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Distributed Computing 1. Introduction Shmuel Zaks ©

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A. Distributed Computing Introduction: model, basic problems leader election, Spanning tree snapshot synchronization Impossibility and lower bounds B. Self stabilization C. ATM networks D. Optical networks

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A. Distributed algorithms days, 1 president4 days, 2 presidents Exercise: find a trade-off between no. of days and no. of presidents. Example 1: synchony 9

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message passing asynchronous Example 2: leader election ? x x x x Exercise: find a better algorithm to find the maximum, prove correctness and analyze performance.

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Example 3: faults Impossibility of consensus The Byzantine Generals Problem

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Example 4: snapshot

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shared memory synchronous Example 5: self stabilization B. Self Stabilization

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Exercise: find an algorithm to do it, prove correctness, analyze performance.

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load = 3 hop count = 2stretch factor = 4/3 (space) (time) Virtual pathVirtual channel C. ATM networks

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Given a network, find an “Optimal Layout” such that: (A) Given an upper bound on the hop count, minimize the load. N=7, Max hop count 1 => Min load 6 Example 6: Optimal designs

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(B) Given an upper bound on the load, minimize the hop count. N=7, Max load 1 => Min hop count 6

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כדור 2 - ממדי ברדיוס 3:

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כדור 2 - ממדי ברדיוס 1 כדור 1 - ממדי ברדיוס 2:

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Sofsem97 18 h=1 l = N-1...

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Sofsem97 19 h=2

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Sofsem97 20 h h-1

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Sofsem97 21 h-1

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Input: Graph G, a vertex v, h, l > 0. Question : is there a VP layout for G, by which v can reach all ther nodes, with hop count bounded by h and load bounded by l ? Example 7: Decision problems

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P P P NP P … … ……… … … … … load hop

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lightpaths p1 p2 Valid coloring D. Optical networks

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ADMs at the endpoints of lightpaths

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W=2, ADM=8 W=3, ADM=7 Minimizing the number of ADMs w/out grooming (g = 1)

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W=2, ADM=8 W=1, ADM=7 g=2 w/ grooming (g > 1)

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W=3, ADM=10 W=2, ADM=8 g=2 w/ grooming

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A solution S is a partition of P into feasible chains and cycles Feasible 1-colorable x u v w b a c d e

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N: The # of paths. SOL: The # of ADMs used by a solution. OPT: The # of ADMs used by an optimal solution. N SOL 2N N OPT 2N SOL/OPT 2 with grooming: SOL/OPT 2 g Example 8: The MIN ADM Problem w, w/out grooming

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NP-Complete g=1: ring – Eilam, Moran, Z., 2002 g>1 ring – Chiu, Modiano, 2000 fixed g path, ring - Shalom, Unger, Z., 2006 star – fixed g>3 ( poly for g=1,2) Shalom, Flammini, Monaco, Moscardelli, Z. Example 9: NP-complete?

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Given an instance G=(V,E) of the Edge Partition in g-regular graphs Problem, we construct a corresponding instance of our problem on star networks: It is possible to show that that there exists a solution using at most 2|E|/g ADMs if and only if the edges of G can be partitioned in g-regular graphs

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The Conflict Graph a b c de a b c de w v x x u x x The Shareability Graph x u v w b a c d e

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a b c de w v x x u x x N = 5 COST = 8 = OPT + 3 = = OPT N

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We show COST <= OPT N...

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Input: a graph and a set of paths Output: a valid coloring a valid coloring with minimum number of ADMs (number of colors can be a parameter) a valid coloring with minimum number of ADMs, given a grooming factor (number of colors can be a parameter) Problems

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Objective function: Minimize the number of ADMs Cases: w/out grooming (g = 1) w/ grooming (g > 1) Results: Complexity Approximability

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Some general topologies (rings, trees) O(log n) Flammini, Moscardelli, Monaco, Shalom, Z., 2005

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Distributed computing Computer networks Communication networks Distributed algorithms Protocols A. Distributed algorithms

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PODC – ACM Symp. On Pronciples of Distributed Systems DISC – International Workshop on Distributed Computing (previously: WDAG) Sirocco – International Colloquium on Structural Information and Communication Complexity

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References podc disc SIROCCO

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1. Introduction 1.1 model

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processors communication problem Communication network

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Anonymous

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12 a e 6 c Unique identities

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d a e b c message passing communication lines, channels topology communication

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ab c d e directed, undirected (message passing)

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message delivery mechanism fifo reliable, no faults finite, arbitrary delay queues of messages (message passing)

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Distributed algorithm, protocol Send a message receive a message do local computation (message passing) Execution

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R4R4 R1R1 R2R2 R3R3 R5R5 e a b d c shared memory

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A B read/write (shared memory)

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synchronization Synchronous, Asynchronous d a e b c

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Synchronous Model ij time t+dtime t Clock Network (synchronization)

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Asynchronous Model ij time t+???time t Clock Network (synchronization)

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Asynchronous Model - many executions Synchronous Model - unique execution rounds (synchronization)

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Asynchronou s Synchronous Shared memory Message passing (synchronization)

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Asynchronous model: for correctness, for upper bound analysis Synchronous model: for lower bound analysis

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Topology Ring d a e b c

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Clique d a e b c (Topology)

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General (Topology)

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Why simple networks? They enable the understanding of many design issues In existing general networks – assume a virtual simple network implemented (e.g. a ring) (Topology)

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Complexity measures Synchronous system time Asynchronous system communication communication (messages, bits) time (synchronous time, longest chain, bounded delay)

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Sense of direction, topological awareness global information number of processors local information topology of the network knowledge of ‘left’ and ‘right’ in a ring network knowledge of neighbour’s identities

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General references lower level protocols general books

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Parallel vs. Distributed computing Parallel computing – given a problem … (ex: sorting) Distributed computing – Given a network … (ex: broadcast)

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(Parallel vs. Distributed computing( Parallel computing : time vs. number of processors Distributed computing: number of messages Complexity goals: Parallel computing: efficiency Distributed computing: correctness

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Partially Synchronous Models Many different models bounded delay networks non accurate clocks eventually synchronous some communication links good failure detectors

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problem, task P1P1 P2P2 P3P3 input output Leader election yes no consensus

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issues design and analysis of algorithms impossibility, lower bounds fault tolerance

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1. Introduction 1.2 basic problems

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problems broadcast snapshot consensus shortest path, maximal flow leader election, breaking symmetry, maximum finding, spanning tree, center termination deadlock

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Example: broadcast d a e b c f

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Broadcast: bfs (breadth-first-search) d a e b c f

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Broadcast: dfs (depth-first-search) d a e b c f

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complexity message complexity 2|E| time complexity 2|E|-1

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broadcast: pi (propogation of information) d a e b c f

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send m to each neighbour stop pi algorithm if receive m along edge e: send m on all edges except e stop

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pi Theorem: The following holds for every execution of the pi algorithm: A processor receives the message m at most once. The execution terminates. each processor receives the message m. The edges on which processors receive m form a spanning tree. The message complexity is 2|E|-|V|+1. The time complexity …

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pif (propogation of information with feedback) shout-echo d a e b c f

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References

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H. Attiya and J. Welch, Distributed Computing, 1995

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References B. Awerbuch, A new distributed depth first search, 1985

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References D. P. Bertsekas and R. Gallager, Data Networks, 1992

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References J. E. Burns, A formal model for message passing systems, TR-91, Indiana University, September 1980.

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References E. Chang, Decentralized algorithms in distributed systems, TR CSRG-103, Department of Computer Science, University of Toronto, October 1979.

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References I. Cidon, Yet another distributed depth first search algorithm, 1993

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References N. Lynch, Distributed Computing, 1992

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References N. Santoro, Sense of direction, topological awareness and communication complexity, SIGACT News, 16, 2, Summer 1984, pp

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References A. Segall, Distributed network protocols, IEEE Trans. on Information Theory, 29 (1), January 1983, pp

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References A. S. Tanenbaum, Computer Networks, 1988

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References G. Tel, Introduction to Distributed Computing, 1992

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21 st PODC, June 2002, Monterey, California Yehuda Lindell, Anna Lysyanskaya, Tal Rabin Sequential Composition of Protocols without Simultaneous Termination Tomasz Jurdzinski, Miroslaw Kutylowski, Jan Zatopianski Efficient algorithms for leader election in radio networks Subhendu Chattopadhyay, Lisa Higham, Karen Seyffarth Dynamic and Self-Stabilizing Distributed Matching

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Matthias Fitzi, Daniel Gottesman, Martin Hirt, Thomas Holenstein, and Adam Smith Byzantine Agreement Secure Against Faulty Majorities From Scratch Kazuo Iwama and Masaki Okita Compact Routing for Average-Case Networks

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22 nd PODC, June 2003, Boston, Massachusetts Amitanand Aiyer, Sanketh Indarapu, Srinathan Kannan, Vinod Vaikuntanathan, C. Pandu Rangan Distributed Consensus in the presence of Sectional Faults Marta Arias, Lenore Cowen, A. Kofi Laing, Compact Roundtrip Routing with Topology-Independent Node Names Matthias Fitzi, Juan Garay Efficient Player-Optimal Protocols for Strong and Differential Consensus

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16 th DISC, October 2002, Toulouse, France Roy Friedman, Achour Mostéfaoui, Sergio Rajsbaum and Michel Raynal Distributed Agreement and its Relation with Error- Correcting Codes Kleoni Ioannidou Transformations of Self-Stabilizing Algorithms Gideon Stupp Stateless Termination Detection Yon Dourisboure and Cyril Gavoille Improved Compact Routing Scheme for Chordal Graphs

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17 th DISC, October 2003, Sorrento, Italy Marcos Aguilera, Burkhard Englert, Eli Gafni Uniform Solvability with a Finite Number of MWMR Registers Partha Dutta, Rachid Guerraoui, Bastian Pochon Tight Lower Bounds on Early Local Decisions in Uniform Consensus Kazuo Iwama, Masaki Okita Compact Routing for Flat Networks Victor Luchangco, Mark Moir, Nir Shavit On the Uncontended Complexity of Consensus

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6 th Sirocco, June 1999, Lacanau- Ocean, France 7 th Sirocco,June 2000, L’Aquila, Italy Paolo Boldi and Sebastiano Vigna, Lower bounds for weak sense of direction A Bui, A.K. Datta, F. Petit and V. Villain, Snap-stabilizing PIF algorithm in trees S. Dobrev, Leader election using any sense of direction

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