CS 312: Algorithm Analysis

Lecture #16: Strongly Connected Components

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Slides by: Eric Ringger, adapting figures from Dasgupta et al.

Objectives

Understand how to linearize a DAG

Introduce the idea of connectedness among vertices in Directed Graphs

Introduce the algorithm for finding Strongly Connected Components

Cycles

How do you detect a cycle in a directed graph?

Property: A directed graph has a cycle if and only if its depth-first search reveals a back edge.

Otherwise: a Directed Acyclic Graph (DAG)

Sources, Sinks, LinearizationDFS Search Forest:

Linearized DAG:

0. Define order; 1. DFS; 2. Read off post-order values in reverse order

Analysis:

Sources, Sinks, Linearization

Property: In a DAG, every edge leads to a vertex with a smaller post number.

Property: Every DAG has at least one source (a node with 0 in-degree).

Property: Every DAG has at least one sink (a node with 0 out-degree).

How to prove these?

Vertex

Many Kinds of Connectedness

Graphs

Undirected Directed

Edge Vertex EdgeConnected components

Biconnected components

Strongly connected components

Similar to biconnected components

Connectivity in Directed Graphs Two nodes u and v of a

directed graph are connected iff there is a path from u to v and a path from v to u.

Each vertex v is connected to itself. Defines an equivalence

relation!

This relation partitions V into disjoint sets that we call strongly connected components. Equivalence classes under

the above relation

Example

Meta-graph:

Meta-Graph

Meta-Graph

Property: Every directed graph is a DAG of its strongly connected components.

How would you find SCCs?

How would you find SCCs?

Idea 1: run DFS, identify back-edges to find cycles, post-process

Idea 2: run DFS, identify back-edges to find cycles, merge cycles How to analyze?

We’re going to go take a different direction …

Finding Strongly Connected Components

To understand the algorithm for finding SCCs, you need to understand the following: Pre and post numbering in DFS Meta-graph created by representing each

SCC with a single meta-node. The meta-graph is a DAG and could therefore

be linearized using DFS.

Finding SCCs

Goal: Given a directed graph, find all of the SCCs Big Idea:

A: Find a (vertex in a) sink SCC B: Work back to find the other SCCs

How do you find a sink SCC? Finding a sink (node) in a DAG is easy But what about when there are cycles?

Problem A seems hard, so solve problem A’ first: Finding a node in a source SCC of a directed graph could

be easier. A’: find a source SCC on the “reverse graph”

Reverse Graph

G To find a sink, identify a source SCC of the reverse graph GR!

GR

Metagraph of GR

Insights Property: The node that receives the highest post

number in a depth-first search must lie in a source strongly connected component.

Property: If C and C’ are strongly connected components, and there is an edge from a node in C to a node in C’, then the highest post number in C is bigger than the highest post number in C’.

Consequence: The strongly connected components can be linearized by arranging them in decreasing order of their post numbers.

Step 1

Reverse the graph to produce GR

Necessary to do the reversal explicitly?

Run depth-first search on GR

Problem 3.4 (ii)

Step 2 How do we continue once a vertex belonging to the sink

component has been discovered?

Recall: The strongly connected components can be linearized by arranging them in decreasing order of their post numbers.

Run DFS on G, keeping track of connected components Process the vertices beginning and breaking ties using the

decreasing order of their post numbers from step 1 Can neglect keeping pre and post numbers this time Can neglect back-, cross-, and forward- edges this time as

well

3 Questions

Is it correct? Relies on the correctness of the properties we

identified earlier How long does it take?

2 DFS passes O(||V|| + ||E||) – again!

Can we do better?

Assignment

HW #11: Practice using the SCC-finding algorithm

Study for Midterm (Study Guide) Testing Center: Tuesday-Thurs (10-12 July) One page of notes hand written or typed by you

Read 4.1-4.7 Shortest Paths Breadth First Search Dijkstra’s Algorithm