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Fault-Tolerant Routing: A Genetic Algorithm and CJC

Arjun RaoCS 717

November 18, 2004

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Next Paper

• [1] Loh, Peter K.K., “Artificial Intelligence Search Techniques as Fault-Tolerant Routing Strategies”

• [2] Loh, Shaw., “A Genetic-Based Fault-Tolerant Routing Strategy for Multiprocessor Networks”

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Our Little Problem…

• AI search techniques topology- and fault-type independent…

• …but non-minimal routes utilized

• Follow-up work shows how genetic algorithms (combined with heuristics) can find minimal routes in presence of network faults

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Genetic Algorithms: Overview• Optimization strategy• Population of potential solutions evolve over

series of generations• Each element of population is chromosome;

each unit of chromosome is gene• Chromosomes undergo crossover and

mutation• Most fit chromosomes selected for next

generation, based upon fitness function

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Abstract Model• Same as before (including definitions of S

and G)

• Pure abstraction suffers from same caveats as before

• Basic idea: Instead of AI search for adaptive route, optimize over population of routes to find best

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Message Packets

• Simplified version:

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Chromosome

• Route Chromosome• Node on route Gene in chromosome

• Length of route Size of chromosome– Chromosome size directly reflects routing

performance!

• Distance traversed basis of fitness

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Population Creation

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Mutation and Crossover

• Mutation: Swap and/or shift• Normal crossover destroys routes, messes

with source and destination; problem w/ different lengths– Use one-point random crossover

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Fitness Function

• F = (Dmax – Droute) / Dmax + – Dmax: Maximum distance between source and

destination– Droute: Distance traveled by specific route : Predefined value to ensure non-zero fitness

• Higher value More fit

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Selection Scheme• Roulette Wheel

– Sum of fitness values * random value from [0,1]– Select chromosomes until fitness sum greater than product

• Tournament Selection– Most fit chromosomes selected

• Stochastic Remainder– Probabilities used to select route

• Which scheme has best performance selecting optimal route?

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Reroute

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Genetic Hybrid Algorithm

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Performance Testing

• RW, TS, SR tested in concert with RR and HCR (previous algorithm was for RR)

• 25-node mesh network• Varying fault percentages• Ten tests, each over 20 generations• Variations in mutation and crossover rates

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Performance Testing (cont.)

• Randomness of RR bad for rerouting• Unsuccessful routes increase with number of

generations

• SR + HCR performed best (prevented premature convergence, maintained good diversity)

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Performance Testing (cont.)

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Comparison With AI-only Strategies

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Next (and Final) Paper

• [3] Loh, Schröder, Hsu., “Fault-Tolerant Routing on Complete Josephus Cubes” (not AI/GA-related but interesting nevertheless)

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What is a Josephus Cube?

• New network topology– Better embedding, communications performance

than hypercube topologies• Can be used for processors, node clusters w/

optical channel architectures– In clusters, fault tolerance sacrificed for scalability– Symmetry (which is good for routing) somewhat

sacrificed as well

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Complete Josephus Cube (CJC)• Augmented (i.e. more links) Josephus Cube• Better fault-tolerance• Routing efficiency maintained!• Properties

– Uniform node degree of log2N + 2– N = 2r, r is order of network– Smaller diameter– Better symmetry– Guaranteed message delivery (w/ up to r + 1 faults)– No deadlocks/livelocks

• We examine CJC as node cluster topology

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Defining the CJC

• CJC(N) is an undirected graph G = (V, E)

• Nodes labeled using function J:– J(1) = 1– J(2i) = 2J(i) – 1, i 1– J(2i + 1) = 2J(i) + 1, i 1

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Defining the CJC (cont.)

• V = {u | J(2r) u J(2r + 1 – 1)}• Three types of edges: H, J, C

– E = EH EJ EC

• (x,y) EH iff H(x,y) = 1, where H is the Hamming distance

• (x,y) EJ iff y = x XOR 6

• (x,y) EC iff y = (x), where finds 2’s complement

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Example: CJC(8), r = 3

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Example: CJC(16), r = 4

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Data Vectors

• Route vector Rv(u) = (Tr+1Tr…T1T0)– Information on paths already traversed– Tr+1 = 1 if |u v| > CEIL(r / 2) else 0

– Tr = 1 if v = u(J), where u(J) is 2’s complement of lowest order bits

– Tk = (uk+1 vk+1), 0 k < r

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Data Vectors (cont.)

• Input link vector Iv(w) = (Lr+1Lr…L1L0)– Li = 0 if message arrives on link i else 1

• Fault status vector Fv(w) = (Sr+1Sr…S1S0)– Sj = 0 if node on link j or link j itself faulty else 1

• Navigation vector Nv(u) = Rv(u) Iv(u) Fv(w) – Designates optimal (or best-possible) paths from

current node

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CJC-FTROUTE() Algorithm1) Extract route vector from packet header2) Update current node’s fault status vector3) If route vector all 0’s, destination reached4) (Re)Initialize input link vector5) Compute navigation vector Nv(u)6) If enabled C link not faulty, set Rv

r+1(u) to 0, take 2’s complement of Rv(u), and route packet

7) Else, do the same with the J link (link r), except complement Rv

0(u) and Rv1(u) only

8) If both C and J links faulty and/or not enabled, find an H link for routing (or misrouting), set Rv

r+1(u) and Rvr(u) to 0, and

either route or discard

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Example 1: CJC(16)

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Example 2: CJC(16)

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Routing With Bounded Faults

• Theorem: Routing between a node pair (x, y) is optimal if FT < (r + 2) which includes faulty J and C incident links at x

• Proof: Let fi faults be encountered at node wi on path of length p. |R(wi)| - fi optimal links left to y; routing from wi to wi+1 optimal if link or |R(wi)| - fi > 0. Generalizing to whole path:

1

0

1

0

|)(|p

iTi

p

ii FfwR

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Maximum Path Distance• Define FTR(u, v) = Set of edges of path from

u to v• Corollary: FTR guarantees path distance of

route upper-bounded by H(x, y) if FT < (r + 2)

• Corollary: FTR guarantees max distance r in order r cluster

• Proof: Follows from above corollary with H(x, y) r

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Final Set of Theorems

• For sub-optimal routing, FTR guarantees max path distance of 2r – H(x, y) + 2 only if FT < (r + 2) (including disabled C and J links at x)

• FTR guarantees deadlock-free routes• FTR guarantees livelock-free routes

– Proof of latter two utilizes notion of virtual networks in CJC