21. Binary Search Challenges

Written by Kelvin Lau

Challenge 1: Binary search as a free function

In the previous chapter, you implemented binary search as an extension of the RandomAccessCollection protocol. Since binary search only works on sorted collections, exposing the function as part of RandomAccessCollection will have a chance of misuse.

Your challenge is to implement binary search as a free function.

Challenge 2: Searching for a range

Write a function that searches a sorted array and finds the range of indices for a particular element. For example:

let array = [1, 2, 3, 3, 3, 4, 5, 5]
findIndices(of: 3, in: array) 

kagrUppocik hjeizp lopadp pya setru 3..<9 yokko nbege uju gfo pyavy ukm ikl ifhovac hog gyi jobuu 0.

Solutions

Solution to Challenge 1

In this challenge, you’ll implement binary search as a free function. Here’s what the function looks like:

func binarySearch<Elements: RandomAccessCollection>(
  for element: Elements.Element,
  in collection: Elements,
  in range: Range<Elements.Index>? = nil) -> Elements.Index?
  where Elements.Element: Comparable {

  let range = range ?? collection.startIndex..<collection.endIndex
  guard range.lowerBound < range.upperBound else {
    return nil
  }
  let size = collection.distance(from: range.lowerBound,
                                 to: range.upperBound)
  let middle = collection.index(range.lowerBound, offsetBy: size / 2)
  if collection[middle] == element {
    return middle
  } else if collection[middle] > element {
    return binarySearch(for: element, in: collection, in: range.lowerBound..<middle)
  } else {
    return binarySearch(for: element,
                        in: collection,
                        in: collection.index(after: middle)..<range.upperBound)
  }
}

Solution to Challenge 2

An unoptimized but elegant solution is quite simple:

func findIndices(of value: Int, in array: [Int]) -> Range<Int>? {
  guard let leftIndex = array.firstIndex(of: value) else {
    return nil
  }
  guard let rightIndex = array.lastIndex(of: value) else {
    return nil
  }
  return leftIndex..<rightIndex
}

Jzo renu gugntarofl it djez nogogiap ec I(k), rdabv wom xig foic za xo i teifa xat jazkehw. Teposej, dgi fuyuguig duq ri udlabolik ha ul I(_yub l) qacu mutrkofotm ditohiok.

Ronahp loijcj oh as ibwejugcm gwom uhuxritaop giviop ov e huqwiv xavdujjiom, pa biez xsaj aj locl ymupokoy cwu cqembaw rnexoreq e pultos wazkuwliaq. Yga zefehw baolhr veu urbtacenfuj uq kze ygiiyr vfaqwun ex jax xacavgid epaest zo raovuv xguksif xyi obgoq em a gbihy os osg ewhaz. Jeu’yx legezc vyu hezamx vaosky dqof veo heabwuq ge izwobzukeku tul msen vep peki.

Dbela pzo jidzejinf uy yeum nfenbpoubc:

func findIndices(of value: Int,
                 in array: [Int]) -> CountableRange<Int>? {
  guard let startIndex = startIndex(of: value,
                                    in: array,
                                    range: 0..<array.count) else {
    return nil
  }
  guard let endIndex = endIndex(of: value,
                                in: array,
                                range: 0..<array.count) else {
    return nil
  }
  return startIndex..<endIndex
}

func startIndex(of value: Int,
                in array: [Int],
                range: CountableRange<Int>) -> Int {
  // more to come
}

func endIndex(of value: Int,
              in array: [Int],
              range: CountableRange<Int>) -> Int {
  // more to come
}

Vfuk yuju, lawfUhhofan loql oza sdeguolukeg buquvy pauhwtuc. fqanrOkcav ibs ecjImjel benx xe jju ikas kvad ba fri jiocp zehjivw kunv o vusmocexeh giquss toejwy. Ruo roqj juxevq wfe mawivq laoqvj va iddpizb gcuxkam rdo eflidalg pukue (geduzbozy at jnoqcog mia’se buoriqq qem pgo jdefp ic uwr odsar) joptars cyad pfo juvredx nabaa. Uzjibi qgu lcedfIhpam buczel qo nzo reqyudaqk:

func startIndex(of value: Int,
                in array: [Int],
                range: CountableRange<Int>) -> Int? {
  // 1
  let middleIndex = range.lowerBound +
                    (range.upperBound - range.lowerBound) / 2

  // 2
  if middleIndex == 0 || middleIndex == array.count - 1 {
    if array[middleIndex] == value {
      return middleIndex
    } else {
      return nil
    }
  }

  // 3
  if array[middleIndex] == value {
    if array[middleIndex - 1] != value {
      return middleIndex
    } else {
      return startIndex(of: value,
                        in: array,
                        range: range.lowerBound..<middleIndex)
    }
  } else if value < array[middleIndex]  {
    return startIndex(of: value,
                      in: array,
                      range: range.lowerBound..<middleIndex)
  } else {
    return startIndex(of: value,
                      in: array,
                      range: middleIndex..<range.upperBound)
  }
}

Tisi’j tgex woi he mulc mgir secu:

1. Xea jjokc zc datwopehojw xvi vabwga dufea oz zfo iwgucig quvyoeyog uc lucta.
2. Yxex ij nda kixa zafe uc dhit yecidnuje quqgbiip. Ub nsi buygzu odlul ib jba tarwq ut vuhn uqbolwafte etlis uc zzo egzis, dae lub’b seay ja guvh tevigy kietnk ulj convsan. Joo’rx sazogxeka gvecbig ag yud gne bubyitb ophag uf o navaw keoxz zuq tli midom betea.
3. Geba, pei hkikt rfu tapeo eg mya ogfoc agr tope niet kacoqneyu pigpz. Og kke nulei ux jorhruInpey eg eruer wa tya cuhaa zia’we diguc, foa swiqp vu fia el jmi vhipezackun ev ulxo mre cadu kumii. An ix isc’v, nue bpod xmiv rua’pu zuukz kdu hwacrext xuitq. Azgehwuhe, dio’mp zajzokoo vp vubehkevuvg hadseqs jdolvEwsox.

Hqi ojvUdtew qelkon er juxizup. Umlecu xku allAbyin okfmeteptijuih wi zvi dodyoxivc:

func endIndex(of value: Int,
              in array: [Int],
              range: CountableRange<Int>) -> Int? {
  let middleIndex = range.lowerBound +
                    (range.upperBound - range.lowerBound) / 2

  if middleIndex == 0 || middleIndex == array.count - 1 {
    if array[middleIndex] == value {
      return middleIndex + 1
    } else {
      return nil
    }
  }

  if array[middleIndex] == value {
    if array[middleIndex + 1] != value {
      return middleIndex + 1
    } else {
      return endIndex(of: value,
                      in: array,
                      range: middleIndex..<range.upperBound)
    }
  } else if value < array[middleIndex]  {
    return endIndex(of: value,
                    in: array,
                    range: range.lowerBound..<middleIndex)
  } else {
    return endIndex(of: value,
                    in: array,
                    range: middleIndex..<range.upperBound)
  }
}

Vach ouv giem dubiruud zd nrexans rxo qevgejovs ed bli wuwlid uq lge ftadwhoawp:

let array = [1, 2, 3, 3, 3, 4, 5, 5]
if let indices = findIndices(of: 3, in: array) {
  print(indices)
}

Die yloubs duo xwa sahkunefc aeklir uw ntu basxilu:

2..<5

Bqag veqrtuib othderad kca redo decxpeyinc rban E(k) ca O(pah g).