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Instructor Neelima Gupta [email protected] Greedy Approach A tool to design algorithms for optimization problems 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. 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. General ‘Greedy’ Approach Step 1: Choose the current best solution. Step 2: Obtain greedy solution on the rest. When to use? There must be a greedy choice to make. The problem must have an optimal substructure. 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. . 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 f j si Example: Consider activities: a1, a2, a3, a4 s1--------f1 s2---------f2 s3------f3 s4------f4 Here a1 is compatible with a3 & a4 a2 is compatible with a3 & a4 But a3 and a4 themselves are not compatible. 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. Lets think of some possible greedy solutions Shortest Job First In the order of increasing start times In the order of increasing finish times Shortest Job First job1 job2 job3 0 1 2 3 4 5 6 7 8 9 Time Thanks to: Navneet Kaur(22), MCA 2012 10 11 12 13 14 15 Shortest Job First job1 job2 job3 0 1 2 3 4 5 6 7 8 9 Time Thanks to: Navneet Kaur(22), MCA 2012 10 11 12 13 14 15 Shortest Job First job1 job2 job3 0 1 2 3 4 5 6 7 SCHEDULE CHOSEN BY THIS APPROACH OPTIMAL SCHEDULE 8 9 Thanks to: Navneet Kaur(22), MCA 2012 10 11 12 13 14 Time 15 Increasing Start Times job1 job2 job3 0 2 4 6 8 10 12 14 16 18 20 Time Thanks to: Navneet Kaur(22), MCA 2012 Increasing Start Times job1 job2 job3 0 2 4 6 8 10 12 14 16 18 20 Time Thanks to: Navneet Kaur(22), MCA 2012 Increasing Start Times job1 job2 job3 0 2 14 4 6 8 10 12 SCHEDULE CHOSEN BY THIS APPROACH OPTIMAL SCHEDULE 16 Thanks to: Navneet Kaur(22), MCA 2012 18 20 Time Increasing Finishing Times i 2 1 3 4 5 Si Fi 2 1 4 5 6 Thanks to Neha (16) Pi 4 5 6 8 9 3 10 4 20 2 Increasing Finishing Times P(1)=10 P(2)=3 P(3)=4 P(4)=20 P(5)=2 0 1 2 3 4 5 6 7 Thanks to Neha (16) 8 9 Time Increasing Finishing Times P(1)=10 P(2)=3 P(4)=20 P(5)=2 0 1 2 3 4 5 6 7 . Thanks to Neha (16) 8 9 Time 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 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 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 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 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 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 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 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 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,…} Activity Selection Problem Proof continued… Clearly, f1 f k thus B is a set of compatible activities, hence an optimal solution too. 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. 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. 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. GREEDY APPROACH Pick the items in the decreasing order of value per unit weight i.e. highest first. Example knapsack capacity 50 Item 2 item 3 Item 1 30 20 10 ● vi = 60 vi = 100 vi/ wi =6 vi/ wi Thanks to: Neha Katyal vi =5 vi/ wi = 120 =4 Example knapsack capacity 50 Item 2 ● 20 item 3 30 10 ● vi = 100 vi/ wi = 5 Thanks to: Neha Katyal vi = 120 vi/ wi = 4 60 Example knapsack capacity 50 item 3 20 30 ● vi = 120 vi/ wi = 4 Thanks to: Neha Katyal + 10 ● 100 60 Example knapsack capacity 50 $80 20/ 30 + 20 100 + 10 60 = 240 Thanks to: Neha Katyal Up Next Dynamic Programming 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 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