Graph Algorithms

Introduction

• Terminology– V, E, directed, adjacent, path, simple path,

cycle, DAG

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Introduction

• Terminology– V, E, directed, adjacent, path, simple path,

cycle, DAG

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Introduction

• Terminology– connected, strongly connected, weakly

connected

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Representation – Adjacency Matrix

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v1 v2 v3 v4 v5 v6 v7 V8

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V7

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Representation – Adjacency Matrix

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v1 v2 v3 v4 v5 v6 v7 V8

v1 1 1 1

v2 1 1 1 1

v3 1 1

v4 1 1 1

v5 1 1 1 1

v6 1 1

v7 1 1 1 1

v8 1 1

Representation – Adjacency List

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Ø

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Topological Ordering

• find vertex with no incoming edges

• print it

• remove it and its edges

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Topological Ordering

• v1

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Topological Ordering

• v1, v2, v3, v7

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Topological Ordering

• v1, v2, v3, v7, v4, v5, v6, v8

• Complexity?

• Improvements?

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Topological Ordering

• Complexity – V2

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

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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

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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

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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

queue – v1

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dv pv

v1 0 v1

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Unweighted Shortest Path

queue – v1

queue – v2, v4, v7

queue – v4, v7, v3, v5

queue – v7, v3, v5

queue – v3, v5, v8

queue – v5, v8, v6

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dv pv

v1 0 v1

v2 1 v1

v3 2 v2

v4 1 v1

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v6 3 v3

v7 1 v1

v8 2 v7

Unweighted Shortest Path

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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 = 0for (;;) 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

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known dv pv

v1 F 0 0

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Dijkstra’s Algorithm

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known dv pv

v1 T 0 0

v2 F 10 v1

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v8 F I 0

known dv pv

v1 T 0 0

v2 F 10 v1

v3 F I 0

v4 F 5 v7

v5 F 10 v7

v6 F I 0

v7 T 3 v1

v8 F 5 v7

Dijkstra’s Algorithm

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known dv pv

v1 T 0 0

v2 F 10 v1

v3 F I 0

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v5 F 10 v7

v6 F I 0

v7 T 3 v1

v8 T 5 v7

known dv pv

v1 T 0 0

v2 F 10 v1

v3 F I 0

v4 T 5 v7

v5 F 10 v7

v6 F I 0

v7 T 3 v1

v8 F 5 v7

Dijkstra’s Algorithm

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known dv pv

v1 T 0 0

v2 T 10 v1

v3 F 12 0

v4 T 5 v7

v5 F 10 v7

v6 F I 0

v7 T 3 v1

v8 T 5 v7

known dv pv

v1 T 0 0

v2 T 10 v1

v3 F 12 v2

v4 T 5 v7

v5 T 10 v7

v6 F 15 v5

v7 T 3 v1

v8 T 5 v7

Dijkstra’s Algorithm

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known dv pv

v1 T 0 0

v2 T 10 v1

v3 T 12 v2

v4 T 5 v7

v5 T 10 v7

v6 T 15 v5

v7 T 3 v1

v8 T 5 v7

known dv pv

v1 T 0 0

v2 T 10 v1

v3 T 12 v2

v4 T 5 v7

v5 T 10 v7

v6 F 15 v5

v7 T 3 v1

v8 T 5 v7

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

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)

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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

1. choose a path from src to sink – augmenting path2. add flow equal to minimum edge on path3. add reverse edges to allow algorithm to undo its

decision4. continue until no augmenting path can be found

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Max-Flow Algorithm

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Max-Flow Algorithm

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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

• Similar to Dijkstra’s1. choose min distance vertex v and mark as

known 2. update distance values for all adjacent vertices

w1. dw = min(dw, cv, w)

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Prim’s Algorithm

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Kruskal’s Algorithm

• choose min cost edge– if it doesn’t create a cycle, add it

• use heap to provide O(ElogE) running time

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Kruskal’s Algorithm

• (v2,v4), (v2,v3), (v4,v7), (v7,v8), (v1,v2), (v3,v6), (v1,v4), (v6,v5)

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Depth-First Search

• Generalization of preorder traversal

dfs(Vertex v)

v.visited = true

for each w adjacent to v

if(!w.visited)

dfs(w)