Find the shortest path from a to z

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Presentation transcript:

Find the shortest path from a to z 3 3 b d g 8 3 2 2 1 h 6 3 a z e 6 1 8 3 9 f i c 1 3

∞ 3 ∞ ∞ 3 b b d d g g 8 3 2 2 1 ∞ h h ∞ 6 3 a z z e e ∞ 6 1 8 3 9 f f Label all vertices as ∞ and put vertex a in the vertex list. ∞ 3 ∞ ∞ 3 b b d d g g 8 3 2 2 1 ∞ h h ∞ 6 3 a z z e e ∞ 6 1 8 3 9 f f i i c c ∞ 1 ∞ 3 ∞ Vertex List: a:0 b: ∞ c: ∞ d: ∞ e: ∞ f: ∞ g: ∞ h: ∞ i: ∞ z: ∞

3 ∞ ∞ 3 b b d d g g 8 3 2 2 1 ∞ h h 6 3 a z z e e ∞ 6 1 8 3 9 f f i i Re-label all vertices that can be reached from a 3 ∞ ∞ 3 2(a) b b d d g g 8 3 2 2 1 ∞ h h 6 3 6(a) a z z e e ∞ 6 1 8 3 9 f f i i c c 1 3(a) ∞ 3 ∞ Vertex List: a:0 b: 2 c: 3 d: ∞ e: 6 f: ∞ g: ∞ h: ∞ i: ∞ z: ∞

3 ∞ ∞ 3 b d d g g 8 3 2 2 1 ∞ h h 6 3 a z z e e ∞ 6 1 8 3 9 f f i i c Choose the unvisited vertex with the smallest label, in this case b. 3 ∞ ∞ 3 2(a) b d d g g 8 3 2 2 1 ∞ h h 6 3 6(a) a z z e e ∞ 6 1 8 3 9 f f i i c c 1 3(a) ∞ 3 ∞ Vertex List: a:0 b: 2 c: 3 d: ∞ e: 6 f: ∞ g: ∞ h: ∞ i: ∞ z: ∞

3 ∞ 3 b d d g g 8 3 2 2 1 ∞ h h 6 3 a z z e e ∞ 6 1 8 3 9 f f i i c c Re-label all vertices that can be reached from b now. 3 ∞ 3 2(a) 5(a,b) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e ∞ 6 1 8 3 9 f f i i c c 1 3(a) ∞ 3 ∞ Vertex List: a:0 b: 2 c: 3 d: 5 e: 6 5 f: ∞ g: ∞ h: ∞ i: ∞ z: ∞

3 ∞ 3 b d d g g 8 3 2 2 1 ∞ h h 6 3 a z z e e ∞ 6 1 8 3 9 f f i i c c Choose the unvisited vertex with the smallest label, now c. 3 ∞ 3 2(a) 5(a,b) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e ∞ 6 1 8 3 9 f f i i c c 1 3(a) ∞ 3 ∞ Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: ∞ g: ∞ h: ∞ i: ∞ z: ∞

3 ∞ 3 b d d g g 8 3 2 2 1 ∞ h h 6 3 a z z e e ∞ 6 1 8 3 9 f f i i c c Re-label, now including c 3 ∞ 3 2(a) 5(a,b) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e ∞ 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 ∞ Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: ∞ h: ∞ i: ∞ z: ∞

3 ∞ 3 b d d g g 8 3 2 2 1 ∞ h h 6 3 a z z e e ∞ 6 1 8 3 9 f f i i c c Choose the smallest unvisited vertex, f 3 ∞ 3 2(a) 5(a,b) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e ∞ 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 ∞ Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: ∞ h: ∞ i: ∞ z: ∞

3 ∞ 3 b d d g g 8 3 2 2 1 ∞ h h 6 3 a z z e e 6 1 8 3 9 f f i i c c 1 Re-label, now using f 3 ∞ 3 2(a) 5(a,b) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e 12(a,c,f) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: ∞ h: 12 i: 7 z: ∞

3 ∞ 3 b d d g g 8 3 2 2 1 ∞ h h 6 3 a z z e e 6 1 8 3 9 f f i i c c 1 Smallest unvisited vertex is now d (alphabetical tie over e) 3 ∞ 3 2(a) 5(a,b) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e 12(a,c,f) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: ∞ h: 12 i: 7 z: ∞

Re-label using d 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e 12(a,c,f) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 12 i: 7 z: ∞

Smallest unvisited vertex is now e 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e 12(a,c,f) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 12 i: 7 z: ∞

Re-label using e, but no vertex gets updated 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e 12(a,c,f) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 12 i: 7 z: ∞

Smallest unvisited vertex is i 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 ∞ h h 6 3 5(a,b) a z z e e 12(a,c,f) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 12 i: 7 z: ∞

Re-label using i 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 h h 6 3 16(a,c,f,i) 5(a,b) a z z e e 12(a,c,f) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 12 i: 7 z: 16

Now, choose g 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 h h 6 3 16(a,c,f,i) 5(a,b) a z z e e 12(a,c,f) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 12 i: 7 z: 16

Re-label, using g 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 h h 6 3 16(a,c,f,i) 5(a,b) a z z e e 9(a,b,d,g) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 12 9 i: 7 z: 16

Now choose h 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 h h 6 3 16(a,c,f,i) 5(a,b) a z z e e 9(a,b,d,g) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 9 i: 7 z: 16

Re-label using h 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 h h 6 3 12(a,b,d,g,h) 5(a,b) a z z e e 9(a,b,d,g) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 9 i: 7 z: 16 12

No vertices left to consider, so we have the shortest path 3 3 2(a) 5(a,b) 8(a,b,d) b d d g g 8 3 2 2 1 h h 6 3 12(a,b,d,g,h) 5(a,b) a z z e e 9(a,b,d,g) 6 1 8 3 9 f f i i c c 1 3(a) 4(a,c) 3 7(a,c,f) Vertex List: a:0 b: 2 c: 3 d: 5 e: 5 f: 4 g: 8 h: 9 i: 7 z: 16 12