1. Improved Randomized Algorithms for Path Problems in Graphs
Surender Baswana
Postdoctoral Researcher
Max Planck Institute for Computer Science

2. Path Problems in Graph
Given a graph, report a path with certain characteristics

3. Path Problems in Graph
Given a graph, report a path with certain characteristics
Transitive closure

4. Path Problems in Graph
Given a graph, report a path with certain characteristics
Transitive closure
All-pairs shortest paths

5. Query paradigm of a path problem
Given a graph G=(V,E)

6. Query paradigm of a path problem
Given a graph G=(V,E)
A sequence of path queries

7. Query paradigm of a path problem
Given a graph G=(V,E)
A sequence of path queries
Answer each path query online

8. Path Problems in static settings
Preprocess a given graph G=(V,E) to form a data-structure that can answer a path query in optimal time.

9. Path Problems in dynamic settings
A graph G=(V,E) ,

10. Path Problems in dynamic settings
A graph G=(V,E) ,
q, q, u , u , q , u , u , u , u , q , q , u , …

11. Path Problems in dynamic settings
AIM : Maintain a data-structure

12. Path Problems in dynamic settings
AIM : Maintain a data-structure
Answer each query in optimal time

13. Path Problems in dynamic settings
AIM : Maintain a data-structure
Answer each query in optimal time
Requires less update time

14. Organization of the talk
Static algorithms
All-pairs approximate shortest paths
Graph Spanners
Dynamic algorithms
Transitive closure
All-pairs shortest paths
Open Problems

15. Organization of the talk
Static algorithms
All-pairs approximate shortest paths
Graph Spanners
Dynamic algorithms
Transitive closure
All-pairs shortest paths
Open Problems

16. All-pairs approximate shortest paths

17. All-Pairs Shortest Paths
Given a graph G=(V,E), preprocess it to compute shortest-path for every pair of vertices u,v ЄV.
|V|=n, |E|=m
δ(u,v) : distance from u to v
One of the most fundamental algorithmic graph problem

18. Existing algorithms for APSP
Floyd and Warshal
O(n3)
Pettie [ICALP 2002, TCS 2004]
O(mn+n2 log log n)
Zwick [ISAAC 2004]
O(n3 (log log n)1/2/log n)

19. Sub-cubic algorithms for APSP
G=(V,E)
Distance matrix
Fast
Matrix Multiplication Subroutine

20. Are there simple and efficient algorithms for APSP with
sub-cubic preprocessing time ?
sub-quadratic space data-structure ?

21. At the cost of approximation
Are there simple and efficient algorithms for APSP with
sub-cubic preprocessing time ?
sub-quadratic space data-structure ?
At the cost of approximation

22. All-pairs approximate shortest Paths
Some error ε in the distance δ*
Additive error ε = k
δ(u,v) ≤ δ*(u,v) ≤ δ(u,v) + k
Multiplicative error ε = t
δ(u,v) ≤ δ*(u,v) ≤ t δ(u,v)

23. Existing algorithms for APASP in undirected graph
Algorithms with additive error
Aingworth et al. [SODA 1996, SICOMP 1999]
Dor et al. [FOCS 1996, SICOMP 1999]
Algorithms with multiplicative error
Cohen [SICOMP 1998]
Cohen & Zwick [SODA 1997, J. Algo. 2001]
Thorup & Zwick [STOC 2001]
Algorithms with additive and multiplicative error
Elkin [PODC 2001]

24. Approximate Distance Oracle [Thorup & Zwick, STOC 2001]
Given a positive integer k
Preprocessing time O(kmn1/k)
Data-structure of size O(kn1+1/k)
Answer any approximate distance query in O(k) time
δ(u,v) ≤ δ*(u,v) ≤ (2k-1) δ(u,v)

25. Approximate Distance Oracle [Thorup & Zwick, STOC 2001]
Given a positive integer k
Preprocessing time O(kmn1/k)
Data-structure of size O(kn1+1/k)
Answer any approximate distance query in O(k) time
δ(u,v) ≤ δ*(u,v) ≤ (2k-1) δ(u,v)
(2k-1)-approx. distance oracle

26. Approximate Distance Oracle [Thorup & Zwick, STOC 2001]
Given a positive integer k
Preprocessing time O(kmn1/k)
Data-structure of size O(kn1+1/k) : Optimal
Answer any approximate distance query in O(k) time
δ(u,v) ≤ δ*(u,v) ≤ (2k-1) δ(u,v)
(2k-1)-approx. distance oracle

27. Approximate distance oracles for un-weighted graphs
Stretch
Size
Preprocessing time
B, Sen
SODA 2004
2k-1
O(kn1+1/k)
min(mn1/k, n2 log n)

28. All-pairs t-approximate shortest paths, for t<3

29. All-pairs 2-approx. shortest path
A : algorithm for (2-ε)-approximate shortest paths
B,C : two nΧn boolean matrices

30. All-pairs 2-approx. shortest path
A : algorithm for (2-ε)-approximate shortest paths
B,C : two nΧn boolean matrices
BΧC
GB,C
B,C
Linear
time A

31. All-pairs 2-approx. shortest path
A : algorithm for (2-ε)-approximate shortest paths
B,C : two nΧn boolean matrices
BΧC
GB,C
B,C
Linear
time A
All-pairs (2- ε)-approx. paths is as difficult as Boolean matrix multiplication

32. All-pairs 2-approx. shortest path [Cohen & Zwick, SICOMP 2000]
G=(V,E) : Undirected weighted graph
Preprocessing time : O(m1/2n3/2)
Space : O(n2)

33. All-pairs 2-approx. shortest paths
2.5
Upper bound ω
2.376
2.0
Lower bound

34. All-pairs 2-approx. shortest paths
2.5
Upper bound ω
2.376
Algorithm 1 Õ(m2/3n + n2)
δ(u,v) ≤ δ*(u,v) ≤ 2δ(u,v) + 1
2.0
Lower bound

35. All-pairs 2-approx. shortest paths
2.5
Upper bound ω
2.376
Algorithm 1 Õ(m2/3n + n2)
δ(u,v) ≤ δ*(u,v) ≤ 2δ(u,v) + 1
2.0
Algorithm 2 Õ(n2)
δ(u,v) ≤ δ*(u,v) ≤ 2δ(u,v) + 3
Lower bound

36. Graph Spanners

37. Graph Spanners Definition :
Given a graph G=(V,E), a sub-graph G=(V,Es) that is
sparse and yet preserves approximate distances pair-wise.

38. Graph Spanners Definition :
Given a graph G=(V,E), a sub-graph G=(V,Es) that is
sparse and yet preserves approximate distances pair-wise.
Multiplicative t-spanner :
δ(u,v) ≤ δs(u,v) ≤ t δ(u,v)

39. δ(u,v) ≤ δs(u,v) ≤ δ(u,v) + t
Graph Spanners
Definition :
Given a graph G=(V,E), a sub-graph G=(V,Es) that is
sparse and yet preserves approximate distances pair-wise.
Multiplicative t-spanner :
δ(u,v) ≤ δs(u,v) ≤ t δ(u,v)
Additive t-spanner :
δ(u,v) ≤ δs(u,v) ≤ δ(u,v) + t

40. Applications of Graph Spanner
Distributed Computing
Design of Synchronizers
Compact routing tables
Computational Biology
Reconstruction of Phylogenetic trees
All-pairs Approximate Shortest Paths

41. Multiplicative Spanners

42. Multiplicative Spanner size versus approximation

43. Multiplicative Spanner size versus approximation

44. Multiplicative Spanner size versus approximation
girth = t
stretch ≥ t-1

45. Multiplicative Spanner size versus approximation
girth = t
stretch ≥ t-1
Erdös[1960], Bondy & Simonovits [1974], Bollobas [1978] :
For all k≥1, there are graphs with Ω(n1+1/k) edges and girth >2k

46. Multiplicative Spanner size versus approximation
For k ≥1, a (2k-1)-spanner may have Ω(n1+1/k) edges

47. Multiplicative Spanner : Results
Stretch
ConstructionTime
Size
Althofer, Das, Dobkin,Joseph, Soares
DCG,1993
2k-1
O(mn1+1/k)
O(n1+1/k)
Thorup, Zwick
STOC,2001
O(mn1/k)
O(kn1+1/k)

48. Multiplicative Spanner : Results
Stretch
ConstructionTime
Size
Althofer, Das, Dobkin,Joseph, Soares
DCG,1993
2k-1
O(mn1+1/k)
O(n1+1/k)
Thorup, Zwick
STOC,2001
O(mn1/k)
O(kn1+1/k)
Can we Compute (2k-1)-spanners in linear time ?

49. Multiplicative Spanner : Results
Stretch
ConstructionTime
Size
Althofer, Das, Dobkin,Joseph, Soares
DCG,1993
2k-1
O(mn1+1/k)
O(n1+1/k)
Thorup, Zwick
STOC,2001
O(mn1/k)
O(kn1+1/k)
B, Sen
ICALP,2003
O(m)

50. Local approach u v
Edge in Spanner
Edge not in Spanner

51. Local approach 2 ≤w ≤w 1
i-1 ≤w ≤w w u v
Pi : For each edge not in Es , there is a path connecting its endpoints
with at-most i edges
none heavier than the edge

52. New Algorithms for (2k-1)-spanner
External-memory :
Integer sorting
Parallel algorithm
O(t) for t > log*n
Distributed Algorithm
O(1) rounds, linear communication complexity

53. Algorithm for 3-spanner

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

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

56. Algorithm for 3-spanner Difficult case : much more than n½ edges
Which n½ edges to select ?

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

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

59. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center

60. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows

61. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows
If v is not adjacent to any sampled vertex

62. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows
If v is not adjacent to any sampled vertex. v

63. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows
If v is not adjacent to any sampled vertex. v

64. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows
If v is adjacent to some sampled vertex

65. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows
If v is adjacent to some sampled vertex. v x
weights

66. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows
If v is adjacent to some sampled vertex. v x
weights

67. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows
If v is adjacent to some sampled vertex. v x
weights

68. Algorithm for 3-spanner Phase 1 : Clustering
Select each vertex r.i. with probability p to be a cluster center
Process each v Є V \S as follows
If v is adjacent to some sampled vertex. v x
weights

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

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

71. Algorithm for 3-spanner Phase 1 : Clustering
G=(V,E) G=(V1,E1)
Every v Є V1 is clustered
For each w-v Є E1
Phase 1 w v o

72. Algorithm for 3-spanner Difficult case : much more than n½ edges
Which n½ edges to select ?

73. Algorithm for 3-spanner Difficult case : much more than n½ edgesv

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

75. Analysis of the algorithm
Correctness ??
Size of the spanner
Running time

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

77. Analysis of the algorithm
Correctness ??
Size of the spanner
n/p Edges added during Phase 2
Running time

78. Analysis of the algorithm
Correctness ??
Size of the spanner
n/p n2p
Running time

79. Analysis of the algorithm
Correctness ??
Size of the spanner
n3/2
Running time

80. Analysis of the algorithm
Correctness ??
Size of the spanner
n3/2
Running time
Modified Radix sort on the label of the edges

81. Spanner has stretch 3 x y

82. Both x and y are clustered
Spanner has stretch 3 x y
Both x and y are clustered

83. Spanner has stretch 3 x y x y

84. Spanner has stretch 3 x y x y

85. Spanner has stretch 3 x y o x y x y

86. Spanner has stretch 3 x y z α o x ß y x y

87. Spanner has stretch 3 x y z α o x ß y x y

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

89. Reviewing 3-spanner algorithm [To build 5-spanner]

90. Reviewing 3-spanner algorithm [To build 5-spanner]
#Vertices
#Clusters V0 n

91. Reviewing 3-spanner algorithm [To build 5-spanner]
#Vertices
#Clusters
n1/2 V1 V0 n

92. Reviewing 3-spanner algorithm [To build 5-spanner]
#Vertices
#Clusters
Must increase the sampling probability
n1/2 V1 V0 n

93. Reviewing 3-spanner algorithm [To build 5-spanner]
#Vertices
#Clusters
n2/3 V1 V0 n

94. Computing 5-Spanner
#Vertices
#Clusters V2
n1/3
n2/3 V1 V0 n

95. Computing 5-Spanner
#Vertices
#Clusters V2
n1/3
n2/3 V1 V0 n

96. Algorithm for (2k-1)-spanner
Vk-1
n1/k V1
n1-1/k V0 n

97. Additive Spanners (un-weighted graphs)

98. Algorithms for Additive Spanners
Aingworth et al. [SODA 1996],
Elkin & Peleg [STOC 2001]
Additive 2-spanner of size O(n3/2)

99. Algorithms for Additive Spanners
Aingworth et al. [SODA 1996],
Elkin & Peleg [STOC 2001]
Additive 2-spanner of size O(n3/2)
B, Kavitha, Mehlhorn, Pettie [SODA 2005]
Additive 6-spanner of size O(n4/3)

100. (α,β)-Spanner (un-weighted graphs)

101. Algorithms for (α,β)-spanner
δ(u,v) ≤ δs(u,v) ≤ α δ(u,v) + β
Elkin & Peleg [STOC 2001]
(1+ε, β)-spanner of size O(n1+γ)

102. Algorithms for (α,β)-spanner
δ(u,v) ≤ δs(u,v) ≤ α δ(u,v) + β
Elkin & Peleg [STOC 2001]
(1+ε, β)-spanner of size O(n1+γ)
β depends inverse exponentially on γ
super polynomially on ε-1

103. Algorithms for (α,β)-spanner
δ(u,v) ≤ δs(u,v) ≤ α δ(u,v) + β
Elkin & Peleg [STOC 2001]
(1+ε, β)-spanner of size O(n1+γ)
B, Kavitha, Mehlhorn, Pettie [SODA 2005]
Linear time algorithm for (k,k-1)-spanner of size O(n1+1/k)

104. Dynamic Algorithms

105. Dynamic Algorithms Transitive Closure
Query time
Update time per edge-del.
Leeuwen & Poutre
TCS 1991
O(1)
O(m)
Demetrescu & Italiano
FOCS 2000
O(n3/m)

106. Dynamic Algorithms Transitive Closure
Query time
Update time per edge-del.
Leeuwen & Poutre
TCS 1991
O(1)
O(m)
Demetrescu & Italiano
FOCS 2000
O(n3/m)
B. & Sen
STOC 2002
O(n2/√m)

107. Dynamic Algorithms Transitive Closure
Query time
Update time per edge-del.
Leeuwen & Poutre
TCS 1991
O(1)
O(m)
Demetrescu & Italiano
FOCS 2000
O(n3/m)
B. & Sen
STOC 2002
O(n2/√m)
Roditty & Zwick
FOCS 2002
O(n)

108. Dynamic Algorithms All-pairs approximate shortest paths
Stretch
Query time
Update time per edge-del.
B. & Sen
SODA 2003 3
O(1)
O(n3/2)

109. Dynamic Algorithms All-pairs approximate shortest paths
Stretch
Query time
Update time per edge-del.
B. & Sen
SODA 2003 3
O(1)
O(n3/2)
Roditty & Zwick
FOCS 2004
1+ε
O(n)

110. Open Problems

111. Open Problems Graph spanners
Additive spanners of size o(n4/3) ?

112. Open Problems Graph spanners
Additive spanners of size o(n4/3) ?
O(n1+1/k) size (2k-1)-spanner for weighted graph inO(m) time?

113. Open Problems All-pairs approximate shortest paths
O(n2+ε) lower bound
O(n2 polylog n) time algorithm without additive error

114. Open Problems All-pairs approximate shortest paths
O(n2+ε) lower bound
O(n2 polylog n) time algorithm without additive error
Approximate Distance Oracles for weighted graphs in (n2) time

115. Open Problems Dynamic algorithms
Fully dynamic all-pairs shortest paths
Update time
Query time
Demetrescu, Italiano
STOC 2003
Õ(n2)
O(1)

116. Open Problems Dynamic algorithms
Fully dynamic all-pairs shortest paths
Update time
Query time
Demetrescu, Italiano
STOC 2003
Õ(n2)
O(1)
Sub-quadratic update time & O(1) query time at the cost of approximation

117. Open Problems Dynamic algorithms
Fully Dynamic algorithm for (2k-1)-spanner ?

118. Thank you