CS 4407 Algorithms Lecture 2: Graphs Reviegprovan/CS4407/2016/L2-Graphs.pdf · CS 4407 Algorithms...

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CS 4407

Algorithms

Lecture 2:

Graphs—Review

Prof. Gregory Provan

Department of Computer Science

University College Cork

CS 4407, Algorithms University College Cork,

Gregory M. Provan

Outline

Motivation

– Importance of graphs for algorithm design

– applications

Overview of algorithms to study

Review of basic graph theory


Gregory M. Provan

Today’s Learning Objectives

Why graphs are useful throughout Computer

Science

Range of applications is large

We use some basic properties of graphs


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

graph problems

Many abstract problems best viewed as graph problems

Extreme: Pointer-based data structures = graphs with extra

information at their nodes


Gregory M. Provan

Examples of Networks

communication

Network

telephone exchanges,

computers, satellites

Nodes Arcs

cables, fiber optics,

microwave relays

Flow

voice, video,

packets

circuits gates, registers,

processors wires current

mechanical joints rods, beams, springs heat, energy

hydraulic reservoirs, pumping

stations, lakes pipelines fluid, oil

financial stocks, currency transactions money

transportation airports, rail yards,

street intersections

highways, railbeds,

airway routes

freight,

vehicles,

passengers

chemical sites bonds energy


Gregory M. Provan

Graph Algorithms Overview

Standard graph algorithms

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

(DFS), heuristic algorithms

– Minimum Spanning Tree

– Shortest Path

– Max-Flow


Gregory M. Provan

Motivations for Definition

Algorithm/Concept Definition BFS, DFS Graph, adjacency structure,

directionality

MST Trees, directionality, Weighted graph

Network Flows Weighted graph

Greedy algorithms Graph, adjacency structure,

directionality

MapReduce Graph, adjacency structure,

directionality

NP-complete: Hamilton cycle Hamilton cycle, paths, cycles

TSP Paths, cycles

graph isomorphism Graph isomorphism


Gregory M. Provan

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


Gregory M. Provan

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.

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.


Gregory M. Provan

Examples of Digraphs & Graphs

Figure B.2


Gregory M. Provan

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 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).


Gregory M. Provan

Combinatorial Facts

In a graph

• 0 |E | C(| V |, 2) = | V | (| V | – 1) / 2 O(| V | 2)

• vV degree(v) = 2 | E |

In a digraph

• 0 | E | | V | 2

• vV in-degree(v) = vV out-degree(v) = | E |

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

otherwise.


Gregory M. Provan

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 length of 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.

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.


Gregory M. Provan

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

directed graph) that visits every vertex exactly

once.


Gregory M. Provan

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 from each other.

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


Gregory M. Provan

Examples of Isomorphic Graphs

Figure B.3


Gregory M. Provan

Graphs, Trees, Forests

Free Tree Forest DAG Trees


Gregory M. Provan

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.

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.


Gregory M. Provan

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


Gregory M. Provan

Graphs: Adjacency Matrix

Example:

1

2 4

3

a

d

b c

A 1 2 3 4

1

2

3 ?? 4


Gregory M. Provan


Example:

1

2 4

3

a

d

b c

A 1 2 3 4

1 0 1 1 0

2 0 0 1 0

3 0 0 0 0

4 0 0 1 0


Gregory M. Provan


How much storage does the adjacency matrix

require?

A: O(V2)

What is the minimum amount of storage

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


Gregory M. Provan


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

– 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 representation


Gregory M. Provan

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}

– Adj[3] = {}

– Adj[4] = {3}

Variation: can also keep

a list of edges coming into vertex

1

2 4

3


Gregory M. Provan

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

out-degree(v) = |E|

takes (V + E) storage (Why?)

– For undirected graphs, # items in adj lists is

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

also (V + E) storage

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


Gregory M. Provan

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 otherwise

If 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 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.


Gregory M. Provan

Example for Graphs

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


Gregory M. Provan

Example for Digraphs

Figure 22.2


Gregory M. Provan

Lecture Summary

Motivation for studying graphs

– Importance of graphs for algorithm design

– applications

Overview of algorithms

– Basic algorithms: DFS, BFS

– More advanced algorithms: flows, tree-decompositions

Review of basic graph theory

CS 4407 Algorithms Lecture 2: Graphs Reviegprovan/CS4407/2016/L2-Graphs.pdf · CS 4407 Algorithms...

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