#### Transcript Arjun - Fault-Tolerant Routing: A Genetic

CS717 Fault-Tolerant Routing: A Genetic Algorithm and CJC Arjun Rao CS 717 November 18, 2004 CS717 Next Paper • [1] Loh, Peter K.K., “Artificial Intelligence Search Techniques as Fault-Tolerant Routing Strategies” • [2] Loh, Shaw., “A Genetic-Based FaultTolerant Routing Strategy for Multiprocessor Networks” CS717 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 CS717 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 CS717 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 CS717 Message Packets • Simplified version: CS717 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 CS717 Population Creation CS717 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 CS717 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 CS717 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? CS717 Reroute CS717 Genetic Hybrid Algorithm CS717 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 CS717 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) CS717 Performance Testing (cont.) CS717 Comparison With AI-only Strategies CS717 Next (and Final) Paper • [3] Loh, Schröder, Hsu., “Fault-Tolerant Routing on Complete Josephus Cubes” (not AI/GA-related but interesting nevertheless) CS717 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 CS717 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 CS717 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 CS717 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 CS717 Example: CJC(8), r = 3 CS717 Example: CJC(16), r = 4 CS717 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 CS717 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 CS717 CJC-FTROUTE() Algorithm 1) 2) 3) 4) 5) 6) Extract route vector from packet header Update current node’s fault status vector If route vector all 0’s, destination reached (Re)Initialize input link vector Compute navigation vector Nv(u) If enabled C link not faulty, set Rvr+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 Rv0(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 Rvr+1(u) and Rvr(u) to 0, and either route or discard CS717 Example 1: CJC(16) CS717 Example 2: CJC(16) CS717 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: p 1 p 1 | R(w ) | f i 0 i i 0 i FT CS717 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 CS717 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