Greedy algorithms make locally optimal choices at each step in the hopes of finding a global optimum. They have two key properties: the greedy choice property, where choices are not reconsidered, and optimal substructure, where optimal solutions contain optimal solutions to subproblems. The activity selection problem of scheduling non-conflicting activities is used as an example. It can be solved greedily by always selecting the activity with the earliest finish time. While greedy algorithms are simple, they do not always find the true optimal solution.

1. Greedy is Good
Greedy Algorithm

2. Introduction
• Greedy algorithm always makes the choice that
looks best at the moment, with hoping that a locally
optimal choice will lead to a global optimum
• Similar to Dynamic Programming, it applies to
Optimization Problem
• It is usually easy to think up and implement
• Most problems for which they work well have two
properties
– Greedy choice property
– Optimal substructure

3. Greedy choice property
• Greedy algorithm never reconsiders its choices
• This is main difference from dynamic programming

4. Optimal substructure
• Optimal solution to the problem contains optimal
solution to the sub-problems

5. Activity Selection problem
• Activity Selection problem is to select the maximum
number of activities that can be performed by a
single person or machine within a time frame , given
a set of activities each marked by start time and
finish time
• Formal definition
– number of activity: n
– start time of activity i is si
– finish time of activity i is fi
– non-conflicting activities i and j: si≥fj or sj≥fi
– Find the maximum set (S) of non-conflicting activities

6. Early Finish Greedy
• Activity Selection problem has optimal substructure
– Assume that activities are sorted by monotonically
increasing finish time
– Aij = Aik ∪ {ak} ∪ Akj
• Select the activity with the earliest finish
• Eliminate the activities that are in conflict
• Repeat until there is no remains

7. Early Finish Greedy

8. Early Finish Greedy

9. Early Finish Greedy

10. Early Finish Greedy

11. Early Finish Greedy

12. Early Finish Greedy
Sort the set of activities by finishing time (f[i])
S=1
f = f[1]
for i=1 to n
if s[i] ≥ f
S=SUi
f = f[i]
end for

13. Cases of failure
• Greedy algorithms don’t always yields on optimal
solution
• Ex) How can a given amount of money be made with
the least number of coins of given denominations?
– Target amount: 6
– Denominations: 1, 3, 4
– Greedy solution: (4, 1, 1)
– Optimal solution: (3, 3)

14. Conclusion
• Greedy algorithms are usually easy to think of, easy
to implement and run fast,
• but it may fail to produce the optimal solution
• Mathematical concepts may give you a recipe for
proving that a problem can be solved with greedy,
but it ultimately comes down to the experience of
the programmer.

15. References
• http://en.wikipedia.org/wiki/Greedy_algorithm
• http://www.topcoder.com/tc?module=Static&d1=tu
torials&d2=greedyAlg
• http://security.re.kr/~sjkim/LectureNotes/SKKU/201
0/CSE3002/Lec13(Alg).pdf