Question

When this process is complete, the array elements that are still set to one indicate that the subscript is a prime number. These subscripts can then be printed. Write a program that uses an array of 1000 elements to determine and print the prime numbers between 2 and \(999 .\) Ignore element 0 of the array. (Bucket Sort) A bucket sort begins with a one-dimensional array of positive integers to be sorted and a two-dimensional array of integers with rows subscripted from 0 to 9 and columns subscripted from 0 to \(n 1\), where \(n\) is the number of values in the array to be sorted. Each row of the twodimensional array is referred to as a bucket. Write a function bucketsort that takes an integer array and the array size as arguments and performs as follows: a. Place each value of the one-dimensional array into a row of the bucket array based on the value's ones digit. For example, 97 is placed in row 7,3 is placed in row 3 and 100 is placed in row \(0 .\) This is called a "distribution pass." b. Loop through the bucket array row by row, and copy the values back to the original array. This is called a "gathering pass." The new order of the preceding values in the one-dimensional array is 100,3 and \(97 .\) c. Repeat this process for each subsequent digit position (tens, hundreds, thousands, etc.). On the second pass, 100 is placed in row 0,3 is placed in row 0 (because 3 has no tens digit) and 97 is placed in row \(9 .\) After the gathering pass, the order of the values in the one-dimensional array is 100,3 and \(97 .\) On the third pass, 100 is placed in row 1,3 is placed in row zero and 97 is placed in row zero (after the 3 ). After the last gathering pass, the original array is now in sorted order. Note that the two-dimensional array of buckets is 10 times the size of the integer array being sorted. This sorting technique provides better performance than a insertion sort, but requires much more memory. The insertion sort requires space for only one additional element of data. This is an example of the spacetime trade-off: The bucket sort uses more memory than the insertion sort, but performs better. This version of the bucket sort requires copying all the data back to the original array on each pass. Another possibility is to create a second two-dimensional bucket array and repeatedly swap the data between the two bucket arrays.

Short answer

Use the Sieve of Eratosthenes to find prime numbers from 2 to 999, marking multiples of each prime as false.

Step-by-step solution

Initialize the Array

Create a boolean array of size 1001, initially set all elements to true. This array will be used to determine prime numbers using the Sieve of Eratosthenes algorithm. The indices of the array represent numbers, and the value at each index represents whether it is a prime number.

Implement Sieve of Eratosthenes

Implement the Sieve of Eratosthenes algorithm. Start with the first prime number, 2. Mark all multiples of 2 (greater than or equal to the square of the number) in the array as false, indicating that they are not prime. Repeat this process for the next number in the array that remains true.

Iterate through Subsequent Numbers

Continue iterating through each number. For numbers still marked as true, treat them as prime numbers, and mark all their multiples as false. Continue this process until you reach the square root of the maximum number (999).

Extract and Print Primes

After going through the numbers, the indices of the array still marked true are the prime numbers. Print these indices, starting from 2 to 999, as these represent the prime numbers in that range.

Key concepts

Prime Numbers

Prime numbers are essential building blocks in mathematics. These are numbers greater than 1 that have no divisors other than 1 and themselves. This unique property makes them fundamental tools in various fields, including cryptography and number theory.
One of the most straightforward ways to determine if a number is prime is to check if it can be divided by any number other than 1 and itself. However, for large numbers, this can be very time-consuming. That's where efficient algorithms like the Sieve of Eratosthenes come in handy.

Array Sorting

Array sorting is a simple yet crucial concept in computer science. It involves arranging the elements of an array in a certain order, typically in ascending or descending order. Sorting is essential because it allows for efficient data management and retrieval.
A common approach to array sorting is through different sorting algorithms, such as quicksort, mergesort, or heapsort. Each algorithm has its pros and cons depending on the scenario, including factors like execution speed and space utilization.

Bucket Sort

Bucket sort is an interesting sorting algorithm that distributes elements into several 'buckets'. These buckets are then sorted individually, either using a different sorting algorithm or by recursively applying the bucket sort.
This sorting method is particularly useful when the input is uniformly distributed over a range. It makes the sorting process faster by breaking it down into simpler tasks. Although it is efficient in terms of sorting time, bucket sort can be memory-intensive since it requires a significant amount of additional space for the buckets.

Space-Time Trade-off

Space-time trade-off is a common consideration in algorithm design, where a balance between memory usage and execution speed is made. It implies that you can save time by using more memory and vice versa.
For example, an algorithm that is fast might consume more space, like bucket sort which needs extra memory for the buckets. Conversely, an algorithm like insertion sort might use minimal space but takes more time for execution. Understanding this trade-off helps in selecting the right algorithm based on specific constraints and requirements.

Algorithm Efficiency

Algorithm efficiency is all about how effectively an algorithm performs, measured by the time it takes to execute and the amount of memory it uses. Efficient algorithms are designed to minimize both.
Efficiency is often reported using "Big O" notation, which describes how the runtime or space requirements grow relative to the input size. Efficient algorithms like the Sieve of Eratosthenes for finding prime numbers demonstrate a lower time complexity, offering better performance for large datasets compared to naive approaches.