Graph Algorithms

Graph Algorithms

Introduction • Terminology • V, E, directed, adjacent, path, simple path, cycle, DAG v3 v2 v1 v5 v6 v4 v8 v7

Introduction • Terminology • connected, strongly connected, weakly connected v5 v5 v8 v8 v7 v7

Representation – Adjacency Matrix v3 v2 v1 v5 v6 v4 v8 v7

Representation – Adjacency List v3 v2 v1 v5 v6 v4 v8 v7 v1 v2 v4 v7 Ø v2 v1 v3 v4 v5 Ø v3 v2 v6 Ø v4 v1 v2 v7 Ø v5 v2 v6 v7 v8 Ø v6 v3 v5 Ø v7 v1 v4 v5 v8 Ø

Topological Ordering • find vertex with no incoming edges • print it • remove it and its edges v3 v2 v1 v5 v6 v4 v8 v7

Topological Ordering • v1 v3 v2 v1 v5 v6 v4 v3 v8 v2 v7 v5 v6 v4 v8 v7

Topological Ordering • v1, v2, v3, v7 v3 v3 v2 v5 v5 v6 v6 v4 v4 v8 v8 v7 v7 v5 v6 v5 v6 v4 v4 v8 v8 v7

Topological Ordering • v1, v2, v3, v7, v4, v5, v6, v8 • Complexity? • Improvements? v6 v5 v5 v6 v6 v4 v8 v8 v8

Topological Ordering v3 • Complexity – V2 • Improvements – as edges are removed, enqueue vertices with 0 indegree • v1, v2, v7, v3, v4, v5, v6, v8 v2 v1 v5 v6 v4 v8 v7

Shortest Path Algorithms • Given as input a weighted graph G=(V, E), and a distinguished vertex, s, find the shortest weighted path from s to every other vertex in G. • Unweighted – every edge has weight 1 • Applications?

Breadth-first Search • Level-order traversal

Unweighted Shortest Path v3 v2 v1 v5 v6 v4 v8 v7 s.dist = 0; for(currdist = 0; currdist < NUM_VERT; currdist++) for each vertex v if(!v.known && v.dist == currdist) v.known = true for each w adjacent to v if (w.dist == INFINITY) w.dist = currdist + 1 w.path = v

Unweighted Shortest Path v3 v2 v1 v5 v6 v4 v8 v7 enqueue(s) s.dist = 0; while(!q.isEmpty()) v = q.dequeue() for each w adjacent to v if(w.dist == INFINITY) w.dist = v.dist + 1 w.path = v q.enqueue(w)

Unweighted Shortest Path v3 v2 queue – v1 v1 v5 v6 v4 v8 v7

Unweighted Shortest Path v3 v2 queue – v1 queue – v2, v4, v7 queue – v4, v7, v3, v5 queue – v7, v3, v5 queue – v3, v5, v8 queue – v5, v8, v6 v1 v5 v6 v4 v8 v7

Unweighted Shortest Path v3 v2 v1 v5 v6 v4 v8 v7 enqueue(s) s.dist = 0; while(!q.isEmpty()) v = q.dequeue() for each w adjacent to v if(w.dist == INFINITY) w.dist = v.dist + 1 w.path = v q.enqueue(w) s.dist = 0; for(currdist = 0; currdist < NUM_VERT; currdist++) for each vertex v if(!v.known && v.dist == currdist) v.known = true for each w adjacent to v if (w.dist == INFINITY) w.dist = currdist + 1 w.path = v

Dijkstra’s Algorithm • Weighted shortest-path first • Greedy algorithm s.dist = 0 for (;;) v = smallest unknown distance vertex if(v == NOT_A_VERTEX) break; v.known = true for each w adjacent to v if(!w.known) if(v.dist + cvw < w.dist) decrease(w.dist to v.dist+cvw) w.path = v

Dijkstra’s Algorithm 2 v3 v2 4 10 6 1 5 7 v1 v5 v6 v4 9 7 2 3 v8 v7 2

Running time – Dijkstra’s • Simple implementation • O(E + V2) • Improvements • Use a priority queue • Smallest unknown distance vertex log V (V times) • decrease log V (E times)

Negative Edge Costs -3 v3 v2 4 3 7 1 5 7 v1 v5 v6 v4 9 -6 2 10 v8 v7 2 enqueue(s) s.dist = 0; while(!q.isEmpty()) v = q.dequeue() for each w adjacent to v if(w.dist > v.dist+cvw) w.dist = v.dist + cvw w.path = v if(w not in q) q.enqueue(w)

Network Flow • Determine maximum flow from source to sink in a directed graph where each edge has given capacity • capacity cv,w is maximum flow for edge (v, w) • total flow coming in must = total flow going out • Example applications?

Max-Flow Algorithm 2 v3 v2 4 10 6 • choose a path from src to sink – augmenting path • add flow equal to minimum edge on path • add reverse edges to allow algorithm to undo its decision • continue until no augmenting path can be found sink src 1 5 7 v1 v5 v6 v4 9 7 2 3 v8 v7 2

Max-Flow Algorithm 2 v3 v2 4 10 6 sink src 1 2 7 v1 v5 v6 v4 3 3 9 4 2 3 v8 v7 2 2 2 v3 v2 2 2 6 8 sink src 1 2 7 v1 v5 v6 v4 3 3 9 4 2 3 v8 v7 2

Max-Flow Algorithm 2 2 v3 v2 4 2 2 6 sink src 1 4 5 7 v1 v5 v6 v4 3 9 4 2 3 v8 v7 2 2 2 v3 v2 4 2 sink src 5 v1 v5 v6 v4 3 3 v8 v7

Algorithm Complexity • O(f * E) • Can be bad if f is large • Improve by choosing augmenting path that increases flow by largest amount

Minimum Spanning Tree • Find a tree (acyclic graph) that covers every vertex and has minimum cost • Number of edges in tree will be V-1

Prim’s Algorithm 2 v3 v2 4 • Similar to Dijkstra’s • choose min distance vertex v and mark as known • update distance values for all adjacent vertices w • dw = min(dw, cv, w) 3 7 1 5 v1 7 v5 v6 v4 9 6 2 10 v8 v7 2

Prim’s Algorithm 2 v3 v2 4 3 1 5 v1 v5 v6 v4 2 v8 v7 2

Kruskal’s Algorithm 2 v3 v2 4 • choose min cost edge • if it doesn’t create a cycle, add it • use heap to provide O(ElogE) running time 3 7 1 5 v1 7 v5 v6 v4 9 6 2 10 v8 v7 2

Kruskal’s Algorithm 2 v3 v2 4 • (v2,v4), (v2,v3), (v4,v7), (v7,v8), (v1,v2), (v3,v6), (v1,v4), (v6,v5) 3 7 1 5 v1 4 v5 v6 v4 9 6 2 10 v8 v7 2

Depth-First Search • Generalization of preorder traversal dfs(Vertex v) v.visited = true for each w adjacent to v if(!w.visited) dfs(w)

Graph Algorithms