graphs. definition zthe graph data structure Õset v of vertices Õcollection e of edges (pairs of...
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GRAPHS
Definition
The Graph Data Structureset V of verticescollection E of edges (pairs of vertices in
V)
Drawing of a Graphvertex <-> circle/ovaledge <-> line connecting the vertex
pair
Sample Uses
Airports and flightsPersons and acquaintance
relationshipsIntersections and streetsComputers and connections; the
Internet
Graph Example
Graph G=(V,E):V={a,b,c,d}, E={(a,b),(b,c),(b,d),(a,d)}
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ab
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Adjacency
Vertices connected by an edge are adjacent
An edge e is incident on a vertex v (and vice-versa) if v is an endpoint of e
Degree of a vertex v, deg(v)number of edges incident on vif directed graph, indeg(v) and outdeg(v)
Graph Properties
Graph G with n vertices and m edges
v in G deg(v) = 2m
v in G indeg(v) = v in G outdeg(v) = mm is O(n2)
m <= n(n-1)/2 if G is undirectedm <= n(n-1) if G is directed
Subgraphs
Graph H a subgraph of graph Gvertex set of H is a subset of vertex set of
Gedge set of H is a subset of edge set of G
Spanning subgraphvertex set of H identical to vertex set of G
Paths and Cycles
(simple) Pathsequence of distinct vertices such that
consecutive vertices are connected through edges
(simple) Cyclesame as path except that first and last
vertices are identicalDirected paths and cycles
edge directions matter
Connectivity
Connected graphthere is a path between any two
vertices
Graph Methods
Container methodsGeneral or Global methods
numVertices(),numEdges(),vertices(),edges()Directed Edge methods
indegree(v), inadjacentVertices(v), inincidentEdges(v)
Update methodsadding/removing vertices or edges
General Methods
numvertices()-returns the number of vertices in G
numedges()- returns the number of edges in G
vertices()- return an enumeration of the vertex positions in G
edges()- returns an enumeration of the edge positions in G
General Methods
Degree (v) - returns the degree of v
adjacentvertices(v)- returns an enumeration of the vertices adjacent to v
incidentedges(v)- returns an enumeration of the edges incident upon v
endvertices(e)- return an array of size 2 storing the end vertices of e
Directed Edges
Directededges()- return an enumeration of all directed edges
indegree(v)- return the indegree of v
outdegree(v)- return the outdegree of v
origin (v)- return the origin of directed edge e
other functions- inadjacentvertices(), outadjacentvertices(), destination()
Updating Graphs
Insertedge(v,w,o)- insert and return an undirected edge between vertices v and w storing the object o at this position
Insertvertex(o)- insert and return a new vertex storing the object o at this position
removevertex (v)- remove the vertex v and all incident edges
removeedge(e)- remove edge e
Implementation of a graph data structure
Graph Implementations
Edge Listvertices and edges are nodesedge nodes refer to incident vertex nodes
Adjacency Listsame as edge list but vertex nodes also
refer to incident edge nodesAdjacency Matrix
2D matrix of vertices; entries are edges
Implementation Examples
Illustrate the following graph using the different implementations
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c
1
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Edge List
1 542 3
A B DC
Adjacency List
1 542 3
A B DC
in out outoutoutin in in
1 45
4 3 2 1 53
2
Adjacency Matrix
a b c d
a
b
c
d
0
0
0
0
0 0
01
0
0
1
1 1
0
1
0
Implementation Examples-Query
Illustrate the following graph using the different implementations
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7
Comparing Implementations
Edge Listsimplest but most inefficient
Adjacency Listmany vertex operations are O(d) time
where d is the degree of the vertexAdjacency Matrix
adjacency test is O(1)addition/removal of a vertex: resize 2D
array
Graph Traversals
Traversal- A systematic procedure for exploring a graph by examining all of its vertices and edges
Consistency- rules to determine the order of visits (top-down, alphabetical,etc.)- make the path consistent with the actual graph (don’t change conventions)
Graph Traversals
Depth First Search (DFS)visit a starting vertex srecursively visit unvisited adjacent
verticesBreadth First Search (BFS)
visit starting vertex svisit unvisited vertices adjacent to s, then
visit unvisited vertices adjacent to them, and so on … (visit by levels)
DFS Traversal
Depth First Search- recursively visit (traverse) unvisited vertices
Dead End- reaching a vertex where all the incident edges go to a previously visited vertex
Back Track- If a dead-end vertex is reached, go to the previous vertex
Traversal Examples
DFS: a,b,c,d,h,e,f,g,l,k,j,i
f
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b
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ij k l
DFS Traversal
Path of the DFS traversal- path=a,b,c,d,h- h is a dead end vertex- back track to d,c,b,a and go to another adjacent (unvisited) vertex -> e- path=a,e,f,g,k,j,i- i is a dead end vertex - back track to all the way back to a (also a dead end vertex)
finished
BFS Traversal
Visit adjacent vertices by levelsdiscovery edges
- edges that lead to an unvisited vertex
cross edge- edges that lead to a previously visited vertex
Traversal Examples
BFS: a,b,e,i,c,f,j,d,g,k,h,l
f
a
b
e
d
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ij k l
DFS Algorithm
Algorithm DFS(v)Input: Vertex vOutput: Depends on the application
mark v as visited; perform processingfor each vertex w adjacent to v do if w is unmarked then DFS(w)
BFS Algorithm
Algorithm BFS(s)Input: Starting vertex sOutput: Depends on the application
// traversal algorithm uses a queue Q// loop terminates when Q is empty
BFS Algorithm continued
Q new Queue()mark s as visited and Q.enqueue(s)while not Q.isEmpty() v Q.dequeue() perform processing on v for each vertex w adjacent to v do if w is unmarked then mark w as visited and Q.enqueue(w)
Query
Let G be a graph whose vertices are the integers 1 through 8 and let the adjacent vertices of each vertex be given by the table below :Vertex Adjacent Vertices1 (2,3,4)2 (1,3,4)3 (1,2,4)4 (1,2,3,6)5 (6,7,8)6 (4,5,7)7 (5,6,8)8 (5,7)
Query
Assume that in the traversal of G, the adjacent vertices of a given vertex are returned in the same order as they are listed in the above table:
a) Draw Gb) Give the sequence of vertices of G visited
using a DFS traversal order starting at vertex 1
c) Give the sequence of vertices visited using a BFS traversal starting at vertex 1
Weighted GraphsOverview
Weighted Graphs
Graph G = (V,E) such that there are weights/costs associated with each edgew((a,b)): cost of edge (a,b)representation: matrix entries <-> costs
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3265
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e35
Problems on Weighted Graphs
Minimum-cost Spanning TreeGiven a weighted graph G, determine a
spanning tree with minimum total edge cost
Single-source shortest pathsGiven a weighted graph G and a source
vertex v in G, determine the shortest paths from v to all other vertices in G
Path length: sum of all edges in path
Weighted Graphs-Concerns
Minimum CostShortest Path
Weighted Graphs
Shortest Path- Dijkstra’s Algorithm
Minimum Spanning Trees- Kruskal’s Algorithm