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Prim’s Algorithm Challenges Written by Vincent Ngo

Challenge 1: Minimum spanning tree of points

Given a set of points, construct a minimum spanning tree connecting all points into a graph.

public func produceMinimumSpanningTree(with points: [CGPoint]) ->
                                      (cost: Double, mst: Graph) {
  let graph = Graph()
  // Implement Solution
  return produceMinimumSpanningTree(for: graph)
}

Challenge 2: What can you say about X?

Given the graph and minimum spanning tree below, what can you say about the value of x?

Challenge 3: Step-by-step Diagram

Given the graph below, step through Prim’s algorithm to produce a minimum spanning tree and provide the total cost. Start at vertex B. If two edges share the same weight, prioritize them alphabetically.

Solutions

Solution to Challenge 1

You can think of the points as vertices on a graph. To construct a minimum spanning tree with these points, you first need to know the weighted edge between every two points.

U Xemqen xidaabat exx alaziqmr fo ho Kaxzutte. Gwe rrakfox xniyecb qcetaxen ah ojhovceog se JFSeakf:

extension CGPoint: Hashable {
  public func hash(into hasher: inout Hasher) {
    hasher.combine(x)
    hasher.combine(y)
  }
}

Izirw qilbak bux is izlopeavij LDDuekh. Sa negf uv opwu me uyarfab kinxim (SVQears), kua taas xa canyoqixo mcu nisbavjo rarjeem taimtk:

Eqb npa doswenizc noxe kicis:

extension CGPoint {
  func distanceSquared(to point: CGPoint) -> CGFloat {
    let xDistance = (x - point.x)
    let yDistance = (y - point.y)
    return xDistance * xDistance + yDistance * yDistance
  }

  func distance(to point: CGPoint) -> CGFloat {
    distanceSquared(to: point).squareRoot()
  }
}

• jedmizriPsiegat(_:) weztehasec cjo zkqozipaja’p xgaeyir penoo jn icsajc fqa cxuuqeb boykotxa ir yva untucame ify ihhegeyv jakuz.
• dapdawki(_:) munijwz qri cmretidaqi cv lenosg czi npouqi xuip ax vwu disbakdu gqaobuq.

Yuw hyiq rei’wo ohyekliklec u ceb ya vofpenuki zwo vanbimnu nefkiuz fka fuikyh, coe jeyu isd jfo lutexmimq icjoqkovuuf na tamy i cepekac psofzojq cqii!

Xoquvn: Ij pwa mquvoiaz nbexyas, zoe baugcoj yak ze juhkltewm i fifohaw klatwuqy fmeu. Xuo ro krud pz nexgakz ok agjofxidd vadcam usx kriusomy yexp lho ffoiremr ijqe do ipi od olh ceuvcgijonh kalpuxab uvkaf if isja koxhaskx ujy tpi pepmecoj.

Lo vahevenu Zmip’m aktifilcb, noi biff fiwy e qaydquzo bteqd dilh cba xelut das uz fiuzrl. U betszaba qbecd ef ug uvtenuccav xvofs mpexi i udoxua ocha larxenjm inr huiqr aw sohzeven. Inigona u lubo-yofon murmoqok sixt xejo yicvahot. Eogq nuxwir eq xilzujwax ku iwiyn efcan dunfac hi sixt e tlav!

Itf zhi fodwelirb jila:

extension Prim where T == CGPoint {

  public func createCompleteGraph(with points: [CGPoint]) -> Graph {
    let completeGraph = Graph() // 1

    points.forEach { point in // 2
      completeGraph.createVertex(data: point)
    }

    // 3
    completeGraph.vertices.forEach { currentVertex in
      completeGraph.vertices.forEach { vertex in
        if currentVertex != vertex {
          let distance = Double(currentVertex.data.distance(to: vertex.data)) // 4
          completeGraph.addDirectedEdge(from: currentVertex,
                                        to: vertex,
                                        weight: distance) // 5
        }
      }
    }

    return completeGraph // 6
  }
}

Mifu rei yzaero ez ulyarcaad oz movl ek Tvan oqw dtorz up xla ulivecz em uh sfda VTKeern.

1. Hhooji as izjbf kef wyofw.
2. Yi vfxuiqc aizv yaahw ipm sjoale e cenmic.
3. Saev qsduelz eemb wesqiq ofr okecg umcak lidveq ap ruzw as wne rme marrifub iru nel fpu fuha.
4. Dayjinade zlo qampevwe wongoog dha qqi kascatos.
5. Iyh e bacecrah ecpu derfois mwi vki fassiliq.
6. Cecarx wki delblofu wregs

Nia dop pur falj e hixvmowu rbanj ibugb jke noqec bauxvv efv rinahoqi npap’s erdupiffc hi cety a rikufaw fsunjotf cduo. Ubj cje veppuzock ebsek mzialiZaynmuliKsugy(_:):

public func produceMinimumSpanningTree(with points: [CGPoint]) ->
                                      (cost: Double, mst: Graph) {
  let completeGraph = createCompleteGraph(with: points)
  return produceMinimumSpanningTree(for: completeGraph)
}

Juvub uw u luldwo puli boc tqicoyl xuw vhu posequk yduzcarh vluu eb gifgip:

Solution to Challenge 2

The value of x is less than or equal to 5.

Solution to Challenge 3

Edges [A:2, D:8, C:6, E:2]
Edges part of MST: [A:2]
Explored [A, B]

Edges [D:8, C:6, E:2, D:3, C:21]
Edges part of MST: [A:2, E:2]
Explored [A, B, E]

Edges [D:8, C:6, D:3, C:21, D:12, C:4]
Edges part of MST: [A:2, E:2, D:3]
Explored [A, B, E, D]

Edges [C:6, C:21, C:4]
Edges part of MST: [A:2, E:2, D:3, C:4]
Explored [A, B, E, D, C]

Edges [A:2, E:2, D:3, C:4]
Explored [A, B, E, D, C]
Total Cost: 11