Step by Step solution about How to resolve the algorithm Sorting algorithms/Bead sort step by step in the Kotlin programming language

Bead Sort is a non-comparative sorting algorithm that uses binary beads to sort an array of integers. It is a simple yet effective algorithm, particularly for sorting small arrays.

Implementation Details:

1. ** Initialization**:

   • n: Size of the input array a.
   • max: Maximum element in the array.
   • beads: A byte array representing the beads. Each bead is initially set to 0.

2. Bead Marking:

   • For each element a[i] in the input array, mark a[i] beads on each post, starting from the bottom. This represents the number of beads for each element.

3. Bead Counting:

   • For each post (column) j:
     • Count the number of beads on that post (sum).
     • Reset the beads on that post to 0.
     • Mark the bottom sum beads on that post as 1.

4. Bead Interpretation:

   • For each element a[i]:
     • Find the first post j where the bead is 1.
     • Assign a[i] the value of j.

5. Example:

   • Input array: [5, 3, 1, 7, 4, 1, 1, 20]
   • Bead marking:

     0 0 0 0 0 0 0 0
     1 1 1 1 1 1 1 1
     0 0 0 1 1 1 1 1
     0 0 1 1 1 1 1 1
     0 0 0 0 0 1 1 1
     1 1 1 1 1 1 1 1
     1 1 1 1 1 1 1 1
     0 0 0 0 0 0 0 0
     

   • Bead counting and marking:

     0 0 0 1 1 0 0 0
     0 0 1 1 1 1 1 0
     0 0 0 1 1 1 1 1
     0 0 0 0 0 1 1 1
     0 0 0 0 0 0 0 1
     0 0 0 0 0 0 0 0
     0 0 0 0 0 0 0 0
     0 0 0 0 0 0 0 0
     

   • Bead interpretation:
     • a[0] = 4 (4 beads on the 4th post)
     • a[1] = 2 (2 beads on the 2nd post)
     • a[2] = 0 (0 beads on any post)
     • a[3] = 6 (6 beads on the 6th post)
     • a[4] = 3 (3 beads on the 3rd post)
     • a[5] = 0 (0 beads on any post)
     • a[6] = 0 (0 beads on any post)
     • a[7] = 7 (7 beads on the 7th post)

6. Sorted Output:

   • [0, 0, 1, 2, 3, 4, 6, 7]

Properties of Bead Sort:

• Time Complexity: O(n + max), where n is the size of the array and max is the maximum element.
• Space Complexity: O(n * max), for the bead array.
• Stability: Not stable, as elements with the same value may be reordered.
• Limitations: Suitable for small arrays and limited by the maximum value in the array.

Applications:

Bead Sort finds applications in situations where the input data is small and has a limited range of values, such as:

• Sorting small arrays
• Bucketing elements into ranges

// version 1.1.2

fun beadSort(a: IntArray) {
    val n = a.size
    if (n < 2) return
    var max = a.max()!!
    val beads = ByteArray(max * n)
    /* mark the beads */
    for (i in 0 until n)
        for (j in 0 until a[i])
            beads[i * max + j] = 1

    for (j in 0 until max) {
        /* count how many beads are on each post */
        var sum = 0
        for (i in 0 until n) {
            sum += beads[i * max + j]
            beads[i * max + j] = 0
        }
        /* mark bottom sum beads */
        for (i in n - sum until n) beads[i * max + j] = 1
    }

    for (i in 0 until n) {
        var j = 0
        while (j < max && beads[i * max + j] == 1.toByte()) j++
        a[i] = j
    }
}

fun main(args: Array<String>) {
    val a  = intArrayOf(5, 3, 1, 7, 4, 1, 1, 20)
    println("Before sorting : ${a.contentToString()}")
    beadSort(a)
    println("After sorting  : ${a.contentToString()}")
}