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Transcript of data structures and algorithms 2marksunit 4

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UNIT IV
GRAPHS
2MARK QUESTION AND ANSWERS:
1. 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 an edge x E 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 outdegree 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.

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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? IT 2201 DATA STRUCTURES AND ALGORITHMS II IT
R.Barona, AP/IT
20
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

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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
21. Define graph traversals?
Traversing a graph is an efficient way to visit each vertex and edge exactly once.
22. List the two important key points of depth first search.
i) If path exists from one node to another node, walk across the edge exploring
the edge.
ii) If path does not exist from one specific node to any other node, return to the
previous node where we have been before backtracking.
23. What do you mean by breadth first search (BFS)?
BFS performs simultaneous explorations starting from a common point and
spreading out independently.
24. Differentiate BFS and DFS.
No. DFS BFS
1. Backtracking is possible from a
dead end
Backtracking is not possible

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2. Vertices from which exploration is
incomplete are processed in a
The vertices to be explored are
organized as a
3. Search is done in one particular
direction
The vertices in the same level are
maintained
25. What do you mean by tree edge?
If w is undiscovered at the time vw is explored, then vw is called a tree edge and
v becomes the parent of w.
26. What do you mean by back edge?
If w is the ancestor of v, then vw is called a back edge.
27. Define biconnectivity.
A connected graph G is said to be biconnected, if it remains connected after removal of
any one vertex and the edges that are incident upon that vertex. A connected graph is
biconnected, if it has no articulation points.
28. What do you mean by articulation point
If a graph is not biconnected, the vertices whose removal would disconnect the
graph are known as articulation points.
29. What do you mean by shortest path?
A path having minimum weight between two vertices is known as shortest path,
in which weight is always a positive number.
30. Define Activity node graph.
Activity node graphs represent a set of activities and scheduling constraints. Each node

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represents an activity (task), and an edge represents the next activity.
31. Define adjacency list.
Adjacency list is an array indexed by vertex number containing linked lists. Each node Vi
the ith array entry contains a list with information on all edges of G that leave Vi. It is used to
represent the graph related problems
DIAGRAMMATIC REPRESENTATIONS OF GRAPH:
Vertex
A vertex (also called a node) is a fundamental part of a graph. It can have a name, whichwe will call the key. A vertex may also have additional information. We will call thisadditional information the payload.
Edge
An edge (also called an arc) is another fundamental part of a graph. An edge connects twovertices to show that there is a relationship between them. Edges may be oneway or twoway. If the edges in a graph are all oneway, we say that the graph is a directed graph, ora digraph. The class prerequisites graph shown above is clearly a digraph since you musttake some classes before others.
Weight
Edges may be weighted to show that there is a cost to go from one vertex to another. Forexample in a graph of roads that connect one city to another, the weight on the edge mightrepresent the distance between the two cities.
With those definitions in hand we can formally define a graph. A graph can be represented
by G where G=(V,E). For the graph G, V is a set of vertices and E is a set of edges. Each edge isa tuple (v,w) where w,vV. We can add a third component to the edge tuple to represent a weight.
A subgraph s is a set of edges e and vertices v such that eEand vV.
shows another example of a simple weighted digraph. Formally we can represent this graph as the
set of six vertices:
V={V0,V1,V2,V3,V4,V5}
and the set of nine edges:
E={(v0,v1,5),(v1,v2,4),(v2,v3,9),(v3,v4,7),(v4,v0,1),(v0,v5,2),(v5,v4,8),(v3,v5,3),(v5,v2
,1)}

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Path
A path in a graph is a sequence of vertices that are connected by edges. Formally we would
define a path as w1,w2,...,wn such that (wi,wi+1)E for all 1in1. The unweighted
path length is the number of edges in the path, specifically n1. The weighted path length is
the sum of the weights of all the edges in the path. For example inFigure 2the path
from V3 to V1 is the sequence of vertices (V3,V4,V0,V1). The edges
are {(v3,v4,7),(v4,v0,1),(v0,v1,5)}.
Cycle
A cycle in a directed graph is a path that starts and ends at the same vertex. For example,
inFigure 2the path (V5,V2,V3,V5) is a cycle. A graph with no cycles is called an acyclic
graph. A directed graph with no cycles is called a directed acyclic graph or a DAG. We will
see that we can solve several important problems if the problem can be represented as a
DAG.
An Adjacency Matrix
One of the easiest ways to implement a graph is to use a twodimensional matrix. In this matrix
implementation, each of the rows and columns represent a vertex in the graph. The value that is
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stored in the cell at the intersection of row v and column w indicates if there is an edge from
vertex v to vertex w. When two vertices are connected by an edge, we say that they are adjacent
An Adjacency Matrix Representation for a Graph
An Adjacency ListA more spaceefficient way to implement a sparsely connected graph is to use an adjacency list. In
an adjacency list implementation we keep a master list of all the vertices in the Graph object and
then each vertex object in the graph maintains a list of the other vertices that it is connected to. In
our implementation of the Vertex class we will use a dictionary rather than a list where the dictionary
keys are the vertices, and the values are the weights.Figure 4illustrates the adjacency list
representation for the graph inFigure 2.
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An Adjacency List Representation of a Graph
The advantage of the adjacency list implementation is that it allows us to compactly represent a
sparse graph. The adjacency list also allows us to easily find all the links that are directly connected
to a particular vertex.
Topological Sorting
A topological sort takes a directed acyclic graph and produces a linear ordering of all its vertices
such that if the graph G contains an edge (v,w)then the vertex v comes before the vertex w in the
ordering. Directed acyclic graphs are used in many applications to indicate the precedence of
events. Making pancakes is just one example; other examples include software project schedules,
precedence charts for optimizing database queries, and multiplying matrices.
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Figure 33: A Graph G
Figure 34: Its Transpose GT
Dijkstras Algorithm
The algorithm we are going to use to determine the shortest path is called Dijkstras algorithm.
Dijkstras algorithm is an iterative algorithm that provides us with the shortest path from oneparticular starting node to all other nodes in the graph. Again this is similar to the results of a breadth
first search.
PROBLEM SOLVING:

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Figure 3: Tracing Dijkstras Algorithm
Figure 4: Tracing Dijkstras Algorithm

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Figure 5: Tracing Dijkstras Algorithm
Figure 6: Tracing Dijkstras Algorithm

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Figure 7: Tracing Dijkstras Algorithm
Figure 8: Tracing Dijkstras Algorithm

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Prims Spanning Tree Algorithm
Figure 11: Tracing Prims Algorithm

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Figure 12: Tracing Prims Algorithm
Figure 13: Tracing Prims Algorithm

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Figure 14: Tracing Prims Algorithm
Figure 15: Tracing Prims Algorithm

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Figure 16: Tracing Prims Algorithm
Figure 17: Tracing Prims Algorithm

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SummaryIn this chapter we have looked at the graph abstract data type, and some implementations of a
graph. A graph enables us to solve many problems provided we can transform the original problem
into something that can be represented by a graph. In particular, we have seen that graphs areuseful to solve problems in the following general areas.
Breadth first search for finding the unweighted shortest path. Dijkstras algorithm for weighted shortest path. Depth first search for graph exploration. Strongly connected components for simplifying a graph. Topological sort Minimum weight spanning trees