CS660: Timing Code

CS660 Combinatorial Algorithms
Fall Semester, 1996
Timing Code

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San Diego State University -- This page last updated Monday, 09 September, 1996

Contents of Timing Code

                                                Timing Code Slide # 1

References

Introduction to Mathematical Statistics, 3rd Edition, Hogg, Craig, 1970, Chapter 6

                                                Timing Code Slide # 2

Timing Analysis

Timing in C on Rohan

main()
{

int k, iterations;
for (iterations = 0; iterations < 50; iterations++)
{ start(); /* start the timer */
for (k = 0; k < 2000000; k++) /* do some work */ k = k;
stop(); /* stop the timer */

printf("Time taken: %ld\n", report());
};

}
Result on Rohan
Time Frequency Occurred
30 2
31 2
32 9
33 10
34 11
35 9
36 5
37 1
39 1

                                                Timing Code Slide # 3

Source for Timing C Code on Rohan
#include <stdio.h>
#include <sys/times.h>
#include <limits.h>

static struct tms _start; /* Stores the starting time*/
static struct tms _stop; /* Stores the ending time*/

int start()
{
times(&_start);
}

int stop()
{
times(&_stop);
}

unsigned long report()
{
return _stop.tms_utime - _start.tms_utime;
}

main()
{
int k, iterations;

for (iterations = 0; iterations < 50; iterations++)
{
start();: /* start the timer */

for (k = 0; k < 2000000; k++): /* do some work */
k = k;

stop();: /* stop the timer */

printf("Time taken: %ld\n", report());
};

}

                                                Timing Code Slide # 4

Handling Measurement Errors [1]

Repeat a measurement n times
Let the measurements be labeled

Let

and

The confidence interval for the true measurement is[2]:

The value of t determine the probability the measurement is in the interval

When n >= 50
Probability value of t

50%	80%	90%	95%	99%
0.67	1.28	1.64	1.96	2.58

In Example

, s*s = 3.15, s = 1.78 selecting t = 1.96 we get

95% confidence interval is (33.20, 34.20) for actual loop time

                                                Timing Code Slide # 5

Student t table - When n < 50

r = n-1	90%	95%	99%
1	6.314	12.706	63.657
2	2.920	4.303	9.925
3	2.353	3.182	5.841
4	2.132	2.776	4.604
5	2.015	2.571	4.032

6	1.943	2.447	3.707
7	1.895	2.365	3.499
8	1.860	2.306	3.355
9	1.833	2.262	3.250
10	1.812	2.228	3.169

20	1.725	2.086	2.845
30	1.697	2.042	2.750

                                                Timing Code Slide # 6

Estimating Complexity from Timing Results

Fun with Functions
Let f(n) = 3n*n + 4n + 5 and g(n) = 3n*n

Fact: g(n) is an approximation of f(n)

Notation: f(n) = g(n) +

n	f(n)	g(n)	% error
1	12	3	75.00%
10	345	300	13.04%
20	1285	1200	6.61%
30	2825	2700	4.42%
40	4965	4800	3.32%
50	7705	7500	2.66%
60	11045	10800	2.22%
70	14985	14700	1.90%
80	19525	19200	1.66%
90	24665	24300	1.48%
100	30405	30000	1.33%
200	120805	120000	0.67%
300	271205	270000	0.44%

                                                Timing Code Slide # 7

Eyeballing Complexity
Let

then

Timing Results

N	Bubble	Insertion
100	1	1
200	5	3
400	19	11
800	79	42
1600	317	166

                                                Timing Code Slide # 8

Plotting Complexity Cubic or Quadratic[3]?

                                                Timing Code Slide # 9

Plotting Complexity Engineers Method (Modified)
Let

then

Let b = 2 and

then

                                                Timing Code Slide # 10

Plotting Complexity Transform the Axis
Let

and

(or

) then:

g(J) = f(

) = a(

)^k= aJ

So g(J) is linear!

                                                Timing Code Slide # 11

Example

n	f(n) =5n*n+n + 3	J=n*n
1	9	1
10	513	100
20	2023	400
30	4533	900
40	8043	1600
50	12553	2500
60	18063	3600

                                                Timing Code Slide # 12

Which is Quadratic?

                                                Timing Code Slide # 13

Mathematical Analysis and Timing Code

Bubble sort worst case is

( n*n)

Complexity is an*n+ bn + c

Timing Results Worst Case

N	Bubble Sort
400	20
500	31
600	45
700	61
800	79

Least Squares fit of data to an*n+ bn + c
Bubble sort worst case is 0.0001143n*n + 0.01084n - 2.738

Predicted vs. Actual Time for Bubble Sort

N	Actual	Predicted	% Error
900	105	99.601	5.14%
1000	124	122.402	1.29%
1100	149	147.489	1.01%
2000	496	476.142	4.00%
2400	713	681.646	4.40%

CS660 Combinatorial Algorithms Fall Semester, 1996 Timing Code

Contents of Timing Code

References

Timing Analysis

Handling Measurement Errors[1]

Estimating Complexity from Timing Results

Mathematical Analysis and Timing Code

CS660 Combinatorial Algorithms
Fall Semester, 1996
Timing Code

Handling Measurement Errors [1]