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Graph spanners : static, dynamic and fault tolerant

Graph Spanners. Definition : Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties . Graph Spanners. Definition : Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial propertiessparse

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Graph spanners : static, dynamic and fault tolerant

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    1. Graph spanners : static, dynamic and fault tolerant Surender Baswana Department of CSE IIT Kanpur

    2. Graph Spanners Definition : Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties

    3. Graph Spanners Definition : Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties sparse

    4. Graph Spanners Definition : Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties sparse preserves approximate distances pair-wise.

    5. Graph Spanners Definition : Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties sparse preserves approximate distances pair-wise. d(u,v) = ds(u,v) = t d(u,v) for some constant t = 1

    6. Graph Spanners Definition : Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties sparse preserves approximate distances pair-wise. d(u,v) = ds(u,v) = t d(u,v) for some constant t = 1

    7. Communication network : Motivation for spanners

    8. Communication network : Motivation for spanners

    9. Communication network : Motivation for spanners Minimizing the total cost : sparseness is desirable

    10. Communication network : Motivation for spanners Minimizing the total cost : sparseness is desirable

    11. Communication network : Motivation for spanners Minimizing the pair-wise distances : small stretch is desirable

    12. Communication network : Motivation for spanners Minimizing the pair-wise distances : small stretch is desirable

    13. Graph spanners A trade off between sparseness and stretch

    14. Graph spanners A trade off between sparseness and stretch Sparse d(u,v) = ds(u,v) = t d(u,v)

    15. Graph spanners A trade off between sparseness and stretch Sparse d(u,v) = ds(u,v) = t d(u,v) t-Spanner

    16. Aim : The sparsest spanner of a weighted graph with stretch t.

    17. Organization of the talk Optimal size of a t-spanner Spanners : Static Spanner Dynamic spanners Fault Tolerant Spanners A simple and linear time algorithm for static spanner (how problem itself guides to its solution) A simple algorithm for fault-tolerant spanner

    18. Optimal size of a t-spanner

    19. Optimal size of a t-spanner

    20. Optimal size of a t-spanner

    21. Optimal size of a t-spanner

    22. Optimal size of a t-spanner

    23. Optimal size of a t-spanner

    24. Optimal size of a t-spanner

    25. Optimal size of a t-spanner Let k be any positive integer There are graphs whose (2k-1)-spanner (a 2k-spanner) must have O(n1+1/k) edges

    26. Optimal size of a t-spanner Let k be any positive integer There are graphs whose (2k-1)-spanner (a 2k-spanner) must have O(n1+1/k) edges

    27. Static spanners

    28. Aim of an Algorithmist To design an algorithm A such that

    29. Dynamic spanners

    30. A dynamic spanner an initial graph G=(V,E) followed by an online sequence of updates: i,i,d,i,d,i,i,d,d,i,i,d, ....

    31. A dynamic spanner an initial graph G=(V,E) followed by an online sequence of updates: i,i,d,i,d,i,i,d,d,i,i,d, .... Aim : To maintain a t-spanner in online manner efficiently

    32. Dynamic algorithms for graph spanners

    33. Fault Tolerant spanners

    34. Why do we need a fault tolerant spanner ? For networks where Failures occur rarely There is simultaneous repair going on At any time, there are at most f edges which may be dead.

    35. What is a fault tolerant spanner ? A subgraph Es is called f-fault tolerant spanner if for any set F of f edges from E, Es -F is a (2k-1)-spanner of E-F

    36. Results on size of fault tolerant spanners

    37. A simple and linear time algorithm for static spanner

    38. Earlier algorithms for graph spanners

    39. Can we compute a (2k-1)-spanner in O(m) time ?

    40. Local approach Let G=(V,ES) be a spanner of G=(V,E)

    41. Local approach Let G=(V,ES) be a spanner of G=(V,E)

    42. Local approach Let G=(V,ES) be a spanner of G=(V,E)

    43. Local approach Let G=(V,ES) be a spanner of G=(V,E)

    44. Local approach Let G=(V,ES) be a spanner of G=(V,E)

    45. External memory algorithms for (2k-1)-spanner Time complexity : Integer sorting

    46. Distributed Algorithms for (2k-1)-spanner Number of Rounds : O(k) , Communication complexity : O(km) (linear)

    47. Streaming Algorithm for (2k-1)-spanner For Unweighted graphs Number of passes : 1 Processing time per edge : O(1)

    48. Algorithm for 3-spanner

    49. Aims : Given a graph G=(V,E) The number of edges = O(n3/2 ) Stretch = 3 Computation time = O(m)

    50. Algorithm for 3-spanner Easy case : fewer than n½ edges

    51. Algorithm for 3-spanner Difficult case : much more than n½ edges

    52. Algorithm for 3-spanner Difficult case : much more than n½ edges

    53. Algorithm for 3-spanner Phase 1 : Clustering Phase 2 : Adding edges between vertices and clusters

    54. Algorithm for 3-spanner Phase 1 : Clustering Phase 2 : Adding edges between vertices and clusters

    55. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p.

    56. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p. Process each v ? V \S as follows

    57. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p. Process each v ? V \S as follows If v is not adjacent to any sampled vertex

    58. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p . Process each v ? V \S as follows If v is not adjacent to any sampled vertex.

    59. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p . Process each v ? V \S as follows If v is not adjacent to any sampled vertex. add all its edges.

    60. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p. Process each v ? V \S as follows If v is not adjacent to any sampled vertex. add all its edges. If v is adjacent to some sampled vertex

    61. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p . Process each v ? V \S as follows If v is not adjacent to any sampled vertex. add all its edges. If v is adjacent to some sampled vertex.

    62. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p. Process each v ? V \S as follows If v is not adjacent to any sampled vertex. add all its edges. If v is adjacent to some sampled vertex.

    63. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p Process each v ? V \S as follows If v is not adjacent to any sampled vertex. add all its edges. If v is adjacent to some sampled vertex.

    64. Algorithm for 3-spanner Phase 1 : Clustering S : select each vertex independently with probability p Process each v ? V \S as follows If v is not adjacent to any sampled vertex. add all its edges. If v is adjacent to some sampled vertex.

    65. Algorithm for 3-spanner Phase 1 : Clustering G=(V,E) G=(V1,E1)

    66. Algorithm for 3-spanner Phase 1 : Clustering G=(V,E) G=(V1,E1) Every v ? V1 is clustered

    67. Algorithm for 3-spanner Phase 1 : Clustering G=(V,E) G=(V1,E1) Every v ? V1 is clustered Every red edge (w-v) ? E1 is ......

    68. Algorithm for 3-spanner Phase 1 : Clustering G=(V,E) G=(V1,E1) Every v ? V1 is clustered Every red edge (w-v) ? E1 is at-least as heavy as (v-o)

    69. Algorithm for 3-spanner Phase 1 : Clustering G=(V,E) G=(V1,E1) Every v ? V1 is clustered Every red edge (w-v) ? E1 is at-least as heavy as (v-o)

    70. Algorithm for 3-spanner Difficult case : much more than n½ edges

    71. Algorithm for 3-spanner Difficult case : much more than n½ edges

    72. Algorithm for 3-spanner Phase 2 : adding edges between vertices and clusters

    73. Analysis of the algorithm Size of the spanner Edges added during Phase 1 + Edges added during Phase 2 Correctness ??

    74. Analysis of the algorithm Size of the spanner Edges added during Phase 1 + Edges added during Phase 2 Correctness ??

    75. Analysis of the algorithm Size of the spanner Edges added during Phase 1 + Edges added during Phase 2 n/p Correctness ??

    76. Analysis of the algorithm Size of the spanner Edges added during Phase 1 + Edges added during Phase 2 n/p + n2p Correctness ??

    77. Analysis of the algorithm Size of the spanner Edges added during Phase 1 + Edges added during Phase 2 n/p + n2p = n3/2 , for p = 1/vn Correctness ??

    78. Spanner has stretch 3 Property P3 holds

    79. Spanner has stretch 3 Property P3 holds

    80. Spanner has stretch 3 Property P3 holds

    81. Spanner has stretch 3 Property P3 holds

    82. Spanner has stretch 3 Property P3 holds

    83. Spanner has stretch 3 Property P3 holds

    84. Spanner has stretch 3 Property P3 holds

    85. Algorithm for (2k-1)-spanner

    86. Algorithm for (2k-1)-spanner

    87. Algorithm for (2k-1)-spanner Invariant : At level i, we have graph G=(Vi,Ei) Every vertex in Vi is clustered For every edge e ? Ei

    88. Algorithm for fault tolerant spanner

    89. Algorithm for f-edge fault tolerant spanners Given graph (V,E); Es = F; For i=1 to f+1 do { Ei = spanner(E-Es ,2k-1); Es = Es U Ei; } return Es ;

    90. Algorithm for f-edge fault tolerant spanners Given graph G=(V,E); Es = F; For i=1 to f+1 do { Ei = spanner(E-Es ,2k-1); Es = Es U Ei; } return Es ; Observation : Ei and Ej are edge disjoint for i<>j.

    91. Algorithm for f-edge fault tolerant spanners Let F be any set of f edges. Let (u,v) ? E-F ; Aim : There has to be a path of length at most 2k-1 between u and v in E-Es

    92. Algorithm for f-edge fault tolerant spanners Let F be any set of i edges. Let (u,v) ? E-F ; Aim : There is a path of length 2k-1 between u and v in E-Es (follows from observation)

    93. Summary Simple and efficient algorithms exist for graph spanners Static spanners Dynamic spanners Fault tolerant spanners Open problems (exist for additive spanners)

    94. Thank you

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