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 s_i and a finish time f_i, where s_i ≤f_i. If selected activity "i" take place meanwhile the half-open time interval [s_i,f_i). Activities i and j are compatible if the intervals (s_i, f_i) and [s_i, f_i) do not overlap (i.e. i and j are compatible if s_i ≥f_i or s_i ≥f_i). The activity-selection problem chosen the maximum- size set of mutually consistent activities.

Algorithm Of Greedy- Activity Selector:

GREEDY- ACTIVITY SELECTOR (s, f)
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

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: