Single-Source Shortest Paths (25/24)

1
Single-Source Shortest Paths (25/24)

• HW 25-2 and 25-3 p. 546/24-2 and 24-3 p.615
• Given a graph G(V,E) and w E ? ?
• weight of ltv1,...,vkgt is w(p) ?
  w(vi,vi1)
• Single-source shortest-paths
• find a shortest path from a source s to every
  vertex v ?V
• Single pair shortest path problem asks for a u-v
  shortest path for some pair of vertices
• All pairs shortest-paths problem will be next time

2
Shortest Paths (25/24)

• Predecessor subgraph for restoring shortest paths
• Shortest-paths tree rooted at source s
• Th Subpath of a shortest path is a shortest path
• Triangle Inequality
• ?(u,v) ? ?(u,x) ?(x,v)
• Well defined some paths may not exist if there
  is a negative-weight cycle in graph

u
v
x
lt0
3
Bellman-Ford (25.3/24.1)

• Most basic (BFS based) algorithm, shortest paths
  (tree) easy to reconstruct.
• for each v ?V do dv ? ? ds ? 0
• Relaxation
• for i 1,...,V-1 do
• for each edge (u,v) ? E do
• dv ? mindv, duw(u,v)
• Negative cycle checking
• for each v ?V do if dvgt du w(u,v) then no
  solution
• Finally dv ?(s,v)

4
Bellman-Ford Analysis (25.3/24.1)

• Runtime O(VE)
• Correctness
• Lemma dv ? ?(s,v)
• Initially true
• Let dv du w(u,v)
• by triangle inequality for first violation
  dv lt ?(s,v) ? ?(s,u)w(u,v) ?
  d(u)w(u,v)
• After V-1 passes all d values are ?s if there
  are no negative cycles
• s ? v1 ? v2 ? ... ? v (some shortest path)
• After i-th iteration ds,vi is correct and
  final

5
Dag Shortest Paths (25.4/24.2)

• DAG shortest paths
• Bellman-Ford O(VE)
• Topological sort O(VE) (DFS)
• Will never relaxed edges out of vertex v until
  have done all edges in v
• Runtime O(VE)
• Application PERT (program evaluation and review
  technique)
• Critical path is the longest path through the dag
• Negating the edge weights and running dag
  shortest paths algorithm

6
Dijkstras Shortest Paths (25.2/24.3)

• Better than BF since non-negative weights.
• Like BFS but uses priority queue.
• for each v ?V do dv ? ? ds ? 0
• S ? ? Q ? V
• While Q ? ? do
• u ? Extract-Min(Q)
• S ? S u
• for v adjacent to u do
• dv ? mindv, duw(u,v)
• (relaxation Decrease-Key)

7
d
?
3
17
b
c
e
?
?
?
13
9
12
6
22
11
7
a
12
?
14
f
0
8
10
8
?
4
?
20
?
i
h
10
g
8
1
For each v ? V do dv lt- ? ds lt- 0S lt-
?, Q lt- V
?
j
8
d
?
3
17
b
c
e
12
?
?
13
9
12
6
22
11
7
a
12
?
14
f
0
8
10
8
?
4
?
20
8
i
h
10
g
8
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
1
?
j
9
d
?
3
17
b
c
e
12
?
?
13
9
12
6
22
11
7
a
12
?
14
f
0
8
10
8
?
4
28
20
8
i
h
10
g
8
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
1
?
j
10
d
?
3
17
b
c
e
12
25
?
13
9
12
6
22
11
7
a
12
?
14
f
0
8
10
8
?
4
19
20
i
8
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
?
j
11
d
?
3
17
b
c
e
12
25
?
13
9
12
6
22
11
7
a
12
27
14
f
0
8
10
8
23
4
19
20
8
i
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
j
12
d
?
3
17
b
c
e
12
25
?
13
9
12
6
22
11
7
a
12
27
14
f
0
8
10
8
23
4
19
20
8
i
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
j
13
d
28
3
17
b
c
e
12
25
34
13
9
12
6
22
11
7
a
12
27
14
f
0
8
10
8
23
4
19
20
8
i
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
j
14
d
28
3
17
b
c
e
12
25
34
13
9
12
6
22
11
7
a
12
27
14
f
0
8
10
8
23
4
19
20
8
i
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
j
15
d
28
3
17
b
c
e
12
25
34
13
9
12
6
22
11
7
a
12
27
14
f
0
8
10
8
23
4
19
20
8
i
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
j
16
d
28
3
17
b
c
e
12
25
34
13
9
12
6
22
11
7
a
12
27
14
f
0
8
10
8
23
4
19
20
8
i
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
j
17
d
28
3
17
b
c
e
12
25
34
13
9
12
6
22
11
7
a
12
27
14
f
0
8
10
8
23
4
19
20
8
i
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
j
18
d
28
3
17
b
c
e
12
25
34
13
9
12
6
22
11
7
a
12
27
14
f
0
8
10
8
23
4
19
20
8
i
h
10
g
8
1
While Q ? ? do U Extract-Min(Q) S
S u for v adjacent to u do dv
mindv, duw(u,v)
29
j
19
Dijkstras Runtime (25.2/24.3)

• Extract-Min executed V times
• Decrease-Key executed E times
• Time V?T(Extract-Min) (finddelete O(log
  V))
• E ?T(Decrease-Key) (deleteadd
  O(log V))
• Binary Heap E ? log V (30
  years ago)
• Fibonacci Heap E V ? log V (10 years
  ago)
• Optimal time algorithm found 1/2 year ago. It
  runs in time O(E) (Mikel Thorup)

20
Dijkstras Correctness (25.2/24.3)

• The same Lemma as for BF dv ? ?(s,v)
• Th Whenever u is added to S, du ?(s,u)
• Let u be first s.t. du gt ?(s,u)
• Let y be first ?V-S on actual
• shortest s-u path ? dy ?(s,y)
• For ys predecessor x, dx ?(s,x)
• when put x in S dy gets ?(s,y)
• du gt ?(s,u) ?(s,y) ?(y,u) dy ?(y,u) ?
  dy

S
u
s
y
Q
x