Dijkstras Shortest Path Algorithm Home node tree Guest

Dijkstra's Shortest Path Algorithm Home node tree = { } Guest node tree = { s, 2, 3, 4, 5, 6, 7, t } 0 s 24 2 9 14 18 30 11 5 5 16 20 7 distance label 6 2 6 15 3 44 4 19 6 t 1

Dijkstra's Shortest Path Algorithm H={ } G = { s, 2, 3, 4, 5, 6, 7, t } Starting from origin 0 s 24 2 9 14 18 30 11 5 5 16 20 7 distance label 6 2 6 15 3 44 4 19 6 t 2

Dijkstra's Shortest Path Algorithm H={s} G = { 2, 3, 4, 5, 6, 7, t } decrease label by updating X 9 0 s 24 2 9 18 X 14 14 30 11 5 5 16 20 7 distance label 15 X 6 2 6 15 3 44 4 19 6 t 3

Dijkstra's Shortest Path Algorithm H={s} G = { 2, 3, 4, 5, 6, 7, t } Find the guest node with minimal label X 9 0 s 24 2 9 30 11 5 5 16 20 distance label 15 X 6 2 6 7 3 18 X 14 15 44 4 19 6 t 4

Dijkstra's Shortest Path Algorithm H = { s, 2 } G = { 3, 4, 5, 6, 7, t } Add node 2 into home node tree X 9 0 s 24 2 9 18 X 14 30 11 5 5 16 20 7 15 X 6 2 6 15 3 44 4 19 6 t 5

Dijkstra's Shortest Path Algorithm H = { s, 2 } G = { 3, 4, 5, 6, 7, t } decrease label by updating X 33 X 9 0 s 24 2 9 18 X 14 30 11 5 5 16 20 7 15 X 6 2 6 15 3 44 4 19 6 t 6

Dijkstra's Shortest Path Algorithm H = { s, 2 } G = { 3, 4, 5, 6, 7, t } X 33 X 9 0 s 24 2 9 Find guest node with minimal 18 label X 14 6 11 5 5 16 20 7 15 X 6 2 30 15 3 44 4 19 6 t 7

Dijkstra's Shortest Path Algorithm H= { s, 2, 6 } G = { 3, 4, 5, 7, t } decrease label by updating 32 Add node 6 into home node tree X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 11 5 5 16 20 7 15 X 6 2 6 15 3 44 4 19 6 t 8

Dijkstra's Shortest Path Algorithm H = { s, 2, 6 } G = { 3, 4, 5, 7, t } 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 11 5 5 16 20 7 15 X 6 2 6 15 3 44 Find guest node closest 4 19 6 t 9

Dijkstra's Shortest Path Algorithm H = { s, 2, 6, 7 } G = { 3, 4, 5, t } 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 16 20 7 6 2 6 15 3 44 4 19 6 t 59 X 10

Dijkstra's Shortest Path Algorithm Find guest node with minimal distance H= { s, 2, 6, 7 } G = { 3, 4, 5, t } 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 16 20 7 6 2 6 15 3 44 4 19 6 t 59 X 11

Dijkstra's Shortest Path Algorithm H = { s, 2, 3, 6, 7 } G = { 4, 5, t } Add node 3 into home node tree 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 16 20 7 6 2 6 15 3 44 4 19 6 t 51 59 X X 12

Dijkstra's Shortest Path Algorithm H = { s, 2, 3, 6, 7 } G = { 4, 5, t } 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 20 7 15 X 44 X 35 X 5 5 6 2 6 15 3 Find node with minimal distance label 44 11 16 4 19 6 t 51 59 X X 13

Dijkstra's Shortest Path Algorithm H = { s, 2, 3, 5, 6, 7 } G = { 4, t } 32 Add node 5 into home node tree X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 16 20 7 6 2 6 15 3 44 45 X 4 19 6 t 50 51 X 59 X X 14

Dijkstra's Shortest Path Algorithm H = { s, 2, 3, 5, 6, 7 } G = { 4, t } 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 45 X 4 19 Find node with 6 minimal distance 16 label 20 7 6 2 6 15 3 44 t 50 51 X 59 X X 15

Dijkstra's Shortest Path Algorithm H = { s, 2, 3, 4, 5, 6, 7 } G={t} 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 16 20 7 6 2 6 15 3 44 45 X 4 19 6 t 50 51 X 59 X X 16

Dijkstra's Shortest Path Algorithm H = { s, 2, 3, 4, 5, 6, 7 } G={t} 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 16 20 7 6 2 6 15 3 44 Find node with minimal distance label 45 X 4 19 6 t 50 51 X 59 X X 17

Dijkstra's Shortest Path Algorithm H = { s, 2, 3, 4, 5, 6, 7, t } G={} 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 16 20 7 6 2 6 15 3 44 45 X 4 19 6 t 50 51 X 59 X X 18

Dijkstra's Shortest Path Algorithm H = { s, 2, 3, 4, 5, 6, 7, t } G={} Backtracking for finding shortest path 32 X 33 X X 9 0 s 24 2 9 18 X 14 30 44 X 35 X 5 5 15 X 11 16 20 7 6 2 6 15 3 44 45 X 4 19 6 t 50 51 X 59 X X 19

Label Setting Algorithm (Computer Science Notation) S : = ; T : = N; //setup two node sets, S: Home, T: Guest d(i) : = ∞ for i N; d(s) : = 0; //initialize distance label p(s) : = 0; //initialize predecessor while S ≠N do // perform the following operation until all the nodes are permanently labeled begin (node selection, also called FINDMIN) let i ∈T be a node for which d(i) = min {d(j) : j ∈ T}; // select the node i with the minimum label in the temporarily labeled set S : = S {i}; // add node i into permanently labeled set T: = T –{i}; // remove node i from temporarily labeled set Update(i) // update distance labels for all outgoing arcs from i end; 20

A Generic Shortest Path Algorithm (Operations Research Notation) Algorithm LABEL CORRECTING; begin n n d(1) : = 0 and Pred(1) : = ø; d(j) : = for each j N – {1}; while some arc (i, j) satisfies d(j) > d(i) + cij do begin d(j) : = d(i) + cij; – Pred(j) : = i; – end; n 21