Dijkstra’s algorithm
N: set of nodes for which shortest path already found
Initialization: (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)
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

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
Dijkstra's Shortest Path Algorithm
Find shortest path from s to t.
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Dijkstra's Shortest Path Algorithm
S={ }
PQ = { s, 2, 3, 4, 5, 6, 7, t }
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distance label
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Dijkstra's Shortest Path Algorithm
S={ }
PQ = { s, 2, 3, 4, 5, 6, 7, t }
delmin
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Dijkstra's Shortest Path Algorithm
S={s}
PQ = { 2, 3, 4, 5, 6, 7, t }
decrease key
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distance label
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Dijkstra's Shortest Path Algorithm
S={s}
PQ = { 2, 3, 4, 5, 6, 7, t }
delmin
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distance label
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Dijkstra's Shortest Path Algorithm
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
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Dijkstra's Shortest Path Algorithm
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
decrease key
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Dijkstra's Shortest Path Algorithm
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
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delmin
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 6 }
PQ = { 3, 4, 5, 7, t }
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 6 }
PQ = { 3, 4, 5, 7, t }
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delmin
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 6, 7 }
PQ = { 3, 4, 5, t }
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 6, 7 }
PQ = { 3, 4, 5, t }
delmin
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 6, 7 }
PQ = { 4, 5, t }
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X X
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 6, 7 }
PQ = { 4, 5, t }
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delmin
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X X
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 5, 6, 7 }
PQ = { 4, t }
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X X
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 5, 6, 7 }
PQ = { 4, t }
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delmin
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X X
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 4, 5, 6, 7 }
PQ = { t }
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X 9
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X X
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 4, 5, 6, 7 }
PQ = { t }
32
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 33
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X 9
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s
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X 35
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delmin
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X X
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 4, 5, 6, 7, t }
PQ = { }
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 33
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X 9
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s
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X 35
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45 X
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t
50 51
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X 59
X X
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Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 4, 5, 6, 7, t }
PQ = { }
32
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 33
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X 9
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s
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X 35
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X X
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Modified Dijkstra’s algorithm
Dijkstra-aux (G, target-node,sub-path)
N: set of nodes for which shortest path already found
Initialization: Start with node s= (pop sub-path)//last node on sub-path

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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 jV’, 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)


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For each node j  N
Dj = min (Dj, Di+Cij)
Go to Step A
Modified Dijkstra’s k-Path algorithm
Dijkstra-recurse (G, target-node, Path, count)

Do while count< k and Path  
–

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New-Path = Dijkstra-aux (G, target-node, Path)// min-cost path to targetnode
 If New-Path   //another min-cost path
– count=count+1; Path-set=Path-setNew-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 algorithm
Dijkstra (G, target-node)
Initialization: Start with node s= source node

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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  
–

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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-setNew-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 algorithm
Dijkstra-recurse (G, target-node, Path, count)

Do while count< k and Path  
–


New-Path = Dijkstra-aux (G, target-node, Path)// min-cost path to targetnode
 If New-Path   //another min-cost path
– count=count+1; Path-set=Path-setNew-Path
 Else New-Path=Dijkstra-aux (G(V,E), target-node, pop-Path, count)
End while
Return Path-set