dynamic program

1. NADAR SARASWATHI COLLEGE OF
ARTS AND SCIENCE,
VADAPUTHUPATTI,THENI.
DEPARTEMENT OF COMPUTER SCIENCE
DYNAMIC PROGRAMMING
(The GENERAL METHOD).
SUBMITTED BY:
S.AISHWARYA LAKSHMI I-MSC(CS)

2. THE GENERAL METHOD
 Dynamic programming approach is similar to divide and conquer in
breaking down the problem into smaller and yet smaller possible sub-
problems.
 But unlike, divide and conquer, these sub-problems are not solved
independently.
 Rather, results of these smaller sub-problems are remembered and
used for similar or overlapping sub-problems.
 Dynamic programming is used where we have problems, which can be
divided into similar sub-problems, so that their results can be re-
used.

3. THE GENERAL METHOD
 Mostly, these algorithms are used for optimization. Before solving the in-
hand sub-problem, dynamic algorithm will try to examine the results of the
previously solved sub-problems.
 The solutions of sub-problems are combined in order to achieve the best
solution.
So we can say that −
 The problem should be able to be divided into smaller overlapping sub-
problem.
 An optimum solution can be achieved by using an optimum solution of smaller
sub-problems.
 Dynamic algorithms use Memoization.

4. Knapsack Problem using Dynamic
Programming
 Given a set of items, each having different weight and value or profit
associated with it.
 Find the set of items such that the total weight is less than or equal
to a capacity of the knapsack and the total value earned is as large as
possible.
 The knapsack problem is useful in solving resource allocation problem.
 Let X = < x1, x2, x3, . . . . . , xn> be the set of n items. Sets W = <w1, w2,
w3, . . . , wn> and V = < v1, v2, v3, . . . , vn> are weight and value
associated with each item in X. Knapsack capacity is M unit.

5. Knapsack Problem using Dynamic
Programming
 The knapsack problem is to find the set of items which maximizes the
profit such that collective weight of selected items does not cross
the knapsack capacity.
 Select items from X and fill the knapsack such that it would
maximize the profit.
 Knapsack problem has two variations.
 0/1 knapsack, that does not allow breaking of items. Either add an
entire item or reject it. It is also known as a binary knapsack.
 Fractional knapsack allows breaking of items. Profit will be earned
proportionally.

6. Optimal Merge Patterns
 Given n number of sorted files, the task is to find the minimum
computations done to reach the Optimal Merge Pattern.
 When two or more sorted files are to be merged altogether to form
a single file, the minimum computations are done to reach this file are
known as Optimal Merge Pattern.

7. Optimal Merge Patterns
 If more than 2 files need to be merged then it can be done in pairs. For example, if need to
merge 4 files A, B, C, D.
 First Merge A with B to get X1, merge X1 with C to get X2, merge X2 with D to get X3 as
the output file.
 An optimal merge pattern corresponds to a binary merge tree with minimum weighted
external path length.
 The function tree algorithm uses the greedy rule to get a two- way merge tree for n files.
 The algorithm contains an input list of n trees. There are three field child, rchild, and
weight in each node of the tree.
 Initially, each tree in a list contains just one node. This external node
has lchild and rchild field zero whereas weight is the length of one of the n files to be
merged.

8. Shortest Path
 This problem is similar to finding number of path from source to destination
with k edges in which we have to find total number of paths.
 Now for this problem the shortest path will be one of the total number of
paths with lowest weight cost.
 To understand this problem, let's take an example of directed graph with 6
vertices {0, 1, 2, 3, 4, 5}, and 9 weighted edges between them.

9. 0-1 Knapsack
 Given weights and values of n items, put these items in a knapsack of
capacity W to get the maximum total value in the knapsack.
 In other words, given two integer arrays val[0..n-1] and wt[0..n-1]
which represent values and weights associated with n items
respectively.
 Also given an integer W which represents knapsack capacity, find out
the maximum value subset of val[] such that sum of the weights of
this subset is smaller than or equal to W.
 You cannot break an item, either pick the complete item or don’t pick
it .

10. THANK YOU!!!