,
(3)
xv ∈ {0, 1} v ∈ V. We construct a feasible solution to the dual of the linear programming relaxation of (IP). The linear programming relaxation is wv xv Min v∈V
Subject to: ds (v)xv ≥ b(S) + τ (S) (LP) v∈S
S ⊆ V, E[S] = 0
xv ≥ 0
and its dual is Max
v∈V
(b(S) + τ (S))ys
S
Subject to: ds (v)ys ≤ wv (D)
v∈V
S:v∈S
ys ≥ 0 S ⊆ V, E[S] = 0. Then we give a primal-dual 2-approximation algorithm as follows: Algorithm (G = (V, E)): 1) y = 0;l = 0 2) F = Φ 3) V = V ;E = E 4) While F is not a WVC for G a) l = l + 1 b) Recursively remove degree one vertices and incident edges from V and E c) Increase yV until ∃vl ∈ V , s.t. T :vl ∈T dT (vl )yT = wvl d) F = F ∪ {vl } e) Remove vl from V and attached edges from E . 5) For (j = l;j > 0;j − −) a) if F − {vj } is a WVC then F = F − {vj } 6) F = F It is not hard to see that this algorithm is effectively equivalent to the following: start with F = 0 and the graph G. Recursively remove any degree one vertices and associated edges from the graph. Pick the vertex v that achieves the minimum ε = minv∈S wv /d(v). Add v to F , and set wu = wu − εd(u) for all u ∈ V . Remove v from the graph, and repeat until F is a WVC. A straightforward implementation of
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this algorithm takes O(mn) time, where m is the number of edges in the graph and n is the number of vertices. Then we prove that the algorithm is a 2-approximation algorithm. Theorem 3: The primal-dual algorithm constructs a WVC and a solution y feasible for (D) such that
wv ≤ 2
v∈F
(b(S) + τ (S))yS − 2
S
yS .
(5)
S
Hence the algorithm is a 2-approximation algorithm. Proof: We reduce the proof of the theorem to inequality (3). By construction of the algorithm, v∈F
wv =
v∈F S:v∈S
ds (v)ys =
In this section, we present simulation results of comparing the performance of the various algorithms that solve the weak vertex cover problem. The main objective of the simulations is to demonstrate that our proposed algorithmic solutions are not only theoretically sound but also they could give significant benefits over naive solutions in practice for a wide variety of realistic network topologies. Simulations are based on network topologies generated using the Waxman model [16] and Power-Law [17] model. The result show the performance advantage of our heuristics when comparing them with others algorithms. A. Waxman Model
ds (v)ys .
(6)
S v∈S∩F
Thus if we can show that for any ys > 0 then
V. S IMULATIONS
ds (v) ≤ 2(b(S) + τ (S)) − 2,
(7)
v∈(S∩F )
then the theorem statement will follow. In order to apply (3), it is sufficient to argue that S ∩ F is a minimal WVC for the graph G[S]. By construction of the algorithm F is a minimal WVC. And at the point in time when the algorithm chooses the vertex vl , none of the vertices in F ∩ S is currently in F ; they are added at some later point in the algorithm. Therefore, because the final step of the algorithm deletes redundant vertices in the reverse of the order in which they were added, F − F , for the current value of F , must be a minimal WVC for the current graph (V , E ). Thus, we have that F ∩ S is a minimal WVC for G[S]. IV. W EAK V ERTEX C OVER W ITH B LACKOUT V ERTICES In some scenarios, the monitoring agent could not deploy in some network nodes; for example, some routers could not support SNMP protocol. We designate these network nodes in which could not be activated SNMP agent and call it blackout vertices. Blackout vertices could be handled by assigning them infinite weight. In fact, it exists a simply reduction from the weak vertex cover with blackout vertices problem to the weak vertex cover problem. Due to this reduction, we could solve the weak vertex cover with blackout vertices problem by using the approximation algorithms for the weak vertex cover problem. Let B be the vertices of the blackout vertices. We could bypass the blackout vertices one by one as follows: For each blackout vertex, v ∈ B, connect edges between every two vertices that have a path of length two connecting them in which v is the middle vertex. Remove the blackout vertex from the graph, and continue with the next vertex in B. It is not hard to see that a subset of vertices, U , is a weak vertex cover set that does not contain any vertex from B, if and only if it is a weak vertex cover set of the graph obtained by the reduction.
We generate 500-node network topologies using the Waxman model, which is a popular topology model for networking research. Different network topologies are generated by varying three parameters: (1)n, the number of nodes in the network graph; (2)α, a parameter that controls the density of short edges in the networks; and (3)β, a parameter that controls the average node degree. We compare the performance of three algorithms: the 2approximation algorithms for Vertex Cover [12], the greedy algorithm with approximation ratio 2(1+ln d) for Weak Vertex Cover [9], and our primal-dual algorithm. The comparison is in terms of the number of nodes that need to run SNMP [1] in order to measure the bandwidth of each link in the generated network graphs. We denote the number of SNMP WV C , and N2W V C activations for these algorithms by N2V C , Ngreedy respectively. Table I presents one set of simulation results; we have obtained similar results for other parameter settings. The third and fourth columns in the table represent the maximum and average degree of the nodes in the generated network graph respectively. Our results indicate that using our approximation algorithm can reduce the number of SNMP activations as much as 73% over the naive approach which activate an SNMP agent on every network node [7]. And the result of our algorithm is better than the other two algorithms. TABLE I C OMPARISONS OF M ONITORING A LGORITHMS n
α
β
500 500 500
0.1 0.4 0.4
0.08 0.02 0.08
Max Deg 11 12 33
Ave Deg 2.916 3.444 6.184
WV C N2V C Ngreedy N2W V C
N2W V C n
297 324 384
0.298 0.322 0.430
257 290 334
149 161 215
B. Power-Law Model Power-Law model can be used to generate route-level topologies using BRITE [18], which is the best known power-law-based topology generator. BRITE generates different topologies by changing the values of the following parameters: (1) HS, Size of one side of the plane; (2) LS, Size of one side of a high level square; (3)N P , Node Placement;
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(4) m, Number of links added per new node; and (5) IG, Incremental Growth. We start with average degree m = 2 (i.e. tree topology) and increase the degree gradually until m = 10 (i.e. average degree of 10). We also compare the performance of three algorithms: the 2-approximation algorithms for Vertex Cover [12], the greedy algorithm with approximation ratio 2(1+ln d) for Weak Vertex Cover [9], and our primal-dual algorithm. The comparison is in terms of the number of nodes that need to run SNMP [1] in order to measure the bandwidth of each link in the generated network graphs. We denote the number of SNMP WV C , and N2W V C activations for these algorithms by N2V C , Ngreedy respectively. Table II presents one set of simulation results; we have obtained similar results for other parameter settings. Our results indicate that using our approximation algorithm is better than the other algorithms. TABLE II C OMPARISONS OF M ONITORING A LGORITHMS n 500 500 500 500 500
Ave Deg 2 4 6 8 10
N2V C 307 337 352 379 402
WV C Ngreedy 264 281 301 325 347
N2W V C 204 221 243 265 282
N2W V C n
0.408 0.442 0.486 0.530 0.562
VI. C ONCLUSION In this paper, we have addressed the problem of efficiently monitoring link bandwidth based on flow-conservation law. This problem could be abstracted to Weak Vertex Cover problem, which is NP-hard. An approximation preserving reduction from the vertex cover problem to the weak vertex cover problem is demonstrated. It implies that it is difficult to get an approximation algorithm with approximation ratio lower than 2 for the weak vertex cover problem. We have proposed a primal-dual algorithm with approximation ratio 2 to solve Weak Vertex Cover problem. In fact, we could get different primal-dual algorithms by using different integer programming formulation. We also discuss the problem of weak vertex cover with blackout vertices. Due to a simply reduction, we could use approximation algorithms of the weak vertex cover problem to solve the problem of weak vertex cover with blackout vertices. Our work is helpful to solve Weak Vertex Cover problem and its application in monitoring link bandwidth. Further research would be conducted to exploit knowledge of traffic flows in the network to further reduce the required overhead for monitoring link bandwidth.
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ACKNOWLEDGMENTS The research reported here has been supported by the National Natural Science Foundation of China(No.60373023).
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