module Algorithms::Sort
rdoc
This module implements sorting algorithms. Documentation is provided for each algorithm. MIT License Copyright (c) 2009 Kanwei Li Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
Public Class Methods
Bubble sort: A very naive sort that keeps swapping elements until the container is sorted. Requirements: Needs to be able to compare elements with <=>, and the [] []= methods should be implemented for the container. Time Complexity: О(n^2) Space Complexity: О(n) total, O(1) auxiliary Stable: Yes
Algorithms::Sort.bubble_sort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
# File lib/algorithms/sort.rb, line 37 def self.bubble_sort(container) loop do swapped = false (container.size-1).times do |i| if (container[i] <=> container[i+1]) == 1 container[i], container[i+1] = container[i+1], container[i] # Swap swapped = true end end break unless swapped end container end
Comb sort: A variation on bubble sort that dramatically improves performance. Source: yagni.com/combsort/ Requirements: Needs to be able to compare elements with <=>, and the [] []= methods should be implemented for the container. Time Complexity: О(n^2) Space Complexity: О(n) total, O(1) auxiliary Stable: Yes
Algorithms::Sort.comb_sort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
# File lib/algorithms/sort.rb, line 60 def self.comb_sort(container) container gap = container.size loop do gap = gap * 10/13 gap = 11 if gap == 9 || gap == 10 gap = 1 if gap < 1 swapped = false (container.size - gap).times do |i| if (container[i] <=> container[i + gap]) == 1 container[i], container[i+gap] = container[i+gap], container[i] # Swap swapped = true end end break if !swapped && gap == 1 end container end
# File lib/algorithms/sort.rb, line 299 def self.dualpivot(container, left=0, right=container.size-1, div=3) length = right - left if length < 27 # insertion sort for tiny array container.each_with_index do |data,i| j = i - 1 while j >= 0 break if container[j] <= data container[j + 1] = container[j] j = j - 1 end container[j + 1] = data end else # full dual-pivot quicksort third = length / div # medians m1 = left + third m2 = right - third if m1 <= left m1 = left + 1 end if m2 >= right m2 = right - 1 end if container[m1] < container[m2] dualpivot_swap(container, m1, left) dualpivot_swap(container, m2, right) else dualpivot_swap(container, m1, right) dualpivot_swap(container, m2, left) end # pivots pivot1 = container[left] pivot2 = container[right] # pointers less = left + 1 great = right - 1 # sorting k = less while k <= great if container[k] < pivot1 dualpivot_swap(container, k, less += 1) elsif container[k] > pivot2 while k < great && container[great] > pivot2 great -= 1 end dualpivot_swap(container, k, great -= 1) if container[k] < pivot1 dualpivot_swap(container, k, less += 1) end end k += 1 end # swaps dist = great - less if dist < 13 div += 1 end dualpivot_swap(container, less-1, left) dualpivot_swap(container, great+1, right) # subarrays dualpivot(container, left, less-2, div) dualpivot(container, great+2, right, div) # equal elements if dist > length - 13 && pivot1 != pivot2 for k in less..great do if container[k] == pivot1 dualpivot_swap(container, k, less) less += 1 elsif container[k] == pivot2 dualpivot_swap(container, k, great) great -= 1 if container[k] == pivot1 dualpivot_swap(container, k, less) less += 1 end end end end # subarray if pivot1 < pivot2 dualpivot(container, less, great, div) end container end end
# File lib/algorithms/sort.rb, line 385 def self.dualpivot_swap(container, i, j) container[i], container[j] = container[j], container[i] end
Dual-Pivot Quicksort is a variation of Quicksort by Vladimir Yaroslavskiy. This is an implementation of the algorithm as it was found in the original research paper:
iaroslavski.narod.ru/quicksort/DualPivotQuicksort.pdf
Mirror: codeblab.com/wp-content/uploads/2009/09/DualPivotQuicksort.pdf
“This algorithm offers O(n log(n)) performance on many data sets that cause other quicksorts to degrade to quadratic performance, and is typically faster than traditional (one-pivot) Quicksort implementations.”
-- http://download.oracle.com/javase/7/docs/api/java/util/Arrays.html
The algorithm was improved by Vladimir Yaroslavskiy, Jon Bentley, and Joshua Bloch, and was implemented as the default sort algorithm for primatives in Java 7.
Implementation in the Java JDK as of November, 2011: www.docjar.com/html/api/java/util/DualPivotQuicksort.java.html
It is proved that for the Dual-Pivot Quicksort the average number of comparisons is 2*n*ln(n), the average number of swaps is 0.8*n*ln(n), whereas classical Quicksort algorithm has 2*n*ln(n) and 1*n*ln(n) respectively. This has been fully examined mathematically and experimentally.
Requirements: Container should implement pop and include the Enumerable module. Time Complexity: О(n log n) average, О(n log n) worst-case Space Complexity: О(n) auxiliary
Stable: No
Algorithms::Sort.dualpivotquicksort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
# File lib/algorithms/sort.rb, line 294 def self.dualpivotquicksort(container) return container if container.size <= 1 dualpivot(container, 0, container.size-1, 3) end
Heap sort: Uses a heap (implemented by the Containers
module) to sort the collection. Requirements: Needs to be able to compare elements with <=> Time Complexity: О(n^2) Space Complexity: О(n) total, O(1) auxiliary Stable: Yes
Algorithms::Sort.heapsort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
# File lib/algorithms/sort.rb, line 106 def self.heapsort(container) heap = Containers::Heap.new(container) ary = [] ary << heap.pop until heap.empty? ary end
Insertion sort: Elements are inserted sequentially into the right position. Requirements: Needs to be able to compare elements with <=>, and the [] []= methods should be implemented for the container. Time Complexity: О(n^2) Space Complexity: О(n) total, O(1) auxiliary Stable: Yes
Algorithms::Sort.insertion_sort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
# File lib/algorithms/sort.rb, line 121 def self.insertion_sort(container) return container if container.size < 2 (1..container.size-1).each do |i| value = container[i] j = i-1 while j >= 0 and container[j] > value do container[j+1] = container[j] j = j-1 end container[j+1] = value end container end
# File lib/algorithms/sort.rb, line 251 def self.merge(left, right) sorted = [] until left.empty? or right.empty? left.first <= right.first ? sorted << left.shift : sorted << right.shift end sorted + left + right end
Mergesort: A stable divide-and-conquer sort that sorts small chunks of the container and then merges them together. Returns an array of the sorted elements. Requirements: Container should implement [] Time Complexity: О(n log n) average and worst-case Space Complexity: О(n) auxiliary Stable: Yes
Algorithms::Sort.mergesort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
# File lib/algorithms/sort.rb, line 243 def self.mergesort(container) return container if container.size <= 1 mid = container.size / 2 left = container[0...mid] right = container[mid...container.size] merge(mergesort(left), mergesort(right)) end
Quicksort: A divide-and-conquer sort that recursively partitions a container until it is sorted. Requirements: Container should implement pop and include the Enumerable module. Time Complexity: О(n log n) average, O(n^2) worst-case Space Complexity: О(n) auxiliary Stable: No
Algorithms::Sort.quicksort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
def self.quicksort(container)
return [] if container.empty? x, *xs = container quicksort(xs.select { |i| i < x }) + [x] + quicksort(xs.select { |i| i >= x })
end
# File lib/algorithms/sort.rb, line 175 def self.partition(data, left, right) pivot = data[front] left += 1 while left <= right do if data[frontUnknown] < pivot back += 1 data[frontUnknown], data[back] = data[back], data[frontUnknown] # Swap end frontUnknown += 1 end data[front], data[back] = data[back], data[front] # Swap back end
def self.quicksort(container, left = 0, right = container.size - 1)
if left < right middle = partition(container, left, right) quicksort(container, left, middle - 1) quicksort(container, middle + 1, right) end
end
# File lib/algorithms/sort.rb, line 201 def self.quicksort(container) bottom, top = [], [] top[0] = 0 bottom[0] = container.size i = 0 while i >= 0 do l = top[i] r = bottom[i] - 1; if l < r pivot = container[l] while l < r do r -= 1 while (container[r] >= pivot && l < r) if (l < r) container[l] = container[r] l += 1 end l += 1 while (container[l] <= pivot && l < r) if (l < r) container[r] = container[l] r -= 1 end end container[l] = pivot top[i+1] = l + 1 bottom[i+1] = bottom[i] bottom[i] = l i += 1 else i -= 1 end end container end
Selection sort: A naive sort that goes through the container and selects the smallest element, putting it at the beginning. Repeat until the end is reached. Requirements: Needs to be able to compare elements with <=>, and the [] []= methods should be implemented for the container. Time Complexity: О(n^2) Space Complexity: О(n) total, O(1) auxiliary Stable: Yes
Algorithms::Sort.selection_sort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
# File lib/algorithms/sort.rb, line 88 def self.selection_sort(container) 0.upto(container.size-1) do |i| min = i (i+1).upto(container.size-1) do |j| min = j if (container[j] <=> container[min]) == -1 end container[i], container[min] = container[min], container[i] # Swap end container end
Shell sort: Similar approach as insertion sort but slightly better. Requirements: Needs to be able to compare elements with <=>, and the [] []= methods should be implemented for the container. Time Complexity: О(n^2) Space Complexity: О(n) total, O(1) auxiliary Stable: Yes
Algorithms::Sort.shell_sort [5, 4, 3, 1, 2] => [1, 2, 3, 4, 5]
# File lib/algorithms/sort.rb, line 143 def self.shell_sort(container) increment = container.size/2 while increment > 0 do (increment..container.size-1).each do |i| temp = container[i] j = i while j >= increment && container[j - increment] > temp do container[j] = container[j-increment] j -= increment end container[j] = temp end increment = (increment == 2 ? 1 : (increment / 2.2).round) end container end