Transcript Graph Representation (23.1/22.1) • HW: problem 23.3, p.496 • G=(V, E) -graph:

Graph Representation (23.1/22.1)
• HW: problem 23.3, p.496
• G=(V, E) -graph:
– V = V(G) - vertices;
– E = E(G) - edges (connecting pairs of vertices)
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• Adjacency-list
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• Adjacency-matrix
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Depth-First Search (23.3/22.3)
• Methodically explore all vertices and edges
All vertices are White, t = 0
For each vertex u do if u is White then Visit(u)
Procedure Visit(u)
color u Gray; d[u]  t  t +1
for each v adjacent to u do
if v is White then Visit(v)
color u Black
f [u]  t  t +1
Gray vertices =
stack of recursive calls
Depth-First Search
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Depth-First Search (23.3/22.3)
• Runtime of DFS = O(V+E)
– once per vertex
– once per edge
• Kinds of edges:
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tree edge
back edge
forward edge
cross edge
(gray to gray)
(gray to gray)
(gray to black)
(gray to black)
• G is undirected  only tree and back
• Undirected G is acyclic  no back edges
– If a graph is acyclic can be found in O(V) time
Topological Sort (23.4/22.4)
• DAG = directed acyclic graph
– has levels or depth
– cannot return up
• Topological Sort(G)
– call DFS(G) to compute f[v]
– sort according to finishing times
• Directed graph G is acyclic  no back edges
Breadth-First Search (23.2/22.2)
• BFS discovers all vertices at distance k before any vertices
at distance k+1.
• Initialization
– assigns  to all vertices: w(v)= 
– color all white
• Color s gray, w(s)=0, enqueue s in Q
• for the head u of Q
– for each v adjacent to u,
• d[v]=d[u]+1
• enqueue v in Q
– dequeue Q
– color u black
Breadth-First Search
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
Breadth-First Search (23.2/22.2)
• Run-time = O(V+E)
• Shortest-path tree
– the final weight is the minimum distance
– keep predecessors and get the shortest path
– all BFS shortest paths make a tree