1.1 Data Structure and Algorithm Lecture 11 Application of BFS  Shortest Path Topics Reference: Introduction to Algorithm by Cormen Chapter 25: Single-Source Shortest Paths

1.2 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm Dijkstra’s algorithm solves the single- source shortest paths problem on a weighted, directed graph G =(V,E) fro the case in which all edge weights are nonnegative. We assume that w(u,v) >=0 for each edge (u,v)

1.3 Data Structure and Algorithm Algorithm Dijkstra(G,w,s)  Initialize(G,s)  S = 0  Q = V[G]  While Q != 0 do  U = ExtractMin(Q)  S = S union {u}  For each vertex v of adj[u] do  Relax(u,v,w) Complexty: Using priority queue:O (V 2 +E) Using heap as priority queue: O( (V+E) lg 2 V) cost of building heap is O(V) cost of ExtracMin is O(lgV) and there are |V| such operations cost of relax is O(lgV) and there are |E| such operations.

1.4 Data Structure and Algorithm Algorithm( Cont.) Initialize(G,s)  for each vertex of V[G] do  d[v] = infinity  Л[v] = NIL  d[s] = 0 relax(u,v,w)  if d[v] >d[u] +w(u,v) then  d[v] = d[u] + w(u,v)  Л[v] = u

1.5 Data Structure and Algorithm Complexity Using priority queue:O (V2 +E) Using heap as priority queue: O( (V+E) lg2V)  cost of building heap is O(V)  cost of ExtracMin is O(lgV) and there are |V| such operations  cost of relax is O(lgV) and there are |E| such operations.

1.6 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm Find shortest path from s to t. s 3 t

1.7 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t        0 distance label S = { } Q = { s, 2, 3, 4, 5, 6, 7, t }

1.8 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t        0 distance label S = { } Q = { s, 2, 3, 4, 5, 6, 7, t } ExtractMin()

1.9 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 distance label S = { s } Q = { 2, 3, 4, 5, 6, 7, t } decrease key  X   X X

1.10 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 distance label S = { s } Q = { 2, 3, 4, 5, 6, 7, t }  X   X X ExtractMin()

1.11 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2 } Q = { 3, 4, 5, 6, 7, t }  X   X X

1.12 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2 } Q = { 3, 4, 5, 6, 7, t }  X   X X decrease key X 32

1.13 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2 } Q = { 3, 4, 5, 6, 7, t }  X   X X X 32 ExtractMin()

1.14 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 6 } Q = { 3, 4, 5, 7, t }  X   X X X X

1.15 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 6 } Q = { 3, 4, 5, 7, t }  X   X X X X ExtractMin()

1.16 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 6, 7 } Q = { 3, 4, 5, t }  X   X X X X 35 X 59 X

1.17 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 6, 7 } Q = { 3, 4, 5, t }  X   X X X X 35 X 59 X ExtractMin

1.18 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 3, 6, 7 } Q = { 4, 5, t }  X   X X X X 35 X 59 XX 51 X 34

1.19 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 3, 6, 7 } Q = { 4, 5, t }  X   X X X X 35 X 59 XX 51 X 34 ExtractMin

1.20 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 3, 5, 6, 7 } Q = { 4, t }  X   X X X X 35 X 59 XX 51 X 34 X 50 X 45

1.21 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 3, 5, 6, 7 } Q = { 4, t }  X   X X X X 35 X 59 XX 51 X 34 X 50 X 45 ExtractMin

1.22 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 3, 4, 5, 6, 7 } Q = { t }  X   X X X X 35 X 59 XX 51 X 34 X 50 X 45

1.23 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 3, 4, 5, 6, 7 } Q = { t }  X   X X X X 35 X 59 XX 51 X 34 X 50 X 45 ExtractMin

1.24 Data Structure and Algorithm Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 3, 4, 5, 6, 7, t } Q = { }  X   X X X X 35 X 59 XX 51 X 34 X 50 X 45 ExtractMin

1.25 Data Structure and Algorithm The Bellman-Form Algorithm BELLMAN-FORD(G,w,s)  Initialize()  for i = 1 to |V[G]| -1 do  for each edge (u,v) of E[G] do  relax(u,v,w)  for each edge (u,v) of E[G] do  if d[v] > d[u] + w(u,v) then  return FALSE  return TRUE Complexity O(VE)

1.26 Data Structure and Algorithm Example:Bellman-Ford z u x v y 0    

1.27 Data Structure and Algorithm Example:Bellman-Ford z u x v y 0 6  

1.28 Data Structure and Algorithm Example:Bellman-Ford z u x v y

1.29 Data Structure and Algorithm Example:Bellman-Ford z u x v y

1.30 Data Structure and Algorithm Example:Bellman-Ford z u x v y