Greedy Algorithms - Analysis of Algorithms - Lecture Slides, Slides of Design and Analysis of Algorithms

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Greedy Algorithms

Greedy Algorithms

• Similar to dynamic programming.
• Used for optimization problems.
• Idea
  • When we have a choice to make, make the one that looks best right now. Make a locally optimal choice in hope of getting a globally optimal solution.
• Greedy algorithms don’t always yield an optimal solution. But sometimes they do. We’ll see a problem for which they do. Then we’ll look at some general characteristics of when greedy algorithms give optimal solutions.

Activity selection

• Note
• Could have many other objectives:
  • Schedule room for longest time.
  • Maximize income rental fees.
• Assume that activities are sorted by finish

time: f 1 ≤ f 2 ≤ f 3 ≤... ≤ fn-1 ≤ fn.

Example

• S sorted by finish time:

Optimal substructure of activity

selection

• Let Aij be a maximum-size set of mutually

compatible activities in S ij.

• Let ak ∈ Aij be some activity in Aij. Then we

have two subproblems:

• Find mutually compatible activities in Sik (activities that start after a (^) i finishes and that finish before a (^) k starts).
• Find mutually compatible activities in Skj (activities that start after a (^) k finishes and that finish before a (^) j starts).

Optimal substructure of activity

selection

• Let
• A (^) ik = A (^) ij ∩ Sik = activities in A (^) ij that finish before a (^) k starts ;
• A (^) kj = A (^) ij ∩ Skj = activities in A (^) ij that start afer a (^) k finishes :
• Then A (^) ij = A (^) ik ∪{a (^) k } ∪A (^) kj
• ⇒ ⎮A (^) ij⎮= ⎮A (^) ik⎮ + ⎮A (^) kj⎮+ 1.
• Claim
• Optimal solution A (^) ij must include optimal solutions for the two subproblems for S (^) ik and S (^) kj.

One recursive solution

• Since optimal solution A (^) ij must include optimal solutions to the subproblems for S (^) ik and S (^) kj, could solve by dynamic programming.
• Let c[i, j] = size of optimal solution for Sij. Then
• c[i, j] = c[i, k] + c[k, j] + 1:
• But we don’t know which activity a (^) k to choose, so we have to try them all:
• Could then develop a recursive algorithm and memoize it. Or could develop a bottom-up algorithm and fill in table entries.
• Instead, we will look at a greedy approach.

Making the greedy choice

• Choose an activity to add to optimal solution before solving subproblems. For activity-selection problem, we can get away with considering only the greedy choice: the activity that leaves the resource available for as many other activities as possible.
• Question: Which activity leaves the resource available for the most other activities?
• Answer: The first activity to finish. (If more than one activity has earliest finish time, can choose any such activity.)
• Since activities are sorted by finish time, just choose activity a 1.

Making the greedy choice

• Making greedy choice of a 1 ⇒ S 1 remains as

only subproblem to solve.

• By optimal substructure, if a 1 is in an optimal

solution, then an optimal solution to the

original problem consists of a 1 plus all

activities in an optimal solution to S 1.

• But need to prove that a 1 is always part of

some optimal solution.

Theorem

• If S (^) k is nonempty and a (^) m has the earliest finish time in S (^) k, then a (^) m is included in some optimal solution.
• Proof Let Ak be an optimal solution to S (^) k, and let a (^) j have the earliest finish time of any activity in Ak. If a (^) j = a (^) m , done. Otherwise, let A’k = Ak – {a (^) j} ∪ {a (^) m} but with a (^) m substituted for a (^) j.
• Claim
• Activities in A’k are disjoint.
• Proof Activities in Ak are disjoint, a (^) j is first activity in Ak to finish, and f (^) m ≤ f (^) j. (proves claim)
• Since |A’k| = |Ak|, conclude that A’k is an optimal solution to S (^) k, and it includes a (^) m. (proves theorem)

Recursive greedy algorithm

• Start and finish times are represented by arrays s

and f , where f is assumed to be already sorted in monotonically increasing order.

• To start, add fictitious activity a 0 with f 0 = 0, so

that S 0 = S, the entire set of activities.

• Procedure REC-ACTIVITY-SELECTOR takes as

parameters the arrays s and f, index k of current subproblem, and number n of activities in the original problem.

Recursive greedy algorithm

Iterative greedy algorithm

• Can convert the recursive algorithm to an iterative

one. It’s already almost tail recursive.

Greedy strategy

• The choice that seems best at the moment is the one we go with.
• What did we do for activity selection?

1. Determine the optimal substructure.
2. Develop a recursive solution.
3. Show that if we make the greedy choice, only one subproblem remains.
4. Prove that it’s always safe to make the greedy choice.
5. Develop a recursive greedy algorithm.
6. Convert it to an iterative algorithm.