1. Application: Efficiency of Algorithms I
Lecture 40
Section 9.3
Thu, Mar 31, 2005

2. Estimating Run Times
Suppose the run-time of a program has growth rate (f(x)), where x is the input size.
Then the run-time is (approx.) of the form
t = cf(x)
for some real number c.

3. Estimating Run Times
Suppose the program runs in t0 seconds when the input size is x0. Then
c = t0/f(x0).
Thus, the run time is given by

4. Example: Estimating Run Time
Suppose a program has growth rate (x2) and that the program runs in t0 = 5 s when the input size is x0 = 100.
Then the run time is given by
t = 5(x2/1002) s.
Thus, if input size were 500, then the run time would
t = 55002/1002 = 125 s.

5. Example: Estimating Run Time
Suppose a program has growth rate
(x log x).
and that the program runs in t0 = 5 s when the input size is x0 = 100.
Then the run time is

6. Example: Estimating Run Time
Thus, if input size were 500, then the run time would be

7. Comparison of Growth Rates
Consider programs with growth rates of (1), (log x), (x), (x log x), and (x2).
Assume that each program runs in 1 s when the input size is 100.
Calculate the run times for input sizes 102, 103, 104, 105, 106, and 107.

8. Comparison of Growth Ratesx
(1)
(log x)
(x)
(x log x)
(x2)
102
1 s
1.0 s
103
1.5 s
10 s
15 s
100 s
104
2.0 s
200 s
10 ms
105
2.5 s
1 ms
2.5 ms
1 sec
106
3.0 s
30 ms
1.7 min
107
3.5 s
100 ms
350 ms
2.8 hrs

9. Comparison with (2x) Now consider a program with growth rate (2x).
Assume that the program runs in 1 s when the input size is 100.
Calculate the run times for input sizes 100, 110, 120, 130, 140, and 150.

10. Comparison with (2x) x (x2) (2x) 100 1 s 110 1.2 s 1.0 ms 120
1.0 sec
130
1.7 s
18 min
140
2.0 s
13 days
150
2.3 s
36 yrs :
1000
100 s
2.7 x y

11. Comparison with (2x) x (x2) (2x) 100 1 s 110 1.2 s 1.0 ms 120
1.0 sec
130
1.7 s
18 min
140
2.0 s
13 days
150
2.3 s
36 yrs :
1000
100 s
2.7 x y x
(1)
(log x)
(x)
(x log x)
(x2)
102
1 s
1.0 s
103
1.5 s
10 s
15 s
100 s
104
2.0 s
200 s
10 ms
105
2.5 s
1 ms
2.5 ms
1 sec
106
3.0 s
30 ms
1.7 min
107
3.5 s
100 ms
350 ms
2.8 hrs

12. Comparison with (2x) x (x2) (2x) 100 1 s 110 1.2 s 1.0 ms 120
1.0 sec
130
1.7 s
18 min
140
2.0 s
13 days
150
2.3 s
36 yrs :
1000
100 s
2.7 x y x
(1)
(log x)
(x)
(x log x)
(x2)
102
1 s
1.0 s
103
1.5 s
10 s
15 s
100 s
104
2.0 s
200 s
10 ms
105
2.5 s
1 ms
2.5 ms
1 sec
106
3.0 s
30 ms
1.7 min
107
3.5 s
100 ms
350 ms
2.8 hrs

13. Comparison with (x!) Now consider a program with growth rate (x!).
Assume that the program runs in 1 s when the input size is 100.
Calculate the run times for input sizes 100, 101, 102, 103, 104, and 105.

14. Comparison with (x!) x (2x) (x!) 100 1 s 101 2 s 101 s 102 4 s
10 ms
103
8 s
1.1 sec
104
16 s
1.8 min
105
32 s
3.2 hrs :
110
1.0 ms
5.4 x 106 y

15. Comparison with (x!) x (2x) (x!) 100 1 s 101 2 s 101 s 102 4 s
10 ms
103
8 s
1.1 sec
104
16 s
1.8 min
105
32 s
3.2 hrs :
110
1.0 ms
5.3 x 106 y x
(x2)
(2x)
100
1 s
110
1.2 s
1.0 ms
120
1.4 s
1.0 sec
130
1.7 s
18 min
140
2.0 s
13 days
150
2.3 s
36 yrs :
1000
100 s
2.7 x y

16. Comparison with (x!) x (2x) (x!) 100 1 s 101 2 s 101 s 102 4 s
10 ms
103
8 s
1.1 sec
104
16 s
1.8 min
105
32 s
3.2 hrs :
110
1.0 ms
5.3 x 106 y x
(x2)
(2x)
100
1 s
110
1.2 s
1.0 ms
120
1.4 s
1.0 sec
130
1.7 s
18 min
140
2.0 s
13 days
150
2.3 s
36 yrs :
1000
100 s
2.7 x y

17. Tractable vs. Intractable
Algorithms that are O(xd), for some d > 0, are considered tractable.
Algorithms that are not O(xd), for some d > 0, are considered intractable.
Of course, all algorithms are do-able for small input sizes.
It is practical to factor small integers (< 50 digits).
It is impractical to factor large integers (> 200 digits).