1. Graph Search Algorithms
COSC 3101E, PROF. J. ELDER

2. Graph a b Node ~ city or computer Edge ~ road or data cable c
Undirected or Directed
A surprisingly large number of computational problems can be expressed as graph problems.
COSC 3101E, PROF. J. ELDER

3. Directed and Undirected Graphs
(c) The subgraph of the graph in part (a) induced by the vertex set {1,2,3,6}.
(b) An undirected graph G = (V,E), where V = {1,2,3,4,5,6} and
E = {(1,2), (1,5), (2,5), (3,6)}. The vertex 4 is isolated.
A directed graph G = (V, E), where V = {1,2,3,4,5,6} and
E = {(1,2), (2,2), (2,4), (2,5), (4,1), (4,5), (5,4), (6,3)}.
The edge (2,2) is a self-loop.
COSC 3101E, PROF. J. ELDER

4. Trees A tree is a connected, acyclic, undirected graph.
Forest
Graph with Cycle
A tree is a connected, acyclic, undirected graph.
A forest is a set of trees (not necessarily connected)
COSC 3101E, PROF. J. ELDER

5. Running Time of Graph Algorithms
Running time often a function of both |V| and |E|.
For convenience, drop the | . | in asymptotic notation, e.g. O(V+E).
COSC 3101E, PROF. J. ELDER

6. End of Lecture 10
Oct 16, 2007

7. Representations: Undirected Graphs
Adjacency List
Adjacency Matrix
COSC 3101E, PROF. J. ELDER

8. Representations: Directed Graphs
Adjacency List
Adjacency Matrix
COSC 3101E, PROF. J. ELDER

9. Breadth-First Search
Goal: To recover the shortest paths from a source node s to all other reachable nodes v in a graph.
The length of each path and the paths themselves are returned.
Notes:
There are an exponential number of possible paths
This problem is harder for general graphs than trees because of cycles! ? s
COSC 3101E, PROF. J. ELDER

10. Breadth-First Search Idea: send out search ‘wave’ from s.
Keep track of progress by colouring vertices:
Undiscovered vertices are coloured black
Just discovered vertices (on the wavefront) are coloured red.
Previously discovered vertices (behind wavefront) are coloured grey.
COSC 3101E, PROF. J. ELDER

11. Found Not Handled Queue
BFS
Found Not Handled Queue
First-In First-Out (FIFO) queue
stores ‘just discovered’ vertices s b a e d g c f j i h m k l
COSC 3101E, PROF. J. ELDER

12. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s b a s
d=0 e d g c f j i h m k l
COSC 3101E, PROF. J. ELDER

13. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=0 e a
d=1 d d g g c f b j i h m k l
COSC 3101E, PROF. J. ELDER

14. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a a
d=1 d e d g g b c f j i h m k l
COSC 3101E, PROF. J. ELDER

15. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=1 d e d g g b c f c
d=2 j f
d=2 i h m k l
COSC 3101E, PROF. J. ELDER

16. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=1 e d g g b c f c
d=2 j f
d=2 m i e h m k l
COSC 3101E, PROF. J. ELDER

17. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=1 e d g b c f c
d=2 j f
d=2 m i e h j m k l
COSC 3101E, PROF. J. ELDER

18. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=1 e d g c f c
d=2 j f
d=2 m i e h j m k l
COSC 3101E, PROF. J. ELDER

19. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a c
d=2 e d f m g e c f j j
d=2 i h m k l
COSC 3101E, PROF. J. ELDER

20. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=2 e d f m g e c f j j h
d=3
d=2 i i h m
d=3 k l
COSC 3101E, PROF. J. ELDER

21. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=2 e d m g e c f j j h
d=3
d=2 i i h m
d=3 k l
COSC 3101E, PROF. J. ELDER

22. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=2 e d g e c f j j h
d=3
d=2 i i h l m
d=3 k l
COSC 3101E, PROF. J. ELDER

23. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=2 e d g c f j j h
d=3
d=2 i i h l m
d=3 k l
COSC 3101E, PROF. J. ELDER

24. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=2 e d g c f j h
d=3
d=2 i i h l m
d=3 k l
COSC 3101E, PROF. J. ELDER

25. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a h
d=3 i e d l g c f j
d=2 i h m
d=3 k l
COSC 3101E, PROF. J. ELDER

26. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=3 i e d l g k
d=4 c f j
d=2 i h m
d=3 k
d=4 l
COSC 3101E, PROF. J. ELDER

27. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=3 e d l g k
d=4 c f j
d=2 i h m
d=3 k
d=4 l
COSC 3101E, PROF. J. ELDER

28. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=3 e d g k
d=4 c f j
d=2 i h m
d=3 k
d=4 l
COSC 3101E, PROF. J. ELDER

29. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a k
d=4 e d g c f j
d=2 i h m
d=3 k
d=4 l
COSC 3101E, PROF. J. ELDER

30. Found Not Handled Queue
BFS
Found Not Handled Queue
d=0 s
d=1 b a
d=4 e
d=5 d g c f j
d=2 i h m
d=3 k
d=4 l
COSC 3101E, PROF. J. ELDER

31. Breadth-First Search Algorithm
BLACK
RED
GRAY
Q is a FIFO queue.
Each vertex assigned finite d value at most once.
Q contains vertices with d values {i, …, i, i+1, …, i+1}
d values assigned are monotonically increasing over time.
COSC 3101E, PROF. J. ELDER

32. Breadth-First-Search is Greedy
Vertices are handled:
in order of their discovery (FIFO queue)
Smallest d values first
COSC 3101E, PROF. J. ELDER

33. Correctness d Basic Steps: s v & there is an edge from u to v u
The shortest path to u has length d
There is a path to v with length d+1.
COSC 3101E, PROF. J. ELDER

34. Correctness
Vertices are discovered in order of their distance from the source vertex s.
When we discover v, how do we know there is not a shorter path to v?
Because if there was, we would already have discovered it! s d v u
COSC 3101E, PROF. J. ELDER

35. Correctness
Two-step proof:
On exit:
COSC 3101E, PROF. J. ELDER

36. u v s
COSC 3101E, PROF. J. ELDER

37. BLACK s v u
RED
BLACK
RED
GRAY
COSC 3101E, PROF. J. ELDER

38. x s v u
Contradiction!
COSC 3101E, PROF. J. ELDER

39. Correctness
COSC 3101E, PROF. J. ELDER

40. Progress? On every iteration one vertex is processed (turns gray).
BLACK
RED
GRAY
COSC 3101E, PROF. J. ELDER

41. Running Time
BLACK
RED
BLACK
RED
GRAY
COSC 3101E, PROF. J. ELDER

42. Optimal Substructure Property
The shortest path problem has the optimal substructure property:
Every subpath of a shortest path is a shortest path.
The optimal substructure property
is a hallmark of both greedy and dynamic programming algorithms.
allows us to compute both shortest path distance and the shortest paths themselves by storing only one d value and one predecessor value per vertex. u v s
shortest path
COSC 3101E, PROF. J. ELDER

43. Recovering the Shortest Path
For each node v, store predecessor of v in (v). s v u
(v)
Predecessor of v is
(v) = u.
COSC 3101E, PROF. J. ELDER

44. Recovering the Shortest Path
COSC 3101E, PROF. J. ELDER

45. Colours are actually not required
COSC 3101E, PROF. J. ELDER

46. Depth First Search (DFS)
Idea:
Continue searching “deeper” into the graph, until we get stuck.
If all the edges leaving v have been explored we “backtrack” to the vertex from which v was discovered.
Does not recover shortest paths, but can be useful for extracting other properties of graph, e.g.,
Topological sorts
Detection of cycles
Extraction of strongly connected components
COSC 3101E, PROF. J. ELDER

47. Depth-First Search
Explore every edge, starting from different vertices if necessary.
As soon as vertex discovered, explore from it.
Keep track of progress by colouring vertices:
Black: undiscovered vertices
Red: discovered, but not finished (still exploring from it)
Gray: finished (found everything reachable from it).
COSC 3101E, PROF. J. ELDER

48. Found Not Handled Stack <node,# edges>
DFS
Note: Stack is Last-In First-Out (LIFO)
Found Not Handled Stack <node,# edges> d f s / b a e d g c f j i h m k l
COSC 3101E, PROF. J. ELDER

49. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s / 1/ b a e d g c f j i
s,0 h m k l
COSC 3101E, PROF. J. ELDER

50. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s / 1/ 2/ b a e d g c f j i
a,0
s,1 h m k l
COSC 3101E, PROF. J. ELDER

51. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s / 1/ 2/ 3/ b a e d g c f j
c,0 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

52. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s / 1/ 2/ 3/ 4/ b a e d g c f
h,0 j
c,1 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

53. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s / 1/ 2/ 3/ 5/ 4/ b a e d g c
k,0 f
h,1 j
c,1 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

54. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s / 1/ 2/ 3/
5/6 4/ b a e d g c f
h,1 j
c,1
Path on Stack i
a,1
s,1 h m
Tree Edge k l
COSC 3101E, PROF. J. ELDER

55. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s / 1/ 2/ 3/
5/6
4/7 b a e d g c f j
c,1 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

56. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
5/6
4/7 b a e d g c f
i,0 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

57. DFS Found Not Handled Stack <node,# edges> s b a e d g c f i,1 j
Cross Edge to handled node: d[h]<d[i] s 8/ 1/ / 2/ 3/
5/6
4/7 b a e d g c f
i,1 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

58. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
5/6
4/7 b a e d g c f
i,2 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

59. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/ 9/
5/6
4/7 b a e d g c
l,0 f
i,3 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

60. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/ 9/
5/6
4/7 b a e d g c
l,1 f
i,3 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

61. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
9/10
5/6
4/7 b a e d g c f
i,3 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

62. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
11/
9/10
5/6
4/7 b a e d g c
g,0 f
i,4 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

63. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
11/
12/
9/10
5/6
4/7 b a e d g
j,0 c
g,1 f
i,4 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

64. DFS Found Not Handled Stack <node,# edges> s b a e d g j,1 c g,1
Back Edge to node on Stack: s 8/ 1/ / 2/ 3/
11/
12/
9/10
5/6
4/7 b a e d g
j,1 c
g,1 f
i,4 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

65. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
11/
12/
13/
9/10
5/6
4/7 b a e d
m,0 g
j,2 c
g,1 f
i,4 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

66. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
11/
12/
13/
9/10
5/6
4/7 b a e d
m,1 g
j,2 c
g,1 f
i,4 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

67. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
11/
12/
13/14
9/10
5/6
4/7 b a e d g
j,2 c
g,1 f
i,4 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

68. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
11/
12/15
13/14
9/10
5/6
4/7 b a e d g c
g,1 f
i,4 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

69. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f
i,4 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

70. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
17/
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c
f,0 f
i,5 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

71. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
17/
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c
f,1 f
i,5 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

72. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s 8/ 1/ / 2/ 3/
17/18
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f
i,5 j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

73. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ / 2/ 3/
17/18
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f j
c,2 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

74. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges>
Forward Edge: d[f]>d[c] s
8/19 1/ / 2/ 3/
17/18
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f j
c,3 i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

75. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ / 2/
3/19
17/18
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f j i
a,1
s,1 h m k l
COSC 3101E, PROF. J. ELDER

76. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ / 2/
3/19
17/18
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f j i
a,2
s,1 h m k l
COSC 3101E, PROF. J. ELDER

77. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f j i
s,1 h m k l
COSC 3101E, PROF. J. ELDER

78. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
21/
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f j i
d,0
s,2 h m k l
COSC 3101E, PROF. J. ELDER

79. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
21/
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f j i
d,1
s,2 h m k l
COSC 3101E, PROF. J. ELDER

80. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
21/
11/16
12/15
13/14
9/10
5/6
4/7 b a e d g c f j i
d,2
s,2 h m k l
COSC 3101E, PROF. J. ELDER

81. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
21/
11/16
12/15
13/14
9/10
5/6
4/7
22/ b a e d g c f j
e,0 i
d,3
s,2 h m k l
COSC 3101E, PROF. J. ELDER

82. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
21/
11/16
12/15
13/14
9/10
5/6
4/7
22/ b a e d g c f j
e,1 i
d,3
s,2 h m k l
COSC 3101E, PROF. J. ELDER

83. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
21/
11/16
12/15
13/14
9/10
5/6
4/7
22/23 b a e d g c f j i
d,3
s,2 h m k l
COSC 3101E, PROF. J. ELDER

84. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
21/24
11/16
12/15
13/14
9/10
5/6
4/7
22/23 b a e d g c f j i
s,2 h m k l
COSC 3101E, PROF. J. ELDER

85. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/ /
2/20
3/19
17/18
21/24
11/16
12/15
13/14
9/10
5/6
4/7
22/23 b a e d g c f j i
s,3 h m k l
COSC 3101E, PROF. J. ELDER

86. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/
25/
2/20
3/19
17/18
21/24
11/16
12/15
13/14
9/10
5/6
4/7
22/23 b a e d g c f j i
b,0
s,4 h m k l
COSC 3101E, PROF. J. ELDER

87. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/
25/
2/20
3/19
17/18
21/24
11/16
12/15
13/14
9/10
5/6
4/7
22/23 b a e d g c f j i
b,1
s,4 h m k l
COSC 3101E, PROF. J. ELDER

88. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/
25/
2/20
3/19
17/18
21/24
11/16
12/15
13/14
9/10
5/6
4/7
22/23 b a e d g c f j i
b,2
s,4 h m k l
COSC 3101E, PROF. J. ELDER

89. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
8/19 1/
25/
2/20
3/19
17/18
21/24
11/16
12/15
13/14
9/10
5/6
4/7
22/23 b a e d g c f j i
b,3
s,4 h m k l
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90. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges> s
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s,4 h m k l
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91. End of Lecture 11
Oct 18, 2007

92. Found Not Handled Stack <node,# edges>
DFS
Found Not Handled Stack <node,# edges>
Tree Edges
Back Edges
Forward Edges s
Finished!
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Cross Edges b a e d g c f j i h m k l
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93. Classification of Edges in DFS
Tree edges are edges in the depth-first forest Gπ. Edge (u, v) is a tree edge if v was first discovered by exploring edge (u, v).
Back edges are those edges (u, v) connecting a vertex u to an ancestor v in a depth-first tree.
Forward edges are non-tree edges (u, v) connecting a vertex u to a descendant v in a depth-first tree.
Cross edges are all other edges. They can go between vertices in the same depth-first tree, as long as one vertex is not an ancestor of the other. s a c h k f i l m j e b g d
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94. Classification of Edges in DFS
Tree edges: Edge (u, v) is a tree edge if v was black when (u, v) traversed.
Back edges: (u, v) is a back edge if v was red when (u, v) traversed.
Forward edges: (u, v) is a forward edge if v was gray when (u, v) traversed and d[v] > d[u].
Cross edges (u,v) is a cross edge if v was gray when (u, v) traversed and d[v] < d[u].
Classifying edges can help to identify
properties of the graph, e.g., a graph is
acyclic iff DFS yields no back edges. s a c h k f i l m j e b g d
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95. Undirected Graphs
In a depth-first search of an undirected graph, every edge is either a tree edge or a back edge.
Why?
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96. Undirected Graphs
Suppose that (u,v) is a forward edge or a cross edge in a DFS of an undirected graph.
(u,v) is a forward edge or a cross edge when v is already handled (grey) when accessed from u.
This means that all vertices reachable from v have been explored.
Since we are currently handling u, u must be red.
Clearly v is reachable from u.
Since the graph is undirected, u must also be reachable from v.
Thus u must already have been handled: u must be grey.
Contradiction! u v
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97. Depth-First Search Algorithm
BLACK
RED
BLACK
GRAY
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98. Depth-First Search Algorithm
BLACK
RED
BLACK
GRAY
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99. End of Lecture 12
Oct 23, 2007

100. Topological Sorting (e.g., putting tasks in linear order)
An application of Depth-First Search

101. Linear Order underwear socks pants shoes underwear pants socks shoes
socks underwear pants shoes
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102. Linear Order underwear socks pants shoes Too many video games?
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103. Linear Order a b h c i d j e k g f l ….. l (V) (V2)
Precondition: A Directed Acyclic Graph (DAG)
Post Condition: Find one valid linear order b h c i d j e k
Algorithm:
Find a terminal node (sink).
Put it last in sequence.
Delete from graph & repeat g
We can do better!
(V) (V2) f l
….. l
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104. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f g f l e d
….. f
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105. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g l g l e f d
When node is popped off stack, insert at front of linearly-ordered “to do” list.
Linear Order:
….. f
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106. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g g f l e d
Linear Order:
l,f
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107. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l e d
Linear Order:
g,l,f
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108. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l d
Linear Order:
e,g,l,f
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109. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l
Linear Order:
d,e,g,l,f
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110. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g k j f l i
Linear Order:
d,e,g,l,f
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111. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g j f l i
Linear Order:
k,d,e,g,l,f
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112. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l i
Linear Order:
j,k,d,e,g,l,f
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113. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l
Linear Order:
i,j,k,d,e,g,l,f
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114. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g c f l b
Linear Order:
i,j,k,d,e,g,l,f
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115. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l b
Linear Order:
c,i,j,k,d,e,g,l,f
COSC 3101E, PROF. J. ELDER

116. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l
Linear Order:
b,c,i,j,k,d,e,g,l,f
COSC 3101E, PROF. J. ELDER

117. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g h f l a
Linear Order:
b,c,i,j,k,d,e,g,l,f
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118. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l a
Linear Order:
h,b,c,i,j,k,d,e,g,l,f
COSC 3101E, PROF. J. ELDER

119. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l
Linear Order:
a,h,b,c,i,j,k,d,e,g,l,f
Done!
COSC 3101E, PROF. J. ELDER

120. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Proof:
Case 1: u goes on stack first before v.
Because of edge,
v goes on before u comes off
v comes off before u comes off
v goes after u in order. 
Consider each edge v … u … u v u… v…
COSC 3101E, PROF. J. ELDER

121. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Proof:
Case 1: u goes on stack first before v.
Case 2: v goes on stack first before u.
v comes off before u goes on.
v goes after u in order. 
Consider each edge u … v … u v u… v…
COSC 3101E, PROF. J. ELDER

122. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Proof:
Case 1: u goes on stack first before v.
Case 2: v goes on stack first before u.
v comes off before u goes on.
Case 3: v goes on stack first before u.
u goes on before v comes off.
Panic: u goes after v in order. 
Cycle means linear order is impossible 
Consider each edge u …
The nodes in the stack form a path starting at s. v … u v u… v…
COSC 3101E, PROF. J. ELDER

123. Found Not Handled Stack
Linear Order
Found Not Handled Stack
Alg: DFS a b h c i d j e k g f l
Linear Order:
a,h,b,c,i,j,k,d,e,g,l,f
Done!
COSC 3101E, PROF. J. ELDER

124. Back to Shortest Path
BFS finds the shortest paths from a source node s to every vertex v in the graph.
Here, the length of a path is simply the number of edges on the path.
But what if edges have different ‘costs’? 7 1 5 v v 2 s s 2 3
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125. Single-Source (Weighted) Shortest Paths

126. The Problem
What is the shortest driving route from Toronto to Ottawa? (e.g. MAPQuest, Microsoft Streets and Trips)
Input:
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127. Example
Single-source shortest path search induces a search tree rooted at s.
This tree, and hence the paths themselves, are not necessarily unique.
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128. Shortest path variants
Single-source shortest-paths problem: – the shortest path from s to each vertex v. (e.g. BFS)
Single-destination shortest-paths problem: Find a shortest path to a given destination vertex t from each vertex v.
Single-pair shortest-path problem: Find a shortest path from u to v for given vertices u and v.
All-pairs shortest-paths problem: Find a shortest path from u to v for every pair of vertices u and v.
COSC 3101E, PROF. J. ELDER

129. Negative-weight edges
OK, as long as no negative-weight cycles are reachable from the source.
If we have a negative-weight cycle, we can just keep going around it, and get w(s, v) = −∞ for all v on the cycle.
But OK if the negative-weight cycle is not reachable from the source.
Some algorithms work only if there are no negative-weight edges in the graph.
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130. Optimal substructure
Lemma: Any subpath of a shortest path is a shortest path
Proof: Cut and paste.
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131. Cycles Shortest paths can’t contain cycles:
Already ruled out negative-weight cycles.
Positive-weight: we can get a shorter path by omitting the cycle.
Zero-weight: no reason to use them  assume that our solutions won’t use them.
COSC 3101E, PROF. J. ELDER

132. Output of a single-source shortest-path algorithm
For each vertex v in V:
d[v] = δ(s, v).
Initially, d[v]=∞.
Reduce as algorithm progresses.
But always maintain d[v] ≥ δ(s, v).
Call d[v] a shortest-path estimate.
π[v] = predecessor of v on a shortest path from s.
If no predecessor, π[v] = NIL.
π induces a tree — shortest-path tree.
COSC 3101E, PROF. J. ELDER

133. All shortest-paths algorithms start with the same initialization:
INIT-SINGLE-SOURCE(V, s)
for each v in V
do d[v]←∞
π[v] ← NIL
d[s] ← 0
COSC 3101E, PROF. J. ELDER

134. Relaxing an edge
Can we improve shortest-path estimate for v by going through u and taking (u,v)?
RELAX(u, v,w)
if d[v] > d[u] + w(u, v) then
d[v] ← d[u] + w(u, v)
π[v]← u
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135. General single-source shortest-path strategy
Start by calling INIT-SINGLE-SOURCE
Relax Edges
Algorithms differ in the order in which edges are taken and
how many times each edge is relaxed.
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136. Example: Single-source shortest paths in a directed acyclic graph (DAG)
Since graph is a DAG, we are guaranteed no negative-weight cycles.
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137. Algorithm
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138. Example
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139. Example
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140. Example
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141. Example
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142. Example
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143. Example
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144. Correctness: Path relaxation property (Lemma 24.15)
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145. Correctness of DAG Shortest Path Algorithm
Because we process vertices in topologically sorted order, edges of any path are relaxed in order of appearance in the path.
Edges on any shortest path are relaxed in order.
By path-relaxation property, correct.
COSC 3101E, PROF. J. ELDER

146. Example: Dijkstra’s algorithm
Applies to general weighted directed graph (may contain cycles).
But weights must be non-negative.
Essentially a weighted version of BFS.
Instead of a FIFO queue, uses a priority queue.
Keys are shortest-path weights (d[v]).
Maintain 2 sets of vertices:
S = vertices whose final shortest-path weights are determined.
Q = priority queue = V-S.
COSC 3101E, PROF. J. ELDER

147. Dijkstra’s algorithm
Dijkstra’s algorithm can be viewed as greedy, since it always chooses the “lightest” vertex in V − S to add to S.
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148. Dijkstra’s algorithm: Analysis
Using minheap, queue operations takes O(logV) time
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149. Key:
Example
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150. Example
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151. Example
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152. Example
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153. Example
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154. Example
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155. Correctness of Dijkstra’s algorithm
Loop invariant: d[v] = δ(s, v) for all v in S.
Initialization: Initially, S is empty, so trivially true.
Termination: At end, Q is empty S = V  d[v] = δ(s, v) for all v in V.
Maintenance:
Need to show that
d[u] = δ(s, u) when u is added to S in each iteration.
d[u] does not change once u is added to S.
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156. Correctness of Dijkstra’s Algorithm: Upper Bound Property
Proof:
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157. Correctness of Dijkstra’s Algorithm
Handled
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158. Correctness of Dijkstra’s Algorithm
Handled
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159. Correctness of Dijkstra’s algorithm
Loop invariant: d[v] = δ(s, v) for all v in S.
Maintenance:
Need to show that
d[u] = δ(s, u) when u is added to S in each iteration.
d[u] does not change once u is added to S.  ?
COSC 3101E, PROF. J. ELDER

160. End of Lecture 13
Oct 25, 2007