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
12.
Early Finish GreedySort the set of activities by finishing time (f[i])S=1f = 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.
Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.
Be the first to comment