Sample Practice Problems on Complexity Analysis of Algorithms

Prerequisite: Asymptotic Analysis, Worst, Average and Best Cases, Asymptotic Notations, Analysis of loops.

Problem 1: Find the complexity of the below recurrence: 

              { 3T(n-1), if  n>0,
T(n) =   { 1, otherwise

Solution:

Let us solve using substitution.

T(n) = 3T(n-1)
       = 3(3T(n-2)) 
       = 3²T(n-2)
       = 3³T(n-3)
       ... 
       ...
       = 3ⁿT(n-n)
       = 3ⁿT(0) 
       = 3ⁿ

This clearly shows that the complexity of this function is O(3ⁿ).

Problem 2: Find the complexity of the recurrence: 

             { 2T(n-1) - 1, if n>0,
T(n) =   { 1, otherwise

Solution:

Let us try solving this function with substitution.

T(n) = 2T(n-1) - 1
       = 2(2T(n-2)-1)-1 
       = 2²(T(n-2)) - 2 - 1
       = 2²(2T(n-3)-1) - 2 - 1 
       = 2³T(n-3) - 2² - 2¹ - 2⁰
.....
       .....
       = 2ⁿT(n-n) - 2^n-1 - 2^n-2 - 2^n-3
..... 2² - 2¹ - 2⁰

       = 2ⁿ - 2^n-1 - 2^n-2 - 2^n-3
..... 2² - 2¹ - 2⁰
= 2ⁿ - (2ⁿ-1) 

[Note: 2^n-1 + 2^n-2 + ...... +  2⁰ = 2ⁿ - 1]

T(n) = 1

Time Complexity is O(1). Note that while the recurrence relation looks exponential
he solution to the recurrence relation here gives a different result.

Problem 3: Find the complexity of the below program: 

Solution: Consider the comments in the following function. 

Time Complexity: O(n), Even though the inner loop is bounded by n, but due to the break statement, it is executing only once.

Problem 4: Find the complexity of the below program: 

Solution: Consider the comments in the following function. 

Time Complexity: O(n log²n).

Problem 5: Find the complexity of the below program: 

Solution: Consider the comments in the following function. 

Time Complexity: O(n²logn).

Problem 6: Find the complexity of the below program: 

Solution: We can define the terms 's' according to relation s_i = s_i-1 + i. The value of 'i' increases by one for each iteration. The value contained in 's' at the i^th iteration is the sum of the first 'i' positive integers. If k is total number of iterations taken by the program, then while loop terminates if: 1 + 2 + 3 ....+ k = [k(k+1)/2] > n So k = O(√n).
Time Complexity: O(√n).

Problem 7: Find a tight upper bound on the complexity of the below program: 

Solution: Consider the comments in the following function. 

Time Complexity: O(n⁵)

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