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An Activity Selection Problem

The activity selection problem is a mathematical optimization problem. Our first illustration is the problem of scheduling a resource among several challenge activities. We find a greedy algorithm provides a well designed and simple method for selecting a maximum- size set of manually compatible activities.

Suppose S = {1, 2....n} is the set of n proposed activities. The activities share resources which can be used by only one activity at a time, e.g., Tennis Court, Lecture Hall, etc. Each Activity "i" has start time si and a finish time fi, where si ≤fi. If selected activity "i" take place meanwhile the half-open time interval [si,fi). Activities i and j are compatible if the intervals (si, fi) and [si, fi) do not overlap (i.e. i and j are compatible if si ≥fi or si ≥fi). The activity-selection problem chosen the maximum- size set of mutually consistent activities.

Algorithm Of Greedy- Activity Selector:

1. n ← length [s]
2. A ← {1}
3. j ← 1.
4. for i ← 2 to n
5. do if si ≥ fi
6. then A ← A ∪ {i}
7. j ← i
8. return A

Example: Given 10 activities along with their start and end time as

S = (A1 A2 A3 A4 A5 A6 A7 A8 A9 A10)
Si = (1,2,3,4,7,8,9,9,11,12)
fi = (3,5,4,7,10,9,11,13,12,14)

Compute a schedule where the greatest number of activities takes place.

Solution: The solution to the above Activity scheduling problem using a greedy strategy is illustrated below:

Arranging the activities in increasing order of end time

Activity Selection Problem
Activity Selection Problem

Now, schedule A1

Next schedule A3 as A1 and A3 are non-interfering.

Next skip A2 as it is interfering.

Next, schedule A4 as A1 A3 and A4 are non-interfering, then next, schedule A6 as A1 A3 A4 and A6 are non-interfering.

Skip A5 as it is interfering.

Next, schedule A7 as A1 A3 A4 A6 and A7 are non-interfering.

Next, schedule A9 as A1 A3 A4 A6 A7 and A9 are non-interfering.

Skip A8 as it is interfering.

Next, schedule A10 as A1 A3 A4 A6 A7 A9 and A10 are non-interfering.

Thus the final Activity schedule is:

Activity Selection Problem

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