1. MCA 301: Design and Analysis of Algorithms
Instructor
Neelima Gupta

2. Greedy Approach A tool to design algorithms for optimization problems
Table Of Contents
Greedy Approach
A tool to design algorithms for optimization problems

3. What is greedy approach?
Choosing a current best solution without worrying about future. In other words the choice does not depend upon future sub-problems.

4. What is greedy approach?
Such algorithms are locally optimal,
For some problems, as we will see shortly, this local optimal is global optimal also and we are happy.

5. General ‘Greedy’ Approach
Step 1:
Choose the current best solution.
Step 2:
Obtain greedy solution on the rest.

6. When to use? There must be a greedy choice to make.
The problem must have an optimal substructure.

7. Activity Selection Problem
Given a set of activities, S = {a1, a2, …, an} that need to use some resource.
Each activity ai has a possible start time si & finish time fi, such that 0  si < fi < 
We need to allocate the resource in a compatible manner, such that the number of activities getting the resource is maximized.
The resource can be used by one and only one activity at any given time. .

8. Activity Selection Problem
Two activities ai and aj are said to be compatible, if the interval they span do not overlap. ..i.e. fi  sj or fj  si
Example:
Consider activities: a1, a2, a3, a4
s f1
s f2
s f3
s f4
Here a1 is compatible with a3 & a4
a2 is compatible with a3 & a4
But a3 and a4 themselves are not compatible.

9. Activity Selection Problem
Solution: Applying the general greedy algorithm
Select the current best choice, a1 add it to the solution set.
Construct a subset S’ of all activities compatible with a1, find the optimal solution of this subset.
Join the two.

10. Lets think of some possible greedy solutions
Shortest Job First
In the order of increasing start times
In the order of increasing finish times

11. Shortest Job First job1 job2 job3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Time
Thanks to: Navneet Kaur(22), MCA 2012

12. Shortest Job First job1 job2 job3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Time
Thanks to: Navneet Kaur(22), MCA 2012

13. Shortest Job First job1 job2 job3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 2 3 4 5 6 7 8 9 10 11 12 13 14 15
SCHEDULE CHOSEN BY THIS APPROACH
Time
OPTIMAL SCHEDULE
Thanks to: Navneet Kaur(22), MCA 2012

14. Increasing Start Times
job1
job2
job3 2 4 6 8 10 12 14 16 18 20
Time
Thanks to: Navneet Kaur(22), MCA 2012

15. Increasing Start Times
job1
job2
job3 2 4 6 8 10 12 14 16 18 20
Time
Thanks to: Navneet Kaur(22), MCA 2012

16. Increasing Start Times
job1
job2
job3 2 4 6 8 10 12 14 16 18 20
SCHEDULE CHOSEN BY THIS APPROACH
Time
OPTIMAL SCHEDULE
Thanks to: Navneet Kaur(22), MCA 2012

17. Increasing Finishing Times
i Si Fi Pi
Thanks to Neha (16)

18. Increasing Finishing Times
P(1)=10
P(2)=3
P(3)=4
P(4)=20
P(5)=2 1
Time 2 3 4 5 6 7 8 9
Thanks to Neha (16)

19. Increasing Finishing Times
P(1)=10
P(2)=3
P(3)=4
P(4)=20
P(5)=2 1
Time 2 3 4 5 6 7 8 9 .
Thanks to Neha (16)

20. ACTIVITY SELECTION PROBLEM
We include a₁ in the solution. And then recurse on S′ = {aₓ ԑ S-{a₁} : aₓ is compatible with a₁} where S is input set of activities.
Thanks to: Navneet Kaur(22), MCA 2012

21. ACTIVITY SELECTION PROBLEM
CLAIM: If B′ is an optimal solution of S′, then B=B′  {a1} is an optimal solution of S. PROOF: Suppose  an imaginary solution B″, which is optimal and includes a1 . Suppose length of B″, i.e., |B″| = k″
Thanks to: Navneet Kaur(22), MCA 2012

22. ACTIVITY SELECTION PROBLEM
Now, we have to prove two things: I. B is feasible. II. |B| = k″ OR we can prove that |B′| = k″ - 1
Thanks to: Navneet Kaur(22), MCA 2012

23. ACTIVITY SELECTION PROBLEM
Proof of I. --- B′ is a subset of S′. And S′ is compatible with a₁ . Hence, B is feasible.
Thanks to: Navneet Kaur(22), MCA 2012

24. ACTIVITY SELECTION PROBLEM
Proof of II. --- Consider the set B″ - {a₁} i.) Can |B′| ≥ k″ ? If yes, then |B′  {a₁}| ≥ k″ + 1
Thanks to: Navneet Kaur(22), MCA 2012

25. ACTIVITY SELECTION PROBLEM
But, this is contradiction to a problem that B″ is optimal because |B″| = k″ And if the size of optimal solution is k″, then we cannot have a solution of size greater than k″ and this is giving a solution of size k″+1, which is not possible. Hence, statement (i) is wrong.
Thanks to: Navneet Kaur(22), MCA 2012

26. ACTIVITY SELECTION PROBLEM
(ii) Can |B′| < k″ - 1 ? Consider B″- {a₁}. This is a feasible solution of S′. This implies that OPT(S′) ≥ k″ - 1 Hence, Statement (ii) is wrong.
Thanks to: Navneet Kaur(22), MCA 2012

27. ACTIVITY SELECTION PROBLEM
From (i) and (ii), we get |B′| = k″ - 1 This implies that |B| = k″. Hence, B is optimal.
Thanks to: Navneet Kaur(22), MCA 2012

28. Activity Selection Problem
Statement:
 an optimal solution to a problem that contains a1
Proof:
Let A = {ak,…} be an optimal solution. Let ak be the first activity in A i.e. the finishing time of ak is the least.
Construct another solution:
B = A – {ak}  {a1} = {a1,…}

29. Activity Selection Problem
Proof continued…
Clearly, f1  fk thus B is a set of compatible activities, hence an optimal solution too.

30. Activity Selection Problem
Statement:
The solution is globally optimal.
Proof:
Suppose B = {a1…} has an optimal solution containing k+1 elements. (a1 being the first element)
Clearly, B – {a1} has an optimal solution with k elements.

31. Activity Selection Problem
Proof continued…
Now, suppose for B’ = B - {a1}  another optimal solution containing more than k elements.
Then we can construct another optimal solution
B* = B’  {a1} with more than k+1 elements.
This is a contradiction to our assumption of an optimal solution with k+1 elements.

32. FRACTIONAL KNAPSACK PROBLEM
Given a set S of n items, with value vi and weight wi and a knapsack with capacity W.
Aim: Pick items with maximum total value but with weight at most W. You may choose fractions of items.

33. GREEDY APPROACH
Pick the items in the decreasing order of value per unit weight i.e. highest first.

34. Example Item 2 item 3 vi = 60 vi = 100 vi = 120
knapsack capacity 50
Item item 3
Item 1
vi = vi = vi = 120
vi/ wi = vi/ wi = vi/ wi = 4 30 20 10
Thanks to:
Neha Katyal

35. Example Item 2 item 3 vi = 100 vi = 120 vi/ wi = 5 vi/ wi = 4 30 20 10
knapsack capacity 50
Item item 3 60
vi = vi = 120
vi/ wi = vi/ wi = 4 30 20 10
Thanks to:
Neha Katyal

36. Example item 3 vi = 120 20 vi/ wi = 4 30 10 Thanks to: Neha Katyal 100
knapsack capacity 50
item 3
100 + 60
vi = 120
vi/ wi = 4 20 30 10
Thanks to:
Neha Katyal

37. Example $80 + = 240 20/30 20 10 Thanks to: Neha Katyal 100
knapsack capacity 50
$80 +
100 60
= 240
20/30 20 10
Thanks to:
Neha Katyal

38. Up Next
Dynamic Programming

39. ACTIVITY SELECTION PROBLEM
Options that could be followed while scheduling the jobs:
Shortest Job First
Eg. Three jobs to be scheduled:
Job1- start=5, end=10
Job2- start=1, end=7
Job3- start=8, end=15
Our shortest job first would schedule just job1
But the optimal algorithm would have scheduled 2 jobs - job2 and job3.
So this approach is not working.
Thanks to: Navneet Kaur(22), MCA 2012

40. Next option that could be followed while scheduling the jobs:
Smallest start time first
Eg. Three jobs to be scheduled:
Job1- start=1, end=20
Job2- start=2, end=7
Job3- start=8, end=15
Our smallest start time first would schedule just job1
But the optimal algorithm would have scheduled 2 jobs - job2 and job3.
So this approach is also not working.
Thanks to: Navneet Kaur(22), MCA 2012