Graph Data Structure-Notes

8/11/2019 Graph Data Structure-Notes

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Graph Data Structure

Recall the ADT binary tree: a tree structure used mainly to represent 1 to 2 relations,i.e. each item has at most two immediate successors.

Limitations of tree structures:

An item in a tree can appear in one position only An item has exactly one predecessor (parent node) unless it is the root (none) the

only available links are parent/child linksE.g. : Attempting to represent a transport link data with a tree structure:

Graph

A graph is a pair (V, E), where V is a set of nodes, called vertices and E is a collection

(can be duplicated) of pairs of vertices, called edges.

Vertices and edges are data structures and store elements

Example:

A vertex represents an airport and stores the three-letter airport code

An edge represents a flight route between two airports and stores the mileage of the route


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Applications

Electronic circuits

o Printed circuit board

o

Integrated circuit

Transportation networks

o Highway network

o Flight network

Computer networks

Local area network

Internet

Web

Databases

o Entity-relationship diagram


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Graph Terminology

A graphis a pair (V, E), where V is a set of nodes, called verticesand E is a collection (can be

duplicated) of pairs of vertices, called edges.Edges, also called arcs, are represented by (u, v) and are either:

Directed if the pairs are ordered (u, v)

u the originv the destination

Undirected if the pairs are unordered

Then a graph can be:

Directed graph (di-graph) if all the edges are directedUndirected graph (graph) if all the edges are undirected

Mixed graph if edges are both directed or undirected


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Path is a sequence of alternating vertices and edges such that each successive vertex isconnected by the edge. Frequently only the vertices are listed especially if there are no parallel

edges.PathOf length k from u to u' in G = (V, E) is a sequence of vertices < v0, v1, ..., vk > where u =

v0 and u' = vk, and (vi -1, vi) E, for i = 1, 2, ..., k |P| =Path Length - equals the number of edges in the path

There is always a 0-length path from u to u

Reachable - given u and u', u' is reachable from u if there exists a path P from u to

u'Simple path is a path with distinct vertices.

Directed path is a path of only directed edgesDirected cycle is a cycle of only directed edges.

Sub-graph is a subset of vertices and edges.All of these graphs are sub-graphs of the first graph.

Cycle is a path that starts and ends at the same vertex.In Directed Graphs:

a cycle path(circuit) is a path < v0, v1, ..., vk> such that v0= vkand |P| 1

asimple cycleis when v0, v1, ..., vkin P are distinct, does not contain the same

edge more than once. aself loopis a cycle of length 1, and a directed graph with no self loops is

referred to as simpleDirected Acyclic Graphs(DAG) - a directed graph with no cycles (circuits)

In Undirected Graphs:

a cycle path(circuit) is a path < v0, v1, ..., vk> such that v0= vkand |P| 1

asimple cycleis when v0, v1, ..., vkin P are distinct and |P| 3 (1,2,1) is not a simple cycle.

Undirected Acyclic Graph- an undirected graph with no simple cycles

(circuits)


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Simple graphs have noparallel edges or self-loops

Connected graph v/s Strongly connected graph v/s Complete graph

Connectedis usually associated with undirected graphs (two way edges): there is a path

between every two nodes.

Strongly connectedis usually associated with directed graphs (one way edges): there is aroutebetween every two nodes.

Complete graphsare undirected graphs where there is an edgebetween every pair ofnodes.


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Connected component is the maximal connected sub-graph of an unconnected graph.Following is a graph with three connected component.

Strongly connected components- if the directed graph is not strongly connected, then it has

strongly connected components where each of these strongly connected components are made upof a subset of vertices from V, and each component is strongly connected.

Main Methods of the Graph ADT

Constructor operations to construct a graph:

i. createGraph( )// post: creates an empty graph

ii. addNode(newItem)// post: adds newItem in a graph. No change if newItem already exists.

iii. addEdge(Node1, Node2)// post: adds an edge between two existing nodes (Node1 and Node2)

Predicate operations to test graph

i. isEmpty( )// post: determines whether a graph is empty

i i. isLink(Node1, Node2 )// post: returns true if edge (Node1, Node2) is present in the graph

Selector operations to select items of a queue

i. deleteNode( Node)

// post: remove a node and any edges between this node and other nodes.ii. deleteEdge(Node1, Node2 )// post: delete the edge between the two given nodes Node1, Node2.


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Graph Representations

While several representations for graphs are possible, we shall study only the three most

commonly used: adjacency matrices, adjacency lists and adjacency multi-lists. Once again, thechoice of a particular representation will depend upon the application one has in mind and the

functions one expects to perform on the graph.

Adjacency Matrix

Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let

the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i to vertex j.Adjacency matrix for undirected graph is always symmetric. Adjacency Matrix is also used to

represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j withweight w.

[A weighted graph is a digraph which has values attached to the directed edges. These valuesrepresent the costof travelling from one node to the next. The cost can be measured in many

terms, depending upon the application. For instance, the 3 digraphs below all represent the samething; a graph of airport terminals and flight lines between those terminals. The only difference is

the meaning of the weights of the edges: Graph A represents the flight distance betweenterminals; graph B represents the average flight time in minutes; and graph C represents the

dollar cost of a plane ticket between the hubs. Looking at graphs A and B, you can see that therelative costof a direct flight from hub 1 to 3 is cheaper in terms of time and miles than the more

roundabout route of travel from hub 1 to 2 to 3. However, in graph C, the ticket costs of aroundabout route are cheaper than the direct flight.]

The adjacency matrix representations for different graphs are as follows.


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Pros: Representation is easier to implement and follow. Removing an edge takes O(1) time.Queries like whether there is an edge from vertex u to vertex v are efficient and can be done

O(1).

Cons: Consumes more space O(V^2). Even if the graph is sparse (contains less number of

edges), it consumes the same space. Adding a vertex is O(V^2) time.

Adjacency List:

An array of linked lists is used. Size of the array is equal to number of vertices. Let the array bearray[]. An entry array[i] represents the linked list of vertices adjacent to the ith vertex. This

representation can also be used to represent a weighted graph. The weights of edges can bestored in nodes of linked lists. Following is adjacency list representation of the following graph.

Pros:Saves space O(|V|+|E|) . In the worst case, there can be C(V, 2) number of edges in a graph

thus consuming O(V^2) space. Adding a vertex is easier.

Cons:Queries like whether there is an edge from vertex u to vertex v are not efficient and can bedone O(V).


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Inverse Adjacency List

By name itself, it looks something opposite of adjacency list. So, in the inverse adjacency list,

again the graph is represented by n linked lists, but here the nodes in the list i represents thevertices which are adjacent from (not to) the vertex i. This representation is used to find in-

degree of a node.

Adjacency Multi-listsIn the adjacency list representation of an undirected graph each edge ( vi,vj) is represented by two

entries, one on the list for vi and the other on the list for vj. As we shall see, in some situations itis necessary to be able to determine the second entry for a particular edge and mark that edge as

already having been examined. This can be accomplished easily if the adjacency lists are actuallythere will be exactly one node, but this node will be in two lists, i.e., the adjacency lists for each

of the two nodes it is incident to. The node structure now becomes where

M is a one bit mark field that may be used to indicate whether or not the edge has been

examined. The storage requirements are the same as for normal adjacency lists except for theaddition of the mark bitM.


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Graph Traversal

BFS and DFS algorithm are to be covered from Fundamentals of Data Structure in C byHorowitz and Sahni

Application of DFS

Depth-first search (DFS) is an algorithm (or technique) for traversing a graph.

Following are the problems that use DFS as a building block

1) For an unweighted graph, DFS traversal of the graph produces the minimum spanningtree and all pair shortest path tree

2) Path FindingWe can specialize the DFS algorithm to find a path between two given vertices u and z.

i) Call DFS(G, u) with u as the start vertex.ii) Use a stack S to keep track of the path between the start vertex and the current vertex.

iii) As soon as destination vertex z is encountered, return the path as thecontents of the stack

3)

Topological Sorting(https://www.cs.usfca.edu/~galles/visualization/TopoSortDFS.html )

i) Call DFS(G, u) with u as the start vertex.ii) Note starting time of a node explored for each recursive call.

iii) Mark finish time when function call returns for each node and add node at the front oflist.

iv) Call DFS for all unvisited manner and repeat step i) to iii).v) Output list.
https://www.cs.usfca.edu/~galles/visualization/TopoSortDFS.htmlhttps://www.cs.usfca.edu/~galles/visualization/TopoSortDFS.htmlhttps://www.cs.usfca.edu/~galles/visualization/TopoSortDFS.htmlhttps://www.cs.usfca.edu/~galles/visualization/TopoSortDFS.html


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4) Solving puzzles with only one solution, such as mazes.

5) Finding Strongly Connected Components of a graph

A directed graph is strongly connected if there is a path between all pairs of vertices. Astrongly connected component (SCC) of a directed graph is a maximal strongly

connected subgraph. For example, there are 3 SCCs in the following graph.

We can find all strongly connected components in O(V+E) time using Kosarajusalgorithm. Following is detailed Kosarajus algorithm.

1)Create an empty stack S and do DFS traversal of a graph. In DFS traversal, aftercalling recursive DFS for adjacent vertices of a vertex, push the vertex to stack.

2)Reverse directions of all arcs to obtain the transpose graph.3)One by one pop a vertex from S while S is not empty. Let the popped vertex be v.

Take v as source and do DFS. The DFS starting from v prints strongly connectedcomponent of v.

Application of BFS

Breadth-first search (BFS) is an algorithm (or technique) for traversing a graph.

Following are the problems that use BFS as a building block

1) Finding Strongly Connected Components of a graph// Apply same tech described above, except use bfs algorithm instead of dfs

2) FordFulkerson method for computing the maximum flow in a flow network

3) Testing bipartiteness

A bipartite graph, also called a bigraph, is a set of graph vertices decomposed into

two disjoint sets such that no two graph vertices within the same set are adjacent.ORA Bipartite Graph is a graph whose vertices can be divided into two independent

sets, U and V such that every edge (u, v) either connects a vertex from U to V or a vertexfrom V to U. In other words, for every edge (u, v), either u belongs to U and v to V, or ubelongs to V and v to U. We can also say that there is no edge that connects vertices ofsame set.


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Following is a simple algorithm to find out whether a given graph is Birpartite ornot using Breadth First Search (BFS).1. Assign RED color to the source vertex (putting into set U).2. Color all the neighbors with BLUE color (putting into set V).3. Color all neighbors neighbor with RED color (putting into set U). 4. This way, assign color to all vertices such that it satisfies all the constraints of m waycoloring problem where m = 2.5. While assigning colors, if we find a neighbor which is colored with same color ascurrent vertex, then the graph cannot be colored with 2 vertices (or graph is not Bipartite)

4)

Finding single source shortest path based on path length.

Note: Now, discuss BFS and DFS advantage and disadvantage and common

differences.

Graph Data Structure-Notes

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