Greedy Algorithm

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Greedy Algorithm
Instructor Yao-Ting Huang
Bioinformatics Laboratory, Department of Computer
Science Information Engineering, National Chung
Cheng University.
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Greedy Algorithms

• A greedy algorithm always makes the choice that
  looks best at the moment
• The hope a locally optimal choice will lead to a
  globally optimal solution.
• For some problems, it works.
• Dynamic programming can be overkill
• Greedy algorithms tend to be easier to code

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The Knapsack Problem

• The 0-1 knapsack problem
• The thief must choose among n items, where the
  ith item worth vi dollars and weighs wi pounds
• Carrying at most W pounds, maximize value
• Note assume vi, wi, and W are all integers
• 0-1 b/c each item must be taken or left in
  entirety
• A variation, the fractional knapsack problem
• Thief can take fractions of items
• Think of items in 0-1 problem as gold ingots, in
  fractional problem as buckets of gold dust

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Solving The Knapsack Problem

• The optimal solution to the fractional knapsack
  problem can be found with a greedy algorithm
• How?
• The optimal solution to the 0-1 problem cannot be
  found with greedy strategy.
• Greedy strategy take in order of dollars/pound
• Example 3 items weighing 10, 20, and 30 pounds,
  knapsack can hold 50 pounds

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The Greedy Strategy
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The Knapsack Problem Greedy Vs. Dynamic

• The fractional problem can be solved greedily.
• The 0-1 problem cannot be solved with a greedy
  approach.
• It can be solved with dynamic programming.

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Activity-Selection

• The activity-selection problem is to select a
  maximum-size subset of mutually compatible
  activities.
• Example

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Activity-Selection

• Formally
• Given a set S of n activities
• si start time of activity i
• fi finish time of activity i
• Find max-size subset A of compatible activities

• Assume that f1 ? f2 ? ? fn

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Activity SelectionRepeated Subproblems

• Consider a recursive algorithm that tries all
  possible compatible subsets to find a maximal
  set, and notice repeated subproblems

S1?A?
yes
no
S2?A?
S-12?A?
no
no
yes
yes
S-1,2
S
S-2
S
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The optimal substructure of the
activity-selection problem

• Sijak?S fi ? sk lt fk ? sj,
• Sij is the subset of activities in S that can
  start after activity ai finishes and finish
  before activity aj starts.
• f00 and sn1 ?. Then S S0,n1, and the
  ranges for i and j are given by 0 ? i, j ? n1.
• Let Aik and Akj be optimal solutions
• Aij could be Aik ?ak?Akj

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A recursive solution
Let ci, j size of maximum-size subset of
mutually compatible activities in Sij

• Dynamic programming?

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Converting a dynamic-programming solution to a
greedy solution

• Consider any nonempty subproblem Sij, and let am
  be the activity in Sij with the earliest finish
  time
• We will show
• 1. Activity am is used in some maximum-size
  subset of mutually compatible activities of Sij.
• 2. The subproblem Sim is empty, so that choosing
  am leaves the subproblem Smj as the only one that
  may be nonempty.

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Converting a dynamic-programming solution to a
greedy solution

• Let am be the activity in Sij with the earliest
  finish time
• We wanna prove the subproblem Sim is empty.
• Suppose Sim is nonempty.
• There is some activity ak between i and m.
• ak is earlier than am.
• Contradiction.

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Converting a dynamic-programming solution to a
greedy solution

• Let am be the activity in Sij with the earliest
  finish time
• Activity am is used in some maximum-size subset
  of mutually compatible activities of Sij.
• Let Aij be a maximum-size subset of activities in
  Sij
• Let ak is the first activity in Aij
• Suppose ak ? am
• Consider Aij - ak ? am
• fm lt fk
• This new set has the same size as Aij

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Converting a dynamic-programming solution to a
greedy solution

• Consider any nonempty subproblem Sij, and let am
  be the activity in Sij with the earliest finish
  time
• 1. Activity am is used in some maximum-size
  subset of mutually compatible activities of Sij.
• 2. The subproblem Sim is empty, so that choosing
  am leaves the subproblem Smj as the only one that
  may be nonempty.

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Activity SelectionA Greedy Algorithm

• So actual algorithm is simple
• Sort the activities by finish time
• Schedule the first activity (i.e., earliest
  finish time)
• Then schedule the next activity in sorted list
  which starts after previous activity finishes
• Repeat until no more activities
• Intuition is even more simple
• Always pick the shortest activity available at
  the time

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An iterative greedy algorithm

• GREEDY-ACTIVITY-SELECTOR(s, f)
• 1 n ? lengths
• 2 a ? a1
• 3 i ? 1
• 4 for m ? 2 to n
• 5 do if sm ? fi
• 6 then A ? A ? am
• 7 i ? m
• 8 return A

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Example on 11 Activities
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