User:Earlster/Sandbox

Earlster/Sandbox

Class	Sorting algorithm
Data structure	Array
Optimal	O(n)

This is a WIP

Elements are distributed among bins

Then, elements are sorted within each bin

Proxmap Sort, or Proxmap, is a sorting algorithm that works by partitioning an array into a number of buckets. Each bucket is then sorted individually by insertion sort. Since bucket sort is not a comparison sort, the Ω(n log n) lower bound is inapplicable. The computational complexity estimates involve the number of buckets.

Bucket sort works as follows:

1. Set up an array of initially empty "buckets."
2. Scatter: Go over the original array, putting each object in its bucket.
3. Insertion Sort each non-empty bucket.
4. Gather: Visit the buckets in order and put all elements back into the original array.

Pseudocode

function proxmap-sort(array, n) is
  buckets ← new array of n empty lists
  for i = 0 to (length(array)-1) do
    insert array[i] into buckets[msbits(array[i], k)]
  for i = 0 to n - 1 do
    next-sort(buckets[i])
  return the concatenation of buckets[0], ..., buckets[n-1]

Here array is the array to be sorted and n is the number of buckets to use. The function msbits(x,k) returns the k most significant bits of x (floor(x/2^(size(x)-k))); different functions can be used to translate the range of elements in array to n buckets, such as translating the letters A–Z to 0–25 or returning the first character (0–255) for sorting strings. The function next-sort is a sorting function; using bucket-sort itself as next-sort produces a relative of radix sort; in particular, the case n = 2 corresponds to quicksort (although potentially with poor pivot choices).

Optimizations

A common optimization is to put the elements back in the original array first, then run insertion sort over the complete array; because insertion sort's runtime is based on how far each element is from its final position, the number of comparisons remains relatively small, and the memory hierarchy is better exploited by storing the list contiguously in memory.

Variants

Generic bucket sort

The most common variant of bucket sort operates on a list of n numeric inputs between zero and some maximum value M and divides the value range into n buckets each of size M/n. If each bucket is sorted using insertion sort, the sort can be shown to run in expected linear time (where the average is taken over all possible inputs). However, the performance of this sort degrades with clustering; if many values occur close together, they will all fall into a single bucket and be sorted slowly.

References

• Thomas A. Standish "Using O(n) ProxmapSort and O(1) ProxmapSearch to Motivate CS2 Students (Part I)" from Donald Bren School of Information and Computer Sciences at UC Irvine.
• Norman Jacobson "A Synopsis of ProxmapSort & ProxmapSearch" from Donald Bren School of Information and Computer Sciences at UC Irvine.

External links

• http://www.cs.uah.edu/~rcoleman/CS221/Sorting/ProxMapSort.html