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Combinatorial Group Testing in Telecommunications II.

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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

 We deal with two type of faults

 Physical failts (has a specific location)  Logical faults (Murphy’s law)

Renesys analysis of the

• perating

routers during Sandy hurricane

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Fast Failure Localization

 „Compressed sensing”

 Optical signal is sent along a test-trail  sub 50ms

STTL SNFC CHCG NYCM LSAN LSVG SLKC DNVR KSCY TULS CLEV STLS WASH BSTN CHRL DTRT TRNT ATLN IPLS HSTN DLLS ELPS NSVL MIAM MPLS NWOR

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 The network topology is known

 G=(V,E) 2-connected

Separating system of cycles

Localize Single Link Failure with Monitoring Cycles (m-cycles)

1 2 3 Alarm code table

c2 c1 c0

0-1 0 0 1 0-2 0 1 0 0-3 0 1 1 1-2 1 0 0 1-3 1 0 1 2-3 1 1 0 c0 c1 c2 0-3 0 1 1

#monitors= 3 Cover length = 9

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• 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

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Non-zero codes

 If a node has degree 2, the neighbouring links cannot

be distinguished with cycles:

 Using monitoring-trails instead of cycles

Monitoring-trails (M-Trail)

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2 3 1 4 (b) An m-trail solution t1 t0 t2 (0,1) 1 1 5 (0,2) 1 1 1 7 (0,3) 1 4 (1,2) 1 1 3 (1,3) 1 1 6 (2,4) 1 1 (3,4) 1 2 t2 t1 t0 Link (c) Alarm code table Decimal (a) m-trail R T a b c d e

No optical loopback switching

• 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.

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 Bm-trail is a connected sub-graph

 Euler constraint is relaxed

Bi-directional M-Trails (BM-Trails)

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.

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 Given: an undirected 2 connected graph

• 1. bm-trail – connected components
• 2. m-trail – trail (Euler sub-graph)
• 3. m-cycle – closed trail

 List of failures:

• A. Single link

B.

Dual, triple link failures (Dense SRLG)

• C. Single and some multi-link failures (Sparse

SRLG)

 Goal: find a minimum number of m-trail/m-

cycle/bm-trail in the graph, such that there are no pair of failure event with exactly the same m- trail/m-cycle/bm-trail passing through.

Problem definition

#monitors log (# Failures +1) 

1 2 3

001 010 011 100 101 110

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 Number of bm-trails is #links/2  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]

Ring networks with single link failure

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f e n

Polynomial time constructions

 Complete graph m-trail:  Complete graph bm-trail:  2D-grid bm-trail:  C1,2 circulant graph m-trail:

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 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)

Multiple Failures

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1 2 3 Alarm code table

c2 c1 c0

0-1 0 0 1 0-2 0 1 0 0-3 0 1 1 1-2 1 0 0 1-3 1 0 1 2-3 1 1 0

c0 c1 c2

1 1 0-2 0-1

Constraints on the length of an m-trail

 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

• f 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

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Separating systems

Let H be finite set of elements. The system A ⊆ 2𝐼 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. Let m and k be positive integers, such that 𝑙 <

𝑛 2 . Let us

denote tha smallest size of a separating system A ⊆ 2 𝑛

• f 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).

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The minimum number of tests with bounded size

 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).

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Illustration of the proofs

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1 𝑗𝑔 𝑗𝑢𝑓𝑛 𝑗𝑡 𝑗𝑜 𝑢ℎ𝑓 𝑢𝑓𝑡𝑢 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓

items tests

Each row has at most k elements

• 1. We want to increase the weight of each row.
• 2. We have more 0 than 1 in each row as 𝑙 <

𝑛 2.

• 3. There is going to be an item who has no pair.

1 We can if the new code is not assigned to any other item. Each item may have a pair- item, that has the same code except the last bit. 1 1 1 1 1 1 1

How to compute n(m,k) ?

 We construct a matrix with unique column vectors, and

minimum total weigth

items tests = n

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1 1 1 1 1 1 1 1 1 1 1 The last code has weight j+1

Each row has at most k elements Total number

• f 1s

Find smallest 𝑘 such that 𝑏 < 𝑜 𝑘 + 1 This can be constructed such that every row has k elements

Test size vs number of tests

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k m=100 items

Number of tests

log2 𝑛 𝐼(𝑙 𝑛) ≤

 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

Signaling free failure localization

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Distributed failure localization

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T3 T1 T4 T2

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 The number of

alarms is no longer a concern

 Minimize the total

length of m-trails

Node failures only

Distributed failure localization

 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

• n its status

 The larger the test the weaker separates the set of

items.

Localize node failures as well

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 Let us define a matrix with n columns (number of

tests) and m=|V| rows

Lower bound on the number of tests

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3 1 2

T3 T1 T4 T2

T1 T2 T3 T4 Node 0

1/3 1/3 1/3

Node 1

1/3 1/3 1/3

Node 2

1/3 1/3 1/3

Node 3

1/3 1/3 1/3

 The average size of the test at node v is  We have

where m is the number of items (nodes)

The average test size at a node

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Number of tests at v

The inequality of harmonic and arithmetic mean

The bound is applied to matrix ω

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T1 T2 T3 T4 Node 0

1/3 1/3 1/3

Node 1

1/3 1/3 1/3

Node 2

1/3 1/3 1/3

Node 3

1/3 1/3 1/3

Lower bound on v

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The minimum of function g()

Where is the minimum?

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The minimum of function g()

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 Find the maximum of

After derivate

 To localize node failure at every node

Lower bound on the number of tests

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We have construction with 2

Interesting related question: When does group testing make sense?

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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