Greedy Algorithms

1
Greedy Algorithms

• Like Dynamic Programming, greedy algorithms are
  used for optimization.
• The basic concept is that when we have a choice
  to make, make the one that looks best right now.
• That is, make a locally optimal choice in hope of
  getting a globally optimal solution.
• Greedy algorithms wont always yield an optimal
  solution, but sometimes, as in the activity
  selection problem, they do.

2
Activity-Selection Problem

• Sa1,a2, . . . ,an is set of activities wanting
  same resource.
• Each ai has start time si and finish time fi such
  that
• 0 si lt fi lt 8
• ai takes place in the half-open time interval si
  , fi).
• ai and aj are compatible if si , fi) and sj ,
  fj) do not overlap, (i.e. si ³ fj or sj ³ fi ).

3
Sample Set of Activities

• Sample set S (sorted by finish time)

4
Optimal Substructure of Activity-Selection Problem

• Define Sij ak ? S fi sk lt fk sj.
• (subset of activities in S that can start after
• ai finishes and finish before aj starts)
• To avoid special cases, add fictitious activities
  a0 and an1 and define f0 0 and sn1 8.
• Assume activities are sorted in order of finish
  time
• f0 f1 f2 fn lt fn1
• Thus space of subproblems is limited to selecting
  maximum-size subset from Sij for 0 i lt j n
  1. (All other Sij are empty.)

5
Optimal Substructure (continued)

• Suppose optimal solution Aij to Sij includes ak.
• Then solutions Aik to Sik and Akj to Skj must be
  optimal as well.

6
Recursive Solution

• 0
  if Sij Ø
• ci, j
• max ci, k ck, j 1 if
  Sij ? Ø
• iltkltj

7
Simplifying the Recursive Solution

• Consider any nonempty subproblem Sij, and let am
  be the activity in Sij with the earliest finish
  time
• fm min fk ak ? Sij.
• Then
• 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.

8
Simplifying the Recursive Solution (continued)

• Proof of Part 1
• Suppose Aij is a maximum-size subset of mutually
  compatible activities of Sij, and let us order
  the activities in Aij in order of finish time.
  Let ak be the first activity in Aij.
• Case 1 ak am
• Then am is used in some maximum-size
• subset of mutually compatible activities
• of Sij.

9
Simplifying the Recursive Solution (continued)

• Case 2 ak ? am
• Construct subset Aij Aij ak ? am. The
  activities in Aij are disjoint since those in
  Aij are disjoint, ak is the first activity in Aij
  to finish, and fm fk. Noting that Aij has the
  same number of activities as Aij, we see that
  Aij is a maximum-size subset of mutually
  compatible activities of Sij that includes am.

10
Simplifying the Recursive Solution (continued)

• Proof of Part 2
• Suppose Sim is nonempty. I.e., there is some
  activity ak such that fi sk lt fk sm lt fm.
  Then ak is also in Sij and has an earlier finish
  time than am. This contradicts our choice of am.
  Therefore Sim is empty.

11
Simplifying the Recursive Solution (continued)

• Pattern to the subproblems
• Original problem S S0, n1
• Choose am1 as activity in S with earliest finish
  time. (I.e., m1 1 since activities are sorted.)
• Next subproblem Sm1, n1
• Choose am2 as activity in Sm1, n1 with the
  earliest finish time. (Notice its not
  necessarily true that m2 2.)
• Next subproblem Sm2, n1
• Form of each subproblem Smi, n1 for some
  activity ami

12
Recursive Greedy Algorithm

• Assume that start and finish times are stored in
  arrays s and f .
• Let i and j be indices of subproblem to be
  solved.
• Assume activities are sorted in order of
  increasing finish times
• f1 f2 . . . fn

13
Recursive Greedy Algorithm (continued)

• RECURSIVE-ACTIVITY-SELECTOR (s, f, i, j)
• 1 m ? i 1
• 2 while m lt j and sm lt fi ? Find first
  activity in Sij
• 3 do m ? m 1
• 4 if m lt j
• 5 then return am ? RECURSIVE-ACTIVITY-SELEC
  TOR (s, j, m, j)
• else return Ø
• Initial call RECURSIVE-ACTIVITY-SELECTOR (s, f,
  0, n1)

14
k sk fk
0 - 0
a0
a1
1 1 4 2 3 5 3 0 6 4
5 7
RECURSIVE-ACTIVITY-SELECTOR (s, f, 0, 12)
a0
m 1
a2
RECURSIVE-ACTIVITY-SELECTOR (s, f, 1, 12)
a1
a3
a1
a4
a1
m 4
continued on next slide
time
0 1 2 3 4 5
6 7 8 9 10
11 12 13 14
15
k sk fk
RECURSIVE-ACTIVITY-SELECTOR (s, f, 4, 12)
a5

• 5 3 8
• 6 5 9
• 7 6 10
• 8 11

a1
a4
a6
a1
a4
a7
a1
a4
a8
a1
a4
m 8
continued on next slide
time
0 1 2 3 4 5
6 7 8 9 10
11 12 13 14
16
RECURSIVE-ACTIVITY-SELECTOR (s, f, 8, 12)
k sk fk
a9

• 8 12
• 2 13
• 12 14
• 12 8 -

a1
a4
a8
a10
a1
a4
a8
a11
a1
a4
a8
m 11
RECURSIVE-ACTIVITY-SELECTOR (s, f, 11, 12)
a1
a4
a8
a11
time
0 1 2 3 4 5
6 7 8 9 10
11 12 13 14
17
Iterative Greedy Algorithm

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

18
Elements of the Greedy Strategy

• Greedy-choice property
• A globally optimal solution can be arrived at by
  making a locally optimal (greedy) choice.
• Optimal substructure
• A problem exhibits optimal substructure if an
  optimal solution to the problem contains within
  it optimal solutions to subproblems.