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Fundamentals of Algorithms-2

.Write the steps involved in the general method for

Branch and Bound? Explain?

Branch and Bound strategy is a general algorithm for

finding optimal solutions of various optimization

problems, especially in discrete and combinatorial

optimization. It consists of a systematic enumeration of

all candidate solutions, where large subsets of fruitless

candidates are discarded en masse, by using upper and

lower estimated bounds of the quantity being optimized.

In the following, we have n kinds of items, 1 through n.

Each kind of item i has a value vi and a weight wi. We

usually assume that all values and weights arenonnegative. To simplify the representation, we can also

assume that the items are listed in increasing order of

weight. The maximum weight that we can carry in the bag

is W.

The most common formulation of the problem is the 0-1

knapsack problem, which restricts the number xi of copies

of each kind of item to zero or one. Mathematically the

0-1-knapsack problem can be formulated as:

maximize

subject to

Thebounded knapsack problemrestricts the number xi of

copies of each kind of item to a maximum integer value

ci. Mathematically the bounded knapsack problem can be

formulated as:

maximize

subject to

The unbounded knapsack problem(UKP) places no upper

bound on the number of copies of each kind of item.

Of particular interest is the special case of the problem

with these properties:

it is a decision problem,

it is a 0-1 problem,

for each kind of item, the weight equals the value:

wi = vi.

1. Algorithm U Bound(cp, cw, k, m)

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2. // cp is the current profit total, cw is the current

3. // weight total; k is the index of the last removed

4. // item; and m is the knapsack size

5. // w[i] and p[i] are respectively the weight and

profit

6. // of the ith object7. {

8. b:=cp; c:=cw;

9. for I:=k+1 to n do

10. {

11. if (c+w[i] m) then

12. {

13. c:c+w[i]; b:=b-p[i];

14. }

15. }

16. return b;

17. }

18.

19.

20.

One early application of knapsack algorithms

was in the construction and scoring of tests in which the

test-takers have a choice as to which questions they

answer. On tests with a homogeneous distribution of point

values for each question, it is a fairly simple processto provide the test-takers with such a choice. For

example, if an exam contains 12 questions each worth 10

points, the test-taker need only answer 10 questions to

achieve a maximum possible score of 100 points. However,

on tests with a heterogeneous distribution of point

valuesthat is, when different questions or sections are

worth different amounts of points it is more difficult

to provide choices. Feuerman and Weiss proposed a system

in which students are given a heterogeneous test with a

total of 125 possible points. The students are asked to

answer all of the questions to the best of their

abilities. Of the possible subsets of problems whose

total point values add up to 100, a knapsack algorithm

would determine which subset gives each student the

highest possible score

2.Explain the NP hard and the NP complete problems?

The best algorithms for

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their

solutions have computing times that cluster into two

groups. The first group consists of problems whose

solution are bounded by polynomials of small degree.

The second group is made up to problems whose best-known

algorithms are non-polynomial.

Examples we have seen include the traveling salesperson

and the knapsack problems for which the best algorithms

given in this text have complexities O (n22n) and O

(2n/2) respectively.

In the quest to develop efficient algorithms, no one has

been able to develop a polynomial time algorithm for any

problem in the second group. This is very important

because algorithms whose computing times are greater

then, polynomial very quickly require such vast amounts

of time to execute that even moderate-size problems

cannot be solved.

The theory of NP-completeness, which we present here,

does not provide a method of obtaining polynomial time

algorithms for problems in the second group. Nor does it

say that algorithms of this complexity do not exist.

Instead, what we do is show that many of the problems for

which there are 4 no known polynomial time algorithms are

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computationally related. In fact, we establish two

classes of problems. These are given names, NP-hard and

NP-Complete.

A problem that is NP-Complete has the property that

it can be solved in polynomial time if and only if all

other NP-Complete problems can also be solved in

polynomial time. If an NP-hard problem can be solved in

polynomial time, then all NP-Complete problems can be

solved in polynomial time. All NP complete problems are

NP-hard, but some NP-hard problems are not known to be

NP-complete.

Although one can define many distinct problem classes

having the properties stated above for the NP- hard and

NP-complete classes, the classes we study are related to

non-deterministic computations. The relationship of these

classes to non deterministic computations together with

the apparent power of non- determinism lead to the

intuitive conclusion that no NP- complete or NP- hard

problem is polynomially solvable.

3.Briefly explain the concept of trees?

A tree is a connected graph without any circuits. The

graph in Fig. , for instance, is a tree. Trees with one,

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two, three and four vertices are shown in Fig. . A graph

must have at least one vertex, and therefore so must a

tree. Some authors allow the null tree, a tree without

any vertices. It follows immediately from the definition

that a tree has to be a simple graph, that is, having

neither a self-loop nor parallel edges (because they

both form circuits).

Trees appear in numerous instances. The genealogy of a

family is often represented by means of tree (in fact

the term tree comes from family tree)

A river with its tributaries and subtributaries can be

represented by a tree. The sorting of mail according to

zip code and the punched cards is done according to a

tree (called decision tree or sorting tree).

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