GraphsRepresenting and Using Graphs

Graphs

What is a graph?

A graph is an abstract data structure used to represent a set of objects, some of which are connected by links.

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Graphs

Key Terminology

Vertex: The objects that a graph is composed of. e.g. A, B. Alternate name: Node

Edge: The links between the vertices. Alternate name: Arc Degree: The number of edges connected to a vertex. e.g. A

has degree 3, B degree 2. Neighbour: Vertex X is a neighbour of vertex Y if there is an

edge that directly links X and Y. Path: A sequence of edges leading from one vertex to

another.

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Graphs

Directed and Undirected

Graphs can be directed or undirected.

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Directed Graph (Digraph): A graph in which a direction is associated with each edge between a pair of vertices.

Undirected Graph: A graph in which no direction is associated with the edges.

Graphs

Weighted Graphs

Sometimes a value is associated with each edge in a graph. These values are known as weights and this type of graph is known as a weighted graph or labelled graph.

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Graphs

Trees A tree is a special type of graph which is

undirected, connected and has no cycles:

A rooted tree has one vertex designated as the root which is usually drawn at the top.

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A Not a tree as contains a cycle.

Not a tree as not connected.

Not a tree as directed.

Connected, undirected, no cycles: A TREE

Graphs

Practical Applications of Graphs

Generally want to find least distance / minimal cost.

Application Vertices Edges

Modelling the hyperlink structure of a website.

Web pages Hyperlinks

Map of road network used for route planning.

Locations Roads

Social analysis e.g. the relationships between people.

People Friendships

Mapping breeding/migration patterns in biology.

Locations Migration routes

Planning component layout on microchips to minimise chip size.

Components Wires

Computer network traffic routing. Routers Comms Lines

Map of rail network used for fare calculation.

Stations Railway Lines

Mapping a maze. Junctions Passageways

Graphs

Representing Graphs

Graphs can be represented in a programming language as a data structure.

The two standard methods of representing a graph are:

o Adjacency List

o Adjacency Matrix

Graphs

Adjacency Matrix

A 2-dimensional array (matrix) is used to represent the graph.

The array indices in each dimension run from 1 to n, where n is the number of vertices in the graph.

The value stored at position [ i , j ] is a count of how many edges there are that directly connect vertices i and j.

For a simple graph (undirected, no loops, no more than one edge between two vertices) this value will always be 0 or 1.

Graphs

Example Adjacency Matrices

1 2 3 4 5

1 0 1 1 1 1

2 1 0 0 0 0

3 1 0 0 0 0

4 1 0 0 0 1

5 1 0 0 1 0

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1 2 3 4 5

1 0 1 1 0 1

2 0 0 0 0 0

3 0 0 0 0 0

4 1 0 0 0 1

5 0 0 0 0 0

Undirected Graph

Directed Graph

Graphs

Adjacency List

For each vertex, a list is maintained of the adjacent vertices, i.e. those vertices that are directly connected to the vertex by an edge.

Adjacent vertices are also known as neighbours.

The adjacency list could be represented as a linked list.

Graphs

Example Adjacency Lists

Vertex

Adjacent

1 2,3,4,5

2 1

3 1

4 1,5

5 1,4

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Vertex

Adjacent

1 2,3,5

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4 1,5

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Undirected Graph

Directed Graph

Graphs

Are Lists or Matrices Better? In determining whether a list or matrix is

better, two factors need to be considered:oHow much storage space the data structure will

require.oHow quickly the data structure can be modified or

checked to e.g. Add a new edge or test if an edge exists between two vertices.

The best option depends upon how the list will be used.

Graphs

Are Lists or Matrices Better? An adjacency matrix is more appropriate

when:o there are many edges between vertices,owhen edges may be frequently changed,owhen presence/absence of specific edges needs

to be tested frequently. Adjacency list most appropriate when:o there are few edges between vertices (when

graph is sparse),owhen edges change infrequently,owhen presence/absence of specific edges does

not need to be tested frequently.

Graphs

Are Lists or Matrices Better? A commonly held misconception is that the number of

vertices in a graph determines whether a list or matrix is most appropriate because an adjacency matrix would take up too much storage space for a large number of vertices.

This is not true!o A graph with a large number of vertices and a small number of

edges could be represented in less storage space using an adjacency list as this would avoid storing lots of 0s indicating no edges that would be required in the equivalent matrix.

o A graph with a large number of vertices and a large number of edges could be represented in less storage space using an adjacency matrix as many edges would need to be represented and the matrix would avoid the overhead of storing pointers and node numbers that a list would require.

So, the influencing factor with regard to storage space is always the number of edges even for large numbers of vertices.

Graphs

Searching Graphs

A commonly used operation on a graph is to search for an item within the graph.

To do this a systematic way of exploring each vertex (without getting stuck) is required.

Two common methods of doing this are a depth first search and a breadth first search.

We will look at examples of these methods on the next few slides.

Each example explores the entire graph to illustrate the search order. In reality, for efficiency, an algorithm would terminate as soon as the item that was being searched for was found.

Graphs

Depth First Search

Philosophy: oTravel as far into a graph as possible,

backtracking to vertices which haven’t been explored yet only when a ‘dead end’ has been reached.

Process:oChoose a vertex to start from (usually the root

vertex if the graph is a rooted tree).oExplore as far as possible along each edge until a

vertex is reached from which it is no longer possible to follow an unexplored edge.

oBacktrack to the most recently visited node that hasn’t yet been fully explored.

oRepeat the process.

Graphs

Depth First Search (Tree)

This animation shows a depth first search of a rooted tree.

The search starts at the root node (1).

It is simpler to search a tree than a graph as there are no loops.

Depth first searching is often implemented using recursion.

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Graphs

Depth First Search (Graph) A graph that is not a

tree may contain loops (3453...) so algorithm must avoid getting trapped in these when searching.

Avoid this by using two flags to mark nodes as:o Discoveredo Completely Explored

Backtrack when a discovered node is encountered.

Watch the animation to see a search progress.

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Discovered Completely Explored

Graphs

Breadth First Search

Philosophy:o Look at the vertices that are closest to the starting vertex

first, exploring all of the neighbours of a vertex before going deeper into the graph.

Process:o Choose a vertex to start from.o Visit all of the vertices that are neighbours of this vertex

(and have not already been discovered).o Mark each vertex as discovered as soon as it is visited.o Then visit all of the vertices that are neighbours of this

first set of neighbours and so on. A queue can be used to keep track of vertices

that have been visited which still have neighbours that haven’t been explored yet.

Graphs

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Breadth First Search (Tree/Graph)

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This animation shows a breadth first search.

The queue stores the discovered nodes that are waiting to be visited.

The algorithm tracks discovered nodes so works for graphs as well as trees.

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Queue

Vertex currentlybeing processed

Discovered Currently Processing

Graphs

Shortest Path

Another common graphing problem is to find the shortest path between two vertices.

A shortest path is usually looked for in a weighted graph, where a cost is associated with traversing each edge.

For example, in a satellite navigation device, a road network might be represented by a graph where each vertex is a road junction and each edge is a road.

A time is associated with traversing each edge (road). The optimal route between two locations would be the

path which produced the lowest total time when the times associated with each edge on the route are added up.

Graphs

Dijkstra’s Algorithm

Dijkstra’s algorithm can be used to find the shortest path from a vertex to any other vertex.

The algorithm tries out every possible route. Sadly, a simple implementation of Dijkstra’s

algorithm has time complexity O(v2) where v is the number of vertices in the graph.

More sophisticated implementations are more time efficient but for large graphs, such as a map, it is not feasible to use Dijkstra’s algorithm.

Graphs

More Efficient Shortest Path Therefore, a more sophisticated algorithm such

as the A* algorithm must be used for large graphs like a satnav map.

Unlike Djkstra’s algorithm, the A* algorithm does not search the entire graph to find the shortest route, it uses a heuristic to disregard vertices which are unlikely to be of interest because they seem to lie in the wrong direction.

The A* algorithm is therefore far more time efficient than Dijkstra’s algorithm.

Graphs

Other Graph Problems to Explore

Euler Circuit Problem: Determine if a graph has an Euler circuit, i.e. a path from a vertex back to itself that traverses each edge exactly once.

Travelling Salesman: Given a list of cities and the distances between them, find the shortest possible tour that visits each city exactly once.

Graphs

End of Presentati

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Graphs

Trees or Not Solutions

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Not a tree as not connected. Not a tree as contains circuit.

Not a tree as contains circuit. A tree.

Q1)

Graphs

Adjacency Matrix Solution

Q2) Adjacency matrix for graph:

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A 0 1 0 1 1 0

B 1 0 1 0 1 0

C 0 1 0 0 0 0

D 1 0 0 0 1 0

E 1 1 0 1 0 0

F 0 0 0 0 0 0

Graphs

Adjacency List Solution

Vertex

Adjacent

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Q3) One possible drawing of the graph:

Graphs

Matrix or List Solution

Q4) An adjacency matrix is more appropriate when:

there are many edges between vertices,

when edges may be frequently changed,

when presence/absence of specific edges needs to be tested frequently.

Graphs

Depth First Search (Tree) Solution

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Q5) Search Order:

1 2 4 5 6 7 8 3

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Graphs

Depth First Search (Graph) Solution

Q6) Search Order:

1 2 5 6 7 4 3

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Graphs

Breadth First Search (Tree) Solution

Q7) Search Order:

1 2 4 3 5 6 7 8

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Graphs

Weighted Graphs Solution

Q8) Shortest Path:

C D A E B Length 10 + 12 + 7 + 10 = 39

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