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ITEC 2620M
Introduction to Data Structures
Instructor: Prof. Z. Yang
Course Website:
http://people.math.yorku.ca/~zyang/it
ec2620m.htm
Office: TEL 3049
Graphs
Key Points of this Lecture
• Graph Algorithms
– Definitions, representations, analysis
– Shortest paths
– Minimum-cost spanning tree
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Basic Definitions
• A graph G = ( V, E ) consists of a set of vertices V and a set
of edges E – each edge E connects a pair of vertices in V.
• Graphs can be directed or undirected
– redraw above with arrows – first vertex is source
• Graphs may be weighted
– redraw above with weights, combine definitions
• A vertex vi is adjacent to another vertex vj if they are
connected by an edge in E. These vertices are neighbours
• A path is a sequence of vertices in which each vertex is
adjacent to its predecessor and successor
• The length of a path is the number of edges in it
– The cost of a path is the sum of edge weights in the path
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Basic Definitions (Cont’d)
• A cycle is a path of length greater than one that begins and
ends at the same vertex
• A simple cycle is a cycle of length greater than three that
does not visit any vertex (except the start/finish) more than
once
• Two vertices are connected if there as a path between them
• A subset of vertices S is a connected component of G if
there is a path from each vertex vi to every other distinct
vertex vj in S
• The degree of a vertex is the number of edges incident to it
– the number of vertices that it is connected to
• A graph is acyclic if it has no cycles (e.g. a tree)
– A directed acyclic graph is called a DAG or digraph
(useful for unidirectional relationships
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Representations
• The adjacency matrix of graph G = ( V, E ) for
vertices numbered 0 to n-1 is an n x n matrix M
where M[i][j] is 1 if there is an edge from vi to
vi, and 0 otherwise.
• The adjacency list of graph G = ( V, E ) for
vertices numbered 0 to n-1 consists of an
array of n linked lists. The ith linked list
includes the node j if there is an edge from vi
to vi.
• Example
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Comparisons and Analysis
• Space
– adjacency matrix uses O(|V|2) space (constant)
– adjacency list uses O(|V| + |E|) space (note: pointer overhead)
• better for sparse graphs (graphs with few edges)
• Access Time
– Is there an edge connecting vi to vj?
• adjacency matrix – O(1)
• adjacency list – O(d)
– Visit all edges incident to vi
• adjacency matrix – O(n)
• adjacency list – O(d)
– Primary operation of algorithm and density of graph determines
more efficient data structure
• complete graphs should use adjacency matrix
• traversals of sparse graphs should use adjacency list
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Spanning Tree and Shortest Paths
• Minimum-Cost Spanning Tree
– assume weighted (undirected) connected graph
– use Prim’s algorithm (a greedy algorithm)
• from visited vertices, pick least-cost edge to an unvisited
vertex
• Shortest Paths
– assume weighted (undirected) connected graph
– use Dijkstra’s algorithm (a greedy algorithm)
• build paths from unvisited vertex with least current cost
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