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Intro Searching Model Results Conclusion Distributed Chasing of Network Intruders elia Blin 1 Pierre Fraigniaud 2 Nicolas Nisse 3 L Sandrine Vial 1 1 Universit e dEvry, France 2 CNRS, Universit e Paris-Sud, France 3 Universit e

  1. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  2. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  3. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  4. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  5. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  6. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  7. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  8. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  9. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  10. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  11. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  12. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  13. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  14. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  15. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  16. Intro Searching Model Results Conclusion Definitions Connected Cost of Connectedness : Example in a tree mcs(T) ≥ 4 > s(T) 10/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  17. Intro Searching Model Results Conclusion Definitions Connected Cost of connectedness : Related Work In terms of number of searchers For any tree T , s ( T ) ≤ mcs ( T ) ≤ 2 s ( T ) − 2 (tight). Barri` ere, Fraigniaud, Santoro and Thilikos. [WG, 2003] For any graph G , s ( G ) ≤ mcs ( G ) ≤ (1 + log n ) s ( G ) Fraigniaud and Nisse [LATIN, 2006] Complexity of monotone connected Graph Searching NP-complete in general case. Gustedt. [DAM, 1993] Linear in case of trees. Barri` ere, Flocchini, Fraigniaud and Santoro. [SPAA, 2002] 11/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  18. Intro Searching Model Results Conclusion Search Network Agents Plan Introduction 1 Graph Searching 2 Our model 3 Monotone Connected Search Asynchronous Anonymous Network Mobile Agents Our Results 4 Conclusion 5 12/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  19. Intro Searching Model Results Conclusion Search Network Agents Our Goal For any connected, asynchronous, and anonymous network G , and any v 0 ∈ V ( G ), we propose a distributed algorithm that enables clearing G in a connected way, using searchers starting from v 0 , and initially unaware of G . 13/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  20. Intro Searching Model Results Conclusion Search Network Agents Monotone connected graph searching Alternative definition v 0 ∈ V ( G ) is the homebase of the searchers. Initially, any searcher is placed at v 0 . One single operation is allowed : move a searcher along an edge if it does not implie recontamination. Remarks The homebase remains clear during the whole strategy. mcs ( G , v 0 ) 14/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  21. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  22. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  23. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  24. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  25. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  26. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  27. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  28. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  29. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  30. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u mcs(G,u)=2 v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  31. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  32. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  33. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  34. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  35. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  36. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  37. Intro Searching Model Results Conclusion Search Network Agents Importance of the homebase u mcs(G,v)=3 v 15/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  38. Intro Searching Model Results Conclusion Search Network Agents Asynchronous Network v v 16/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  39. Intro Searching Model Results Conclusion Search Network Agents Asynchronous Network v v 16/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  40. Intro Searching Model Results Conclusion Search Network Agents Asynchronous Network v v 16/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  41. Intro Searching Model Results Conclusion Search Network Agents Asynchronous Network v v 16/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  42. Intro Searching Model Results Conclusion Search Network Agents Asynchronous Network v v 16/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  43. Intro Searching Model Results Conclusion Search Network Agents Asynchronous Network The searchers cannot distinguish one graph from the other. The two red searchers have the same local behaviour. An extra searcher will be called in both cases. v v 16/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  44. Intro Searching Model Results Conclusion Search Network Agents Cost of asynchronicity More generally There exist graphs G such that, any distributed asynchronous graph searching protocol requires mcs ( G ) + 1 searchers to clear G in a connected monotone way. [FHL05] Coordinator The extra searcher, the coordinator is used to synchronize the other searchers. 17/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  45. Intro Searching Model Results Conclusion Search Network Agents Anonymous Network Unknown unknown topology unknown size (no upper bound) Anonymous No vertex labeling Local edge labeling Local Memory whiteboards are specific zone of local node memory, accessible in fair mutual exclusion. 18/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  46. Intro Searching Model Results Conclusion Search Network Agents Example of an anonymous graph 2 1 3 3 4 1 2 4 3 2 1 19/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  47. Intro Searching Model Results Conclusion Search Network Agents Mobile Agents Searchers autonomous mobile computing entities with distincs IDs, running the same algorithm, Mealy automata. 20/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  48. Intro Searching Model Results Conclusion Search Network Agents Mobile Agents The decision of a searcher... leaving a node via some specific port, switching state, writing on the whiteboard, ... is local and depends on : current state, content of the node’s whiteboard, incoming port number. 20/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  49. Intro Searching Model Results Conclusion Search Network Agents Related Work Polynomial algorithms in specific topologies Trees. Barri` ere, Flocchini, Fraigniaud and Santoro. [SPAA, 2002] Hypercubes. Flocchini, Huang and Luccio. [IPDPS, 2005] Chordal rings, Tori, Meshes... For each of these classes of graphs : 1 mcs ( G ) + 1 searchers are used ; 2 each searcher possesses O (log n ) bits of memory ; 3 the size of the node’s whiteboard is O (log n ) bits. 21/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  50. Intro Searching Model Results Conclusion Search Network Agents Related Work Polynomial algorithms in specific topologies Trees. Barri` ere, Flocchini, Fraigniaud and Santoro. [SPAA, 2002] Hypercubes. Flocchini, Huang and Luccio. [IPDPS, 2005] Chordal rings, Tori, Meshes... For each of these classes of graphs : 1 mcs ( G ) + 1 searchers are used ; 2 each searcher possesses O (log n ) bits of memory ; 3 the size of the node’s whiteboard is O (log n ) bits. 21/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  51. Intro Searching Model Results Conclusion Theorem Algo Plan Introduction 1 Graph Searching 2 Our model 3 Our Results 4 Theorem Our Algorithm Conclusion 5 22/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  52. Intro Searching Model Results Conclusion Theorem Algo Theorem For any connected, asynchronous, and anonymous network G , and any u 0 ∈ V ( G ), we propose an distributed algorithm that enables clearing G using searchers starting from the homebase u 0 , and initially unaware of G . 1 It uses at most k = mcs ( G , u 0 ) + 1 searchers if mcs ( G , u 0 ) > 1, and k = 1 searcher if mcs ( G , u 0 ) = 1 ; 2 Every searcher involved in the search strategy computed uses O (log k ) bits of memory ; 3 During the execution, at most O ( m log n ) bits of information are stored at every whiteboard. 23/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  53. Intro Searching Model Results Conclusion Theorem Algo Theorem For any connected, asynchronous, and anonymous network G , and any u 0 ∈ V ( G ), we propose an distributed algorithm that enables clearing G using searchers starting from the homebase u 0 , and initially unaware of G . 1 It uses at most k = mcs ( G , u 0 ) + 1 searchers if mcs ( G , u 0 ) > 1, and k = 1 searcher if mcs ( G , u 0 ) = 1 ; 2 Every searcher involved in the search strategy computed uses O (log k ) bits of memory ; 3 During the execution, at most O ( m log n ) bits of information are stored at every whiteboard. 23/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  54. Intro Searching Model Results Conclusion Theorem Algo Properties the strategy is computed online by the searchers themselves. after the execution of our algorithm, an “optimal” monotone connected search strategy for G is written on the vertices’whiteboards in a distributed way. in the class of graphs with bounded mcs , searchers can be implemented by finite automata. 24/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  55. Intro Searching Model Results Conclusion Theorem Algo Principles of our algorithm The Algorithm Initially, one searcher stands at v 0 , k = 1 While the graph is not clear : Try all monotone connected search strategies using k searchers ; If the graph is not clear, call a new searcher ; Predicate At the end of each loop, the k searchers are standing at v 0 . 25/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  56. Intro Searching Model Results Conclusion Theorem Algo Basic Idea to order the possible strategies using k searchers ; to try all the strategies in the increasing order ; either a strategy clears the graph → OK ; or after trying all the strategies, the graph remains contaminated → another searcher is required. 26/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  57. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  58. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move through any port. 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  59. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move through any port. 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  60. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move to help another searcher . 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  61. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move to help another searcher and then to clear an edge . 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  62. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy (1) A searcher at a clear (or guarded) vertex can move to help another searcher and then to clear an edge . 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  63. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy (2) A searcher at a vertex with only one contaminated port can move to clear an edge . 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  64. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy (2) A searcher at a vertex with only one contaminated port can move to clear an edge . 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  65. Intro Searching Model Results Conclusion Theorem Algo Valid moves Two kind of moves are compatible with a monotone connected strategy Such a configuration is the result of a failing strategy No moves are possible. 27/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  66. Intro Searching Model Results Conclusion Theorem Algo Order on moves and strategies Representation of valid moves ( i , j , p ) denotes : ”searcher i joins searcher j and the smallest searcher follows the port p to clear an edge” ( i , i , p ) denotes : ”searcher i follows the port p to clear the corresponding edge” The moves are ordered in the lexicographic order. Order on the sequences of valid moves A sequence of valid moves corresponds to a partial monotone connected search strategy. The sequence are ordered in the lexicographic order. 28/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  67. Intro Searching Model Results Conclusion Theorem Algo The centralized Algorithm k ≥ 1 being given, the centralized algorithm tries every strategy using k searchers, starting from v 0 . Initially, all searchers are at v 0 ; Valid moves are performing one by one ; the smallest possible move is performed, if no valid move is possible, the last move performed is backtracked. A virtual stack contains the sequence of valid moves that have leaded to the current situation. 29/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  68. Intro Searching Model Results Conclusion Theorem Algo The centralized Algorithm : Example number of searcher(s) : k=1 list of searcher(s) : 1 1 Blue number of free searcher(s) : 0 3 1 | | 2 2 1 3 | | | | | | 2 2 | | v 0 1 1 | | 30/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

  69. Intro Searching Model Results Conclusion Theorem Algo The centralized Algorithm : Example number of searcher(s) : k=2 list of searcher(s) : 1 1 Blue < Green number of free searcher(s) : 0 3 1 | | 2 2 1 3 | | | | | | 2 2 | | v 0 1 1 | | 30/36 L´ elia Blin, Pierre Fraigniaud, Nicolas Nisse, Sandrine Vial Distributed Chasing of Network Intruders

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