[New Algorithm] Introsort sorting algorithm

Introsort/HeapSort.swift

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+import Foundation
+
+private func shiftDown<T>(_ elements: inout [T], _ index: Int, _ range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) {
+    let countToIndex = elements.distance(from: range.lowerBound, to: index)
+    let countFromIndex = elements.distance(from: index, to: range.upperBound)
+
+    guard countToIndex + 1 < countFromIndex else { return }
+
+    let left = elements.index(index, offsetBy: countToIndex + 1)
+    var largest = index
+    if areInIncreasingOrder(elements[largest], elements[left]) {
+        largest = left
+    }
+
+    if countToIndex + 2 < countFromIndex {
+        let right = elements.index(after: left)
+        if areInIncreasingOrder(elements[largest], elements[right]) {
+            largest = right
+        }
+    }
+
+    if largest != index {
+        elements.swapAt(index, largest)
+        shiftDown(&elements, largest, range, by: areInIncreasingOrder)
+    }
+
+}
+
+private func heapify<T>(_ list: inout [T], _ range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) {
+    let root = range.lowerBound
+    var node = list.index(root, offsetBy: list.distance(from: range.lowerBound, to: range.upperBound)/2)
+
+    while node != root {
+        list.formIndex(before: &node)
+        shiftDown(&list, node, range, by: areInIncreasingOrder)
+    }
+}
+
+public func heapsort<T>(for array: inout [T], range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) {
+    var hi = range.upperBound
+    let lo = range.lowerBound
+    heapify(&array, range, by: areInIncreasingOrder)
+    array.formIndex(before: &hi)
+
+    while hi != lo {
+        array.swapAt(lo, hi)
+        shiftDown(&array, lo, lo..<hi, by: areInIncreasingOrder)
+        array.formIndex(before: &hi)
+    }
+}

Introsort/InsertionSort.swift

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+import Foundation
+
+public func insertionSort<T>(for array: inout [T], range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) {
+    guard !range.isEmpty else { return }
+
+    let start = range.lowerBound
+    var sortedEnd = start
+
+    array.formIndex(after: &sortedEnd)
+    while sortedEnd != range.upperBound {
+        let x = array[sortedEnd]
+
+        var i = sortedEnd
+        repeat {
+            let predecessor = array[array.index(before: i)]
+
+            guard areInIncreasingOrder(x, predecessor) else { break }
+            array[i] = predecessor
+            array.formIndex(before: &i)
+        } while i != start
+
+        if i != sortedEnd {
+            array[i] = x
+        }
+        array.formIndex(after: &sortedEnd)
+    }
+
+}

Introsort/IntroSort.swift

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+import Foundation
+
+public func introsort<T>(_ array: inout [T],  by areInIncreasingOrder: (T, T) -> Bool) {
+    //The depth limit is as best practice 2 * log( n )
+    let depthLimit = 2 * floor(log2(Double(array.count)))
+
+    introSortImplementation(for: &array, range: 0..<array.count, depthLimit: Int(depthLimit), by: areInIncreasingOrder)
+}
+
+///This method is recursively executed for each partition result of the quicksort part of the algorithm
+private func introSortImplementation<T>(for array: inout [T], range: Range<Int>, depthLimit: Int, by areInIncreasingOrder: (T, T) -> Bool) {
+    if array.distance(from: range.lowerBound, to: range.upperBound) < 20 {
+        //if the partition count is less than 20 we can sort it using insertion sort. This algorithm in fact performs well on collections
+        //of this size, plus, at this point is quite probable that the quisksort part of the algorithm produced a partition which is
+        //nearly sorted. As we knoe insertion sort tends to O( n ) if this is the case.
+        insertionSort(for: &array, range: range, by: areInIncreasingOrder)
+    } else if depthLimit == 0 {
+        //If we reached the depth limit for this recursion branch, it's possible that we are hitting quick sort's worst case.
+        //Since quicksort degrades to O( n^2 ) in its worst case we stop using quicksort for this recursion branch and we switch to heapsort.
+        //Our preference remains quicksort, and we hope to be rare to see this condition to be true
+        heapsort(for: &array, range: range, by: areInIncreasingOrder)
+    } else {
+        //By default we use quicksort to sort our collection. The partition index method chose a pivot, and puts all the
+        //elements less than pivot on the left, and the ones bigger than pivot on the right. At the end of the operation the
+        //position of the pivot in the array is returned so that we can form the two partitions.
+        let partIdx = partitionIndex(for: &array, subRange: range, by: areInIncreasingOrder)
+
+        //We can recursively call introsort implementation, decreasing the depthLimit for the left partition and the right partition.
+        introSortImplementation(for: &array, range: range.lowerBound..<partIdx, depthLimit: depthLimit &- 1, by: areInIncreasingOrder)
+        introSortImplementation(for: &array, range: partIdx..<range.upperBound, depthLimit: depthLimit &- 1, by: areInIncreasingOrder)
+    }
+}

Introsort/Introsort.playground/Contents.swift

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+//: Playground - noun: a place where people can play
+
+import Foundation
+
+func introsort<T>(_ array: inout [T],  by areInIncreasingOrder: (T, T) -> Bool) {
+    //The depth limit is as best practice 2 * log( n )
+    let depthLimit = 2 * floor(log2(Double(array.count)))
+
+    introSortImplementation(for: &array, range: 0..<array.count, depthLimit: Int(depthLimit), by: areInIncreasingOrder)
+}
+
+///This method is recursively executed for each partition result of the quicksort part of the algorithm
+private func introSortImplementation<T>(for array: inout [T], range: Range<Int>, depthLimit: Int, by areInIncreasingOrder: (T, T) -> Bool) {
+    if array.distance(from: range.lowerBound, to: range.upperBound) < 20 {
+        //if the partition count is less than 20 we can sort it using insertion sort. This algorithm in fact performs well on collections
+        //of this size, plus, at this point is quite probable that the quisksort part of the algorithm produced a partition which is
+        //nearly sorted. As we knoe insertion sort tends to O( n ) if this is the case.
+        insertionSort(for: &array, range: range, by: areInIncreasingOrder)
+    } else if depthLimit == 0 {
+        //If we reached the depth limit for this recursion branch, it's possible that we are hitting quick sort's worst case.
+        //Since quicksort degrades to O( n^2 ) in its worst case we stop using quicksort for this recursion branch and we switch to heapsort.
+        //Our preference remains quicksort, and we hope to be rare to see this condition to be true
+        heapsort(for: &array, range: range, by: areInIncreasingOrder)
+    } else {
+        //By default we use quicksort to sort our collection. The partition index method chose a pivot, and puts all the
+        //elements less than pivot on the left, and the ones bigger than pivot on the right. At the end of the operation the
+        //position of the pivot in the array is returned so that we can form the two partitions.
+        let partIdx = partitionIndex(for: &array, subRange: range, by: areInIncreasingOrder)
+
+        //We can recursively call introsort implementation, decreasing the depthLimit for the left partition and the right partition.
+        introSortImplementation(for: &array, range: range.lowerBound..<partIdx, depthLimit: depthLimit &- 1, by: areInIncreasingOrder)
+        introSortImplementation(for: &array, range: partIdx..<range.upperBound, depthLimit: depthLimit &- 1, by: areInIncreasingOrder)
+    }
+}
+
+var unsorted = randomize(n: 1000)
+
+let swiftSorted = unsorted.sorted()
+introsort(&unsorted, by: <)
+
+print("does it work? \(unsorted == swiftSorted)")

Introsort/Introsort.playground/Sources/HeapSort.swift

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+import Foundation
+
+private func shiftDown<T>(_ elements: inout [T], _ index: Int, _ range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) {
+    let countToIndex = elements.distance(from: range.lowerBound, to: index)
+    let countFromIndex = elements.distance(from: index, to: range.upperBound)
+
+    guard countToIndex + 1 < countFromIndex else { return }
+
+    let left = elements.index(index, offsetBy: countToIndex + 1)
+    var largest = index
+    if areInIncreasingOrder(elements[largest], elements[left]) {
+        largest = left
+    }
+
+    if countToIndex + 2 < countFromIndex {
+        let right = elements.index(after: left)
+        if areInIncreasingOrder(elements[largest], elements[right]) {
+            largest = right
+        }
+    }
+
+    if largest != index {
+        elements.swapAt(index, largest)
+        shiftDown(&elements, largest, range, by: areInIncreasingOrder)
+    }
+
+}
+
+private func heapify<T>(_ list: inout [T], _ range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) {
+    let root = range.lowerBound
+    var node = list.index(root, offsetBy: list.distance(from: range.lowerBound, to: range.upperBound)/2)
+
+    while node != root {
+        list.formIndex(before: &node)
+        shiftDown(&list, node, range, by: areInIncreasingOrder)
+    }
+}
+
+public func heapsort<T>(for array: inout [T], range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) {
+    var hi = range.upperBound
+    let lo = range.lowerBound
+    heapify(&array, range, by: areInIncreasingOrder)
+    array.formIndex(before: &hi)
+
+    while hi != lo {
+        array.swapAt(lo, hi)
+        shiftDown(&array, lo, lo..<hi, by: areInIncreasingOrder)
+        array.formIndex(before: &hi)
+    }
+}

Introsort/Introsort.playground/Sources/InsertionSort.swift

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+import Foundation
+
+public func insertionSort<T>(for array: inout [T], range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) {
+    guard !range.isEmpty else { return }
+
+    let start = range.lowerBound
+    var sortedEnd = start
+
+    array.formIndex(after: &sortedEnd)
+    while sortedEnd != range.upperBound {
+        let x = array[sortedEnd]
+
+        var i = sortedEnd
+        repeat {
+            let predecessor = array[array.index(before: i)]
+
+            guard areInIncreasingOrder(x, predecessor) else { break }
+            array[i] = predecessor
+            array.formIndex(before: &i)
+        } while i != start
+
+        if i != sortedEnd {
+            array[i] = x
+        }
+        array.formIndex(after: &sortedEnd)
+    }
+
+}

Introsort/Introsort.playground/Sources/Partition.swift

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+import Foundation
+
+public func partitionIndex<T>(for elements: inout [T], subRange range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) -> Int {
+    var lo = range.lowerBound
+    var hi = elements.index(before: range.upperBound)
+
+    // Sort the first, middle, and last elements, then use the middle value
+    // as the pivot for the partition.
+    let half = elements.distance(from: lo, to: hi) / 2
+    let mid = elements.index(lo, offsetBy: half)
+
+    sort3(in: &elements, a: lo, b: mid, c: hi, by: areInIncreasingOrder)
+    let pivot = elements[mid]
+
+    while true {
+        elements.formIndex(after: &lo)
+        guard findLo(in: elements, pivot: pivot, from: &lo, to: hi, by: areInIncreasingOrder) else { break }
+        elements.formIndex(before: &hi)
+        guard findHi(in: elements, pivot: pivot, from: lo, to: &hi, by: areInIncreasingOrder) else { break }
+        elements.swapAt(lo, hi)
+    }
+
+
+    return lo
+}
+
+private func findLo<T>(in array: [T], pivot: T, from lo: inout Int, to hi: Int, by areInIncreasingOrder: (T, T)->Bool) -> Bool {
+    while lo != hi {
+        if !areInIncreasingOrder(array[lo], pivot) {
+            return true
+        }
+        array.formIndex(after: &lo)
+    }
+    return false
+}
+
+private func findHi<T>(in array: [T], pivot: T, from lo: Int, to hi: inout Int, by areInIncreasingOrder: (T, T)->Bool) -> Bool {
+    while hi != lo {
+        if areInIncreasingOrder(array[hi], pivot) { return true }
+        array.formIndex(before: &hi)
+    }
+    return false
+}

Introsort/Introsort.playground/Sources/Randomize.swift

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+import Foundation
+
+public func randomize(n: Int) -> [Int] {
+    var unsorted = [Int]()
+    for _ in 0..<n {
+        unsorted.append(Int(arc4random()))
+    }
+    return unsorted
+}

Introsort/Introsort.playground/Sources/Sort3.swift

@@ -0,0 +1,21 @@
+import Foundation
+
+public func sort3<T>(in array: inout [T], a: Int, b: Int, c: Int, by areInIncreasingOrder: (T, T) -> Bool) {
+    switch (areInIncreasingOrder(array[b], array[a]),
+            areInIncreasingOrder(array[c], array[b])) {
+    case (false, false): break
+    case (true, true): array.swapAt(a, c)
+    case (true, false):
+        array.swapAt(a, b)
+
+        if areInIncreasingOrder(array[c], array[b]) {
+            array.swapAt(b, c)
+        }
+    case (false, true):
+        array.swapAt(b, c)
+
+        if areInIncreasingOrder(array[b], array[a]) {
+            array.swapAt(a, b)
+        }
+    }
+}

Introsort/Introsort.playground/contents.xcplayground

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+<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
+<playground version='5.0' target-platform='ios'>
+    <timeline fileName='timeline.xctimeline'/>
+</playground>

Introsort/Partition.swift

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+import Foundation
+
+public func partitionIndex<T>(for elements: inout [T], subRange range: Range<Int>, by areInIncreasingOrder: (T, T) -> Bool) -> Int {
+    var lo = range.lowerBound
+    var hi = elements.index(before: range.upperBound)
+
+    // Sort the first, middle, and last elements, then use the middle value
+    // as the pivot for the partition.
+    let half = elements.distance(from: lo, to: hi) / 2
+    let mid = elements.index(lo, offsetBy: half)
+
+    sort3(in: &elements, a: lo, b: mid, c: hi, by: areInIncreasingOrder)
+    let pivot = elements[mid]
+
+    while true {
+        elements.formIndex(after: &lo)
+        guard findLo(in: elements, pivot: pivot, from: &lo, to: hi, by: areInIncreasingOrder) else { break }
+        elements.formIndex(before: &hi)
+        guard findHi(in: elements, pivot: pivot, from: lo, to: &hi, by: areInIncreasingOrder) else { break }
+        elements.swapAt(lo, hi)
+    }
+
+
+    return lo
+}
+
+private func findLo<T>(in array: [T], pivot: T, from lo: inout Int, to hi: Int, by areInIncreasingOrder: (T, T)->Bool) -> Bool {
+    while lo != hi {
+        if !areInIncreasingOrder(array[lo], pivot) {
+            return true
+        }
+        array.formIndex(after: &lo)
+    }
+    return false
+}
+
+private func findHi<T>(in array: [T], pivot: T, from lo: Int, to hi: inout Int, by areInIncreasingOrder: (T, T)->Bool) -> Bool {
+    while hi != lo {
+        if areInIncreasingOrder(array[hi], pivot) { return true }
+        array.formIndex(before: &hi)
+    }
+    return false
+}