Explanation of Activity Selection Problem with an example

1. AN ACTIVITY-SELECTION PROBLEM
BY SUMITA DAS
Created by Sumita Das

2. An Activity-selection problem
 Suppose we have a set of activities S={a1,a2,….,an} (Total n
activities) that wish to use a common resource which can serve
only one activity at a time.
 Mutually exclusive access to single resource.
 Each activity ai is having start time si and finish time fi, where
0<=si<=fi<∞
 Goal: To find a maximum-size subset of mutually compatible
activities
 Compatible Activities: Activities ai and aj are compatible if si>=fj
or sj>=fi
i.e. Second Activity starts before first activity finishes.
Created by Sumita Das

3. Let set S consists of following activities which is sorted in
monotonically increasing order of finish time.
k 0 1 2 3 4 5 6 7 8 9 10 11
sk - 1 3 0 5 3 5 6 8 8 2 12
fk 0 4 5 6 7 8 9 10 11 12 14 16
Created by Sumita Das

4. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
a11
Created by Sumita Das

5. RECURSIVE-ACTIVITY-SELECTOR( s, f, k, n)
1. m= k +1
2. while m <= n and s[m] < f[k]
3. m=m +1
4. if m <= n
5. then return {am}U RECURSIVE-ACTIVITY- SELECTOR
(s, f, m, n)
6. else return Ø
A Recursive Greedy Algorithm
Created by Sumita Das

6. s- array of start time of activities
f- array of finish time of activities
k- index of subproblem Sk
Where Sk is the set of activities that start after activity ak
finishes.
n- size of original problem
 Add a fictitious activity a0 with f0=0
 So Subproblem S0 is entire set of activities S.
 Initial call RECURSIVE-ACTIVITY-SELECTOR( s, f, 0,
n) solves the entire problem
Created by Sumita Das

7. RECURSIVE-ACTIVITY-SELECTOR( s, f, 0, 11)
1. m= k +1
2. while m <= n and s[m] < f[k]
3. m=m +1
4. if m <= n
5. then return {am}U RECURSIVE-ACTIVITY- SELECTOR
(s, f, m, n)
6. else return Ø
k=0 m=0+1=1
1<=11 and s[1]<f[0]
1<=11
1<0 false. While loop exits
Return a1 U RECURSIVE-
ACTIVITY- SELECTOR (s, f,
1, 11)
k 0 1 2 3 4 5 6 7 8 9 10 11
sk - 1 3 0 5 3 5 6 8 8 2 12
fk 0 4 5 6 7 8 9 10 11 12 14 16
Created by Sumita Das

8. RECURSIVE-ACTIVITY-SELECTOR( s, f, 1, 11)
1. m= k +1
2. while m <= n and s[m] < f[k]
3. m=m +1
4. if m <= n
5. then return {am}U RECURSIVE-ACTIVITY- SELECTOR
(s, f, m, n)
6. else return Ø
k=1 m=1+1=2
2<=11 and s[2]<f[1] 3<4 true
m=2+1=3
3<=11 and s[3]<f[1] 0<4 true
m=3+1=4
4<=11 and s[4]<f[1] 5<4 false. While loop exits4<=11
Return a4 U RECURSIVE-
ACTIVITY- SELECTOR (s, f,
4, 11)
k 0 1 2 3 4 5 6 7 8 9 10 11
sk - 1 3 0 5 3 5 6 8 8 2 12
fk 0 4 5 6 7 8 9 10 11 12 14 16
Created by Sumita Das

9. RECURSIVE-ACTIVITY-SELECTOR( s, f, 4, 11)
1. m= k +1
2. while m <= n and s[m] < f[k]
3. m=m +1
4. if m <= n
5. then return {am}U RECURSIVE-ACTIVITY- SELECTOR
(s, f, m, n)
6. else return Ø
k=4 m=4+1=5
5<=11 and s[5]<f[4] 3<7 true
m=5+1=6
6<=11 and s[6]<f[4] 5<7 true
m=6+1=7
7<=11 and s[7]<f[4] 6<7 true
m=7+1=8
8<=11 and
s[8]<f[4] 8<7 false.While loop exits8<=11
Return a8 U RECURSIVE-
ACTIVITY- SELECTOR (s, f,
8, 11)
k 0 1 2 3 4 5 6 7 8 9 10 11
sk - 1 3 0 5 3 5 6 8 8 2 12
fk 0 4 5 6 7 8 9 10 11 12 14 16
Created by Sumita Das

10. RECURSIVE-ACTIVITY-SELECTOR( s, f, 8, 11)
1. m= k +1
2. while m <= n and s[m] < f[k]
3. m=m +1
4. if m <= n
5. then return {am}U RECURSIVE-ACTIVITY- SELECTOR
(s, f, m, n)
6. else return Ø
k=8 m=8+1=9
9<=11 and s[9]<f[8] 8<11 true
m=9+1=10
10<=11 and s[10]<f[8] 2<11 true
m=10+1=11
11<=11 and s[11]<f[8] 12<11 falseWhile
loop exits
11<=11
Return a11 U RECURSIVE-
ACTIVITY- SELECTOR (s, f,
11, 11)
k 0 1 2 3 4 5 6 7 8 9 10 11
sk - 1 3 0 5 3 5 6 8 8 2 12
fk 0 4 5 6 7 8 9 10 11 12 14 16
Created by Sumita Das

11.  RECURSIVE-ACTIVITY- SELECTOR (s, f, 11, 11)
returns Ø.
 Resulting set of selected activities is {a1, a4 a8, a11}
Created by Sumita Das

12. References
 Introduction to Algorithms 2nd ,Cormen, Leiserson,
Rivest and Stein, The MIT Press, 2001.
 Introduction to Design & Analysis Computer Algorithm
3rd, Sara Baase, Allen Van Gelder, Adison-Wesley, 2000.
Created by Sumita Das