Given an \(n\) -element array \(X\), Algorithm D calls Algorithm \(\mathrm{E}\) on each element \(X[i] .\) Algorithm \(\mathrm{E}\) runs in \(O(i)\) time when it is called on element \(X[i] .\) What is the worst-case running time of Algorithm D?

Time complexity is a fundamental concept in algorithm analysis. It measures the amount of time an algorithm takes to complete based on the size of the input. This measurement helps us understand how an algorithm will perform as the input size increases.
There are several commonly used notations to express time complexity, such as:

• O(1): Constant time complexity, where the time taken does not change with the input size.
• O(n): Linear time complexity, where the time taken grows linearly with the input size.
• O(n^2): Quadratic time complexity, where the time taken grows quadratically as the input size increases.

For our exercise, Algorithm E's time complexity is given as O(i), meaning it takes more time to process larger index elements. When analyzing an algorithm, understanding its time complexity is crucial to predict its performance and scalability.

The worst-case scenario in algorithm analysis refers to the maximum time an algorithm could take to complete. This scenario helps in planning for the most resource-intensive case, ensuring the algorithm can handle it efficiently.
In our exercise, we are asked to find the worst-case running time of Algorithm D, which calls Algorithm E on each element of an array X of size n. Knowing that Algorithm E runs in O(i) time for element X[i], we need to consider the sum of times for each element in the array. It is essential to examine the worst-case scenario to guarantee the algorithm's performance even under the most demanding conditions.
Worst-case analysis is especially important in real-time systems, where meeting deadlines is critical. By knowing the worst-case running time, one can ensure the algorithm meets performance requirements under all conditions.

Summation formulas are mathematical tools used to sum a series of numbers. They help streamline the process of adding elements, especially for large sequences.
For our exercise, we need to determine the total running time of Algorithm D, which involves summing the running times of Algorithm E for each element from 1 to n. The running time for Algorithm E when called on element X[i] is O(i). Thus, we sum up these times:
\[ \text{Total time} = O(1) + O(2) + ... + O(n) \]
To find this sum, we use the summation formula for the first n integers:

\[ 1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}\ \]
Using this formula, the worst-case running time of Algorithm D can be approximated as:

\[ O(1 + 2 + 3 + ... + n) = O \bigg( \frac{n(n+1)}{2} \bigg) = O(n^2) \]
This simplification shows that the worst-case running time of Algorithm D is O(n^2). Summation formulas are instrumental in simplifying and understanding such time complexities.