How does the bead sort algorithm function?

The bead sort algorithm functions by visualising the elements of the list as beads on vertical rods.

Bead sort, also known as gravity sort, is a natural sorting algorithm. It is both a distribution sort and a comparison sort, but it is not a stable sort. The algorithm visualises the elements of the list as beads on vertical rods, where each bead represents one unit. The rods are then shaken so that the beads fall down, stacking on top of each other. The height of each stack then represents the sorted list.

The algorithm begins by mapping each element of the input list to a rod, with the height of each rod corresponding to the value of the element. The rods are then 'shaken' (in a metaphorical sense), causing the 'beads' to fall to the bottom. The beads stack on top of each other, with the highest stacks forming on the left and the lowest on the right. This process is repeated until all the beads have fallen. The height of each stack is then read off, forming the sorted list.

Bead sort can be implemented with arrays and linked lists. In an array implementation, the array is traversed from left to right, with each element being replaced by a list of beads (represented as 1s). The array is then 'shaken' by summing the columns and replacing the original array with the sums. In a linked list implementation, each node contains a counter and a pointer to the next node. The list is traversed, and for each node, a new node is created with a counter equal to the value of the current node and inserted into the sorted list at the appropriate position.

Bead sort is a simple and intuitive algorithm, but it is not often used in practice due to its poor time complexity. The time complexity of bead sort is O(n^2), making it inefficient for large lists. However, it can be useful for educational purposes, as it provides a clear visualisation of the sorting process.