Two-week ISTE workshop on Effective teaching/learning of computer programming Dr Deepak B Phatak Subrao Nilekani Chair Professor Department of CSE, Kanwal Rekhi Building IIT Bombay Lecture 7, Efficiency of programs Thursday 1 July 2010

Overview  Efficiency of programs Notion of time complexity  Example Estimating value of PI Calculating terms of Fibonacci series [Some slides courtesy Prof Milind Sohoni]

Computational time  Although the computer works very fast, it does take a finite amount of time for every computation  If we do not carefully design our program, we may force the machine to do many more computations than necessary to solve the problem  The order of magnitude of time required to execute any program is called its “Time Complexity”  We should design our algorithm (steps in our program) such that the execution time is minimized

Example

Example …

Estimating value of PI #include using namespace std; // estimate pi int main(){ float pi; int i, j, N, count; cout << “Give value of N\n”; cin >> N;

Estimating PI … count=0; for (i=1;i<=N;i=i+1){ for (j=1;j<=N;j=j+1){ if (i*i+j*j<=N*N) count=count+1; }; pi=4.0*count/(N*N); cout << pi << “\n”; return 0; }

Estimating PI … count=0; for (i=1;i<=N;i=i+1){ for (j=1;j<=N;j=j+1){ if (i*i+j*j<=N*N) count=count+1; }; pi=4.0*count/(N*N); cout << pi << “\n”; return 0; }

Quiz Q. We have declared our variables i, j, N as integers, the effect on the estimation of Pi using pi=4.0*count/(N*N); will be A.Negligible, we have declared pi as float B.Very large, because the division operation in the final formula [count/(N*N)] is of the type integer divided by integer C.Very large because the values of some terms may be beyond the limits of integer representation D.None of these

Modified program version 1 #include using namespace std; // estimate pi int main() { float pi; float i,j,N; long count; cout << "N? " ; cin >> N;

Modified program version 1 … count=0; for (i=1;i<=N;i=i+1){ for (j=1;j<=N;j=j+1){ if (i*i+j*j<=N*N) count=count+1; }; pi=4.0*count/(N*N); cout << "Value of Pi is: " << pi << "\n"; return 0; }

Execution results cpp]$ c++ calculatepi.cpp -o pi cpp]$./pi N? 1000 Value of Pi is: cpp]$./pi N? Value of Pi is:

Determining computational efficiency  Having ensured that our program works correctly, we now wish to determine how long it takes to run the program for different values of N  we want to find the execution time using some utility program which internally keeps time while the Operating System (OS) runs our program  UNIX OS and its variants provide a utility program called ‘time’ ALL OS’s provide such utilities Special software used for simulating large number of concurrent users Loadrunner, J-meter,...

Measuring execution time  We can use this command to run our program under its control cpp]$ time./pi N? Value of Pi is: real 0m3.690s user 0m1.076s sys 0m0.012s  What do these different values signify?

Explanations  Real time means the total clock time which the program took to execute from start to finish Includes time spent by us in giving input values  user time and sys (system) time are two independent time counts User time is actually the computing time taken by the computer to execute the program Sys time is the time which was spent by the supervising OS for executing our program  User time represents our algorithmic time complexity

Execution results cpp]$ c++ calculatepi.cpp -o pi cpp]$ time./pi N? 1000 Value of Pi is: real 0m2.496s user 0m0.000s sys 0m0.000s cpp]$ time./pi N? Value of Pi is: real 0m3.690s user 0m1.076s sys 0m0.012s

Execution results … cpp]$ time./pi N? Value of Pi is: real 0m7.181s user 0m4.304s sys 0m0.000s cpp]$ time./pi N? Value of Pi is: real 0m29.742s user 0m26.714s sys 0m0.000s cpp]$

Further reduction in execution time  We observe that the major computation is happening during evaluation of if condition  if (i*i+j*j<=N*N) count=count+1  This computation is done within a nested iteration each running N times. (so N 2 times)  We notice that while the values of i and j are changing every time, that of N is fixed during all iterations So why calculate N*N again and again?

Modified program version 2 … int n2 = N*N; count=0; for (i=1;i<=N;i=i+1){ for (j=1;j<=N;j=j+1){ if (i*i+j*j<=n2) count=count+1; }; pi=4.0*count/n2;

Execution results … cpp]$ c++ calculatepiv2.cpp -o v2 cpp]$ time./v2 N? Value of Pi is: real 0m7.821s user 0m4.308s sys 0m0.012s cpp]$ time./v2 N? Value of Pi is: real 0m31.313s user 0m26.898s No appreciable change, why? sys 0m0.012s

Quiz Q. The execution time for each of the two versions is not appreciably different because: A Multiplication does not take very large time, it is the division operation and the addition operation which is time consuming B since i and j are varying, computing i*i, j*j each takes much longer than N*N C Our program somehow figures that N is not changing, so it calculates N*N only once and uses that value repeatedly D I do not know and also cannot guess

Further reduction in execution time  We observe that the major computation is happening during evaluation of if condition if (i*i+j*j<=N*N) count=count+1  This computation is done within a nested iteration, each running N times. (so N 2 times)  We notice that the value of N is fixed during all iterations, so why calculate N*N again and again?  Instead of considering the square, we can consider only a triangle (half of square) with area Pi/8 This will reduce computations by half

Modified program version 3 … n2 = N*N; count=0; for (i=1;i<=N;i=i+1){ for (j=i;j<=N;j=j+1){ if (i*i+j*j<=n2) count=count+1; }; pi=8.0*count/n2; cout << endl << "Value of Pi is: “ cout << pi << "\n"; return 0;

Execution results … cpp]$ time./v3 N? Value of Pi is: real 0m4.301s user 0m2.132s sys 0m0.008s cpp]$ time./v3 N? Value of Pi is: real 0m17.216s user 0m13.461s sys 0m0.012s Appreciable reduction

Time complexity of an algorithm  There are two ways of looking at the time taken by a program to execute.  Micro-view We look at the time taken by each instruction in the program to execute, and the number of times that instruction is executed. While the machine executes instructions in few tens of microseconds, or even in hundreds of nanoseconds, every instruction takes a different amount of time depending upon its nature A ‘unit’ time can be used to represent comparative execution times

Some hypothetical comparative execution times  Assignment 1 unit  Addition/subtraction 2  Multiplication 3  Division 5  Comparison 3  Floating point operation 5 times int  Some other operations which we have not yet studied, such as reading from disk, or assignment to an array element, take longer time Calculating value of index expression, e.g. A[3*i+j]

Time complexity of an algorithm …  If there is an iteration which executes N times, where N is an input value If N largely determines the amount of computations in a program, we call it the ‘size’ of the problem Number of computations are expressed in terms of N. For our Pi calculations, these may be, say, 24 N N  In general aN 2 + bN + c

Time complexity, Macro view  When comparing two algorithms, we can get such expressions for each, and then compare for specific values of N  Consider prog1, and prog2, with time complexity: Prog1: 24 N N Prog2: N  Prog1 will run faster than prog2 up to a certain N (What is that value?)  But for all values of N greater than this threshold, prog2 will always run faster

Time complexity, Macro view …  It is customary to compare algorithms on the basis of their behaviour for very large values of N [ Limit as N infinity]  Thus Prog1 is order N 2, or O(N 2 ) Prog2 is order N, or O(N)  In general, our effort should be Try to reduce the order of complexity Within the same order, try to reduce the coefficients of the expression

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