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Fault-Tolerant Routing: A Genetic Algorithm and CJC

Arjun Rao

CS 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() Algorithm

1) 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 vector

5) Compute navigation vector Nv(u)

6) If enabled C link not faulty, set Rvr+1(u) to 0, take 2’s

complement of Rv(u), and route packet7) Else, do the same with the J link (link r), except

complement Rv0(u) and Rv

1(u) only8) If both C and J links faulty and/or not enabled, find an H link

for routing (or misrouting), set Rvr+1(u) and Rv

r(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