Graph Algorithms Shortest path problems

Graph Algorithms Shortest path problems

Graph Algorithms Shortest path problems

Graph Algorithms Single-Source Shortest Paths Given graph (directed or undirected) G = (V,E) with weight function w: E  R and a vertex s  V, find for all vertices v  V the minimum possible weight for path from s to v. We will discuss two general case algorithms: Dijkstra's (positive edge weights only) Bellman-Ford (positive end negative edge weights) If all edge weights are equal (let's say 1), the problem is solved by BFS in  (V+E) time.

Graph Algorithms Dijkstra’s Algorithm - Relax Relax(vertex u, vertex v, weight w) if d[v] > d[u] + w(u,v) then d[v]  d[u] + w(u,v) p[v]  u

Graph Algorithms Dijkstra’s Algorithm - Idea

Graph Algorithms Dijkstra’s Algorithm - SSSP-Dijkstra SSSP-Dijkstra(graph (G,w), vertex s) InitializeSingleSource(G, s) S   Q  V[G] while Q  0 do u  ExtractMin(Q) S  S  {u} for v  Adj[u] do Relax(u,v,w) InitializeSingleSource(graph G, vertex s) for v  V[G] do d[v]   p[v]  0 d[s]  0

Graph Algorithms Dijkstra’s Algorithm - Example

Graph Algorithms Dijkstra’s Algorithm - Example 0    

Graph Algorithms Dijkstra’s Algorithm - Example  

Graph Algorithms Dijkstra’s Algorithm - Example  

Graph Algorithms Dijkstra’s Algorithm - Example

Graph Algorithms Dijkstra’s Algorithm - Example

Graph Algorithms Dijkstra’s Algorithm - Example

Graph Algorithms Dijkstra’s Algorithm - Example

Graph Algorithms Dijkstra’s Algorithm - Example

Graph Algorithms Dijkstra’s Algorithm - Example

Graph Algorithms Dijkstra’s Algorithm - Complexity SSSP-Dijkstra(graph (G,w), vertex s) InitializeSingleSource(G, s) S   Q  V[G] while Q  0 do u  ExtractMin(Q) S  S  {u} for u  Adj[u] do Relax(u,v,w) InitializeSingleSource(graph G, vertex s) for v  V[G] do d[v]   p[v]  0 d[s]  0 Relax(vertex u, vertex v, weight w) if d[v] > d[u] + w(u,v) then d[v]  d[u] + w(u,v) p[v]  u  (V)  (1) ?  (E) times in total executed  (V) times

Graph Algorithms Dijkstra’s Algorithm - Complexity InitializeSingleSource T I (V,E) =  (V) Relax T R (V,E) =  (1)? SSSP-Dijkstra T(V,E) = T I (V,E) +  (V) + V  (log V) + E T R (V,E) = =  (V) +  (V) + V  (log V) + E  (1) =  (E + V log V)

Graph Algorithms Dijkstra’s Algorithm - Complexity

Graph Algorithms Dijkstra’s Algorithm - Correctness

Graph Algorithms Dijkstra’s Algorithm - negative weights?

Graph Algorithms Bellman-Ford Algorithm - negative cycles?

Graph Algorithms Bellman-Ford Algorithm - Idea

Graph Algorithms Bellman-Ford Algorithm - SSSP-BellmanFord SSSP-BellmanFord(graph (G,w), vertex s) InitializeSingleSource(G, s) for i  1 to |V[G]  1| do for (u,v)  E[G] do Relax(u,v,w) for (u,v)  E[G] do if d[v] > d[u] + w(u,v) then return false return true

Graph Algorithms Bellman-Ford Algorithm - Example

Graph Algorithms Bellman-Ford Algorithm - Example 0    

Graph Algorithms Bellman-Ford Algorithm - Example  

Graph Algorithms Bellman-Ford Algorithm - Example

Graph Algorithms Bellman-Ford Algorithm - Example

Graph Algorithms Bellman-Ford Algorithm - Example

Graph Algorithms Bellman-Ford Algorithm - Complexity SSSP-BellmanFord(graph (G,w), vertex s) InitializeSingleSource(G, s) for i  1 to |V[G]  1| do for (u,v)  E[G] do Relax(u,v,w) for (u,v)  E[G] do if d[v] > d[u] + w(u,v) then return false return true executed  (V) times  (E)  (1)

Graph Algorithms Bellman-Ford Algorithm - Complexity InitializeSingleSource T I (V,E) =  (V) Relax T R (V,E) =  (1)? SSSP-BellmanFord T(V,E) = T I (V,E) + V E T R (V,E) + E = =  (V) + V E  (1) + E = =  (V E)

Graph Algorithms Bellman-Ford Algorithm - Correctness

Graph Algorithms Bellman-Ford Algorithm - Correctness

Graph Algorithms Bellman-Ford Algorithm - Correctness

Graph Algorithms Bellman-Ford Algorithm - Correctness

Graph Algorithms Shortest Paths in DAGs - SSSP-DAG SSSP-DAG(graph (G,w), vertex s) topologically sort vertices of G InitializeSingleSource(G, s) for each vertex u taken in topologically sorted order do for each vertex v  Adj[u] do Relax(u,v,w)

Graph Algorithms Shortest Paths in DAGs - Example

Graph Algorithms Shortest Paths in DAGs - Example  0 

Graph Algorithms Shortest Paths in DAGs - Example  0 

Graph Algorithms Shortest Paths in DAGs - Example  0 

Graph Algorithms Shortest Paths in DAGs - Example 6 

Graph Algorithms Shortest Paths in DAGs - Example 664 

Graph Algorithms Shortest Paths in DAGs - Example 664 

Graph Algorithms Shortest Paths in DAGs - Example 654 

Graph Algorithms Shortest Paths in DAGs - Example 654 

Graph Algorithms Shortest Paths in DAGs - Example 653 

Graph Algorithms Shortest Paths in DAGs - Example 653 

Graph Algorithms Shortest Paths in DAGs - Complexity T(V,E) =  (V + E) +  (V) +  (V) + E  (1) =  (V + E) SSSP-DAG(graph (G,w), vertex s) topologically sort vertices of G InitializeSingleSource(G, s) for each vertex u taken in topologically sorted order do for each vertex v  Adj[u] do Relax(u,v,w)

Graph Algorithms Application of SSSP - currency conversion

Graph Algorithms Application of SSSP - currency conversion

Graph Algorithms Application of SSSP - currency conversion

Graph Algorithms Application of SSSP - constraint satisfaction

Graph Algorithms Application of SSSP - constraint satisfaction

Graph Algorithms Application of SSSP - constraint satisfaction

Graph Algorithms Application of SSSP - constraint satisfaction

Graph Algorithms Application of SSSP - constraint satisfaction

Graph Algorithms Application of SSSP - constraint satisfaction

Graph Algorithms All-Pairs Shortest Paths Given graph (directed or undirected) G = (V,E) with weight function w: E  R find for all pairs of vertices u,v  V the minimum possible weight for path from u to v.

Graph Algorithms Floyd-Warshall Algorithm - Idea

Graph Algorithms Floyd-Warshall Algorithm - Idea

Graph Algorithms Floyd-Warshall Algorithm - Idea d s,t (i) – the shortest path from s to t containing only vertices v 1,..., v i d s,t (0) = w(s,t) d s,t (k) = w(s,t)if k = 0 min{d s,t (k-1), d s,k (k-1) + d k,t (k-1) }if k > 0

Graph Algorithms Floyd-Warshall Algorithm - Algorithm FloydWarshall(matrix W, integer n) for k  1 to n do for i  1 to n do for j  1 to n do d ij (k)  min(d ij (k-1), d ik (k-1) + d kj (k-1) ) return D (n)

Graph Algorithms Floyd-Warshall Algorithm - Example  -4  0  17  40  2  -50   60 W

Graph Algorithms Floyd-Warshall Algorithm - Example 038  -4  0  17  40  2  -50   D (0)  (0)

Graph Algorithms Floyd-Warshall Algorithm - Example 038  -4  0  17  40   D (1)  (1)

Graph Algorithms Floyd-Warshall Algorithm - Example  0  17   D (2)  (2)

Graph Algorithms Floyd-Warshall Algorithm - Example  0  17   D (3)  (3)

Graph Algorithms Floyd-Warshall Algorithm - Example D (4)  (4)

Graph Algorithms Floyd-Warshall Algorithm - Example D (5)  (5)

Graph Algorithms Floyd-Warshall Algorithm - Extracting the shortest paths

Graph Algorithms Floyd-Warshall Algorithm - Complexity T(V,E) =  (n 3 ) =  (V 3 ) FloydWarshall(matrix W, integer n) for k  1 to n do for i  1 to n do for j  1 to n do d ij (k)  min(d ij (k-1), d ik (k-1) + d kj (k-1) ) return D (n) 3 for cycles, each executed exactly n times

Graph Algorithms All-Pairs Shortest Paths -Johnson's algorithm

Graph Algorithms All-Pairs Shortest Paths - Reweighting

Graph Algorithms All-Pairs Shortest Paths - Reweighting

Graph Algorithms All-Pairs Shortest Paths - Reweighting