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CS2251 DESIGN AND ANALYSIS OF ALGORITHMS
Question Bank
UNIT - I1.What is an algorithm?
An algorithm is a sequence of unambiguous instructions for
solving a problem, i.e., for obtaininga required output for any
legitimate input in finite amount of time.
2. What are the three different algorithms used to find the gcd
of two numbers?
The three algorithms used to find the gcd of two numbers are
1. Euclids algorithm
2. Consecutive integer checking algorithm
3. Middle school procedure
3.What are the fundamental steps involved in algorithmic problem
solving?
The fundamental steps are
1 .Understanding the problem
2. Ascertain the capabilities of computational device
3. Choose between exact and approximate problem solving
4. Decide on appropriate data structures
5. Algorithm design techniques
6. Methods for specifying the algorithm
7. Proving an algorithms correctness
8. Analyzing an algorithm
9 Coding an algorithm
4. What is an algorithm design technique?
An algorithm design technique is a general approach to solving
problems algorithmically that isapplicable to a variety of problems
from different areas of computing.
5. What is pseudocode?
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A pseudocode is a mixture of a natural language and programming
language constructs tospecify an algorithm. A pseudocode is more
precise than a natural language and its usage oftenyields more
concise algorithm descriptions.
5. What are the types of algorithm efficiencies?
The two types of algorithm efficiencies are
1. Time efficiency: indicates how fast the algorithm runs
2.Space efficiency: indicates how much extra memory the
algorithm needs
6. Mention some of the important problem types?
Some of the important problem types are as follows
1.Sorting
2. Searching
3. String processing
4. Graph problems
5. Combinatorial problems
6. Geometric problems
7. Numerical problems
7. What are the classical geometric problems?
The two classic geometric problems are
1.The closest pair problem: given n points in a plane find the
closest pair among them
2. The convex hull problem: find the smallest convex polygon
that would include all the pointsof a given set.
8. What is the basic operation of an algorithm and how is it
identified?
The most important operation of the algorithm is called the
basic operation of the algorithm, theoperation that contributes the
most to the total running time. It can be identified easily because
itis usually the most time consuming operation in the algorithms
innermost loop.
9. What is the running time of a program implementing the
algorithm?
The running time T(n) is given by the following formula
T(n) copC(n)
cop is the time of execution of an algorithms basic operation on
a particular computer and C(n)is the number of times this operation
needs to be executed for the particular algorithm.
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10. What are exponential growth functions?
The functions 2n and n! are exponential growth functions,
because these two functions grow sofast that their values become
astronomically large even for rather smaller values of n.
11. What is worst-case efficiency?
The worst-case efficiency of an algorithm is its efficiency for
the worst-case input of size n,which is an input or inputs of size
n for which the algorithm runs the longest among all possibleinputs
of that size.
12. What is best-case efficiency?
The best-case efficiency of an algorithm is its efficiency for
the best-case input of size n, whichis an input or inputs for which
the algorithm runs the fastest among all possible inputs of that
size
13. What is average case efficiency?
The average case efficiency of an algorithm is its efficiency
for an average case input of size n. Itprovides information about
an algorithm behavior on a typical or random input.
14. What is amortized efficiency?
In some situations a single operation can be expensive, but the
total time for the entire sequenceof n such operations is always
significantly better that the worst case efficiency of that
singleoperation multiplied by n. this is called amortized
efficiency.
15. Define O-notation?
A function t(n) is said to be in O(g(n)), denoted by t(n)
O(g(n)), if t(n) is bounded above bysome constant multiple of g(n)
for all large n, i.e., if there exists some positive constant c
andsome nonnegative integer n0 such that
T(n) cg(n) for all n n0
16. Define -notation?
A function t(n) is said to be in (g(n)), denoted by t(n) (g(n)),
if t(n) is bounded below bysome constant multiple of g(n) for all
large n, i.e., if there exists some positive constant c andsome
nonnegative integer n0 such that
T(n) cg(n) for all n n0
17. Define -notation?
A function t(n) is said to be in (g(n)), denoted by t(n) (g(n)),
if t(n) is bounded both above& below by some constant multiple
of g(n) for all large n, i.e., if there exists some
positiveconstants c1 & c2 and some nonnegative integer n0 such
that c2g(n) t(n) c1g(n) for all n n0
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18. Mention the useful property, which can be applied to the
asymptotic notations and its use?
If t1(n) O(g1(n)) and t2(n) O(g2(n)) then t1(n)+t2(n) max
{g1(n),g2(n)} this property is also true for and notations. This
property will be useful in analyzing algorithms that comprise o
19. What is the recurrence relation to find out the number of
multiplications and
the initial condition for finding the n-th factorial number?
The recurrence relation and initial condition for the number of
multiplications Is M(n)=M(n-1)+1for n>0
M(0)=0f two consecutive executable parts.
20. Write the general plan for analyzing the efficiency for
recursive algorithms.
The various steps include ._
1.Decide on a parameter indicating inputs size.
2.Identify the algorithms basic operation.
3.Check whether the number of times the basic operation is
executed
depends on size of input. If it depends on some additional
property the
worst, average and best-case efficiencies have to be
investigated
separately.
4.Set up a recurrence relation with the appropriate initial
condition , for
the number of times the basic operation is executed.
5.Solve the recurrence or at least ascertain the orders of
growth of its solution.
16 MARKS
1.Discuss the Asymptotic notations with efficiency classes and
examples
2. (a). Define the asymptotic notations used for best case
average case and worst case
analysis of algorithm. (8)
(b)Write an algorithm for finding maximum element of an array;
perform best and
average case complexity with appropriate order notations.
(8)
3. Write an algorithm to find mean and variance of an array
perform best, worst and
average case complexity, defining the notations used for each
type of analysis.
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4.Using the recursive algorithm ,explain how the tower of Hanoi
problem can be solved. What will be timea nd space complexity for
the algorithms.
5. Explain linear search with example
UNIT-II
1.Give the general plan for divide-and-conquer algorithms.
The general plan is as follows
1. A problems instance is divided into several smaller instances
of the same problem, ideallyabout the same size
2. The smaller instances are solved, typically recursively
3. If necessary the solutions obtained are combined to get the
solution of the original problem
2.What is the general divide-and-conquer recurrence
relation?
An instance of size n can be divided into several instances of
size n/b, with a of them needingto be solved. Assuming that size n
is a power of b, to simplify the analysis, the followingrecurrence
for the running time is obtained:
T(n) = aT(n/b)+f(n)
Where f(n) is a function that accounts for the time spent on
dividing the problem into smallerones and on combining their
solutions.
3. Define mergesort.
Mergesort sorts a given array A[0..n-1] by dividing it into two
halves a[0..(n/2)-1] and A[n/2..n-1] sorting each of them
recursively and then merging the two smaller sorted arrays into a
singlesorted one.
4. What is the difference between quicksort and mergesort?
Both quicksort and mergesort use the divide-and-conquer
technique in which the given array ispartitioned into subarrays and
solved. The difference lies in the technique that the arrays
arepartitioned. For mergesort the arrays are partitioned according
to their position and in quicksortthey are partitioned according to
the element values.
5. What is binary search?
Binary search is a remarkably efficient algorithm for searching
in a sorted array. It works bycomparing a search key K with the
arrays middle element A[m]. if they match the algorithm
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stops; otherwise the same operation is repeated recursively for
the first half of the array if K A[m].
6. What are the classic traversals of a binary tree?
The classic traversals are as follows ._Preorder traversal: the
root is visited before left & rightsubtrees ._Inorder
traversal: the root is visited after visiting left subtree and
before visiting right subtree ._Postorder traversal: the root is
visited after visiting the left andright subtrees
7. what is knapsack problem ?
knapsack problem is a problem in combinatorial optimization:
Given a set of items, each with a weightand a value, determine the
count of each item to include in a collection so that the total
weight is less thanor equal to a given limit and the total value is
as large as possible. It derives its name from the problemfaced by
someone who is constrained by a fixed-size knapsack and must fill
it with the most useful items.
8. Give the formula used to find the upper bound for knapsack
problem.
A simple way to find the upper bound ub is to add v, the total
value
of the items already selected, the product of the remaining
capacity of the
knapsack W-w and the best per unit payoff among the remaining
items, which
is vi+1/wi+1
ub = v + (W-w)( vi+1/wi+1)
9. What is greedy technique?
Greedy technique suggests a greedy grab of the best alternative
available in the hope that asequence of locally optimal choices
will yield a globally optimal solution to the entire problem.The
choice must be made as
follows .
Feasible : It has to satisfy the problems constraints ._
Locally optimal : It has to be the best local choice among all
feasible
choices available on that step. .
Irrevocable : Once made, it cannot be changed on a subsequent
step of the algorithm
10. Define control abstraction.
A control abstraction we mean a procedure whose flow of control
is clear but whose primaryoperations are by other procedures whose
precise meanings are left undefined.
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11. What is the substitution method?
One of the methods for solving any such recurrence relation is
called the substitution method
12. What is the maximum and minimum problem?
The problem is to find the maximum and minimum items in a set of
n
elements.Though this problem may look so simple as to be
contrived, it allows us to demonstrate divideand conquer in simple
setting
13. Write a algorithm for straightforward maximum and
minimum
algorithm straight MaxMin(a,n,max,min)
//set max to the maximum and min to the minimum of a[1:n]
{
max := min: = a[i];
for i = 2 to n do
{
if(a[i] >max) then max: = a[i];
if(a[i] >min) then min: = a[i];
}
}
14. Write the algorithm for Iterative binary search?
Algorithm BinSearch(a,n,x)
//Given an array a[1:n] of elements in nondecreasing
// order, n>0, determine whether x is present
{
low : = 1;
high : = n;
while (low < high) do
{
mid : = [(low+high)/2];
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if(x < a[mid]) then high:= mid-1;
else if (x >a[mid]) then low:=mid + 1;
else return mid;
}
return 0;
}
15. Describe the recurrence relation of merge sort?
If the time for the merging operation is proportional to n, then
the computing
time of merge sort is described by the recurrence relation
n = 1, a a constant
T(n) = a
2T (n/2) + n n >1, c a constant
16. Write any two characteristics of Greedy Algorithm?
o To solve a problem in an optimal way construct the solution
from given set of
candidates.
o As the algorithm proceeds, two other sets get accumulated
among this one set
contains the candidates that have been already considered and
chosen while
the other set contains the candidates that have been considered
but rejected.
17. Give the recurrence relation of divide-and-conquer?
The recurrence relation is
T(n) = g(n)
T(n1) + T(n2) + ----+ T(nk) + f(n)
18. Give computing time for Bianry search?
The computing time of binary search by giving formulas that
describe the best,
average and worst cases. Successful searches q(1) q(logn)
q(Logn) best average worst
unsuccessful searches q(logn) best, average, worst.
19. Write the Control abstraction for Divide-and conquer.
Algorithm D And(r)
{
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if small(p) then return S(r);
else
{
divide P into smaller instance _ 1, _ 2, _ k, k 1;
Apply D and C to each of these subproblems
Return combine (DAnd C(_1) DAnd C(_2),----, DAnd ((_k));
}
}
20. Write the Anlysis for the Quick sot.
In analyzing QUICKSORT, we can only make the number of element
comparisions c(n). It is easy to seethat the frequency count of
other operations is of the same order as C(n).
21. What is the Greedy choice property?
solution can arrive at by making a locally optimal choice.
cannot depend on any future choices or on solution to the sub
problem.
16 MARKS
1.. Explain Knapsack Problem (16).
2. Explain the algorithm for maximum and minimum numbers in an
array.
(16)
3. (a) Give a detailed note on Divide and Conquer
techniques.(6)
(b). Sort the following set of elements using merge sort
(10)
12,24,8,71,4,23,6,89,56
4. Write An algorithm for searching an element using Binary
search
method. Give an example. (16)
5. (a) write a pseudo code for a divide and conquer algorithm
for
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merging two sorted arrays into a single sorted one. Explain with
an
example. (12)
(b) Setup an solve a recurrence relation for the number of
key
comparisons made by the above pseudo code. (4)
6. Explain in detail merge sort. Illustrate the algorithm with a
numeric example. Providecomplete analysis of the same
UNIT -III
1.What is the use of TVSP?
In places where the loss exceeds the tolerance level boosters
have to the
placed. Given a network and loss tolerance level the tree vertex
splitting problems is
to determine an optimal placement of boosters.
Thus it is always safe to make greedy choice.
choice are empty.
SNSCT Department of Compute Science and Engineering Page 10
2. Write the specification of TVSP
Let T= (V, E, W) be a weighted directed binary tree where
V_ vertex set
E_ edge set
W_ weight function for the edge.
W is more commonly denoted as w (i,j) which is the weight of the
edge _E.
3. Define feasible solution for TVSP.
Given a weighted tree T(V,E,W) and a tolerance limit _ any
subset X of V is a
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feasible solution if d(T/X).
4. Define optimal solution for TVSP.
An optimal solution is one in which the number of nodes in X is
minimized
5. Write the difference between the Greedy method and Dynamic
programming.
Greedy method Dynamic programming
Only one sequence of decision is Many number of decisions
are
generated. generated1.
Greedy method Dynamic Programming
It does not guarantee to give an It definitely gives an
optimal
solution always. optimal solution always.
6. Define dynamic programming.
Dynamic programming is an algorithm design method that can be
used when a
solution to the problem is viewed as the result of sequence of
decisions.
7. What are the features of dynamic programming?
their values.
considered.
8. Write the general procedure of dynamic programming.
The development of dynamic programming algorithm can be broken
into a
sequence of 4 steps.
1. Characterize the structure of an optimal solution.
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2. Recursively define the value of the optimal solution.
3. Compute the value of an optimal solution in the bottom-up
fashion.
4. Construct an optimal solution from the computed
information.
9. Define principle of optimality.
It states that an optimal sequence of decisions has the property
that whenever
the initial stage or decisions must constitute an optimal
sequence with regard to
stage resulting from the first decision.
10. Give an example of dynamic programming and explain.
An example of dynamic programming is knapsack problem. The
solution to
the knapsack problem can be viewed as a result of sequence of
decisions. We have
to decide the value of xi for 1
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The formula to calculate optimal solution is
g0(m)=max{g1, g1(m-w1)+p1}.
14. Write some applications of traveling salesperson
problem.
assembly line.
the same set of machines.
15. Give the time complexity and space complexity of traveling
salesperson
problem.
16.Define finish time
The finish time fi (S) of job i is the time at which all tasks
of job i have been
completed in schedule S.The finish time F(S) of schedule S is
given by F(S)=max{ fi
(S)} 1
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19. Define preemptive optimal finish time.
Preemptive optimal finish time scheduling for a given set of
tasks is a
preemptive schedule S for which F (S) is minimum over all
preemptive schedules S.
20. Define state space of the problem.
All the paths from the root of the organization tree to all the
nodes is called as
state space of the problem
16MARKS
1.How will you construct a optimal search tree with example.
(16)
2. Explain the Multistage graph with example. (16)
3. Explain the 0/1 knapsack with an algorithm. (16)
4. Describe the Traveling salesman problem & discuss how to
solve it usingDynamic Programming(16)
5. Solve the all-pairs shorest path problem for the digraph with
any example?
UNIT IV
1.,What are the factors that influence the efficiency of the
backtracking algorithm?
The efficiency of the backtracking algorithm depends on the
following four
factors. They are:
i. The time needed to generate the next xk
ii. The number of xk satisfying the explicit constraints.
iii. The time for the bounding functions Bk
iv. The number of xk satisfying the Bk.
2.Define Branch-and-Bound method.
The term Branch-and-Bound refers to all the state space methods
in which all
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children of the E-node are generated before any other live node
can become the E- node.
3.What are the searching techniques that are commonly used in
Branch-and-Bound
method.
The searching techniques that are commonly used in
Branch-and-Bound method
are:
i. FIFO
ii. LIFO
iii. LC
iv. Heuristic search
4.State 8 Queens problem.
The problem is to place eight queens on a 8 x 8 chessboard so
that no two queen
attack that is, so that no two of them are on the same row,
column or on the diagonal.
5.State Sum of Subsets problem.
Given n distinct positive numbers usually called as weights ,
the problem calls for finding all
the combinations of these numbers whose sums are m.
6. State m colorability decision problem.
Let G be a graph and m be a given positive integer. We want to
discover whether the nodes of
G can be colored in such a way that no two adjacent nodes have
the same color yet only m
colors are used.
7.Define chromatic number of the graph.
The m colorability optimization problem asks for the smallest
integer m for which the
graph G can be colored. This integer is referred to as the
chromatic number of the graph.
8.Define Backtracking.
Backtracking is to build up the solution vector one component at
a time and to use
modified criterion function Pi(x1,..,xi) (sometimes called
bounding function) to test whether
the vector being formed has any chance of success. Desired
solution expressed as an ntuple
(x1,x2,,xn) where xi are chosen from some set Si.
If |Si|=mi, m=m1m2..mn candidates are possible
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Yielding the same answer with far fewer than m trials
Advantage : if it is realized that the partial vector (x1,..,xi)
can in no way lead to an
optimum solution, then mi+1mn possible test vectors can be
ignored entirely.
9.Give the categories of the problem in backtracking.
- Whether there is any feasible solution.
- Whether there exists any best solution.
- Finds all possible feasible solution.
10. List down the examples of backtracking.
-Queens problem
-Sum problem
11. What are the two types of constraints used in
Backtracking?
12. Define implicit constraint.
rules that determine which of the tuples in the solution
space of I that satisfy the criterion function.
9. Define explicit constraint.
Explicit constraints are rules that restrict each xi to take on
values only from a given set.
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I
constraints
xi >= 0, or Si = {all nonnegative real numbers}
xi = {0, 1} or Si = { 0, 1 }
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16. Define m color ability decision problem.
Let G be a graph and m be a given positive integer. The nodes of
G can be colored in such
a way that no two adjacent nodes have the same color yet only m
colors are used. This is
termed the m-colorability decision problem. If d is the degree
of the given graph, then it
can be colored with d+ 1 color. The m-colorability optimization
problem asks for the
smallest integer m for which the graph G can be colored. This
integer is referred to as the
chromatic number of the graph. The color of each node is
indicated next to it. If three
colors are needed to color the graph then the graphs chromatic
number is 3.
17.Define promising and non promising node.
Promising Node:
A node in a state space tree is said to be promising if it
corresponds to a
partially constructed solution that may still lead to a complete
solution.
Non Promising node:
A node in a state space tree is said to be non-promising if it
corresponds to
a partially constructed solution that would not be able to lead
to a complete
solution further.
18. How can you represent the solution for 8 queens problem?
All solutions represented as 8-tuples (x1, x2,, x8) where xi is
the column on which
queen i is placed.
Constraints are,
Explicit constraints
Si={1,2,,8}
Implicit constraints
1. No two xis can be the same column or row (By this, solution
space
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reduced from 88 to 8!)
2. No two queens can be on the same diagonal
19. Define problem state & state space?
is each node in the depth-first search tree
20. Define sum of subsets problem?
In the Sum-of-Subsets problem, there are n positive integers
(weights) wi and a positive
integer W.
The goal is to find all subsets of the integers that sum to
W.
For example, n = 4, w = (11, 13, 24, 7), and m = 31, the desired
subsets are (11, 13, 7)
and (24, 7)
The solution vectors can also be represented by the indices of
the numbers as (1, 2, 4)
and (3, 4)
_ All solutions are k-tuples, 1
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problems.s=(1,3,4,5) & d=11 (16)
6. Explain subset-sum problem and discuss the possible solution
strategies using
backtracking. (16)
7. Explain 8-Queens problem with an algorithm. Explain why
backtracking is defined as
a default procedure of last resort for solving problems.
(10+6)
8. Using Backtracking enumerate how can you solve the following
problems
(a) 8-queens problem (8)
(b) Hamiltonian circuit problem (8)
UNIT-V
1. Define graph traversal.
Searching a vertex in a graph can be solved by starting at
vertex V and
systematically searching the graph G for vertices that can be
reached from V.
It starts at initial vertex and visits each and every vertex
exactly once and finally reaches
end vertex.
2. Mention the types of traversal.
(i)
(ii)Depth first search traversal.
3. What is meant by BFS traversal?
(i)
(ii)
(iii)
(iv) ocess is repeated until no more vertexes is left
4. What is the time complexity of BFS?
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(i)
(ii)If the graph is represented by adjacency List: O (|V|+ |E|
).
5. Define depth first search traversal.
(i)
(ii)
(iii)
(iv)
(v) e search terminates when all reached vertices have been
fully explored.
6. What is the time complexity of DFS?
(i)
(ii)
7.Define connected component.
(i)
path from U to V.
(ii)
8. How do identify connected component using breath first
search?
(i)
(ii)
(iii)
9. Define spanning tree.
(i) nimal sub graph, G', of G such that V (G') = V (G) and G'
is
connected
(ii)
(iii) -1edges, and all
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connected graphs with n -1 edges are trees
(iv)A spanning tree has n -1 edge
10.Define bi connected graph and bi connected component.
(i)
(ii)A bi connected component of a connected undirected graph is
a maximal bi
connected sub graph of G
11.What is tree edge and cross edge?
In the breath first search forest, whenever a new unvisited
vertex is reached for the first
time, it is attached as a child to the vertex from which it is
being reached. Such an edge is
called tree edge. If an edge is leading to a previously visited
vertex other than its
immediate predecessor, that is noted as a cross edge.
12.Define branch and bound.
Branch-and-bound refers to all state space search methods in
which all children of an Enode
are generated before any other live node can become the
E-node.
13.Define live node, E-node and dead node.
(i)
generated.
(ii)E-node is a live node whose children are currently being
explored. In other words,
an E-node is a node currently being expanded.
(iii)Dead node is a generated node that is not to be expanded or
explored any further.
All children of a dead node have already been expanded.
14.What are the types search strategies in branch and bound?
(i)
(ii)
(iii)
15. Compare the FIFO and LIFO search.
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(i)
(ii)
(iii)
(iv)
16. What is meant by Class NP (Non Polynomial)?
Class P consists of problems whose solutions are bounded by the
non polynomial.
Examples are Traveling salesperson problem O(n22n), knapsack
problem O(2n/2)
17. What are the two classes of non polynomial time
problems?
(i)NP- hard
(ii) -complete
18. What is meant by NP hard and NP complete problem?
(i) -Hard Problem: A problem L is NP-hard if any only if
satisfiability reduces to L.
(ii) - Complete: A problem L is NP-complete if and only if L is
NP-hard and L NP.
(iii) -hard problems that are not NP-complete.
19. Define intelligent function.
Let g(x) be an estimate of the additional effort needed to reach
an answer from node x, x
is assigned a rank using a function c (.) such that
^c (x) = f(h(x)) + g(x)
Where h(x) is the cost of reaching x from root and f (.) is any
non decreasing function.
20.What is meant by class P (Polynomial)?
Class P consists of problems whose solutions are bounded by the
polynomial of small
degree. Examples are Binary search O(log n), sorting O(n log n),
and matrix
multiplication 0(n 2.81).
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16 MARKS
1. Define spanning tree? Discuss the design steps in prims
algorithm to construct
minimum spanning tree with example. (16)
2. Explain the method of binding the minimum spanning tree for a
connected graph
using prims algorithm. (16)
3. Define spanning tree? Discuss the design steps in kruskal
algorithm to construct
minimum spanning tree with example (16)
4. Compare and contrast the depth first search and birth first
search. How do they fit
in to the decrease and conquer strategies. (16)
5. Explain NP-hard and NP complete problems with example.
(16)
6. Explain connected components and bi-connected components with
psecdocode.
(16)
7. Give a suitable example and explain the birth first search
and depth first search
algorithm. (16)
8. What is branch and bound? Explain detail. (16)
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