Transcript 슬라이드 1 - Go into The Algorithm

16. Greedy Algorithms
Heejin Park
College of Information and Communications
Hanyang University
Contents
Introduction
An activity selection problem
Elements of the greedy strategy
Huffman codes
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Introduction
A greedy algorithm always makes the choice that looks best at
the moment.
It makes a locally optimal choice in the hope that this choice
will lead to a globally optimal solution.
It makes the choice before the subproblems are solved.
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An activity selection problem
An activity selection problem
 To select a maximum-size subset of mutually compatible
activities.
 For example
 Given n classes and 1 lecture room,
 to select the maximum number of classes
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An activity selection problem

A set of activities: S = {a1, a2, ..., an}

Each activity ai has its start time si and finish time fi.
 0 ≤ si < f i < ∞

Activity ai takes place during [si, fi)

Activities ai and aj are compatible
if the intervals [si, fi) and [sj fj) do not overlap.
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An activity selection problem
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An activity selection problem
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An activity selection problem
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{a1, a4, a8, a11}: A largest set of mutually compatible activities
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An activity selection problem
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An activity selection problem
Optimal substructure
 Sij denote the set of activities between ai and aj and
compatible with ai and aj.
 Activities start after ai finishes and finish before aj
starts.
Sij  {ak  S : fi  sk  f k  s j }

For example, S18 = {a4}
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An activity selection problem
Optimal substructure
 We define a0 and an+1 such that f0 = 0 and sn+1 = ∞.
 S = S0,n+1 for 0 ≤ i, j ≤ n + 1.
 Assume that activities are sorted in increasing order of
finish time.
f 0  f1  f 2    f n  f n1
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An activity selection problem
Optimal substructure
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Aij denote an optimal solution to Sij for i ≤ j.
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If Aij includes ak, Aij = Aik U {ak} U Akj
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An activity selection problem
Optimal substructure
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c[i, j]: The number of activities in Aij.
if Sij= Ø
0
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c[i, j ]  max{c[i, k ]  c[k , j ]  1} if Sij≠ Ø
ik  j
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 ak Sij
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An activity selection problem
Greedy algorithm
 Consider any nonempty Sij, and let am be the activity in Sij
with the earliest finish time: fm = min {fk : ak ∈Sij}.
1. Activity am is in some Aij.
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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An activity selection problem
Activity am is in some Aij.
 ak: the first activity in Aij
 If ak = am, done.
 If ak ≠ am , remove ak from Aij add am. The resulting Aij is
another optimal solution because fm ≤ fk.
am
ak
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An activity selection problem
The subproblem Sim is empty, so that choosing am leaves the
subproblem Smj as the only one that may be nonempty.

Sim is empty because am has the earlier finish time in Sij.
am
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Greedy algorithm
 Select the earliest finishing activity one by one.
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