Greedy method

Greedy method in Data Structure and Algorithms

1. Data Structures and Algorithm

2. Prepared by Mrs.P.Anusha M.Sc (IT)., M.Phil.,D.P.T.T .,( Ph.D )., Assistant professor, Department of Information Technology, Bon secours college for women, Thanjavur .

3. GREEDY METHOD

4. Greedy Method General Approach Given a set of n inputs Find a subset called feasible solution of the n inputs subject to some constraints and satisfying a given objective function If the objective function is maximized or minimized the feasible solution is optimal It is locally optimal method

5. Components of Greedy Algorithm Greedy algorithms have the following five components − A candidate set − A solution is created from this set. A selection function − Used to choose the best candidate to be added to the solution. A feasibility function − Used to determine whether a candidate can be used to contribute to the solution. An objective function − Used to assign a value to a solution or a partial solution. A solution function − Used to indicate whether a complete solution has been reached.

6. Greedy method control Abstraction for the subset paradigm

7. Algorithm Step 1: Choose an input from the input set, based on some criterion. If no more input exit. Step 2 : Check whether the chosen input yields to a feasible solution. If no, discard the input and go to step 1 Step 3 : Include the input into the solution vector and update the objective function. Go to step 1.

8. Optimal Storage on Tapes There are n programs that are to be stored on a computer tape of length L. Associated with each program i is length L i Assume the tape is initially positioned at the front. If the programs are stored in the order I = i 1 ,i 2 …i n the time t j needed to retrieve program i j If all programs are retrieved equally often, then the Mean Retrieval Time (MRT) = This problem fits the ordering paradigm. Minimizing the MRT is equivalent to minimizing d(I) =

9. Example: Let n = 3, (L 1 , L 2 , L 3 ) = (5, 10, 3) 6 possible orderings. The optimal is 3,1,2.

10. Assigning programs to tapes

11. Job Sequencing with Deadlines Problem Statement In job sequencing problem, the objective is to find a sequence of jobs, which is completed within their deadlines and gives maximum profit. Solution Let us consider, a set of  n  given jobs which are associated with deadlines and profit is earned, if a job is completed by its deadline. These jobs need to be ordered in such a way that there is maximum profit.

12. It may happen that all of the given jobs may not be completed within their deadlines. Assume, deadline of  i th  job  J i  is  d i  and the profit received from this job is  p i . Hence, the optimal solution of this algorithm is a feasible solution with maximum profit. Thus, D( i ) > 0 for 1⩽i⩽n. Initially, these jobs are ordered according to profit, i.e. p 1 ⩾ p 2 ⩾ p 3 ⩾...⩾ p n

13. Analysis In this algorithm, we are using two loops, one is within another. Hence, the complexity of this algorithm is O(n2). Example Let us consider a set of given jobs as shown in the following table. We have to find a sequence of jobs, which will be completed within their deadlines and will give maximum profit. Each job is associated with a deadline and profit. Job J 1 J 2 J 3 J 4 J 5 Deadline 2 1 3 2 1 Profit 60 100 20 40 20

14. Solution To solve this problem, the given jobs are sorted according to their profit in a descending order. Hence, after sorting, the jobs are ordered as shown in the following table. Job J 2 J 1 J 4 J 3 J 5 Deadline 1 2 2 3 1 Profit 100 60 40 20 20

15. From this set of jobs, first we select  J 2 , as it can be completed within its deadline and contributes maximum profit. Next,  J 1  is selected as it gives more profit compared to  J 4 . In the next clock,  J 4  cannot be selected as its deadline is over, hence  J 3  is selected as it executes within its deadline. The job  J 5  is discarded as it cannot be executed within its deadline. Thus, the solution is the sequence of jobs ( J 2 , J 1 , J 3 ), which are being executed within their deadline and gives maximum profit. Total profit of this sequence is 100 + 60 + 20 = 180.

16. Optimal Merge Pattern Problem Statement Merge a set of sorted files of different length into a single sorted file. We need to find an optimal solution, where the resultant file will be generated in minimum time. If the number of sorted files are given, there are many ways to merge them into a single sorted file. This merge can be performed pair wise. Hence, this type of merging is called as 2-way merge patterns.

17. As, different pairings require different amounts of time, in this strategy we want to determine an optimal way of merging many files together. At each step, two shortest sequences are merged. To merge a p-record file and a q-record file requires possibly p + q record moves, the obvious choice being, merge the two smallest files together at each step. Two-way merge patterns can be represented by binary merge trees. Let us consider a set of n sorted files {f 1 , f 2 , f 3 , …, f n }. Initially, each element of this is considered as a single node binary tree. To find this optimal solution, the following algorithm is used.

18. At the end of this algorithm, the weight of the root node represents the optimal cost.

19. Example Let us consider the given files, f 1 , f 2 , f 3 , f 4  and f 5  with 20, 30, 10, 5 and 30 number of elements respectively. If merge operations are performed according to the provided sequence, then M 1  = merge f 1  and f 2  => 20 + 30 = 50 M 2  = merge M 1  and f 3  => 50 + 10 = 60 M 3  = merge M 2  and f 4  => 60 + 5 = 65 M 4  = merge M 3  and f 5  => 65 + 30 = 95 Hence, the total number of operations is 50 + 60 + 65 + 95 = 270

20. Now, the question arises is there any better solution? Sorting the numbers according to their size in an ascending order, we get the following sequence − f 4 , f 3 , f 1 , f 2 , f 5 Hence, merge operations can be performed on this sequence M 1  = merge f 4  and f 3  => 5 + 10 = 15 M 2  = merge M 1  and f 1  => 15 + 20 = 35 M 3  = merge M 2  and f 2  => 35 + 30 = 65 M 4  = merge M 3  and f 5  => 65 + 30 = 95 Therefore, the total number of operations is 15 + 35 + 65 + 95 = 210 Obviously, this is better than the previous one.

21. In this context, we are now going to solve the problem using this algorithm. Initial Set Step-1

22. Step-2 Step-3

23. Step-4 Hence, the solution takes 15 + 35 + 60 + 95 = 205 number of comparisons.

24. Thank you