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ScienceDirect Procedia Computer Science (2016)124–131 000–000 Procedia Computer Science 109C00(2017) Procedia Computer Science 00 (2016) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
8th International Conference on Ambient Systems, Networks and Technologies, ANT 2017 8th International Conference on Ambient Systems, Networks and Technologies, ANT 2017
Route Route selection selection algorithms algorithms utilizing utilizing the the property property of of the the ZDD ZDD for for compressed sensing-based transmissive network tomography compressed sensing-based transmissive network tomography a a,∗ Teruhito Teruhito Naka Nakaa,, Shinsuke Shinsuke Hara Haraa,∗
a Graduate School of Engineering, Osaka City University, Osaka, 5588585, Japan a Graduate School of Engineering, Osaka City University, Osaka, 5588585, Japan
Abstract Abstract When something abnormal suddenly occurs in a network, for instance, an unknown link is disconnected or give an extremely long When we something suddenly occurs a network, link issensing-based disconnected network or give antomography extremely long delay, need toabnormal immediately identify the in abnormal linkfor to instance, remediateanit.unknown Compressed can delay, we identify need to such immediately themeasuring abnormalpacket link totransmission remediate it.behaviors Compressed sensing-based network can efficiently abnormalidentify links, by over fewer end-to-end routes. tomography Its performance efficiently identify such abnormal links, by measuring packet transmission behaviors over fewer end-to-end routes. Its performance largely depends on pre-selection of measurement routes, so some algorithms have been proposed. However, when the network size largely on pre-selection of conventional measurementroute routes, so somealgorithms algorithmstohave proposed. However, when the network size is large,depends it takes enormous time for selection list abeen huge number of all routes between transmitter is large, it takes enormous time for conventional route selection algorithms to list a huge number of all routes between transmitter and receiver nodes and select adequate measurement routes out of them. and nodes select some adequate measurement routes out for of them. Inreceiver this paper, weand propose route selection algorithms compressed sensing-based transmissive network tomography. In this paper, we propose some route selection algorithms for sensing-based network tomography. The proposed algorithms make efficient use of the property of thecompressed Zero-Suppressed Binary transmissive Decision Diagram (ZDD) used in The proposed algorithms make efficient use of the property of the Zero-Suppressed Binary Decision Diagram (ZDD) in the SIMPATH algorithm, so they can efficiently not only list a limited number of measurement route candidates but alsoused select the SIMPATH algorithm, so they canthem. efficiently not only list a limited of measurement route candidates but algorithms also select adequate measurement routes out of Computer simulation results number reveal that, for given networks, the proposed adequate measurement routes out of them. simulation tomography results revealschemes that, forusing giventhe networks, proposed algorithms can efficiently select measurement routes andComputer the delay-difference selected the measurement routes can can efficiently select measurement routes and the delay-difference tomography schemes using the selected measurement routes can effectively identify abnormal links. effectively abnormal links. c 2016 Theidentify � Authors. Published by Elsevier B.V. c 2016 The � Authors. by Elsevier 1877-0509 ©under 2017 responsibility ThePublished Authors. Published byB.V. Elsevier B.V. Chairs. Peer-review of the Conference Program Peer-review under under responsibility responsibility of of the the Conference Program Chairs. Peer-review Keywords: Network tomography; Compressed sensing; Route Selection; ZDD Keywords: Network tomography; Compressed sensing; Route Selection; ZDD
1. Introduction 1. Introduction In wireless communication networks such as sensor networks and mesh networks, when something abnormal In wireless communication networks such as sensor networks and mesh networks, when something abnormal occurs, network managers need to but sometimes cannot quickly identify it. This is because communication nodes occurs, network managers need to but sometimes cannot quickly identify it. This is because communication nodes automatically try to connect to neighboring nodes in better link states, so network managers cannot know where automatically try to connect to neighboring nodes in better link states, so network managers cannot know where the abnormal link is. Network tomography is an efficient technique to estimate link states by measuring packet the abnormal link is. Network tomography is an efficient technique to estimate link states by measuring packet transmission behaviors such as delays and losses over routes between end-nodes 1,2 . For a given network, when transmission behaviors such as delays and losses over routes between end-nodes 1,2 . For a given network, when transmitter nodes (TXs) and receiver nodes (RXs) are selected out of different nodes located at the boundary of the transmitter nodes (TXs) and receiver nodes (RXs) are selected out of different nodes located at the boundary of the , it is referred to as transmissive network tomography. network 3,4 network 3,4 , it is referred to as transmissive network tomography. ∗ ∗
Corresponding author. Tel.: +81-6-6605-2795. Corresponding Tel.: +81-6-6605-2795. E-mail address:author.
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c 2016 The Authors. Published by Elsevier B.V. 1877-0509 � cunder 1877-0509 � 2016responsibility The Authors.ofPublished by Elsevier B.V.Chairs. Peer-review the Conference Program 1877-0509 © 2017responsibility The Authors. by Elsevier Peer-review under of Published the Conference ProgramB.V. Chairs. Peer-review under responsibility of the Conference Program Chairs. 10.1016/j.procs.2017.05.303
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Network tomography is originally proposed to estimate all link states in a network, so a number of route measurements are required. However, as mentioned above, when we are interested only in identification of abnormal links, thanks to the sparse nature of abnormal link occurrence, compressed sensing technique 5,6 can be introduced into it, so we can do it with route measurements whose number is much less than that of links 7,8 . In compressed sensing-based network tomography, how to select measurement routes plays an important role in enhancing its abnormal link identifiability and energy efficiency. Some algorithms have been proposed 9,10 , but unfortunately, they are not scalable. This is because they try to directly select adequate measurement routes out of all routes between TX and RX, so when the network size is large, it takes enormous time to not only list a huge number of all the routes but also select adequate measurement routes out of them. As shown in this paper, when the number of nodes is more than thirty, the algorithms cannot terminate within one week. In this paper, we propose some measurement route selection algorithms for compressed sensing-based network. Unlike the conventional algorithms, the proposed algorithms are scalable; after efficiently listing all routes between TX and RX in the form of the Zero-Suppressed Binary Decision Diagram (ZDD) 11 by the SIMPATH algorithm 12 , they select a limited number of measurement route “candidates” out of all the routes making effective use of the graphical property of the ZDD, that is, “to share all equivalent sub-graphs,” and then select adequate measurement routes out of them. This paper is organized as follows. Section 2 presents the preliminaries. Section 3 clearly states the problem. Section 4 shows our proposed algorithms in detail. Section 5 demonstrates some computer simulation results on the performance evaluation. Finally, Section 6 concludes the paper. 2. Preliminaries 2.1. Compressed Sensing Compressed sensing is an effective theory in signal processing for reconstructing a finite-dimensional sparse vector based on its linear measurements with dimension smaller than that of the unknown sparse vector 5,6 . First of all, let 1 J p p us define the � p norm (p ≥ 1) of a vector x = [x1 x2 · · · x J ]� ∈ R J as |x� p = , where � denotes the i=1 |xi | I×J transpose operator. Next, let us assume that, through a matrix A ∈ R (I < J), we obtain a linear measurement vector y = [y1 y2 · · · yI ]� ∈ RI for x as y = Ax. By picking up the j-th and j’-th column vectors from A, which are denoted by c j and c j� , respectively, we construct the partial matrix as A j j� = [c j c j� ]. Whether or not one can recover a sparse vector x from y by means of compressed sensing can be evaluated by the mutual coherence μ(A) 6 , which is defined as as the maximum value of the absolute normalized correlations ν(A j j� ) (1 ≤ j, j� ≤ J, j j� ): μ(A) =
max �
ν(A j j� ), �
1≤ j, j ≤J, j j
If k
30, they cannot terminate, since the number of all routes between TX and RX explosively increases. Therefore, in this paper, we consider the problem of how to scalably construct 1-identifiable A.
4. Proposed Route Selection Algorithms SIMPATH can efficiently list all routes between TX and RX, but when the network size is large, the number of all the routes becomes huge, so it takes still enormous time to select adequate routes out of them. The proposed algorithm first “lists” a limited number of route candidates out of all routes between TX and RX, and then “selects” adequate routes out of them.
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rβ
rβ
rα 0
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0
(a) First visit at a vertex with a single input branch
(b) First visit at a vertex with multiple input branches
1
A shared sub-graph
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A shared sub-graph
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(c) Later than the first visit at a vertex with multiple input branches
Fig. 4. Three kinds of visits at a vertex e j .
4.1. Route Candidates List Algorithm In the route candidates list process, we sequentially examine all routes one by one and add only adequate routes to a set of route candidates discarding redundant routes. Let us assume an occasion when we have added a route rα to the set of route candidates and now we are examining the adequacy of another route rβ at a node e j . We can see from (1) that, if two routes are different but share some links, then the mutual coherence between them can be one. As the present problem setting, we are interested in constructing a 1-identifiable A, so the condition of μ(A) < 1 is required. This means that, when we have listed rα in the set of route candidates, if we find that rβ partially shares some links with rα , then we can discard rβ . In other words, rβ is redundant in terms of rα . In ZDDs, “all equivalent sub-graphs are shared,” so we can easily find whether rβ is redundant or not, just by checking the number of input branches to e j . Fig. 4 (a), (b) and (c) show three kinds of visits at e j . Fig. 4 (a) shows a case when e j has a single input branch. In this case, the adequacy examination on rβ can always go forward, because the vertexes lower than e j have never been visited. On the other hand, Fig. 4 (b) and (c) show two cases when e j has multiple input branches. In these cases, if the visit is the first (see (b)), the adequacy examination on rβ can go forward, because the vertexes lower than e j also have never been visited. However, if the visit is not the first (see (c)), the examination should immediately stop, because all distinct routes in the sub-graph connecting to RX have been added to the set of route candidates through the process of the adequacy examination on rα . Consequently, we can conclude that the set of route candidates should be constructed only with the routes which do not have different input branches in the ZDD. Algorithm 1 summarizes the Route Candidates List Algorithm, where condition in line 14 is the key idea for discarding redundant routes.
4.2. Routing Matrix Construction Algorithm Algorithm 2 shows the routing matrix construction algorithm using Rcand . As the conventional algorithm 9 , the proposed algorithm first selects node-disjoint routes out of Rcand into an initial set, using an algorithm 15 . Note that its necessity will be examined in computer simulation. Next, until all links are included in at least one route of A, a route R repeatedly gets included in A, which has the maximum number of links which have not been included in A so far constructed. We refer to this as the “unitization process.” If multiple routes have the same maximum number of the links, then we have two kinds of inclusions, that is, the one with the maximum number of links (namely, the “longest” route) gets included, or the one with the minimum number of links (namely, the “shortest” route) gets included. Now that μ(A) = 1 has been achieved, so finally, until μ(A) < 1, a route R repeatedly gets included in A, which has the minimum number of links which give ν(A j j� ) = 1. We refer to this as the “decimation process.” Similar to the unitization process, if multiple routes have the same minimum number of the links, we have also two kinds of inclusions, that is, the longest route gets included, or the shortest route gets included. Before going to the following section of performance evaluation, we refer to the algorithms selecting longest and shortest routes in the unitization and decimation processes with the initial node-disjoint routes selection as the “disjoint/longest” and the “disjoint/shortest,” respectively. On the other hand, we refer to those without the initial node-disjoint routes selection just as the “longest” and “shortest,” respectively.
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Algorithm 1 Route Candidates List Algorithm 1: Input: Z, IB, NV � Z=ZDD for all routes between TX and RX, 2: � IB={ib( j): the number of input branches at the j-th vertex (link), j = 1, 2, . . . , J} 3: � NV={nv( j) = 0: the initial number of visits so far at the j-th vertex (link), j = 1, 2, . . . , J } 4: Output: Rcand � Rcand =Route Candidates Set 5: j = 1 6: L ← ∅ 7: R ← ∅ 8: Rcand ← ∅ 9: for all j = 1, 2, . . . , J do 10: for all b ∈ {0, 1} do 11: if b = 1 exists then � The j-th vertex has 1-branch 12: R ← R ∪ { j} 13: end if 14: if jb is a 1-terminal vertex) then � jb is the vertex index which the b-branch of the j-th vertex points to 15: Rcand ← Rcand ∪ R 16: L←L∪R 17: else 18: if (in( jb ) = 1) ∨ (nv( jb ) = 0) ∨ (R \ L = ∅) then 19: nv( jb ) = 1 � Memorize that the jb -th vertex has been visited 20: end if 21: end if 22: end for 23: end for Algorithm 2 Routing Matrix Construction Algorithm 1: Input: Rcand � Rcand =Route Candidates Set constructed by Algorithm 1 2: Output: A � A=Routing Matrix 3: L ← ∅ 4: Rdisjoint ← NodeDisjoint(Rcand ) � NodeDisjoint(S) : the function to select node-disjoint routes out of S 15 5: A ← Const(0, Rdisjoint ) � Const(M, S) : the function to construct a matrix with M and S 6: Rcand ← Rcand \ Rdisjoint 7: L ← L R� 8: 9:
∀R� ∈Rdisjoint
while |L| < J do R ← arg max |R� \ L|
� Unitization process to include all routes in A
R� ∈Rcand
A ← Const(A, R) Rcand ← Rcand \ R L←L∪R end while while μ(A) ≥ 1.0 do � Decimation process to make A less than 1 15: R ← arg min |{( j, j� |ν(A�j j� ) = 1, A� = Const(A, R� ), 1 ≤ j, j� ≤ J, j j� }| 10: 11: 12: 13: 14:
16: 17: 18:
R� ∈Rcand
A ← Const(A, R) Rcand ← Rcand \ R end while
5. Performance Evaluation We evaluated the performance of the proposed algorithms by computer simulations. The personal computer used for the performance evaluation was equipped with CPU of Intel Xeon E5-2640 v2 2.0 GHz and memory of 64 GB,
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and the four proposed algorithms as well as the conventional algorithm 9 were implemented with C++. We set the performance matrics as the number of route candidates, reduction rate (I/J), interval factor, energy factor, execution time and mutual coherence for the constructed A. First of all, we randomly generated 200 networks which had intermediate nodes all with degree of freedom of more than 2. Table 1 shows the performance evaluation results for Q = 15 and Q = 30. For the case of Q = 15, the conventional algorithm uses all the routes as the route candidates, so the number of route candidates is more than 1,000, but the proposed algorithms can reduce the number of route candidates less than 100 discarding redundant routes. In addition, the proposed algorithms can make their reduction rate and energy factor less than half those of the conventional algorithm. On the other hand, for the case of Q = 30, the conventional algorithm was not able to terminate (within one week), but as is the case for Q = 15, the proposed algorithms can make the number of route candidates less than 100. Next, Table 2 shows the performance evaluation result for the network with Q = 15 and J = 33 in Fig. 1. It took more than 1 minute for the conventional algorithm to construct the routing matrix directly out of all 6,547 routes. On the other hand, the four proposed algorithms can effectively reduce the number of route candidates to 89 (around 13.6 %) and construct routing matrices with the interval and energy factors less than those of the conventional algorithm within 1 minute. All the five routing matrices constructed for the network with Q = 15 and J = 33 satisfy the mutual coherence less than 1, so it is expected that they can perfectly identify the abnormal link if it is guaranteed that there is only one abnormal link in the network. We confirmed it by computer simulation, where we applied the constructed routing matrices to the delay-difference tomography 16,17 ɽFig. 5 shows the perfect identification rate (PIR) against the indentifiability k. Here, the PIR is defined as the rate that both the false positive rate and false negative rate are jointly 0. Note that the false positive rate means the rate that normal links are identified as abnormal links, whereas the false negative rate means that the rate that abnormal links are identified as normal links. For the case of k = 1, the PIR=1 is achieved for all the five algorithms. For k = 2 and k = 3, the performance depends only on the number of measurement routes, namely, the interval factor, so the conventional algorithm outperformed the four proposed algorithms. However, the present design criterion is to ensure the performance for k = 1 making the interval and energy factors smaller, so in this sense, the shortest algorithm or the disjoint/longest algorithm are considered to be advantageous. Unlike the conventional algorithm, the proposed algorithms can reduce the number of route candidates effectively, so the initial node-disjoint routes selection is found not to be advantageous for the proposed algorithms. 6. Conclusions In this paper, we proposed some measurement routes selection algorithms and evaluated their performances by computer simulations. The proposed algorithms can make effective use of the property of the ZDD, that is, “to share all equivalent sub-graphs,” and effectively reduce the number of routes candidates between TX and RX taking into the definition of mutual coherence into consideration. As a result, they can select measurement routes scalably for networks with larger sizes. In the present paper, we listed route candidates after executing SIMPATH and confirmed the scalability, accuracy and effectiveness of the proposed algorithms. Therefore, if we can incorporate the route candidates listing into SIMPATH, it must be more effective. That is one of our future works. Table 1. Performance evaluation results (Q = 15 and 30). No. of Nodes
Q = 15
Q = 30
Scheme
Route candidates
Reduction rate
Energy factor
Route candidates
Reduction rate
Energy factor
Conventional Disjoint/shortest Disjoint/longest Shortest Longest
> 1, 000 < 100 < 100 < 100 < 100
0.598 0.426 0.416 0.427 0.416
118.9 96.7 94.7 102.7 100.9
NA < 100 < 100 < 100 < 100
NA 0.410 0.405 0.423 0.417
NA 336.0 335.6 358.9 358.1
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Table 2. Performance evaluation result for the network in Fig. 1 (Q = 15, J = 33). Scheme
Route candidates
Interval factor
Energy factor
Execution time
Mutual coherence
Conventional Disjoint/shortest Disjoint/longest Shortest Longest
6547 89 89 89 89
23 13 11 10 11
147 108 89 93 106
>1min
