Question

Suppose a very large sorted list is stored in external storage. Assuming that this list cannot be brought into internal memory, develop a searching algorithm that looks for a key in this list. What major factor(s) should be considered when an external search algorithm is developed? Define the major fac\(\operatorname{tor}(s),\) analyze your algorithm, and show the results using order notation.

Short answer

The Binary Search algorithm is an efficient way to search for a key in a large, sorted list stored in external storage. This algorithm consistently halves the search space, thus resulting in a time complexity of \( O(\log n) \). Some major factors to consider when implementing an external search algorithm include the cost of I/O operations, buffering, and disk seek time.

Step-by-step solution

Define Major Factors

Identifying the key considerations for an external search algorithm is critical. The major factors in this case are: 1. The large size of the dataset: Since the dataset is too large to fit into internal memory, the algorithm must be designed to handle this constraint. 2. The sorted nature of the dataset: As the data is already sorted, certain algorithms, like binary search, can be utilized that take advantage of this property.

Develop a Searching Algorithm

Given the characteristics of the dataset (large and sorted), a Binary Search algorithm is a viable choice. In binary search, the algorithm divides the list into two halves, compares the middle element with the key and continues the search in the corresponding half. Binary search can be efficiently applied to a large, sorted list, and it dramatically reduces the search time by repeatedly halving the search space. Let's denote the size of the list with \( n \).\n\nPseudocode for Binary Search Algorithm:\n\nExternal_Binary_Search(list, key)\nBEGIN\n low = 0\n high = n - 1\n WHILE low <= high\n middle = low + (high - low) / 2\n IF list[middle] == key\n RETURN middle\n ELSE IF list[middle] < key\n low = middle + 1\n ELSE\n high = middle - 1\n END WHILE\n RETURN NOT FOUND\nEND

Analyze the Algorithm

The time complexity of the Binary Search algorithm is \( O(\log n) \). This is because with each comparison, the algorithm reduces the search space by half, thus the time complexity is logarithmic. The space complexity is \( O(1) \), meaning that the storage space required does not change with the size of the input list. This makes it apt for an external search.

Major factors to consider during algorithm implementation

Some major factors to consider when implementing an external search algorithm include: 1. Costs of I/O operations: External storage has slower access times than internal memory, so it's important to minimize I/O operations. In our algorithm, the search data is sequentially read from the external storage which reduces the number of I/O operations. 2. Buffering: Since I/O operations are expensive, reading more than one element at a time may give a performance boost. 3. Disk Seek Time: If data is spread across the disk, then disk seek time (time to move disk arm to proper track) comes into picture. We should strive to structure the data on disk in a way to minimize this.

Key concepts

Binary Search

Binary search is an efficient algorithm for finding an element in a sorted array by repeatedly dividing the search interval in half. It's well-suited for large lists because it significantly reduces the number of comparisons needed to find the target value. Here's how it works:

• The search begins by comparing the target value to the middle element of the list.
• If the target value equals the middle element, the search is concluded, as the element is found.
• If the target value is less than the middle element, the search continues in the lower half of the array.
• If the target value is greater than the middle element, the search proceeds with the upper half.

The process repeats until the target is found or the possible search space reduces to zero. The efficiency of binary search makes it a prime candidate for external searches, where data cannot be fully loaded into memory at once.

Time Complexity Analysis

The time complexity of an algorithm refers to the computational time it takes to complete relative to the length of the input. For binary search, this complexity is denoted as \( O(\log n) \). Why \( O(\log n) \)? Because with each comparison, the algorithm effectively halves the remaining search space, which is characteristic of logarithmic time growth. This means even for very large datasets, the number of steps required to find the element grows moderately.In addition to time, we often consider space complexity, which for binary search is \( O(1) \) since the amount of memory used does not depend on the input size. This makes it ideal for searching large data sets stored externally, where memory constraints are often a critical factor.

I/O Operations in Algorithms

I/O operations refer to input and output processes where data is transferred between the internal memory and external storage. The costs of these operations are significant, especially when dealing with large data sets stored externally, as access time is considerably slower compared to internal memory access. When implementing external search algorithms like binary search, it's crucial to minimize I/O operations. Strategies can include:

• Reading larger blocks of data at once to reduce the number of accesses.
• Caching frequently accessed data to minimize repeated read operations.

Additionally, the physical layout of data on the disk should consider minimizing disk seek time, as the mechanical movement required to access data in different locations contributes heavily to I/O delay.

Pseudocode Design

Pseudocode serves as an intermediary between human language and programming constructs, outlining algorithms in an easily understandable manner without strict syntactical rules. When designing the pseudocode for an external search algorithm like binary search, clarity and conciseness are paramount. A well-structured pseudocode for binary search would typically involve:

• Initializations: setting low and high indices that define the current search boundaries.
• A loop that runs as long as the search boundaries are valid, checking the middle element each time.
• Conditional logic to adjust the search boundaries based on comparisons with the middle element.
• Return statements to indicate if the element has been found or not.

With external algorithms, additional considerations include incorporating logic to manage which segments of data to load into memory, based on the current search needs, to optimize performance.