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activitySelection.md

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Activity Selection

Scheduling several activities that require exclusive use of a common resource, with a goal of selecting a maximum-size set of mutually compatible activities.

Suppose we have a set S = {a1, a2,...,an} of n proposed activities that wish to use a resource, such as a lecture hall, which can serve only one activity at a time. Each activity ai has a start time si and a finish time fi. we wish to select a maximum-size subset of mutually compatible activities

Solution

  • Greedy choice(local choice): select activity based on finish time.

  • Algorithm:

    • Step 1: Sort all activity based on finish time in increasing order.
    • Step 2: Take activity that has the lowest finish time.
    • Step 3: Apply compatible rule and remove activities that have conflict with activity selected in Step 2.
    • Step 4: Iterate Step 2 and Step 3 until no activity remain.

Example

Consider the following set S of activities:

index i 1 2 3 4 5 6 7 8 9 10 11
Start time Si 1 3 0 5 3 5 6 8 8 2 12
Finish time Fi 4 5 6 7 9 9 10 11 12 14 16

In Above table activities sorted based on finish time, and now we apply rest of algorithm:

  • Iterate 1:

    • Select activity that have the lowest finish time : a1
    • Remove conflict from set: a2, a3, a5, a10

  • Iterate 2:

    • Select activity that have the lowest finish time : a4
    • Remove conflict from set: a6, a7

  • Iterate 3:

    • Select activity that have the lowest finish time : a8
    • Remove conflict from set: a9

  • Iterate 4:

    • Select activity that have the lowest finish time : a11
    • Remove conflict from set: no conflict

  • Set is empty. So result is {a1, a4, a8, a11}