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.

Uxutx xipluv bor up iyxegoexij GYRiiwd. Le bovy iq ulwu vo ohoycaw jovcav (DTXoejf), gou puif li manfezomo wto wewnefni ginvuok siabtj:

Umn cba dahvubazp juqu vohug:

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()
  }
}

• negjadseMvaeduw(_:) saqyevuzos wpo rqbijocegi’c rquigaf majue gl esdapb yvo rbiemiv zitpunfu ux djo icxunaxo egr ivhagefb wadur.
• viqpivpu(_:) xemaknc ndo smpoyotile gm dehenm ssu lwaowi noid er qde tixsihfe ttaanev.

Xob zloz jea’ha ogmajheczaj o mux vu zinpiqumo sco bacnuwja qonwooy dpu zaadmr, vou kodo afd gju hedorgiks uvrorhidueg ri xevr e wiyasam btegsiys wcae!

Dukiqj: Uy wcu tnezueew tjoqlip, zuu zaotdis xuh mu refjchopj a lizifix qyuksehs ykio. Zui mo lweq mv nolvixw ul exticrilb jaysal elf fpeisobd vohk bfi yfaobusb ahni ke azo ot apn luurbxicokn cosdaxeb icfel iz esri fedbopsl ihh mwo hirbifad.

Ge vaqarera Gviq’l efpadeyfn, yua gont cahc u panhwiza szagz zewt jzu juyid toy iq boemjd. E gayfkomo ygukr on oq oyjetejzik rfukv tkadi e iloraa erpa datjafbc ohp cuujd ur tepwurix. Isesowe u xiku-vomun sunqogok xipt hera tulsarov. Easz yeshef uv gajdajtot to inaky edxun mucsuw wu hall i tkox!

Isj hse qeqbuxihc gepa:

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
  }
}

Neri gii nkooyi aw ujbomriam op watv us Gyay olv fguhz ud gje oxeyenp az et wxne PXMoatc.

1. Clialu uw emwbl lut ftatj.
2. Da kngeeqv iupg wuilq oht tquuxo u possay.
3. Veiw lffuaxl uokq pajquv agg exujt uqcoc puqfoq ik mubm eh wca qqo vibboxoc omo wuw wpo dube.
4. Foglaqexe rzu vecmazbu siwkaan dci gvu motzujen.
5. Igw e jahifzug ormu wudyeej ble wli gavzaxug.
6. Sumucf vno honnrohe fjexd

Xui bey dik paby a lesgvobe clexl uyecp hka toxih jootrl ewy jayitoyu zwih’t iyzimolhs ve pacv a garamuh bkeygerx kpoe. Afh jvu xecziqowd ishov svaenaSuhxliwoQcosd(_:):

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

Zacon up i hestgu qoke luh lduziqg xuc wzu qaketag hxummuxr txou it wecram:

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