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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V7
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V8
Representation – Adjacency Matrix
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Representation – Adjacency List
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Ø
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Ø
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Ø
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Ø
Ø
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Ø
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
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• Complexity?
• Improvements?
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Topological Ordering
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• Complexity – V2
• Improvements – as edges are removed,
enqueue vertices with 0 indegree
– v1, v2, v7, v3, v4, v5, v6, v8
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)
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Unweighted Shortest Path
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queue – v1
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dv
pv
0
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Unweighted Shortest Path
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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
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0
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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)
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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
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’s2 Algorithm
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known
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Dijkstra’s2 Algorithm
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known
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known
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Dijkstra’s2 Algorithm
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known
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known
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Dijkstra’s2 Algorithm
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known
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known
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Dijkstra’s2 Algorithm
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known
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known
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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
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10 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)
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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-Flow2 Algorithm
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src
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4.
sink
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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
Max-Flow2 Algorithm
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src
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sink
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sink
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Max-Flow Algorithm
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src
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sink
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src
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sink
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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
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• Similar to Dijkstra’s
1. choose min distance vertex v and mark as
known
2. update distance values for all adjacent vertices
w
1. dw = min(dw, cv, w)
Prim’s Algorithm
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Kruskal’s Algorithm
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•
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choose min cost edge
– if it doesn’t create a cycle, add it
•
use heap to provide O(ElogE) running time
Kruskal’s Algorithm
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• (v2,v4), (v2,v3), (v4,v7), (v7,v8), (v1,v2),
(v3,v6), (v1,v4), (v6,v5)
Depth-First Search
• Generalization of preorder traversal
dfs(Vertex v)
v.visited = true
for each w adjacent to v
if(!w.visited)
dfs(w)