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Dijkstra’s algorithmN: set of nodes for which shortest path already foundInitialization: (Start with source node s)
N = {s}, Ds = 0, “s is distance zero from itself”
Dj=Csj for all j s, distances of directly-connected
neighbors
Step A: (Find next closest node i) Find i N such that Di = min Dj for j N Add i to N If N contains all the nodes, stop
Step B: (update minimum costs) For each node j N Dj = min (Dj, Di+Cij) Go to Step A
Minimum distance from s to j through node i in N
2
Dijkstra's Shortest Path Algorithm
Find shortest path from s to t.
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
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11
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19
6
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Dijkstra's Shortest Path Algorithm
s
3
t
2
6
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18
2
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15 5
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20
44
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11
6
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6
0
distance label
S = { }PQ = { s, 2, 3, 4, 5, 6, 7, t }
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Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
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0
distance label
S = { }PQ = { s, 2, 3, 4, 5, 6, 7, t }
delmin
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Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
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14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
distance label
S = { s }PQ = { 2, 3, 4, 5, 6, 7, t }
decrease key
X
X
X
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Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
distance label
S = { s }PQ = { 2, 3, 4, 5, 6, 7, t }
X
X
X
delmin
7
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
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6
15
9
14
0
S = { s, 2 }PQ = { 3, 4, 5, 6, 7, t }
X
X
X
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Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
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15 5
30
20
44
16
11
6
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6
15
9
14
0
S = { s, 2 }PQ = { 3, 4, 5, 6, 7, t }
X
X
X
decrease key
X 33
9
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
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24
18
2
9
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15 5
30
20
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16
11
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6
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9
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0
S = { s, 2 }PQ = { 3, 4, 5, 6, 7, t }
X
X
X
X 33
delmin
10
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
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6
15
9
14
0
S = { s, 2, 6 }PQ = { 3, 4, 5, 7, t }
X
X
X
X 33
44X
X
32
11
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
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4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 6 }PQ = { 3, 4, 5, 7, t }
X
X
X
44X
delmin
X 33X
32
12
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
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5
18
2
9
14
15 5
30
20
44
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11
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6
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9
14
0
S = { s, 2, 6, 7 }PQ = { 3, 4, 5, t }
X
X
X
44X
35X
59 X
24
X 33X
32
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Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 6, 7 }PQ = { 3, 4, 5, t }
X
X
X
44X
35X
59 X
delmin
X 33X
32
14
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 3, 6, 7 }PQ = { 4, 5, t }
X
X
X
44X
35X
59 XX 51
X 34
X 33X
32
15
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 3, 6, 7 }PQ = { 4, 5, t }
X
X
X
44X
35X
59 XX 51
X 34
delmin
X 33X
32
24
16
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 3, 5, 6, 7 }PQ = { 4, t }
X
X
X
44X
35X
59 XX 51
X 34
24
X 50
X 45
X 33X
32
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Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
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11
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6
15
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14
0
S = { s, 2, 3, 5, 6, 7 }PQ = { 4, t }
X
X
X
44X
35X
59 XX 51
X 34
24
X 50
X 45
delmin
X 33X
32
18
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 3, 4, 5, 6, 7 }PQ = { t }
X
X
X
44X
35X
59 XX 51
X 34
24
X 50
X 45
X 33X
32
19
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 3, 4, 5, 6, 7 }PQ = { t }
X
X
X
44X
35X
59 XX 51
X 34
X 50
X 45
delmin
X 33X
32
24
20
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 3, 4, 5, 6, 7, t }PQ = { }
X
X
X
44X
35X
59 XX 51
X 34
X 50
X 45
X 33X
32
21
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
S = { s, 2, 3, 4, 5, 6, 7, t }PQ = { }
X
X
X
44X
35X
59 XX 51
X 34
X 50
X 45
X 33X
32
Modified Dijkstra’s algorithmDijkstra-aux (G, target-node,sub-path)N: set of nodes for which shortest path already foundInitialization: Start with node s= (pop sub-path)//last node on sub-path
V’ = V – {sub-path} //search over nodes not already in sub-path N = {s}, Ds = 0 for s sub-path, “s is distance zero from itself”
Dj=Csj for all jV’, j s, distances of directly-connected neighbors
Step A: (Find next closest node i) Find i N such that Di = min Dj for j N Add i to N If N contains j=target-node,
– return N, Csj
– Else return //no path to target-node
Step B: (update minimum costs) For each node j N Dj = min (Dj, Di+Cij) Go to Step A
Modified Dijkstra’s k-Path algorithmDijkstra-recurse (G, target-node, Path, count)
Do while count< k and Path – New-Path = Dijkstra-aux (G, target-node, Path)// min-cost path to
target-node If New-Path //another min-cost path
– count=count+1; Path-set=Path-setNew-Path– E’ = E – {(pop-Path, target-node)//remove edge from graph– New-Path=Dijkstra-aux (G(V,E’), target-node, pop-Path,
count) // graph with edge deleted to prevent finding same path
Else New-Path=Dijkstra-aux (G(V,E), target-node, pop-Path, count)
End while Return Path-set
Modified Dijkstra’s k-Path algorithmDijkstra (G, target-node)Initialization: Start with node s= source node
V’ = V – {s} //search over all nodes Path-set = //set of min-cost paths count=0 //path counter Path = {s} Do while count< k and Path
– New-Path = Dijkstra-aux (G, target-node, Path)// min-cost path to target-node
If New-Path //another min-cost path– count=count+1; Path-set=Path-setNew-Path– E’ = E – {(pop-Path, target-node)//remove edge from
graph– New-Path=Dijkstra-aux (G(V,E’), target-node, pop-
Path )// min-cost path to target-node Else New-Path=Dijkstra-aux (G(V,E), target-node, pop-Path )
End while Return Path-set
25
Modified Dijkstra’s k-Path algorithmDijkstra-recurse (G, target-node, Path, count)
Do while count< k and Path – New-Path = Dijkstra-aux (G, target-node, Path)// min-cost path to
target-node If New-Path //another min-cost path
– count=count+1; Path-set=Path-setNew-Path Else New-Path=Dijkstra-aux (G(V,E), target-node, pop-Path,
count) End while Return Path-set
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