ALGORITHMS - SHORT NOTES

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DESIGN AND ANALYSIS OF ALGORITHMS
1. What is performance measurement?
Performance measurement is concerned with obtaining the space and the
time requirements of a particular algorithm.
2. What is an algorithm?
An algorithm is a finite set of instructions that, if followed, accomplishes a
particular task. In addition, all algorithms must satisfy the following criteria:
1) input
2) Output
3) Definiteness
4) Finiteness
5) Effectiveness.
3. Define Program.
A program is the expression of an algorithm in a programming language.
Sometimes works such as procedure, function and subroutine are used
synonymously program.
4. Write the For LOOP general format.
The general form of a for Loop is
For variable : = value 1 to value 2 step
Step do
{
statement 1
statement n 

}
5. What is recursive algorithm?
An algorithm is said to be recursive if the same algorithm is invoked in the
body. An algorithm that calls itself is Direct recursive. Algorithm A is said to be
indeed recursive if it calls another algorithm, which in turn calls A.
6. What is space complexity?
The space complexity of an algorithm is the amount of memory it needs to
run to completion.
7. What is time complexity?
The time complexity of an algorithm is the amount of computer time it
needs to run to completion.
8. Give the two major phases of performance evaluation
Performance evaluation can be loosely divided into two major phases:
(i) a prior estimates (performance analysis)
(ii) a Posterior testing(performance measurement)

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9. Define input size.
The input size of any instance of a problem is defined to be the number of
words(or the number of elements) needed to describe that instance.
10. Define best-case step count.
The best-case step count is the minimum number of steps that can be
executed for the given parameters.
11. Define worst-case step count.
The worst-case step count is the maximum number of steps that can be
executed for the given parameters.
12. Define average step count.
The average step count is the average number of steps executed an
instances with the given parameters.
13. Define the asymptotic notation “Big on” (0)
The function f(n) = O(g(n)) iff there exist positive constants C and no such
that f(n)C * g(n) for all n, n n0.
14. Define the asymptotic notation “Omega” ( ).
The function f(n) =(g(n)) iff there exist positive constant C and no
such that f(n) C * g(n) for all n, n n0.
15. Define the asymptotic tnotation “theta” ()
The function f(n) =(g(n)) iff there exist positive constant C1, C2, and no
such that C1 g(n)f(n) C2 g(n) for all n, n n0.
16. Define Little “oh”.
The function f(n) = 0(g(n))
iff
Lim f(n) = 0
n - g(n)
17. Define Little Omega.
The function f(n) = (g(n))
iff
Lim f(n) = 0
n - g(n)
18. Write algorithm using iterative function to fine sum of n numbers.
Algorithm sum(a,n)
{
S : = 0.0

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For i=1 to n do
S : - S + a[i];
Return S;
}
19. Write an algorithm using Recursive function to fine sum of n numbers,
Algorithm Rsum (a,n)
{
If(n_ 0) then
Return 0.0;
Else
Return Rsum(a, n- 1) + a(n);
20. Define the divide an conquer method.
Given a function to compute on ‘n’ inputs the divide-and-comquer
strategy suggests splitting the inputs in to’k’ distinct susbsets, 1k n, yielding ‘k’
subproblems. The subproblems must be solved, and then a method must be
found to combine subsolutions into a solution of the whole. If the subproblems
are still relatively large, then the divide-and conquer strategy can possibly be
reapplied.
21. Define control abstraction.
A control abstraction we mean a procedure whose flow of control is clear
but whose primary operations are by other procedures whose precise meanings
are left undefined.
22. Write the Control abstraction for Divide-and conquer.
Algorithm DAnd()
{
if small(p) then return S();
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));
}
}
23. What is the substitution method?
One of the methods for solving any such recurrence relation is called the
substitution method.
24. What is the binary search?
If ‘q’ is always chosen such that ‘aq’ is the middle element(that is,
q=[(n+1)/2), then the resulting search algorithm is known as binary search.

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25. Give computing time for Bianry search?
In conclusion we are now able completely describe the computing time of
binary search by giving formulas that describe the best, average and worst
cases.
Successful searches
(1) (logn) (Logn)
best average worst
unsuccessful searches
(logn)
best, average, worst
26. Define external path length?
The external path length E, is defines analogously as sum of the distance
of all external nodes from the root.
27. Define internal path length.
The internal path length ‘I’ is the sum of the distances of all internal nodes
from the root.
28. 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
divide-and-comquer in simple setting.
29.What is merge sort and give it’s time complexity?
In merge sort, the elements are to be sorted in non-decreasing order. Given a
sequence of n elements i.e. a [1], a [2]…. a [n], the general idea is to imagine
them split
into 2 sets a [1], …a [(n/2)] and a [(n/2) +1], …. a [n].Each set is individually
sorted, and
the resulting sorted sequence are merge to produce a single sorted sequence of
‘n’elements. The time complexity is O (nlogn) for worst case.
30. What is the Quick sort?
n quicksort, the division into subarrays is made so that the sorted
subarrays do not need to be merged later.
30. 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 see that the frequency count of other operations
is of the same order as C(n).
31. Is insertion sort better than the merge sort?
Insertion sort works exceedingly fast on arrays of less then 16 elements,
though for large ‘n’ its computing time is O(n2).

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32. 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];
}
}
33. Give the recurrence relation of divide-and-conquer?
The recurrence relation is
T(n) = g(n)
T(n1) + T(n2) + ----+ T(nk) + f(n)
34. 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];
if(x a[mid]) then high:= mid-1;
else if (x a[mid]) then low:=mid + 1;
else return mid;
}
return 0;
}
35. What are internal nodes?
The circular node is called the internal nodes.
36. Describe the recurrence relation ofr 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
37.What is meant by feasible solution?
Given n inputs and we are required to form a subset such that it satisfies some

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given constraints then such a subset is called feasible solution.
38. Write any two characteristics of Greedy Algorithm?
* To solve a problem in an optimal way construct the solution from given set of
candidates.
* 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.
39. Define optimal solution?
A feasible solution either maximizes or minimizes the given objective function is
called as optimal solution
40. What is Knapsack problem?
A bag or sack is given capacity n and n objects are given. Each object has
weight wi and profit pi .Fraction of object is considered as xi (i.e) 0<=xi<=1 .If
fraction is 1 then entire object is put into sack. When we place this fraction into
the sack we get wixi and pixi.
41. Define weighted tree.
A directed binary tree for which each edge is labeled with a real number (weight)
is called as weighted tree.
42. 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.
43. What is the Greedy choice property?
* The first component is greedy choice property (i.e.) a globally optimal solution
can arrive at by making a locally optimal choice.
* The choice made by greedy algorithm depends on choices made so far but it
cannot depend on any future choices or on solution to the sub problem.
* It progresses in top down fashion.
44. What is greedy method?
Greedy method is the most important design technique, which makes a choice
that looks best at that moment. A given ‘n’ inputs are required us to obtain a
subset that satisfies some constraints that is the feasible solution. A greedy
method suggests that one can device an algorithm that works in stages
considering one input at a time.
45. What are the steps required to develop a greedy algorithm?
* Determine the optimal substructure of the problem.
* Develop a recursive solution.

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* Prove that at any stage of recursion one of the optimal choices is greedy
choice. Thus it is always safe to make greedy choice.
* Show that all but one of the sub problems induced by having made the greedy
choice are empty.
* Develop a recursive algorithm and convert into iterative algorithm.
46. What is activity selection problem?
The ‘n’ task will be given with starting time si and finishing time fi. Feasible
solution is that the task should not overlap and optimal solution is that the task
should be completed in minimum number of machine set.
47. 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 <i,j> _ E.
48. Define forest.
Collection of sub trees that are obtained when root node is eliminated is known
as forest
55. Define optimal solution for TVSP.
An optimal solution is one in which the number of nodes in X is minimized.
56. Write the general algorithm for Greedy method control abstraction.
Algorithm Greedy (a, n)
{
solution=0;
for i=1 to n do
{
x= select(a);
if feasible(solution ,x) then
solution=Union(solution ,x);
}
return solution;
}
57. . Define optimal solution for Job sequencing with deadlines.
Feasible solution with maximum profit is optimal solution for Job sequencing
with deadlines.
58.Write the difference between the Greedy method and Dynamic
programming.
Greedy method Dynamic programming
1.Only one sequence of decision is generated.

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1.Many number of decisions are generated.
2.It does not guarantee to give an optimal solution always.
2.It definitely gives an optimal solution always.
59.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.
60.What are the features of dynamic programming?
_ Optimal solutions to sub problems are retained so as to avoid recomputing
their values.
_ Decision sequences containing subsequences that are sub optimal are not
considered.
_ It definitely gives the optimal solution always.
61.What are the drawbacks of dynamic programming?
_ Time and space requirements are high, since storage is needed for all level.
_ Optimality should be checked at all levels.
62.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.
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.
63.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.
64.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<i<n. First we make a decision on x1 and then on x2
and so on. An optimal sequence of decisions maximizes the object function pixi.
65.Write about optimal merge pattern problem.
Two files x1 and x2 containing m & n records could be merged together to obtain
one merged file. When more than 2 files are to be merged together. The merge
can be
accomplished by repeatedly merging the files in pairs. An optimal merge pattern
tells which pair of files should be merged at each step. An optimal sequence is a
least cost sequence.
66.Explain any one method of finding the shortest path.

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One way of finding a shortest path from vertex i to j in a directed graph G is to
decide which vertex should be the second, which is the third, which is the fourth,
and so on, until vertex j is reached. An optimal sequence of decisions is one that
results in a path of least length.
67.Define 0/1 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. xi is restricted to have the value 0 or
1 and by using the function knap(l, j, y) we can represent the problem as
maximum pi xi subject to wi xi < y where l - iteration, j - number of objects, y –
capacity.
68.What is the formula to calculate optimal solution in 0/1 knapsack
problem?
The formula to calculate optimal solution is
g0(m)=max{g1, g1(m-w1)+p1}.
69.Write about traveling salesperson problem.
Let g = (V, E) be a directed. The tour of G is a directed simple cycle that includes
every vertex in V. The cost of a tour is the sum of the cost of the edges on the
tour. The traveling salesperson problem to find a tour of minimum cost.
70.Write some applications of traveling salesperson problem.
_ Routing a postal van to pick up mail from boxes located at n different sites.
_ Using a robot arm to tighten the nuts on some piece of machinery on an
assembly line.
_ Production environment in which several commodities are manufactured on
the same set of machines.
71.Give the time complexity and space complexity of traveling salesperson
problem.
_ Time complexity is O (n2 2n).
_ Space complexity is O (n 2n).
80. What are the requirements that are needed for performing
Backtracking?
To solve any problem using backtracking, it requires that all the solutions
satisfy a complex set of constraints. They are:
i. Explicit constraints.
ii. Implicit constraints.
81.Define explicit constraint.
They are rules that restrict each xi to take on values only from a give set.
They depend on the particular instance I of the problem being solved. All tuples
that
satisfy the explicit constraints define a possible solution space.

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82. Define implicit constraint.
They are rules that determine which of the tuples in the solution space of I
satisfy the criteria function. It describes the way in which the xi must relate to
each
other.
83.Define state space tree.
The tree organization of the solution space is referred to as state space
tree.
84.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
85.Define answer states.
Answer states are those solution states s for which the path from the root
to s defines a tuple that is a member of the set of solutions of the problem.
86.What are static trees?
The tree organizations that are independent of the problem instance being
solved are called as static tree.
87.What are dynamic trees?
The tree organizations those are independent of the problem instance
being solved are called as static tree.
88.Define a live node.
A node which has been generated and all of whose children have not yet
been generated is called as a live node.
89. Define a E – node.
E – node (or) node being expanded. Any live node whose children are
currently being generated is called as a E – node.
90.Define a dead node.
Dead node is defined as a generated node, which is to be expanded further/
all of whose children have been generated.
91.,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

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iv. -The number of xk satisfying the Bk.
92.Define Branch-and-Bound method.
The term Branch-and-Bound refers to all the state space methods in which
all children of the E-node are generated before any other live node can become
the Enode.
93.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
94.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.
95.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.
96. 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.
97.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.
98. Define a planar graph.
A graph is said to be planar iff it can be drawn in such a way that no two
edges cross each other.
99. What are NP- hard and Np-complete problems?
The problems whose solutions have computing times are bounded by
polynomials of small degree.
100. What is a decision problem?
Any problem for which the answer is either zero or one is called decision
problem.

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101. What is maxclique problem?
A maxclique problem is the optimization problrm that has to determine the size
of a largest clique in Grapg G where clique is the maximal subgraph of a graph.
102. what is approximate solution?
A feasible solution with value close to the value of an optimal solution is called
approximate solution.
ALL THE BEST !!!
1. Explain algorithm specification?
1. pseudo code conventions
1. Comments begin with //.
2. Blocks are indicated and matching braces {}
3. An identifier begins with a letter.

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4. Assignment
5. Two Boolean values
6. Multidimensional arrays
7. Looping
8. Conditional statement
9. Input and Output.
10. Algorithm.
2. Recursive algorithm
An algorithm is said to be recursive if the same algorithm is invoked in the body.
An algorithm that calls itself is Direct recursive. Algorithm A is said to be indeed
recursive if it calls another algorithm, which in turn calls A.
2. Explain all asymptotic notations?
Big Oh:-
The function f(n) = O(g(n)) iff there exist positive constants C and no such
that f(n)C * g(n) for all n, n n0.
Omega:-
The function f(n) =(g(n)) iff there exist positive
constant C and no such that f(n) C * g(n) for all n, n n0.
Theta:-
The function f(n) =(g(n)) iff there exist positive constant C1, C2, and no
such that C1 g(n)f(n) C2 g(n) for all n, n n0.
Little Oh:-
The function f(n) = 0(g(n))
iff
Lim f(n) = 0
n - g(n)
Little omega:-
The function f(n) = (g(n))
iff
Lim g(n) = 0
n - f(n)
3. Explain Performance analysis?
1. Space Complexity:-
The space complexity of an algorithm is the amount of memory it needs to run to
completion.
algorithm using iterative function to fine sum of n numbers
Algorithm sum(a,n)
{
S : = 0.0
For i=1 to n do
S : - S + a[i];
Return S;
}
algorithm using Recursive function to fine sum of n numbers
Algorithm Rsum (a,n)
{

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If(n_ 0) then
Return 0.0;
Else
Return Rsum(a, n- 1) + a(n);
}
2. Time Complexity:-
The time complexity of an algorithm is the amount of computer time it
needs to run to completion.
3. Asymptotic Notation:-
4. Performance measurement:-
5. Practical Complexities:-
4. Explain binary search method?
If ‘q’ is always chosen such that ‘aq’ is the middle element(that is, q=[(n+1)/2),
then the resulting search algorithm is known as binary search.
In conclusion we are now able completely describe the computing time of
binary search by giving formulas that describe the best, average and worst
cases.
Algorithm binsearch(a,n,x)
{
low:=1;
high:=n;
while(low<high) do
{
mid:=(low+high)/2;
if(x<a[mid]) then
high:= mid-1;
else if(x>a[mid]) then
low:= mid+1;
else
return mid;
}
return 0;
}
Successful searches
(1) (logn) (Logn)
best average worst
unsuccessful searches
(logn)
best, average, worst
5. Explain Quick sort?
In quick sort the division into sub arrays is made so that the sorted
subarrays do not need to be merged later.
Algorithm partition(a,m,p)
{
v:=a[m];
I:=m;

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J:=p;
Repeat
{
repeat
I:=I+1;
Until(a[I]>=v);
Repeat
J:=j-1;
Until(a[j]<=v);
If (I<j) then interchange(a,I,j);
{until(I>=j);
a[m]:=a[j];
a[j]:=v;
return j;
}
Algorithm interchange(a,I,j)
{
p:=a[I];
a[I]:=a[j];
a[j]:=p;
}
Algorithm quicksort(p,q)
{
if(p<q) then
{
j:=partition(a,p,q+1);
quicksort(p,j-1);
quicksort(j+1,q);
}
}
In analyzing quick sort we can only make the number of element comparisons
c(n). It is easy to see that the frequency count of other operations is of the same
order as
c(n).
6. Write the algorithm for finding the maximum and minimum and explain
it?
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 converted it allows us to
demonstrate divide and conquer in simple setting.
Iterative algorithm
Algorithm straightmaxmin(a,n,max,min)
{
max:=min:=a[1];
for I:= 2 to n do
{
if(a[I]>max) then max:=a[I];

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if(a[I]<min) then min := a[I];
}
}
Recursive algorithm
Algorithm maxmin(I,j,max,min)
{
if(I==j) then max:=min:=a[I];
else if (I=j-1) then
{
if(a[I]<a[j]) then
{
max:=a[j];
min:=a[I];
}
else
{
max:=a[I];
min:=a[j];
}
}
else
{
mid:=(I+j)/2;
maxmin(I,mid,max,min);
maxmin(mid+1,j,max1,min1);
if ( max< max1) then max:= max1;
if(min > min1) then min:= min1;
}
}
7. Explain the concept of mergesort?
In merge sort the elements are to sorted in non decreasing order. Given a
sequence of n elements that is a[1],a[2]….a[n] the general idea is to magine
them
split into 2 sets a[1],a[2],…..a[n/2] and a[n/2]+1,……,a[n]. each set is
individually sorted and the resulting sorted sequence are merge to produce a
single sorted sequence of n elements. The time complexity is O(n log n) for worst
case.
Insertion sort works exceedingly fast on arrays of less than 16 elements,
though for large n its computing time is O(n2).
The time complexity and space complexity of traveling salesperson problem is
_ Time complexity is O (n2 2n).
_ Space complexity is O (n 2n).
13. Define Algorithm and specify its four areas of study.
An Algorithm is a finite set of instructions tht if followed accopolishes a
particular task. It must have the criterias like”
1. Input, 2. Output, 3. Definiteness, 4. Finiteness, 5. Effectiveness

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The Four areas of study are:-
1. How to devise algorithm:
Creating an algorithm is an art which may never be fully automated. Various
design
techniques yielded good algorithms. Like Divide and conquer, Dynamic
programming, Back tracking etc.,
2. How to validate algorithm:
Once an algorithm is decised it is necessary to show that it computes the
correct answer for all possible legal inputs. We refer to this process as
a;gorithm validation. Second phase is program verification. Solution can be
solved using assertions and then second is using predicate calculus.
3. How to analyze algorithm:
Analysis of alogorithm refers to the task of determining h9ow much
computing time and storage an algorithm requires. This sometimes needs
mathematical skill.
4. How to test a program.
Testing contains two phases. Debugging is find the faulty results occur and if
so correct them. Profiling or performance measurement is the process o9f
executing the correct program on data sets and measuring the time and space
it takes to compute results.
14. What is knapsack problem? Expalin in deatail with alg.
Given n objects and a knwpsack or bag. Object I has a weight wi and the
knapsack has
the capacity m. If a fraction xi 0<xi<1 of object i is placed in to knapsack then a
profit of
pixi earned. The objective is to obtain a filling of the knapsack that maximizes the
total
profit earned. Since the knapsack capacity is m, we rewuire the total weight of all
chosdn
objects to be atmost m. Formally the problem can be stated as
Maximize _pixi
1<i<n
subject to _ wixi <= m
1<i<n
and 0<xi<1, 1<i<n.
The profits and weights are positive numbers.
Algorithm Greedyknapsack(m,n) Needs the objects are considered in order of the
ratio
pi/wi.
Disregarding the time to initially sort the objects each of the three strategy
outlined above
requires O(n) times.
Example: The problem n=3, m= 20 (p1,p2,p3) = (25,24,15) and (w1, w2,w3) =
(18,
15,10) . Out of the feasible solutions the optimal solution is (0, 1, ½ ) which gives
the

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value 31.5.
15. Explain Jobsequencing with deadlines problem with example.
We are given a set of n jobs. Associated with job I is an integer deadline di and
profit pi
> 0. For any job i the profit pi is earned iff the job is completed by its deadline. To
complete a job one has to process the job on a machine one unit of time. Only
one
machine is available for the processing jobs. A fesible solution for this problem is
subsets
J of jobs such that each job in this subset can be completed by its deadline. The
value of
the feasible solution J is the sum of the profits of the jobs in J. An optimal solution
is a
feasible solution with maximum value.
Example : Let n=4, (p1,p2,p3,p4) = (100, 10,15,27) and (d1,d2,d3,d4) = (2,1,2,1).
Out of
The feasible solution and its val ues the optimal one is (1,4) with value 127.
Algorithm JS(d,j,n)
Algorithm Requires us to consider the job in nondecreasing order of pi’s.
Computing time of this algorithm takes O(n2) time.
16. Define Multistage Graphs with forward approach.
A Multistage graph G= (V,E) is a directed graph in which the vertices are
partitioned
into k>=2 disjoint sets Vi, 1<i<k. In addition if <u,v> is an edge in E then u
belongs to
Vi and V belongs to Vi+1 for some I, 1<i<K. The srts V1 and Vk are such that
|v1|=|vk| =1. The vertex s is the source and t the sink. Let c(I,j) be the cost of
edge <I,j>.
the cost of the path from s to t is the sum of the costs of the edges in the path.
The
Multistage graph problem is the minimum cost path from s to t. Each set Vi
defines a
stage in the graph. Because of the constraints on E every path from s to t starts
in stage 1
goes o stage 2 then to stage 3 then to stage 4 and so on and eventually
terminates in stage
K. In the forward approach we obtain
cost(i,j) = min{c(j,l)+cost(i+1,l)}
l €vi+1
<j,l>€ E
Algorithm Fgraph(G,k,n,p)
Time taken is O(|V| + |E|) where V is set of vertices and E is set of edges.
17. Give the Bellman ford algorithm for Single source shortest path.
TO find the single source shortest oath we use Dynamic programming strategy. It
must
be cyclefree shortest path to obtain an algorithm to determine a shortest path
from a

19
source vertex to all remaining vertices in the graph.
When negative edge lengths are permitted we require thet the graph have no
cycles of negative length.
Let dist[u] be the length of a shortest path from the source vertex v to vertex u
under the constraint that the shortest path contains at most L edges.
We make three observations and from it we bring the following
Distk[u] = min {distk-1 [u], min {distk-1 [i] + cost [i,u]}}.
Time taken is O(n3).
18. Define Biconnected components and DFS.
A Graph G is biconnected if and only if it contains no articulatin point. A vertex V
ina graph is said to be an articulation point if and only if the deletion of vertex v
together
with all its edges incident to v disconnects the graph into to two or more non-
empty
components.. A maximal biconeected component is biconnected graph.
The graph G can be transformed into a biconnected graph by using the adge
addition
scheme.
For each articulation point a do
{
let B1, B2,.. BK be the biconnected components containing vertex a.
le vi, vi != a be the vertec in I
Add to G the edges (vi, vi+1 )
}
Doing DFS gives DFN numbers to the nodes. From it the L value is found.
Calculations
and methods are devised to find the articulation point and then the edge
connection is
made to make it biconnected.
The time taken for TVS is O(n)time.
19. Explain branch and bound?
The term Branch-and-Bound refers to all the state space methods in which
all children of the E-node are generated before any other live node can become
the Enode.
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:
v. LC search
vi. 15 – puzzle problem
vii. LC control abstraction
viii. Bounding
ix. FIFO branch and bound
x. LC branch and bound
20. Short note on flow shop scheduling?
The processing of jobs requires the performance of several distinct job. In flow

20
shop we have n jobs each requiring n tasks i.e. T1i, T2i,…...Tmi, 1<i<n.
The conditions of flow shop scheduling are
_ Let Tji denote jth task of the ith job. Task Tij is to be performed on Pj
number of processors where 1<j<m i.e. number of processors will be equal
to number of task
_ Any task Tji must be assigned to the processor Pj.
_ No processor can have more than one task assigned to it at any time. For any
job I processing the task for j>1 cannot be started until T(j-i),i has been
completed.
A nonpreemptive schedule is a schedule in which the processing of a task on any
processor is not terminated until the task is complete.
A preemptive schedule is a schedule in which the processing of a task on any
processor can be terminated before the task is completed.
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<i<n
The mean flow time MFT (S) is defined to be]
MFT (S) = 1 fi(S)
n 1<i<n
Optimal finish time scheduling for a given set of tasks is a nonpreemptive
schedule S for which F (S) is minimum over all nonpreemptive schedules S.
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.
21. Explain backtracking with example?
To solve any problem using backtracking, it requires that all the solutions
satisfy a complex set of constraints. They are:
iii. Explicit constraints.
iv. Implicit constraints.
They are rules that restrict each xi to take on values only from a give set.
They depend on the particular instance I of the problem being solved. All tuples
that
satisfy the explicit constraints define a possible solution space.
They are rules that determine which of the tuples in the solution space of I
satisfy the criteria function. It describes the way in which the xi must relate to
each
other.
The tree organization of the solution space is referred to as state space tree.
Example:-
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.

ALGORITHMS - SHORT NOTES

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