1 2/23/2016 ITCS 6114 Topological Sort Minimum Spanning Trees.

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ITCS 6114

Topological SortMinimum Spanning Trees

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Directed Acyclic Graphs

● A directed acyclic graph or DAG is a directed graph with no directed cycles:

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DFS and DAGs

● Argue that a directed graph G is acyclic iff a DFS of G yields no back edges:■ Forward: if G is acyclic, will be no back edges

○ Trivial: a back edge implies a cycle■ Backward: if no back edges, G is acyclic

○ Argue contrapositive: G has a cycle a back edge Let v be the vertex on the cycle first discovered, and u be the predecessor

of v on the cycle When v discovered, whole cycle is white Must visit everything reachable from v before returning from DFS-Visit() So path from uv is yellowyellow, thus (u, v) is a back edge

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Topological Sort

● Topological sort of a DAG:■ Linear ordering of all vertices in graph G such that

vertex u comes before vertex v if edge (u, v) G● Real-world example: getting dressed

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

Watch

Tie

Jacket

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

Watch

Tie

Jacket

Socks Underwear Pants Shoes Watch Shirt Belt Tie Jacket

Topological Sort Algorithm

Topological-Sort(){

Run DFSWhen a vertex is finished, output itVertices are output in reverse topological order

}

● Time: O(V+E)● Correctness: Want to prove that

(u,v) G uf > vf

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Correctness of Topological Sort

● Claim: (u,v) G uf > vf■ When (u,v) is explored, u is yellow

○ v = yellow (u,v) is back edge. Contradiction (Why?)○ v = white v becomes descendent of u vf < uf

(since must finish v before backtracking and finishing u)

○ v = black v already finished vf < uf

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Minimum Spanning Tree

● Problem: given a connected, undirected, weighted graph:

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● Problem: given a connected, undirected, weighted graph, find a spanning tree using edges that minimize the total weight

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● Which edges form the minimum spanning tree (MST) of the below graph?

H B C

G E D

F

A

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● Answer:

H B C

G E D

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Prim’s Algorithm

MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);

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Prim’s Algorithm


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Run on example graph

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Prim’s Algorithm


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Run on example graph

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Prim’s Algorithm


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Pick a start vertex r

r

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Prim’s Algorithm


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Red vertices have been removed from Q

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Prim’s Algorithm


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Red arrows indicate parent pointers

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Prim’s Algorithm


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Prim’s Algorithm


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Prim’s Algorithm


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Prim’s Algorithm


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Prim’s Algorithm


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Prim’s Algorithm


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Prim’s Algorithm


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Prim’s Algorithm


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Prim’s Algorithm


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Review: Prim’s Algorithm


What is the hidden cost in this code?

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MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; DecreaseKey(v, w(u,v));

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MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; DecreaseKey(v, w(u,v));

How often is ExtractMin() called?How often is DecreaseKey() called?

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What will be the running time?A: Depends on queue binary heap: O(E lg V) Fibonacci heap: O(V lg V + E)

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Single-Source Shortest Path

● Problem: given a weighted directed graph G, find the minimum-weight path from a given source vertex s to another vertex v■ “Shortest-path” = minimum weight ■ Weight of path is sum of edges■ E.g., a road map: what is the shortest path from

Chapel Hill to Charlottesville?

1 2/23/2016 ITCS 6114 Topological Sort Minimum Spanning Trees.

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Transcript of 1 2/23/2016 ITCS 6114 Topological Sort Minimum Spanning Trees.