Network Science Depth-First Search Joao Meidanis University of - - PowerPoint PPT Presentation

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Network Science Depth-First Search Joao Meidanis University of Campinas, Brazil March 24, 2020 Summary Depth-First Search (DFS) Algorithm 1 Example 2 Applications 3 Meidanis (Unicamp) Network Science March 24, 2020 2 / 24 Depth-First

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Network Science

Depth-First Search Joao Meidanis

University of Campinas, Brazil

March 24, 2020

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Summary

1

Depth-First Search (DFS) Algorithm

2

Example

3

Applications

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Depth-First Search (DFS) Algorithm

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Depth-First Search

Needs: adjacency lists Provides: edge classification cycle detection topological sort

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Algorithm — Core

function DFS-Visit(u, Adj) for v in Adj[u] do if v not in parent then parent[v] ← u DFS-Visit(v, Adj) end if end for end function

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Algorithm — Main

function DFS(V, Adj) parent ← {} for v in V do if v not in parent then parent[v] ← None DFS-Visit(v, Adj) end if end for end function

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Example

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

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Example: directed graph a b c d e f g

Running time O(V + E) linear time

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Applications

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Edge classification

Depends on DFS itself, not just graph Types of edges: tree edges forward edges backward edges cross edges

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Algorithm additions

Starting and ending times

useful to classify edges

forward edges: u → v with u v backward edges: u → v with u v cross edges: u → v with u v impossible:

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Undirected graph

no forward edges no cross edges

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Cycle detection

Graph has a cycle ⇐ ⇒ DFS has a backward edge

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Topological sorting

Premises: Acyclic graphs Tasks that depend on one another Results: Topological sort: Safe order for the tasks DFS: reverse order of finishing times

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Cycle detection

DFS has a backward edge = ⇒ Graph has a cycle backward edge: u → v with u v

a b c d e f g

u starts while v active = ⇒ there is a path from v to u

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Cycle detection

Graph has a cycle = ⇒ DFS has a backward edge

vk v0 v1 v2 v3

v0: first visited vertex in cycle v1, v2, v3, . . . , vk: start after v0 v1, v2, v3, . . . , vk: start before v0 finishes Therefore, v0 vk and vk → v0 is a backward edge

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Topological sorting

Example: getting dressed socks shoes trousers belt shirt underwear socks → shoes underwear → trousers shirt → trousers trousers → belt trousers → shoes trousers socks shoes belt underwear shirt

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Topological sorting

Example: getting dressed trousers socks shoes belt underwear shirt shoes, socks, belt, trousers, underwear, shirt

✛

belt, shoes, trousers, shirt, socks, underwear

✛

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