CS 4407

Algorithms

Lecture 5:

Graphs—an Introduction

1

Prof. Gregory ProvanDepartment of Computer Science

University College Cork

Outline

� Motivation

– Importance of graphs for algorithm design

– applications

� Overview of algorithms to study

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� Overview of algorithms to study

� Review of basic graph theory

Today’s Learning Objectives

� Why graphs are useful throughout Computer

Science

� Range of applications is large

� We use some basic properties of graphs

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

Motivation

For theoreticians:

� Graph problems are neat, often difficult, hence interesting

For practitioners:

� Massive graphs arise in networking, web modelling, ...

� Problems in computational geometry can be expressed as

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� Problems in computational geometry can be expressed as

graph problems

� Many abstract problems best viewed as graph problems

� Extreme: Pointer-based data structures = graphs with extra

information at their nodes

Examples of Networks

communication

Network

telephone exchanges,computers, satellites

Nodes Arcs

cables, fiber optics,microwave relays

Flow

voice, video,packets

circuitsgates, registers,processors

wires current

mechanical joints rods, beams, springs heat, energy

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

mechanical joints rods, beams, springs heat, energy

hydraulicreservoirs, pumpingstations, lakes

pipelines fluid, oil

financial stocks, currency transactions money

transportationairports, rail yards,street intersections

highways, railbeds,airway routes

freight,vehicles,passengers

chemical sites bonds energy

Graph Algorithms Overview

� Standard graph algorithms

– Breadth-first search (BFS), Depth-first search

(DFS), heuristic algorithms

– Minimum Spanning Tree

– Shortest Path

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

– Shortest Path

– Max-Flow

– Tree-Decomposition Algorithms

• Convert arbitrary graphs to trees of cliques

Applications

�Networking– Internet, communication, transportation

�VLSI & logic circuit design�Graphics

– surface meshes in CAD/CAM

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

– surface meshes in CAD/CAM

�AI and Robotics applications– path planning for autonomous agents– precedence constraints in scheduling

Graphs

� A collection of vertices or nodes, connected by a collection of edges.

� Useful in many applications where there is some “connection” or “relationship” or “interaction” between pairs of objects.– network communication & transportation

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

– network communication & transportation– VLSI design & logic circuit design– surface meshes in CAD/CAM – path planning for autonomous agents– precedence constraints in scheduling

Basic Definitions

� A directed graph (or digraph) G = (V, E) consists of a finite set V, called vertices or nodes, and E, a finite set of ordered pairs, called edges of G. E is a binary relation on V. Cycles, including self-loops are allowed. Multiple edges are not allowed though; (v, w) and (w, v) are distinct edges.

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� An undirected graph (or simply a graph) G = (V, E) consists of a finite set V of vertices, and a finite set E of unordered pairs of distinct vertices, called edges of G. Cycles are allowed, but not self-loops. Multiple edges are not allowed.

Examples of Digraphs & Graphs

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

Figure B.2

Definitions

� Vertex v is adjacent to vertex u if there is an edge (u, v).� Given an edge e = (u, v) in an undirected graph, u and v are the

endpoints of e, and e is incident on u and on v.� In a digraph with edge e = (u, v), u and v are the origin and

destination. We say that e leaves u and enters v. � A digraph or graph is weighted if its edges are labeled with numeric

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� A digraph or graph is weighted if its edges are labeled with numeric values.

� In a digraph,

– the Out-degree of v is the number of edges coming from v.– the In-degree of v is the number of edges coming into v.

� In a graph, the degree of v is the number of edges incident to v. (The in-degree equals the out-degree).

Combinatorial Facts

� In a graph

• 0 ≤≤≤≤ |E | ≤≤≤≤ C(| V |, 2) = | V | (| V | – 1) / 2 ∈∈∈∈ O(| V | 2)• ∑∑∑∑ v∈∈∈∈V degree(v) = 2 | E |

� In a digraph

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� In a digraph

• 0 ≤≤≤≤ | E | ≤≤≤≤ | V | 2

• ∑∑∑∑ v∈∈∈∈V in-degree(v) = ∑∑∑∑ v∈∈∈∈V out-degree(v) = | E |

A graph is said to be sparse if | E | ∈∈∈∈ O(| V|), and denseotherwise.

Definitions (Path vs. Cycle)

� Path: a sequence of vertices <v0, …, vk> such that (vi-

1, vi) is an edge for i = 1 to k, in a digraph. The lengthof the path is the number of edges, k.

� w is reachable from u if there is a path from u to w. A path is simple if all vertices are distinct.

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

path is simple if all vertices are distinct.

� Cycle: a path in a digraph containing at least one edge and for which v0 = vk. A cycle is simple if, in addition, all vertices are distinct.

� For graphs, the definitions are the same, but a simple cycle must visit ≥≥≥≥ 3 distinct vertices.

Historical Terms For Cycles and Paths

� An Eulerian cycle is a cycle, not necessarily simple,

that visits every edge of a graph exactly once.

A Hamiltonian cycle (or path) is a cycle (path in a

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� A Hamiltonian cycle (or path) is a cycle (path in a

directed graph) that visits every vertex exactly

once.

Definitions (Connectivity)

� A graph is acyclic, if it contains no simple cycles.

� A graph is connected, if every one of its vertices can reach every other vertex. I.e., every pair of vertices is connected by a path.

� The connected components of a graph are equivalence classes of vertices under the “is reachable from” relation.

� A digraph is strongly connected, if every two vertices are reachable

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� A digraph is strongly connected, if every two vertices are reachable from each other.

� Graphs G = (V, E) and G’ = (V’, E’ ) are isomorphic, if ∃∃∃∃ a bijection f : V→→→→ V’ such that u, v∈∈∈∈E iff ( f(u), f(v)) ∈∈∈∈E’ .

Examples of Isomorphic Graphs

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

Figure B.3

Graphs, Trees, Forests

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

Free Tree Forest DAG Trees

DAGs versus Trees

� A tree is a digraph with a non-empty set of nodes such that:– There is exactly one node, the root, with in-degree of 0.

– Every node other than the root has in-degree 1.

– For every node a of the tree, there is a directed path from the root to a.

� Textbook (CLRS) suggests that a tree is an undirected graph, by association with free trees.

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

association with free trees.

� This is a valid approach, if you accept that the existence of the distinguished vertex (root) induces a direction on all the edges of the graph.

� However, we usually think of trees as being DAGs.

� Notice that a DAG may not be a tree, even if a root is designated.

Representing Graphs

� Assume V = {1, 2, …, n}

� An adjacency matrix represents the graph as

a n x n matrix A:

– A[i, j] = 1 if edge (i, j) ∈∈∈∈ E (or weight of edge)

= 0 if edge (i, j) ∉∉∉∉ E

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

= 0 if edge (i, j) ∉∉∉∉ E

Graphs: Adjacency Matrix

� Example:

1a

A 1 2 3 4

1

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

2 4

3

a

d

b c

2

3 ??4

Graphs: Adjacency Matrix

� Example:

1a

A 1 2 3 4

1 0 1 1 0

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

2 4

3

a

d

b c

2 0 0 1 0

3 0 0 0 0

4 0 0 1 0

Graphs: Adjacency Matrix

� How much storage does the adjacency matrix

require?

� A: O(V2)

� What is the minimum amount of storage

needed by an adjacency matrix

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

needed by an adjacency matrix

representation of an undirected graph with 4

vertices?

� A: 6 bits

– Undirected graph →→→→ matrix is symmetric

– No self-loops →→→→ don’t need diagonal

Graphs: Adjacency Matrix

� The adjacency matrix is a dense

representation

– Usually too much storage for large graphs

– But can be very efficient for small graphs

� Most large interesting graphs are sparse

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� Most large interesting graphs are sparse

– E.g., planar graphs, in which no edges cross,

have |E| = O(|V|) by Euler’s formula

– For this reason the adjacency list is often a more

appropriate respresentation

Graphs: Adjacency List

� Adjacency list: for each vertex v ∈∈∈∈ V, store a

list of vertices adjacent to v

� Example:

– Adj[1] = {2,3}

– Adj[2] = {3}1

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

– Adj[2] = {3}

– Adj[3] = {}

– Adj[4] = {3}

� Variation: can also keep

a list of edges coming into vertex

2 4

3

Graphs: Adjacency List

� How much storage is required?

– The degree of a vertex v = # incident edges

• Directed graphs have in-degree, out-degree

– For directed graphs, # of items in adjacency lists is

ΣΣΣΣ out-degree(v) = |E|

takes ΘΘΘΘ(V + E) storage (Why?)

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

takes ΘΘΘΘ(V + E) storage (Why?)

– For undirected graphs, # items in adj lists is

ΣΣΣΣ degree(v) = 2 |E| (handshaking lemma)

also ΘΘΘΘ(V + E) storage

� So: Adjacency lists take O(V+E) storage

Graph Representations

Let G = (V, E) be a digraph.� Adjacency Matrix: a |V | × |V |matrix for 1 ≤≤≤≤ v,w ≤≤≤≤ |V |

A[v, w] = 1, if (v, w) ∈∈∈∈ E and 0 otherwiseIf digraph has weights, store them in the matrix.

� Adjacency List: an array Adj[1…|V|] of pointers where for 1 ≤≤≤≤ v ≤≤≤≤|V |, Adj[v] points to a linked list containing the vertices adjacent to v. If the edges have weights then they may also be stored in the

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

to v. If the edges have weights then they may also be stored in the linked list elements.

� Incidence Matrix: a |V | × |E|matrix, B[i, j], of elements

bij = { -1, if edge j leaves vertex i }bij = { 1, if edge j enters vertex i }bij = { 0, otherwise }

– Note: must have no self-loops.

Example for Graphs

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

NOTE: it is common to include cross links between corresponding edges, when needed to mark the edges previously visited. E.g. (v,w) = (w,v).

Figure 22.1

Example for Digraphs

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

Figure 22.2

Lecture Summary

� Motivation for studying graphs

– Importance of graphs for algorithm design

– applications

� Overview of algorithms

CS 4407, AlgorithmsUniversity College Cork,

Gregory M. Provan

� Overview of algorithms

– Basic algorithms: DFS, BFS

– More advanced algorithms: flows, tree-decompositions

� Review of basic graph theory