Optimization using Greedy Algorithms

1
Optimization using Greedy Algorithms

• Greedy Algorithms produce optimal solutions if
  they have the following characteristics
• Greedy-Choice Property
• Global optimal solutions can be obtained by
  making locally optimal choices
• Optimal Substructure
• Global optimal solution contains optimal
  solutions to subproblems

2
Activity Selection Problem Description (see
section 16.1, CLRS)

• We have a set of activities a1, , an
• We have a resource that can only be used by 1
  activity at a time
• ai starts at time si and finishes at time fi
  (sifi)
• ai and aj are compatible if si ? fj or sj ? fi
• Optimization problem select a maximum-size set
  of mutually compatible activities.

3
Activity Selection Algorithm and Example

• A a1
• j 1
• for(i 1 i lt n i)
• if(si gt fi)
• A A U ai
• j i

4
Matroid Properties(see section 16.4, CLRS)

• A matroid M (S,I) satisfies the following
• I is hereditary if B is in I, and A is a subset
  of B, then A is also in I
• Exchange property If A and B are both in I, and
  A lt B then there must be an element x in B-A
  such that AUx is in I.

5
General Greedy Algorithm for Matroids

• This algorithm will find an optimal subset A of
  the matroid M (S,I) with weight function w.
•  
• A ?
• Sort S into non-increasing order by weight w
• For each x in S (in non-increasing order)
• if A U x is in I
• then A A U x
• return A