CS 4407
Algorithms
Lecture 5:
Graphs—an Introduction
Prof. Gregory Provan
Department of Computer Science
University College Cork
1
Outline
Motivation
– Importance of graphs for algorithm design
– applications
Overview of algorithms to study
Review of basic graph theory
CS 4407, Algorithms
University College Cork,
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
CS 4407, Algorithms
University 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
graph problems
Many abstract problems best viewed as graph problems
Extreme: Pointer-based data structures = graphs with extra
information at their nodes
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Examples of Networks
Network
Nodes
Arcs
Flow
telephone exchanges, cables, fiber optics,
computers, satellites microwave relays
voice, video,
packets
gates, registers,
processors
wires
current
joints
rods, beams, springs
heat, energy
hydraulic
reservoirs, pumping
stations, lakes
pipelines
fluid, oil
financial
stocks, currency
transactions
money
airports, rail yards,
street intersections
highways, railbeds,
airway routes
freight,
vehicles,
passengers
sites
bonds
energy
communication
circuits
mechanical
transportation
chemical
CS 4407, Algorithms
University College Cork,
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
– Tree-Decomposition Algorithms
• Convert arbitrary graphs to trees of cliques
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Applications
Networking
– Internet, communication, transportation
VLSI & logic circuit design
Graphics
– surface meshes in CAD/CAM
AI and Robotics applications
– path planning for autonomous agents
– precedence constraints in scheduling
CS 4407, Algorithms
University College Cork,
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
CS 4407, Algorithms
University College Cork,
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.
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Examples of Digraphs & Graphs
Figure B.2
CS 4407, Algorithms
University College Cork,
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).
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
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
• 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 dense
otherwise.
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Definitions (Path vs. Cycle)
Path: a sequence of vertices <v0, …, vk> such that (vi1, 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.
CS 4407, Algorithms
University College Cork,
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.
CS 4407, Algorithms
University College Cork,
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
∈E iff ( f(u), f(v)) ∈E’.
bijection f : V→ V’ such that u, v∈
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Examples of Isomorphic Graphs
Figure B.3
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Graphs, Trees, Forests
Free Tree
Forest
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
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.
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.
CS 4407, Algorithms
University College Cork,
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
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Graphs: Adjacency Matrix
Example:
A
1
d
b
4
c
3
2
3
1
a
2
1
2
3
4
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
??
4
Graphs: Adjacency Matrix
Example:
1
a
d
2
b
4
c
3
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
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
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
representation of an undirected graph with 4
vertices?
A: 6 bits
– Undirected graph → matrix is symmetric
– No self-loops → don’t need diagonal
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
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
– 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
CS 4407, Algorithms
University College Cork,
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}
1
2
Variation: can also keep
3
a list of edges coming into vertex
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
4
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
Σ out-degree(v) = |E|
takes Θ(V + E) storage (Why?)
– For undirected graphs, # items in adj lists is
Σ degree(v) = 2 |E| (handshaking lemma)
also Θ(V + E) storage
So: Adjacency lists take O(V+E) storage
CS 4407, Algorithms
University College Cork,
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.
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Example for Graphs
Figure 22.1
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).
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Example for Digraphs
Figure 22.2
CS 4407, Algorithms
University College Cork,
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, treedecompositions
Review of basic graph theory
CS 4407, Algorithms
University College Cork,
Gregory M. Provan