Greedy Method

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 Li
• Assume the tape is initially positioned at the
  front. If the programs are stored in the order I
  i1,i2in the time tj needed to retrieve program
  ij
• 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, (L1, L2, L3) (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 ith job Ji is di and the
  profit received from this job is pi.
• Hence, the optimal solution of this algorithm is
  a feasible solution with maximum profit.
• Thus, D(i) gt 0 for 1?i?n.
• Initially, these jobs are ordered according to
  profit,
• i.e. p1? p2? p3?...? pn

13
(No Transcript)
14

• 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 J1 J2 J3 J4 J5
Deadline 2 1 3 2 1
Profit 60 100 20 40 20
15

• 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 J2 J1 J4 J3 J5
Deadline 1 2 2 3 1
Profit 100 60 40 20 20
16

• From this set of jobs, first we select J2, as it
  can be completed within its deadline and
  contributes maximum profit.
• Next, J1 is selected as it gives more profit
  compared to J4.
• In the next clock, J4 cannot be selected as its
  deadline is over, hence J3 is selected as it
  executes within its deadline.
• The job J5 is discarded as it cannot be executed
  within its deadline.
• Thus, the solution is the sequence of jobs (J2,
  J1, J3), which are being executed within their
  deadline and gives maximum profit.
• Total profit of this sequence is 100 60 20
  180.

17

• 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.

18

• 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 f1, f2, f3, , fn.
• Initially, each element of this is considered as
  a single node binary tree.
• To find this optimal solution, the following
  algorithm is used.

19

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

20

• Example
• Let us consider the given files, f1, f2, f3,
  f4 and f5 with 20, 30, 10, 5 and 30 number of
  elements respectively.
• If merge operations are performed according to
  the provided sequence, then
• M1  merge f1 and f2 gt 20 30 50
• M2  merge M1 and f3 gt 50 10 60
• M3  merge M2 and f4 gt 60 5 65
• M4  merge M3 and f5 gt 65 30 95
• Hence, the total number of operations is
• 50 60 65 95 270

21

• 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 -
• f4, f3, f1, f2, f5
• Hence, merge operations can be performed on this
  sequence
• M1  merge f4 and f3 gt 5 10 15
• M2  merge M1 and f1 gt 15 20 35
• M3  merge M2 and f2 gt 35 30 65
• M4  merge M3 and f5 gt 65 30 95
• Therefore, the total number of operations is
• 15 35 65 95 210
• Obviously, this is better than the previous one.

22

• In this context, we are now going to solve the
  problem using this algorithm.

Initial Set
Step-1
23
Step-2
Step-3
24
Step-4

• Hence, the solution takes 15 35 60 95 205
  number of comparisons.

25
Thank you