Question

Complete the following table calculating the average and maximum number of comparisons the linear search will perform, and the maximum number of comparisons the binary search will perform. $$\begin{array}{l|lllll} \hline & 50 & 500 & 10,000 & 100,000 & 10,000,000 \\ \text { Array Size } \rightarrow & \text { Elements } & \text { Elements } & \text { Elements } & \text { Elements } & \text { Elements } \\ \hline \text { Linear Search } & & & & & \\ \begin{array}{l} \text { (Average } \\ \text { Comparisons) } \end{array} & & & & & \\ \hline \text { Linear Search } & & & & & \\ \text { (Maximum } & & & & \\ \text { Comparisons) } & & & & & \\ \hline \text { Binary Search } & & & & & \\ \text { (Maximum } & & & & \\ \text { Comparisons) } & & & & & \\ \hline \end{array}$$

Short answer

Based on the step-by-step solution above, fill in the table with the average and maximum number of comparisons for linear and binary search algorithms for different array sizes. | Array Size (Elements) | Linear Search (Avg Comparisons) | Linear Search (Max Comparisons) | Binary Search (Max Comparisons) | |-----------------------|---------------------------------|---------------------------------|---------------------------------| | 50 | 25 | 50 | 6 | | 500 | 250 | 500 | 9 | | 10,000 | 5,000 | 10,000 | 14 | | 100,000 | 50,000 | 100,000 | 17 | | 10,000,000 | 5,000,000 | 10,000,000 | 24 |

Step-by-step solution

Calculate Linear Search Average Comparisons

For each array size, calculate the average number of comparisons for the linear search algorithm using the formula: $$\frac{n}{2}$$

Calculate Linear Search Maximum Comparisons

For each array size, calculate the maximum number of comparisons for the linear search algorithm using the formula: $$n$$

Calculate Binary Search Maximum Comparisons

For each array size, calculate the maximum number of comparisons for the binary search algorithm using the formula: $$\lceil log_2{n} \rceil$$

Fill in the Table

Using the values calculated in Steps 1 to 3, let's fill in the table: $$\begin{array}{l|lllll} \hline & 50 & 500 & 10,000 & 100,000 & 10,000,000 \\ \text { Array Size } \rightarrow & \text { Elements } & \text { Elements } & \text { Elements } & \text { Elements } & \text { Elements } \\ \hline \text { Linear Search } & & & & & \\ \begin{array}{l} \text { (Average } \\ \text { Comparisons) } \end{array} & 25 & 250 & 5,000 & 50,000 & 5,000,000 \\ \hline \text { Linear Search } & & & & & \\ \text { (Maximum } & & & & \\ \text { Comparisons) } & 50 & 500 & 10,000 & 100,000 & 10,000,000 \\ \hline \text { Binary Search } & & & & & \\ \text { (Maximum } & & & & \\ \text { Comparisons) } & 6 & 9 & 14 & 17 & 24 \\ \hline \end{array}$$

Key concepts

Linear Search

Linear search is a straightforward and intuitive search algorithm. It works by checking each element in a list one by one until it finds the target value or searches through the list entirely. This means it starts from the first element and moves sequentially to the next.

The simplicity of linear search lies in its requirement of no pre-sorting. You just need to traverse through the elements, making it a preferred choice for small or unsorted datasets.

• **Average Comparisons:** For a dataset of size \( n \), linear search will make approximately \( \frac{n}{2} \) comparisons on average, as the target is expected around the middle.
• **Maximum Comparisons:** If the target is not present, or is the last element, it will make around \( n \) comparisons.

Its straightforward execution makes linear search easy to implement, but its inefficiency in large datasets renders it less effective than other better-suited algorithms.

Binary Search

Binary search is a much faster search algorithm but comes with a prerequisite: the dataset must be sorted. It works by repeatedly dividing the search interval in half.

Here's how it operates:

• Start with the middle element of the sorted array.
• If the target value is the middle element, you're done.
• If the target is less than the middle element, narrow the search range to the lower half.
• If the target is more than the middle element, focus on the upper half.
• Repeat until the target is found or the range is empty.

This halving approach drastically reduces the number of comparisons needed.

The maximum number of comparisons needed in binary search for a dataset of size \( n \) is given by \( \lceil \log_2{n} \rceil \), due to this repeated division.

Comparison Complexity

Comparison complexity refers to the number of comparisons an algorithm makes during its execution. This is a key indicator of the algorithm's runtime efficiency, especially in search algorithms.

In a linear search, the comparison complexity is linear or \( O(n) \), as it potentially checks each element once. Compare that to binary search where comparison complexity is logarithmic or \( O(\log n) \).

This complexity difference is significant and highlights why binary search is more efficient than linear search for large datasets when they are sorted. The lower the complexity, the faster the algorithm runs as it reduces the operations needed to achieve the desired outcome.

Algorithm Efficiency

Algorithm efficiency concerns how optimal an algorithm is in terms of time and resources it requires. Two main elements define this: time complexity and space complexity.

**Time Complexity:** This indicates how the execution time increases with the size of the input. A linear search takes \( O(n) \) time, meaning its execution grows linearly with input. Binary search, on the other hand, has a complexity of \( O(\log n) \), which means it grows logarithmically with respect to input size.

**Space Complexity:** Pertains to the memory space needed by the algorithm. Both linear and binary searches primarily operate in constant space, \( O(1) \), since they typically only require a few additional variables, regardless of input size. Efficient algorithms achieve a balance between minimizing time and space complexities.

Computational Analysis

Computational analysis involves evaluating the performance and efficiency of algorithms, like search algorithms. This includes understanding the time and space complexities, which are crucial for assessing an algorithm's scalability.

Through computational analysis, one can predict how an algorithm will perform with increasing data sizes. For instance:

• Linear search, with its \( O(n) \) time complexity, may become impractical for massive datasets.
• Binary search, with \( O(\log n) \) time complexity, remains efficient even as datasets grow substantially, assuming they are sorted.

Analyzing algorithms computationally helps in choosing the right algorithm for the problem at hand, ensuring optimal performance depending on dataset size and constraints.