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hollancehollance
committedJan 30, 2016
Add permutations and combinations
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‎Combinatorics/Combinatorics.playground/Contents.swift

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//: Playground - noun: a place where people can play
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/* Calculates n! */
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func factorial(n: Int) -> Int {
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var n = n
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var result = 1
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while n > 1 {
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result *= n
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n -= 1
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}
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return result
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}
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factorial(5)
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factorial(20)
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/*
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Calculates P(n, k), the number of permutations of n distinct symbols
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in groups of size k.
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*/
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func permutations(n: Int, _ k: Int) -> Int {
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var n = n
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var answer = n
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for _ in 1..<k {
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n -= 1
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answer *= n
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}
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return answer
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}
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permutations(5, 3)
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permutations(50, 6)
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permutations(9, 4)
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/*
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Prints out all the permutations of the given array.
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Original algorithm by Niklaus Wirth.
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See also Dr.Dobb's Magazine June 1993, Algorithm Alley
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*/
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func permuteWirth<T>(a: [T], _ n: Int) {
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if n == 0 {
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print(a) // display the current permutation
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} else {
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var a = a
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permuteWirth(a, n - 1)
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for i in 0..<n {
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swap(&a[i], &a[n])
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permuteWirth(a, n - 1)
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swap(&a[i], &a[n])
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}
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}
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}
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let letters = ["a", "b", "c", "d", "e"]
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print("Permutations of \(letters):")
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permuteWirth(letters, letters.count - 1)
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let xyz = [ "x", "y", "z" ]
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print("\nPermutations of \(xyz):")
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permuteWirth(xyz, 2)
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/*
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Prints out all the permutations of an n-element collection.
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The initial array must be initialized with all zeros. The algorithm
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uses 0 as a flag that indicates more work to be done on each level
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of the recursion.
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Original algorithm by Robert Sedgewick.
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See also Dr.Dobb's Magazine June 1993, Algorithm Alley
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*/
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func permuteSedgewick(a: [Int], _ n: Int, inout _ pos: Int) {
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var a = a
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pos += 1
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a[n] = pos
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if pos == a.count - 1 {
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print(a) // display the current permutation
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} else {
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for i in 0..<a.count {
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if a[i] == 0 {
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permuteSedgewick(a, i, &pos)
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}
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}
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}
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pos -= 1
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a[n] = 0
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}
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print("\nSedgewick permutations:")
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let numbers = [0, 0, 0, 0] // must be all zeros
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var pos = -1
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permuteSedgewick(numbers, 0, &pos)
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/*
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Calculates C(n, k), or "n-choose-k", i.e. how many different selections
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of size k out of a total number of distinct elements (n) you can make.
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*/
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func combinations(n: Int, _ k: Int) -> Int {
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return permutations(n, k) / factorial(k)
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}
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combinations(3, 2)
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combinations(28, 5)
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print("\nCombinations:")
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for i in 1...20 {
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print("\(20)-choose-\(i) = \(combinations(20, i))")
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}
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/* Supporting code because Swift doesn't have a built-in 2D array. */
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struct Array2D<T> {
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let columns: Int
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let rows: Int
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private var array: [T]
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init(columns: Int, rows: Int, initialValue: T) {
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self.columns = columns
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self.rows = rows
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array = .init(count: rows*columns, repeatedValue: initialValue)
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}
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subscript(column: Int, row: Int) -> T {
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get { return array[row*columns + column] }
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set { array[row*columns + column] = newValue }
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}
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}
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/*
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Calculates C(n, k), or "n-choose-k", i.e. the number of ways to choose
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k things out of n possibilities.
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Thanks to the dynamic programming, this algorithm from Skiena allows for
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the calculation of much larger numbers, at the cost of temporary storage
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space for the cached values.
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*/
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func binomialCoefficient(n: Int, _ k: Int) -> Int {
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var bc = Array2D(columns: n + 1, rows: n + 1, initialValue: 0)
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for i in 0...n {
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bc[i, 0] = 1
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bc[i, i] = 1
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}
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if n > 0 {
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for i in 1...n {
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for j in 1..<i {
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bc[i, j] = bc[i - 1, j - 1] + bc[i - 1, j]
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}
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}
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}
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return bc[n, k]
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}
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binomialCoefficient(30, 15)
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binomialCoefficient(66, 33)

‎Combinatorics/Combinatorics.playground/contents.xcplayground

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

‎Combinatorics/Combinatorics.playground/timeline.xctimeline

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<?xml version="1.0" encoding="UTF-8"?>
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<Timeline
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version = "3.0">
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<TimelineItems>
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</TimelineItems>
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</Timeline>

‎Combinatorics/Combinatorics.swift

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/* Calculates n! */
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func factorial(n: Int) -> Int {
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var n = n
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var result = 1
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while n > 1 {
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result *= n
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n -= 1
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}
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return result
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}
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/*
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Calculates P(n, k), the number of permutations of n distinct symbols
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in groups of size k.
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*/
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func permutations(n: Int, _ k: Int) -> Int {
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var n = n
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var answer = n
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for _ in 1..<k {
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n -= 1
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answer *= n
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}
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return answer
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}
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/*
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Prints out all the permutations of the given array.
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Original algorithm by Niklaus Wirth.
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See also Dr.Dobb's Magazine June 1993, Algorithm Alley
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*/
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func permuteWirth<T>(a: [T], _ n: Int) {
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if n == 0 {
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print(a) // display the current permutation
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} else {
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var a = a
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permuteWirth(a, n - 1)
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for i in 0..<n {
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swap(&a[i], &a[n])
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permuteWirth(a, n - 1)
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swap(&a[i], &a[n])
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}
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}
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}
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/*
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Prints out all the permutations of an n-element collection.
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The initial array must be initialized with all zeros. The algorithm
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uses 0 as a flag that indicates more work to be done on each level
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of the recursion.
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Original algorithm by Robert Sedgewick.
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See also Dr.Dobb's Magazine June 1993, Algorithm Alley
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*/
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func permuteSedgewick(a: [Int], _ n: Int, inout _ pos: Int) {
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var a = a
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pos += 1
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a[n] = pos
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if pos == a.count - 1 {
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print(a) // display the current permutation
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} else {
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for i in 0..<a.count {
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if a[i] == 0 {
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permuteSedgewick(a, i, &pos)
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}
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}
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}
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pos -= 1
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a[n] = 0
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}
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/*
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Calculates C(n, k), or "n-choose-k", i.e. how many different selections
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of size k out of a total number of distinct elements (n) you can make.
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Doesn't work very well for large numbers.
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*/
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func combinations(n: Int, _ k: Int) -> Int {
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return permutations(n, k) / factorial(k)
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}
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/*
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Calculates C(n, k), or "n-choose-k", i.e. the number of ways to choose
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k things out of n possibilities.
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Thanks to the dynamic programming, this algorithm from Skiena allows for
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the calculation of much larger numbers, at the cost of temporary storage
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space for the cached values.
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*/
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func binomialCoefficient(n: Int, _ k: Int) -> Int {
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var bc = Array2D(columns: n + 1, rows: n + 1, initialValue: 0)
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for i in 0...n {
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bc[i, 0] = 1
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bc[i, i] = 1
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}
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if n > 0 {
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for i in 1...n {
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for j in 1..<i {
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bc[i, j] = bc[i - 1, j - 1] + bc[i - 1, j]
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}
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}
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}
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return bc[n, k]
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}

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