This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Graphs
Definitions & Properties
Implementations
DepthFirst Search
Topological Sort
BreadthFirst Search
Weighted Graphs
SingleSource Shortest Path on DAGs
General SingleSource Shortest Path (Dijkstra’s Algorithm) CSE 2011
Prof. J. Elder  66  Last Updated: 4/1/10 2:37 PM Properties
Property 1 Notation v deg(v) V number of edges deg(v) Proof: each edge is counted
twice number of vertices E = 2E degree of vertex v Property 2
In an undirected graph with no
selfloops and no multiple
edges
E V (V Example
V = 4
E = 6
deg(v) = 3 1)/2 Proof: each vertex has degree
at most (V 1)
Q: What is the bound for a digraph? A: E V (V
CSE 2011
Prof. J. Elder 1)
 67  Last Updated: 4/1/10 2:37 PM Main Methods of the (Undirected) Graph ADT
Vertices and edges Update methods are positions insertVertex(o): insert a vertex
storing element o store elements insertEdge(v, w, o): insert an
edge (v,w) storing element o Accessor methods
endVertices(e): an array of the
two endvertices of e removeVertex(v): remove vertex
v (and its incident edges) opposite(v, e): the vertex
opposite to v on e removeEdge(e): remove edge e Iterator methods areAdjacent(v, w): true iff v and
w are adjacent
replace(v, x): replace element at
vertex v with x
replace(e, x): replace element at
edge e with x
CSE 2011
Prof. J. Elder  68  incidentEdges(v): edges
incident to v
vertices(): all vertices in the
graph
edges(): all edges in the graph Last Updated: 4/1/10 2:37 PM Running Time of Graph Algorithms Running time often a function of both V and E. For convenience, we sometimes drop the  .  in
asymptotic notation, e.g. O(V+E). CSE 2011
Prof. J. Elder  69  Last Updated: 4/1/10 2:37 PM Implementing a Graph (Simplified) Adjacency List CSE 2011
Prof. J. Elder  70  Adjacency Matrix Last Updated: 4/1/10 2:37 PM ...
View
Full
Document
This note was uploaded on 02/14/2012 for the course CSE 2011Z taught by Professor Elder during the Fall '11 term at York University.
 Fall '11
 Elder
 Data Structures, Sort

Click to edit the document details