How do you analyses an algorithm?

How do you analyses an algorithm?

Algorithm analysis refers to the process of determining how much computing time and storage that algorithms will require. In other words, it’s a process of predicting the resource requirement of algorithms in a given environment. In order to solve a problem, there are many possible algorithms. One has to be able to choose the best algorithm for the problem at hand using some scientific method. To classify some data structures and algorithms as good, we need precise ways of analyzing them in terms of resource requirement. The main resources are:

• Running Time

• Memory Usage

• Communication Bandwidth

Running time is usually treated as the most important since computational time is the most precious resource in most problem domains.

There are two approaches to measure the efficiency of algorithms:

•  Empirical: Programming competing algorithms and trying them on different instances.

•  Theoretical: Determining the quantity of resources required mathematically (Execution time, memory space, etc.) needed by each algorithm.

However, it is difficult to use actual clock-time as a consistent measure of an algorithm’s efficiency, because clock-time can vary based on many things. For example,

•  Specific processor speed

•  Current processor load

•  Specific data for a particular run of the program

o Input Size

o Input Properties 

• Operating Environment

Accordingly, we can analyze an algorithm according to the number of operations required, rather than according to an absolute amount of time involved. This can show how an algorithm’s efficiency changes according to the size of the input.

Complexity Analysis

Complexity Analysis is the systematic study of the cost of computation, measured either in time units or in operations performed, or in the amount of storage space required.

The goal is to have a meaningful measure that permits comparison of algorithms independent of operating platform.

There are two things to consider:

• Time Complexity: Determine the approximate number of operations required to solve a problem of size n.

• Space Complexity: Determine the approximate memory required to solve a problem of size n. Complexity analysis involves two distinct phases:

•  Algorithm Analysis: Analysis of the algorithm or data structure to produce a function T that describes the algorithm in terms of the operations performed in order to measure the complexity of the algorithm.

• Order of Magnitude Analysis: Analysis of the function T (n) to determine the general complexity category to which it belongs. There is no generally accepted set of rules for algorithm analysis. However, an exact count of operations is commonly used.

Analysis Rules:

1. We assume an arbitrary time unit.

2. Execution of one of the following operations takes time 1:

• Assignment Operation

• Single Input/Output Operation

• Single Boolean Operations

• Single Arithmetic Operations

• Function Return

3. Running time of a selection statement (if, switch) is the time for the condition evaluation + the maximum of the running times for the individual clauses in the selection.

4. Loops: Running time for a loop is equal to the running time for the statements inside the loop * number of iterations.

The total running time of a statement inside a group of nested loops is the running time of the statements multiplied by the product of the sizes of all the loops.

For nested loops, analyze inside out.

• Always assume that the loop executes the maximum number of iterations possible.

5. Running time of a function call is 1 for setup + the time for any parameter calculations + the time required for the execution of the function body.

Examples:

1. int count(){ int k=0;

cout<< “Enter an integer”; cin>>n;

for (i=0;i k=k+1; return 0;}

Time Units to Compute

-------------------------------------------------

1 for the assignment statement: int k=0

1 for the output statement.

1 for the input statement. In the for loop:

1 assignment, n+1 tests, and n increments.

n loops of 2 units for an assignment, and an addition. 1 for the return statement.

-------------------------------------------------------------------

T (n)= 1+1+1+(1+n+1+n)+2n+1 = 4n+6 = O(n) 2. int total(int n)

{

int sum=0;

for (int i=1;i<=n;i++) sum=sum+1;

return sum;

}

Time Units to Compute

-------------------------------------------------

1 for the assignment statement: int sum=0 In the for loop:

1 assignment, n+1 tests, and n increments.

n loops of 2 units for an assignment, and an addition. 1 for the return statement.

-------------------------------------------------------------------

T (n)= 1+ (1+n+1+n)+2n+1 = 4n+4 = O(n) 3. void func()

{

int x=0; int i=0; int j=1;

cout<< “Enter an Integer value”; cin>>n;

while (i x++; i++;

}

while (j

{

j++;

}

}

Time Units to Compute

-------------------------------------------------

1 for the first assignment statement: x=0;

1 for the second assignment statement: i=0;

1 for the third assignment statement: j=1;

1 for the output statement.

1 for the input statement.

In the first while loop: 

n+1 tests

n loops of 2 units for the two increment (addition) operations 

In the second while loop:

n tests

n-1 increments

-------------------------------------------------------------------

T (n)= 1+1+1+1+1+n+1+2n+n+n-1 = 5n+5 = O(n) 4. int sum (int n)

{

int partial_sum = 0;

for (int i = 1; i <= n; i++) partial_sum = partial_sum +(i * i * i); return partial_sum;

}

Time Units to Compute

-------------------------------------------------

1 for the assignment.

1 assignment, n+1 tests, and n increments.

n loops of 4 units for an assignment, an addition, and two multiplications. 1 for the return statement.

-------------------------------------------------------------------

T (n)= 1+(1+n+1+n)+4n+1 = 6n+4 = O(n)

Formal Approach to Analysis

In the above examples we have seen that analysis so complex. However, it can be simplified by using some formal approach in which case we can ignore initializations, loop control, and book keeping.

For Loops: Formally

• In general, a for loop translates to a summation. The index and bounds of the summation are the same as the index and bounds of the for loop.

• Suppose we count the number of additions that are done. There is 1 addition per iteration of the loop, hence N additions in total.

Nested Loops: Formally

• Nested for loops translate into multiple summations, one for each for loop.

• Again, count the number of additions. The outer summation is for the outer for loop.

Consecutive Statements: Formally

• Add the running times of the separate blocks of your code

Conditionals: Formally

• If (test) s1 else s2: Compute the maximum of the running time for s1 and s2.