Greedy Algorithms

1. Greedy Algorithms • Reading Material: • Alsuwaiyel’s Book: Section 8.1 • CLR Book (2nd Edition): Section 16.1

2. Greedy Algorithms • Like dynamic programming algorithms, greedy algorithms are usually designed to solve optimization problems • Unlike dynamic programming algorithms, • greedy algorithms are iterative in nature. • An optimal solution is reached from local optimal solutions. • This approach does not work all the time. • A proof that the algorithm does what it claims is needed, and usually not easy to get.

3. Fractional Knapsack Problem • Given n items of sizes s1, s2, …, sn and values v1, v2, …, vn and size C, the problem is to find x1, x2, …, xn that maximize subject to

4. Solution to Fractional Knapsack Problem • Consider yi= vi / si • What is yi? • What is the solution?

5. Activity Selection Problem • Problem Formulation • Given a set of n activities, S = {a1, a2, ..., an}thatrequire exclusive use of a common resource, find the largest possible set of nonoverlapping activities (also called mutually compatible). • For example, scheduling the use of a classroom. • Assume that aineeds the resource during period [si, fi), which is a half-open interval, where • si = start time of the activity, and • fi = finish time of the activity. • Note: Could have many other objectives: • Schedule room for longest time. • Maximize income rental fees.

6. Activity Selection Problem: Example • Assume the following set S of activities that are sorted by their finish time, find a maximum-size mutually compatible set.

8. Solving the Activity Selection Problem • Define Si,j= {akS : fi sk< fksj} • activities that start after aifinishes and finish before ajstarts • Activities in Si,jare compatible with: • Add the following [fictitious] activities a0 = [– , 0) and an+1 = [ , +1) • Hence, S = S0,n+1 and the range of Si,j is 0 i,jn+1

9. Solving the Activity Selection Problem • Assume that activities are sorted by monotonically increasing finish time: • i.e., f0 f1 f2 ... fn <fn+1 • Then, Si,j =  for i  j. Proof: • Therefore, we only need to worry about Si,j where 0 i < jn+1

10. Solving the Activity Selection Problem • Suppose that a solution to Si,jincludes ak. We have 2 sub-problems: • Si,k(start after aifinishes, finish before akstarts) • Sk,j(start after akfinishes, finish before ajstarts) • The Solution to Si,jis (solution to Si,k)  {ak}  (solution to Sk,j) • Since akis in neither sub-problem, and the subproblems are disjoint, |solution to S| = |solution to Si,k|+1+|solution to Sk,j|

11. Recursive Solution to Activity Selection Problem • Let Ai,j= optimal solution to Si,j. So Ai,j= Ai,k {ak} Ak,j, assuming: • Si,jis nonempty, and • we know ak. • Hence,

12. Finding the Greedy Algorithm • Theorem: Let Si,j , and let ambe the activity in Si,jwith the earliest finish time: fm=min { fk: akSi,j}. Then: • amis used in some maximum-size subset of mutually compatible activities of Si,j • Sim= , so that choosing amleaves Sm,jas the only nonempty subproblem.

13. Recursive Greedy Algorithm

14. Iterative Greedy Algorithm

15. Greedy Strategy • Determine the optimal substructure. • Develop a recursive solution. • Prove that at any stage of recursion, one of the optimal choices is the greedy choice. Therefore, it's always safe to make the greedy choice. • Show that all but one of the subproblems resulting from the greedy choice are empty. • Develop a recursive greedy algorithm. • Convert it to an iterative algorithm.

16. Money Change Problem • Given a currency system that has n coins with values v1, v2 , ..., vn, where v1 = 1, the objective is to pay change of value y in such a way that the total number of coins is minimized. More formally, we want to minimize the quantity subject to the constraint Here, x1, x2 , ..., xn, are nonnegative integers (so ximay be zero).

17. Money Change Problem • What is a greedy algorithm to solve this problem? • Is the greedy algorithm optimal?