CS2201 DATA STRUCTURES UNIT V

UNIT VGRAPHSSYLLABUS:

Definitions Topological sort breadth-first traversal - shortest-path algorithms minimum spanning tree Prim's and Kruskal's algorithms Depth-first traversal biconnectivity Euler circuits applications of graphs1.1 DEFINITIONS:

A graph G = (V, B) consists of a set of vertices, V and set of edges E.

Vertices are referred to as nodes and the arc between the nodes are referred to as Edges. Each edge is a pair (v, w) where v, w V. (i.e.) v = V1, w = V2

Fig. 5.1.1

Here V1,V2, V3, V4 are the vertices and (V1, V2), (V2, V3) (V3, V4) (V4, V1) (V2,V4)(V1,V3 ) are edges.

BASIC TERMINOLOGIES Directed Graph (or) Digraph

Directed graph is a graph which consists of directed edges, where each edge in E is unidirectional. It is also referred as Digraph. If(v, w) is a directed edge then

(v, w) # (w, v)

Fig. 5.1.2

Undirected Graph

An undirected graph is a graph, which consists of undirected edges. If(v, w) is an undirected edge then (v,w) = (w, v)

Fig. 5.1.3

Weighted Graph

A graph is said to be weighted graph if every edge in the graph is assigned a weight or value. It can be directed or undirected graph.

Fig. 5.1.4(a)

Fig. 5.1.4(b)

Complete Graph

A complete graph is a graph in which there is an edge between every pair of vertices. A complete graph with n vertices will haven (n - 1)/ 2 edges.

Fig. 5.1.5

Fig. 5.1.5(a) Vertices of a Graph

In fig. 5.1.5

Number of vertics is 4

Number of edges is 6

(i.e) There is a path from every vertex to every other vertex.

A complete digraph is a strongly connected graph.

Strongly Connected Graph

If there is a path from every vertex to every other vertex in a directed graph then it is said to be strongly connected graph. Otherwise, it is said to be weakly connected graph.

Fig. 5.1.6 Strongly Connected Graph Fig. 5.1.7 Weakly Connected Graph

Path

A path in a graph is a sequence of vertices w1,w2,,,,,,wn such that wi,wi+1 E for 1 Num (C) (ie) 7>3 /* if Low (W) >= Num (V) */ the v is an articulation Pt. Therefore C is an articulation point.

Low (E) = Num (D), Hence D is an articulation point.

1.7.2 NP - Complete Problems

A decision problem D is-said to be NP complete if 1. It belongs to class NP. 2. Every problem in NP is polynomially reducible to D.

A problem P1 can be reduced to P2 as follows

Provide a mapping so that any instance of P1 can be transformed to an instance of P2. Solve P2 and then map the answer back to the original. As an example, numbers are entered into a pocket calculator in decimal. The decimal numbers are converted to binary, and all calculations are performed in binary. Then the final answer is converted back to decimal for display. For P1 to be polynomially reducible to P2 all the work associated with the transformations must be performed in polynomial time.

Hamilton Cycle Problem transformed to travelling salesman problem.

Some of the examples of NP - complete problems are Hamilton circuit, Travelling salesman, knapsack, graph coloring, Bin packing and partition problem.

1.8 Euler circuits:

A popular puzzle is to reconstruct these figures using a pen, drawing each line exactly once. The pen may not be lifted from the paper while the drawing is being performed. As an extra challenge, make the pen finish at the same point at which it started. This puzzle has a surprisingly simple solution. Stop reading if you would like to try to solve it.

The first figure can be drawn only if the starting point is the lower left- or right-hand corner, and it is not possible to finish at the starting point. The second figure is easily drawn with the finishing point the same as the starting point, but the third figure cannot be drawn at all within the parameters of the puzzle.

We can convert this problem to a graph theory problem by assigning a vertex to each intersection. Then the edges can be assigned in the natural manner.

After this conversion is performed, we must find a path in the graph that visits every edge exactly once. If we are to solve the "extra challenge," then we must find a cycle that visits every edge exactly once. This graph problem was solved in 1736 by Euler and marked the beginning of graph theory. The problem is thus commonly referred to as an Euler path (sometimes Euler tour) or Euler circuit problem, depending on the specific problem statement. The Euler tour and Euler circuit problems, though slightly different, have the same basic solution.

Conversion of puzzle to graphThe first observation that can be made is that an Euler circuit, which must end on its starting vertex, is possible only if the graph is connected and each vertex has an even degree (number of edges). This is because, on the Euler circuit, a vertex is entered and then left. If any vertex v has odd degree, then eventually we will reach the point where only one edge into v is unvisited, and taking it will strand us at v. If exactly two vertices have odd degree, an Euler tour, which must visit every edge but need not return to its starting vertex, is still possible if we start at one of the odd-degree vertices and finish at the other. If more than two vertices have odd degree, then an Euler tour is not possible.

The observations of the preceding paragraph provide us with a necessary condition for the existence of an Euler circuit. It does not, however, tell us that all connected graphs that satisfy this property must have an Euler circuit, nor does it give us guidance on how to find one. It turns out that the necessary condition is also sufficient. That is, any connected graph, all of whose vertices have even degree, must have an Euler circuit. Furthermore, a circuit can be found in linear time.

We can assume that we know that an Euler circuit exists, since we can test the necessary and sufficient condition in linear time. Then the basic algorithm is to perform a depth-first search. There is a surprisingly large number of "obvious" solutions that do not work. The main problem is that we might visit a portion of the graph and return to the starting point prematurely. If all the edges coming out of the start vertex have been used up, then part of the graph is untraversed. The easiest way to fix this is to find the first vertex on this path that has an untraversed edge, and perform another depth-first search. This will give another circuit, which can be spliced into the original. This is continued until all edges have been traversed.As an example, consider the graph in

Graph for Euler circuit problem

It is easily seen that this graph has an Euler circuit. Suppose we start at vertex 5, and traverse the circuit 5, 4, 10, 5. Then we are stuck, and most of the graph is still untraversed.

Graph remaining after 5, 4, 10, 5

We then continue from vertex 4, which still has unexplored edges. A depth-first search might come up with the path 4, 1, 3, 7, 4, 11, 10, 7, 9, 3, 4. If we splice this path into the previous path of 5, 4, 10, 5, then we get a new path of 5, 4, 1, 3, 7 ,4, 11, 10, 7, 9, 3, 4, 10, 5.

Graph after the path 5, 4, 1, 3, 7, 4, 11, 10, 7, 9, 3, 4, 10, 5Notice that in this graph all the vertices must have even degree, so we are guaranteed to find a cycle to add. The remaining graph might not be connected, but this is not important. The next vertex on the path that has untraversed edges is vertex 3. A possible circuit would then be 3, 2, 8, 9, 6, 3. When spliced in, this gives the path 5, 4, 1, 3, 2, 8, 9, 6, 3, 7, 4, 11, 10, 7, 9, 3, 4, 10, 5.

Graph remaining after the path 5, 4, 1, 3, 2, 8, 9, 6, 3, 7, 4, 11, 10, 7, 9, 3, 4, 10, 5On this path, the next vertex with an untraversed edge is 9, and the algorithm finds the circuit 9, 12, 10, 9. When this is added to the current path, a circuit of 5, 4, 1, 3, 2, 8, 9, 12, 10, 9, 6, 3, 7, 4, 11, 10, 7, 9, 3, 4, 10, 5 is obtained. As all the edges are traversed, the algorithm terminates with an Euler circuit.To make splicing simple, the path should be maintained as a linked list. To avoid repetitious scanning of adjacency lists, we must maintain, for each adjacency list, a pointer to the last edge scanned. When a path is spliced in, the search for a new vertex from which to perform the next dfs must begin at the start of the splice point. This guarantees that the total work performed on the vertex search phase is O(|E|) during the entire life of the algorithm. With the appropriate data structures, the running time of the algorithm is O(|E| + |V|).1.9 Applications of Graphs:Graphs are among the most ubiquitous models of both natural and human-made structures. They can be used to model many types of relations and process dynamics in physical, biological and social systems. Many problems of practical interest can be represented by graphs.

In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. One practical example: The link structure of a website could be represented by a directed graph. The vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B. A similar approach can be taken to problems in travel, biology, computer chip design, and many other fields. The development of algorithms to handle graphs is therefore of major interest in computer science. There, the transformation of graphs is often formalized and represented by graph rewrite systems. They are either directly used or properties of the rewrite systems (e.g. confluence) are studied. Complementary to graph transformation systems focussing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction-safe, persistent storing and querying of graph-structured data.

Graph-theoretic methods, in various forms, have proven particularly useful in linguistics, since natural language often lends itself well to discrete structure. Traditionally, syntax and compositional semantics follow tree-based structures, whose expressive power lies in the Principle of Compositionality, modeled in a hierarchical graph. Within lexical semantics, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; semantic networks are therefore important in computational linguistics. Still other methods in phonology (e.g. Optimality Theory, which uses lattice graphs) and morphology (e.g. finite-state morphology, using finite-state transducers) are common in the analysis of language as a graph. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as TextGraphs, as well as various 'Net' projects, such as WordNet, VerbNet, and others.

Graph theory is also used to study molecules in chemistry and physics. In condensed matter physics, the three dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. For example, Franzblau's shortest-path (SP) rings. In chemistry a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds. This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.

Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore diffusion mechanisms, notably through the use of social network analysis software.

Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist (or habitats) and the edges represent migration paths, or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species.

In mathematics, graphs are useful in geometry and certain parts of topology, e.g. Knot Theory. Algebraic graph theory has close links with group theory.

A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. For example if a graph represents a road network, the weights could represent the length of each road.

A digraph with weighted edges in the context of graph theory is called a network. Network analysis have many practical applications, for example, to model and analyze traffic networks. Applications of network analysis split broadly into three categories:

1. First, analysis to determine structural properties of a network, such as the distribution of vertex degrees and the diameter of the graph. A vast number of graph measures exist, and the production of useful ones for various domains remains an active area of research.

2. Second, analysis to find a measurable quantity within the network, for example, for a transportation network, the level of vehicular flow within any portion of it.

3. Third, analysis of dynamical properties of networks.UNIT-V GRAPHS PART A1.Define Graph?

A graph G consist of a nonempty set V which is a set of nodes of the graph, a set E which is the set of edges of the graph, and a mapping from the set for edge E to a set of pairs of elements of V. It can also be represented as G=(V, E).

2. Define adjacent nodes?

Any two nodes which are connected by an edge in a graph are called adjacent nodes. For example, if and edge xE is associated with a pair of nodes (u,v) where u, v V, then we say that the edge x connects the nodes u and v.

3. What is a directed graph?

A graph in which every edge is directed is called a directed graph.

4. What is a undirected graph?

A graph in which every edge is undirected is called a directed graph.

5. What is a loop?

An edge of a graph which connects to itself is called a loop or sling.

6.What is a simple graph?

A simple graph is a graph, which has not more than one edge between a pair of nodes than such a graph is called a simple graph.

7. What is a weighted graph?

A graph in which weights are assigned to every edge is called a weighted graph.

8. Define out degree of a graph?

In a directed graph, for any node v, the number of edges which have v as their initial node is called the out degree of the node v.

9. Define indegree of a graph?

In a directed graph, for any node v, the number of edges which have v as their terminal node is called the indegree of the node v.

10. Define path in a graph?

The path in a graph is the route taken to reach terminal node from a starting node.

11.What is a simple path?

A path in a diagram in which the edges are distinct is called a simple path. It is also called as edge simple.

12. What is a cycle or a circuit?

A path which originates and ends in the same node is called a cycle or circuit.

13. What is an acyclic graph?

A simple diagram which does not have any cycles is called an acyclic graph.

14. What is meant by strongly connected in a graph?

An undirected graph is connected, if there is a path from every vertex to every other vertex. A directed graph with this property is called strongly connected.

15. When is a graph said to be weakly connected?

When a directed graph is not strongly connected but the underlying graph is connected, then the graph is said to be weakly connected.

16. Name the different ways of representing a graph?

a. Adjacency matrix

b. Adjacency list

17. What is an undirected acyclic graph?

When every edge in an acyclic graph is undirected, it is called an undirected acyclic graph. It is also called as undirected forest.

18. What are the two traversal strategies used in traversing a graph?

a. Breadth first search

b. Depth first search

19. What is a minimum spanning tree?

A minimum spanning tree of an undirected graph G is a tree formed from graph edges that connects all the vertices of G at the lowest total cost.

20. Name two algorithms two find minimum spanning tree

_ Kruskals algorithm

_ prims algorithm

PART B

(BIG QUESTIONS)

1.Explain Shortest path algorithm with example? Refer Page No: 14 to 222.Explain Depth first and breadth first traversal? Refer Page No: 30 to 373.Explain spanning and minimum spanning tree? Refer Page No: 19 to 294.Explain Kruskals and prims algorithm? Refer Page No: 21 to 295.Explain topological sorting? Refer Page No: 7 to 96. What is Topological Sort? Write down the pseudocode to perform topological sort and apply the same to the following graph.

7. Explain the Dijkstras algorithm and find the shortest path from A to all other vertices in the following graph.

8. Explain prims algorithm in detail and find the minimum spanning free for the following graph.

9. Find all the articulation Points in the given graph. Show the depth first spanning tree and the values of Num and Low for each vertex.

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