Combinatorial Group Testing in Telecommunications II. Hosszu va - PowerPoint PPT Presentation

Combinatorial Group Testing in Telecommunications II. Hosszu Éva Budapesti Műszaki és Gazdaságtudományi Egyetem Villamosmérnöki és Informatikai Kar Távközlési és Médiainformatikai Tanszék High-Speed Networks Laboratory (HSNLab) MTA- BME Lendület Jövő Internet kutatócsoport

Network faults 2  We deal with two type of faults  Physical failts (has a specific location) Renesys analysis of the operating routers during Sandy hurricane  Logical faults (Murphy ’s law)

Fast Failure Localization 3  „Compressed sensing”  Optical signal is sent along a test-trail  sub 50ms STTL TRNT BSTN MPLS DTRT CHCG CLEV SLKC NYCM DNVR KSCY WASH IPLS SNFC STLS NSVL LSVG TULS LSAN ATLN CHRL ELPS DLLS NWOR HSTN MIAM

Localize Single Link Failure with Monitoring Cycles (m-cycles) 4  The network topology is known  G=(V,E) 2-connected 0  Separating system of cycles c 0 c 1 c 2 Alarm code table c 1 c 0 0 - 1 0 0 1 3 0 - 2 0 1 0 c 2 0 - 3 0 0 - 3 0 1 1 1 1 1 2 1 - 2 1 0 0 1 - 3 1 0 1 Non-zero #monitors= 3 2 - 3 1 1 0 codes Cover length = 9 B. Wu, L. Yeung, and Pin- Han Ho, “Monitoring Cycle Design for Fast Link Failure Localization in All- Optical Networks”, in IEEE/OSA Journal of Lightwave Technology, Vol. 27, No. 9, May 2009 4

Monitoring-trails (M-Trail) 5  If a node has degree 2, the neighbouring links cannot be distinguished with cycles:  Using monitoring-trails instead of cycles T Link t 2 t 1 t 0 c Decimal 2 t 0 (0,1) 1 0 1 5 a (0,2) 1 1 1 7 b (0,3) 1 0 0 4 R 4 (1,2) 0 1 1 3 d t 1 1 No optical (1,3) 1 1 0 6 loopback t 2 (2,4) 0 0 1 1 switching 0 e (3,4) 0 1 0 2 3 (a) m-trail (c) Alarm code table (b) An m-trail solution B. Wu, Pin-Han Ho, and K. L. Yeung , “Monitoring Trail: On Fast Link Failure Localization in WDM Mesh Networks”, IEEE/OSA Journal of Lightwave Technology , Vol. 27, No. 23, December 2009. 5

Bi-directional M-Trails (BM-Trails) 6  Bm-trail is a connected sub-graph  Euler constraint is relaxed Optical loopback switching N. Harvey, M. Patrascu, Y. Wen, S. Yekhanin , and V. Chan, “Non -Adaptive Fault Diagnosis for All- Optical Networks via Combinatorial Group Testing on Graphs,” in IEEE INFOCOM, 2007, pp. 697 – 705 .

Problem definition 7 0  Given: an undirected 2 connected graph 001 1. bm-trail – connected components 010 011 2. m-trail – trail (Euler sub-graph) 3 101 3. m-cycle – closed trail 110  List of failures: 1 2 100 A. Single link Dual, triple link failures ( Dense SRLG ) B. C. Single and some multi-link failures ( Sparse SRLG)  Goal: find a minimum number of m-trail/m- #monitors  cycle/bm-trail in the graph, such that there are no  log (# Failures +1)  pair of failure event with exactly the same m- trail/m-cycle/bm-trail passing through.

Ring networks with single link failure 8  Number of bm-trails is  #links/2  f n e  To distinguish the failure of link e and f we need an bm-trail terminating in node n .  Each bm-trail can terminate at most two nodes, thus 2*[#bmtrails]  [#nodes]

Polynomial time constructions 9  Complete graph m-trail:  Complete graph bm-trail:  2D-grid bm-trail:  C 1,2 circulant graph m-trail:

Multiple Failures 10  With multi-link failures - Code of a multiple failures is the bitwise OR of the codes of all the items of the failure  Structured version of non-adaptive Combinatorial Group Testing (CGT) c 0 c 2 c 1 0 0 - 1 0 - 1 0 0 1 0 - 2 0 - 2 0 1 0 c 1 0 - 3 0 1 1 c 0 0 1 1 3 1 - 2 1 0 0 c 2 1 - 3 1 0 1 1 2 2 - 3 1 1 0 Alarm code table

Constraints on the length of an m-trail 11  The length of an m-trail is bounded in its physical distance  The links have similar lengths  Investigated for CGT for single failure (The question was raised by A. Rényi 1961): G.O.H. Katona , „On Separating Systems of Finite Set,” Journal of Combinatorial Theroy Vol. 1, No. 2, 1966 R. Ahlswede , „Rate -wise Optimal Non-Sequential Search Strategies under a Cardinality Constraint on the Tests, „ Discrete applied Mathematics 156, 2008

Separating systems 12 Let H be finite set of elements. The system A ⊆ 2 𝐼 is a separitng system if for any 𝑦, 𝑧 ∈ 𝐼 , 𝑦 ≠ 𝑧: ∃𝑈 ∈ 𝐵 , such that 𝑦 ∈ 𝑈, 𝑧 ∉ 𝑈 or 𝑦 ∉ 𝑈, 𝑧 ∈ 𝑈 . 𝑛 Let m and k be positive integers, such that 𝑙 < 2 . Let us denote tha smallest size of a separating system A ⊆ 2 𝑛 of sets of size  exactly k by n(m,k) ,  at most k by n’(m,k) ,  average size at most k by n*(m,k) .

The minimum number of tests with bounded size 13  Theorem:  Theorem: Minimum number of tests of size  exactly k by n(m,k) ,  at most k by n’( m,k) ,  average size at most k by n*(m,k) .

Illustration of the proofs 14 Each row has at items most k elements 1 1 0 0 1 1 1 𝑗𝑔 𝑗𝑢𝑓𝑛 𝑗𝑡 𝑗𝑜 𝑢ℎ𝑓 𝑢𝑓𝑡𝑢 0 0 tests 0 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 0 0 1 1 1 0 We can if the new code is not assigned to any other 1 1. We want to increase the weight of each row. item. 𝑛 Each item may have a pair- 2. We have more 0 than 1 in each row as 𝑙 < 2 . item, that has the same 3. There is going to be an item who has no pair. code except the last bit.

How to compute n(m,k) ? 15  We construct a matrix with unique column vectors, and minimum total weigth items Each row has at 0 1 0 0 0 0 0 0 1 1 most k elements 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 tests Total number 0 0 0 0 0 1 0 0 0 0 The last code has weight j+1 = n of 1s 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 This can be Find smallest 𝑘 such constructed such 𝑜 that every row has that 𝑏 < 𝑘 + 1 k elements

Test size vs number of tests 16 Number of tests m=100 items log 2 𝑛 ≤ 𝐼(𝑙 k 𝑛)

Signaling free failure localization 17  Drawbacks of the alarm dissemination process  Electronic signaling is required  Increase the failure localization latency  Localize failures independently at each node  Any node along an m-trail can obtain the on-off status of the m-trail via optical signal tapping  Alarm dissemination is no longer needed

Distributed failure localization 18 18 T 4 T 3 T 2 T 1  The number of alarms is no longer a concern  Minimize the total length of m-trails

Distributed failure localization Node failures only

Localize node failures as well 20  Each node must be traversed by a unique set of m- trails  Each node has a unique alarm code table  The larger the test the more nodes have information on its status  The larger the test the weaker separates the set of items.

Lower bound on the number of tests 21  Let us define a matrix with n columns (number of tests) and m=|V| rows T 1 T 2 T 3 T 4 3 2 1 / 3 1 / 3 1 / 3 Node 0 0 T 4 T 3 Node 1 1 / 3 1 / 3 1 / 3 0 T 2 T 1 Node 2 0 1 / 3 1 / 3 1 / 3 0 1 Node 3 1 / 3 0 1 / 3 1 / 3

The average test size at a node 22  The average size of the test at node v is Number of tests at v The inequality of harmonic and arithmetic mean  We have where m is the number of items (nodes)

The bound is applied to matrix ω 23 T 1 T 2 T 3 T 4 1 / 3 1 / 3 1 / 3 Node 0 0 Node 1 1 / 3 1 / 3 1 / 3 0 Node 2 0 1 / 3 1 / 3 1 / 3 Node 3 1 / 3 0 1 / 3 1 / 3

Lower bound on  v 24

The minimum of function g() 25 Where is the minimum?

The minimum of function g() 26  Find the maximum of After derivate

Lower bound on the number of tests 27  To localize node failure at every node We have construction with 2

Interesting related question: When does group testing make sense? 28 Solution of the differential equation of the bound

Summary  Fast failure localization in all-optical networks  We applied the theory of non-adaptive group testing  Two lower bounds on network resources  Ahlswede and Katona theorems  General cost function for the tests