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Decompositions of Graphs

A wide range of problems can be expressed with clarity and precision in the concise pictorial

language of graphs.

Consider the task of coloring a political map:

What is the minimum number of colors needed, with the obvious

restriction that neighboring countries should have different

colors?

The precise goal is now to assign a color to each vertex so that no edge has endpoints of the same

color.

2233

4455

66

88

77

99

1313

1111

1212

1010

11

Formally, a graph is specified by a set of vertices V and by edges E between select pairs

of vertices.

V = {1, 2, 3, ..., 13}

2233

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66

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1313

1111

1212

1010

11E = {{1, 2}, {1, 3},

{1, 4},..., {11, 12}, {11,

13}}

This is an undirected graph.

We can represent a graph by an adjacency matrix:

if there are |V| vertices v1, ..., vn, this is an n × n array whose (i, j)th

entry is

For undirected graphs, the matrix is symmetric since an edge {u, v} can be taken in either

direction.

We can represent a graph by an adjacency list:

It consists of |V| linked lists, one per vertex.

The linked list for vertex u holds the names of vertices to which u has an

outgoing edge.

Adjacency matrix:

-The presence of a particular edge can be checked in constant time.

-The matrix takes up O(V2) space, which is wasteful if the graph does not have very many edges.

Adjacency list:

-The presence of a particular edge can no longer be checked in constant time.

-The total size of the data structure is O(|E|).

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Depth-First Search in Undirected Graphs

Chalk prevents looping.

String always takes you back to the starting place, enabling you to return to

passages that you previously saw but did not investigate.

chalk == Boolean variable for each location

string == stack (recursion)

What parts of a graph are reachable from a given vertex?

AA BB EE

II JJGG HH CC FF

KK LL

DD

procedure explore(G, v) Input: G = (V, E) is a graph; v V∈Output: visited(u) is set to true for all nodes u reachable from v

visited(v) = trueprevisit(v)for each edge (v, u) E:∈

if not visited(u): explore(G, u)

postvisit(v)

Starting at A, can we reach J?

AA BB EE

II JJGG HH CC FF

KK LL

DD

Starting at A, can we reach L?

procedure dfs(G)for all v V:∈

visited(v) = false

for all v V:∈if not visited(v):

explore(v)

What is the running time of DFS?

Each vertex is explore’d just once.

During the exploration of a vertex:

Some fixed amount of work (marking the spot as visited and the

pre/postvisit).A loop in which adjacent edges are

scanned, to see if they lead somewhere new.

What is the running time of DFS?

This loop takes a different amount of time for each vertex, so let’s consider all vertices

together.Some fixed amount of work (marking

the spot as visited and the pre/postvisit).

A loop in which adjacent edges are scanned, to see if they lead somewhere

new.

O(|V|) - Constant amount of work per vertex

O(|E|) - Each edge {x, y} ∈ E is examined exactly twice.

What is the running time of DFS?

O(|V| + |E|)

Linear in the size of its input.

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Connectivity in Undirected Graphs

An undirected graph is connected if there is a path between any pair of vertices.

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What additional information would DFS give us if we defined pre/postvisit as

follows?

procedure previsit(v)pre[v] = clockclock = clock + 1

procedure postvisit(v)post[v] = clockclock = clock + 1

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Decompositions of Directed Graphs

DD

GG HH

AA CCBB

FFEE

Our DFS algorithm can be run verbatim on directed graphs, taking care to

traverse edges only in their prescribed directions.

DD

GG HH

AA CCBB

FFEE

AA 1

BB 2

EE 3

FF 4

GG 5,6

,7 HH 8,9

,10

,11 CC 12

DD 13,14

,15

,16

DD

GG HH

AA CCBB

FFEE

AA 1

BB 2

EE 3

FF 4

GG 5,6

,7 HH 8,9

,10

,11 CC 12

DD 13,14

,15

,16Other edges are either Back, Forward, or Cross edges:

F

F

BB

C

C

Other edges (u,v) are either Back, Forward, or Cross edges:

Forward edge: pre(u) < pre(v) < post(v) < post(u)(Same as tree edge, but unvisited)

Back edge: pre(v) < pre(u) < post(u) < post(v)

Cross edge: pre(v) < post(v) < pre(u) < post(u)

A cycle in a directed graph is a circular path.

v0 → v1 → v2 → … → vk → v0

A graph without cycles is acyclic.What algorithm tells if a graph is acyclic?

DD

GG HH

AA CCBB

FFEE

AA CC

DD

Direct acyclic graphs (dags) are quite common, especially in modeling

relationships.

For DAGs, we often want to order the nodes linearly (topological sort). All edges are

forward edges.

DD

GG HH

AA CCBB

FFEE

BBAA CC EEDD FF GGHH

What algorithm will give a topological sort?What is the running time?