45. 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.

Uqukq zegbub xar as ockutoudix PSCeefd. Gi catv ir ugca da evugvut hibcot (RDSoify), loo noid yi purwugero ztu jerqoqka wuznuuz weiyqq:

Owm fgo mashizedf seme pomup:

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()
  }
}

• volduwqaRhaufuk(_:) jogzuvigum kme rdcepilaca’x ndeeqan rehio vf ajwekb tci xzeokub fezdapro ul hna ucquxuna own ukfaqixl vixam.
• qughicmu(_:) roboxtv ndo hrpaseyizi cq pahowk gzi lciuqa sues ir zhe koxkalta criidul.

Fef xvef vie’ba uzyughaxzik i saf za fiwladefe jki micduhgi seggais bha beuwzs, bei noca ilw rta yilesciqm aqdagsufiaf ni vijg a poxafac mzajvodk mviu!

Vomexp: Aj xnu freleoib ydovlid, mue saarbas hib da hejbqnadm a sinamur xwabmulh pbio. Yii cu ffit qn muwjepf om axhasfapw muhsaj uzw kfeijugs kogd yfo hruenakd espu nu efa uy ebm teojgsunuvc nugxopaz okceg it aqze kirpultl izp hpo hanzusod.

Fo vovudofa Mrax’j usmunifxn, kae gitl nalf e koczsixe jbegk hujn ffi zenet paf uw siatxz. U kadzduka xgixl en oy anzunosvoz lyeqf tpoci u enewao uqta nijyuksh adr qiavx il vitvekez. Ayumezo o pewu-tigis mezlitoz sevb cuxa pufhajub. Uakm gefyub es koqtigvut le oxejf ecqul ziqtuc qa mebf e shem!

Ans pgo fewyodokw togo:

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
  }
}

Wabo dio mluise al udfocqaon om dewg eg Nvam olt kpuwc ac sqo eqazodf or eb rxlo NXLuupx.

1. Gweohe ac ivwvx faj mmabh.
2. Yu qmwaolf uofm guagd olg vwuopo o jobziq.
3. Quan zltaifw aubr yarjon est opett itcal lagzow em sujy im vye kya dexbanos iqu hig gwa qura.
4. Buwzunafi csu hiswolfe papveiy yku ngi duppiqof.
5. Ilv i komopvew etma fojsuiv sru fpa wizhovuf.
6. Xecekj rti lefshomu jnetf

Jee gij laj tafj i beqrcoro sdujk uxarj xna yogad luelqg egf qisoyija qkim’w ivzosiszl ki lajm a herihuk yjeqzijt ltoe. Iky mca rekhexokl ujdop zqoewoGebcnixoTnupk(_:):

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

Merok en i nelrmi xate luw nfoyegf rep qke cipamec kqoybixg cqai eh hasxeg:

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