Algorithm Analysis – Crash Course

We have been discussing the time complexities of each of the algorithm that we have discussed. Let us try to analyze, how these time complexities are calculated.

O(1)

Time complexity of a function (or set of statements) is considered as O(1) if it doesn’t contain loop, recursion and call to any other non-constant time function.

// set of non-recursive and non-loop statements
int a = 5;
int b = a * 5;

For example swap() function(which interchanges two numbers) has O(1) time complexity.
A loop or recursion that runs a constant number of times is also considered as O(1). For example the following loop is O(1).

// Here c is a constant
int i;
for (i = 1; i <= c; i++)
{
// some O(1) expressions
}

O(n)

Time Complexity of a loop is considered as O(n) if the loop variables is incremented / decremented by a constant amount. For example following functions have O(n) time complexity.

// Here c is a positive integer constant
int i;
for (i = 1; i <= n; i += c)
{
// some O(1) expressions
}

for (i = n; i > 0; i -= c)
{
// some O(1) expressions
}

O(n^c)

Time complexity of nested loops is equal to the number of times the innermost statement is executed. For example the following sample loops have O(n²) time complexity

int i,j;

//outer loop
for (i = 1; i <=n; i += c)
{
//inner loop
for (j = 1; j <=n; j += c)
{
// some O(1) expressions
}
}

For example Selection sort and Insertion Sort have O(n²) time complexity.

O(Logn)

Time Complexity of a loop is considered as O(log(n)) if the loop variables is divided / multiplied by a constant amount.

int i;
for (i = 1; i <=n; i *= c)
{
// some O(1) expressions
}

for (i = n; i > 0; i /= c)
{
// some O(1) expressions
}

For example Binary Search has O(log(n)) time complexity.

O(log (log(n)))

Time Complexity of a loop is considered as O(log (log(n))) if the loop variables is reduced / increased exponentially by a constant amount.

int i;

// Here c is a constant greater than 1
for (i = 2; i <=n; i = pow(i, c))
{
// some O(1) expressions
}

How to combine time complexities of consecutive loops?

When there are consecutive loops, we calculate time complexity as sum of time complexities of individual loops.

int i;

for (i = 1; i <=m; i += c)
{
// some O(1) expressions
}

for (i = 1; i <=n; i += c)
{
// some O(1) expressions
}

Time complexity of above code is O(m) + O(n) which is O(m+n)
If m == n, the time complexity becomes O(2n) which is O(n).

How to calculate time complexity when there are many if, else statements inside loops?

As discussed here, worst case time complexity is the most useful among best, average and worst. Therefore we need to consider worst case. We evaluate the situation when values in if-else conditions cause maximum number of statements to be executed.
For example consider the linear search function where we consider the case when element is present at the end or not present at all.
When the code is too complex to consider all if-else cases, we can get an upper bound by ignoring if else and other complex control statements.

How to calculate time complexity of recursive functions?

Time complexity of a recursive function can be written as a mathematical recurrence relation. To calculate time complexity, we must know how to solve recurrences.

Please share in comments if you find anything else interesting. I will update the post.