Graph Representation (23.1/22.1)

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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

• 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 du ? 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

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Depth-First Search (23.3/22.3)

• Runtime of DFS O(VE)
• once per vertex
• once per edge
• Kinds of edges
• tree edge (gray to gray)
• back edge (gray to gray)
• forward edge (gray to black)
• cross edge (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

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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 fv
• sort according to finishing times
• Directed graph G is acyclic ? no back edges

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Breadth-First Search (23.2/22.2)

• BFS discovers all vertices at distance k before
  any vertices at distance k1.
• 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,
• dvdu1
• enqueue v in Q
• dequeue Q
• color u black

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Breadth-First Search (23.2/22.2)

• Run-time O(VE)
• Shortest-path tree
• the final weight is the minimum distance
• keep predecessors and get the shortest path
• all BFS shortest paths make a tree