Question

What is the time complexity \(T(n)\) of the nested loops below? For simplicity, you may assume that \(n\) is a power of \(2 .\) That is, \(n=2^{k}\) for some positive integer \(k\) : \\[ \begin{array}{l} i=n_{i} \\ \text { while }(i>=1)\\{ \\ \qquad \begin{array}{c} j=i \\ \text { while }(j<=n) \end{array} \end{array} \\] < body of the inner while loop> \(\quad\) I/ Needs \(\Theta(1)\) \\[ j=2^{*} j \\] } \\[ i=|1 / 2| \\] }

Short answer

The time complexity of the given nested loops is \(\Theta((\log n)^2)\).

Step-by-step solution

Understanding the Algorithm

The provided code snippet includes two while loops – one nested within the other. The outer loop begins with 'i' being equal to \(n\) and ends when 'i' becomes less than 1. Each iteration of the loop halves the value of 'i'. The inner loop begins with 'j' equal to 'i' and ends when 'j' becomes greater than \(n\). Each iteration of the loop doubles the value of 'j'. The body of the inner loop takes \(\Theta(1)\) time, which means it runs in constant time.

Analyzing the Outer Loop

The outer loop halves the value of 'i' at each iteration. Given that 'i' starts at \(n\), and assuming \(n = 2^k\), the worst case number of iterations for this loop would be \(k\), since \(2^k = n\). Hence the outer loop runs in logarithmic time, specifically in \(\Theta(\log n)\).

Analyzing the Inner Loop

The inner loop doubles the value of 'j' at each iteration. So for a given value of 'i', the inner loop would run until 'j' exceeds \(n\). Therefore, the number of iterations for the inner loop would be proportional to \(\log n\) as well. Thus, the inner loop also runs in logarithmic time, specifically \(\Theta(\log n)\).

Time Complexity of Nested Loops

Since the inner loop is nested within the outer loop, and both run in \(\Theta(\log n)\) time, the total time complexity of the two nested loops would be the product of their individual time complexities. Namely, \(\Theta((\log n)^2)\).

Key concepts

Nested Loops

Nested loops occur when one loop runs inside another loop. In the given problem, there are two while loops - the outer and the inner loop.
Nested loops can increase the overall time complexity of a program since the inner loop fully executes for every iteration of the outer loop.

• The outer loop starts with the variable `i` and operates as long as `i` is greater than or equal to 1.
• With each execution of the outer loop block, `i` is halved.
• For every value of `i`, the inner loop begins its execution.

The inner loop initializes with `j` equal to `i` and continues running until `j` is greater than `n`. This nested structure is important for determining the algorithm's overall time complexity.

Logarithmic Time

Operations that run in logarithmic time \( \Theta(\log n)\) are highly efficient, especially when dealing with large data sets. This efficiency arises because logarithmic functions grow very slowly compared to linear, quadratic, or exponential functions.
In our nested loops example, both while loops operate in logarithmic time.

• The outer loop halves `i` in each iteration, continuing until `i` becomes less than 1. Given the initial value of `i`, which is `n`, this results in a logarithmic decrease with respect to the size `n`.
• The inner loop doubles `j` each time it runs. It starts from `j = i` and raises `j` to `n` or beyond, also resulting in logarithmic iterations.

Together, these characteristics determine the algorithm's efficiency in both time and resources. Logarithmic time complexity is represented in Big-O notation as \(O(\log n)\).

Algorithm Analysis

Algorithm analysis is the process of determining how the performance of an algorithm is affected by changes in its input size. This involves assessing the time complexity and space complexity of algorithms.

• Time complexity measures how the runtime of an algorithm increases with input size.
• Space complexity evaluates the amount of memory the algorithm uses as a function of input size.

Our task's main focus is on time complexity, which describes the nested loops' performance in logarithmic time. Analyzing algorithms involves dissecting each part or loop to understand its effect on overall performance. Even though nested loops, at first glance, may appear to have a high cost, their nested iterations both being logarithmic leads to a manageable combined time complexity.
The analysis helps developers optimize code and anticipate how algorithms perform under various conditions.

Theta Notation

Theta notation, represented as \(\Theta(f(n))\), provides a precise way to describe the asymptotic tight bound of a function as input size grows. Whereas Big-O gives just an upper bound, theta notation specifies both an upper and a lower bound for the behavior of an algorithm's time complexity.
In the problem, both inner and outer loops run in \(\Theta(\log n)\) time, meaning they asymptotically behave similarly regarding time complexity.

• The outer loop behavior is defined by its halving of `i`, ensuring its iterations fall within the bounds of \(\Theta(\log n)\).
• The inner loop, by doubling `j`, also exhibits behavior bounded by \(\Theta(\log n)\).

Together, when combined; these loops result in a time complexity of \(\Theta((\log n)^2)\). This double logarithmic behavior emphasizes the accuracy and importance of using theta notation to describe performance limits.