ALG0183 Algorithms & Data Structures by Dr Andy Brooks

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ALG0183 Algorithms & Data Structures

Lecture 18The basics of graphs.

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Watch out for self-loops in graphs.

ALG0183 Algorithms & Data Structures by Dr Andy Brooks

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node or vertex or point of the graph

edge or arc or line of the graph

some terminologynode/hnúturarc/ör

ALG0183 Algorithms & Data Structures by Dr Andy Brooks

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n = 1

n = 2 n = 3

some terminology

A complete ,undirected graph has all possible edges.

n = 4

complete

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An n-vertex, complete, undirected graph has n(n-1)/2 edges.

(excluding self-loops)

ALG0183 Algorithms & Data Structures by Dr Andy Brooks

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London Tube Maphttp://www.tfl.gov.uk/assets/downloads/standard-tube-map.gif

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... graphs are everywhere in the real-world.

A vertex represents a tube station and might store the station name.An edge represents a railway route between tube stations.

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Áfangastaðir Flugfélags Íslands www.flugfelag.is

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... graphs are everywhere in the real-world.

A vertex represents an airport and might store the airport code.An edge represents a flight route between two airports and might store the flight time. But we might need two edges for each flight route because flying with and against the wind should be costed differently and/or because the airport taxes might be different.

destinations/ áfangastaðir

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Graph definitionhttp://www.itl.nist.gov/div897/sqg/dads/HTML/graph.html

Definition: A set of items connected by edges. Each item is called a vertex or node. Formally, a graph is a set of vertices and a binary relation between vertices, adjacency.

Formal Definition: A graph G can be defined as a pair (V,E), where V is a set of vertices, and E is a set of edges between the vertices E {(u,v) | u, ⊆v V}. If the graph is undirected, the adjacency relation defined by the ∈edges is symmetric, or E {{u,v} | u, v V} (sets of vertices rather than ⊆ ∈ordered pairs). If the graph does not allow self-loops, adjacency is irreflexive.

Specialization (... is a kind of me.)directed graph, undirected graph, acyclic graph, directed acyclic graph, planar graph, biconnected graph, connected graph, complete graph, dense graph, sparse graph, hypergraph, multigraph, labeled graph, weighted graph, tree.

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A graph G = (V,E)• G = (V,E)– V is the vertex set, E is the edge set.– Vertices are also called nodes and points.– Edges are also called arcs and lines.

• Each edge connects two different vertices.• An undirected edge has no orientation (u,v).– edge (u,v) is the same as edge (v,u)

• A directed edge has an orientation (u,v).– edge (u,v) is not the same as edge (v,u)

• Undirected graph => no oriented edge.• Directed graph (“digraph”) => every edge has an

orientation.

u v

u v

one-way street

one-way street/einstefnugata

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Some Graphs incomplete &directed

Number of edges in (a) <= n(n-1)/2Number of edges in (c) <= n(n-1)

Unconnected.incomplete &undirected

incomplete &undirected & unconnected

Sahni, © McGraw-Hill

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Some Spanning Trees Connected, undirected subgraphs (no cycles) that include all vertices of the original graph.

n vertices, n-1 edgesSahni, © McGraw-Hill

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Street map applicatione.g. guiding taxi drivers

If we associated lengths with each edge, we would have a weighted digraph and we could try to find the shortest way of getting from intersection i to intersection j.

Sahni, © McGraw-Hill

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Weighted graph exampleThe weights might represent the cost of upgrading roads or laying optical cables between cities.

Sahni, © McGraw-Hill

Question: Is 110 the minimum cost spanning tree? We need to use Kruskal's algorithm .

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• Edges incident on a vertex– a, d, and b are incident on V in (1) and (2)

• Adjacent vertices:– A vertex u is adjacent (or next) to v if there is an edge from v to u.– In (1), U is adjacent to V, but not vice versa.– In (2), U and V are adjacent to each other

• Degree of a vertex– X has degree 2 in (1)– X has degree 5 in (2)– X has in-degree 2 in (1)– X has out-degree 0 in (1)

• Parallel edges– h and i are parallel edges

• Self-loop– j is a self-loop

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some terminologybased on a slide by Tangming Yuan

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• Path– a sequence of vertices connected by

edges– each successive vertex is adjacent to its

predecessor• Simple path

– path such that all its vertices in the path are distinct

• Examples– P1=(V, X, Z) is a simple path– P2=(U, W, X, Y, W, V) is a path that is not

simple

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some terminologybased on a slide by Tangming Yuan

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• Cycle– a path in which the first and final

vertices are the same• Simple cycle

– cycle such that all its vertices and edges are distinct except the first and final vertices

• Examples– C1 = (V, X, Y, W, U, V) is a simple cycle– C2 = (U, W, X, Y, W, V, U) is a cycle that

is not simple

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some terminologybased on a slide by Tangming Yuan

A directed graph having no cycles is called a directed acyclic graph (DAG).

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adjacency-matrix representationhttp://www.itl.nist.gov/div897/sqg/dads/HTML/adjacencyMatrixRep.html

Definition: A representation of a directed graph with n vertices using an n × n matrix, where the entry at (i,j) is 1 if there is an edge from vertex i to vertex j; otherwise the entry is 0. A weighted graph may be represented using the weight as the entry. An undirected graph may be represented using the same entry in both (i,j) and (j,i) or using an upper triangular matrix. (Definition: A matrix that is only defined at (i,j) when i ≤ j.)

example by Dr Srinivasan Ramaswamy

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adjacency-list representationhttp://www.itl.nist.gov/div897/sqg/dads/HTML/adjacencyListRep.html

Definition: A representation of a directed graph with n vertices using an array of n lists of vertices. List i contains vertex j if there is an edge from vertex i to vertex j. A weighted graph may be represented with a list of vertex/weight pairs. An undirected graph may be represented by having vertex j in the list for vertex i and vertex i in the list for vertex j.

The adjacency-list representation is more compact for a sparse matrix. (Definition: A matrix that has relatively few non-zero (or "interesting") entries. It may be represented in much less than n × m space.)

“In most applications, however, a sparse graph is the norm.” Weisse.g. an airport does not have a direct flight route to every other airport...

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Note: Suppose we have a directed graph with four vertices. Here are adjacency-matrix and adjacency-list representations. The arrow (->) means a link in a list.

1 2 3 41 1 1 1 12 1 0 0 03 0 1 0 14 0 1 1 0

1 -> 1 -> 2 -> 3 -> 42 -> 13 -> 2 -> 44 -> 2 -> 3

Big-Oh(space) is|V|2

When all the edges are present |E| = |V|2.

Big-Oh(space) is |V|+|E| -> |E|Algorithms developed for sparse graphs typically use adjacency-list representations.

A graph G = (V,E)

quadratic

linearlist order “unimportant”

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Figure 14.1 A directed graphWeiss ©Addison Wesley

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The shortest path between V0 and V5 has two edges with a cost of 1+8=9.The weighted shortest path between V0 and V5 has three edges and a cost of 1+4+1=6.

example