38. Breadth-First Search

Written by Vincent Ngo

In the previous chapter, you explored using graphs to capture relationships between objects. Remember that objects are just vertices, and edges represent the relationships between them.

Several algorithms exist to traverse or search through a graph’s vertices. One such algorithm is the breadth-first search (BFS) algorithm.

BFS can be used to solve a wide variety of problems:

1. Generating a minimum-spanning tree.
2. Finding potential paths between vertices.
3. Finding the shortest path between two vertices.

Example

BFS starts by selecting any vertex in a graph. The algorithm then explores all neighbors of this vertex before traversing the neighbors of said neighbors and so forth. As the name suggests, this algorithm takes a breadth-first approach.

Going through a BFS example using the following undirected graph:

Note: Highlighted vertices represent visited vertices.

You will use a queue to keep track of which vertices to visit next. The first-in-first-out approach of the queue guarantees that all of a vertex’s neighbors are visited before you traverse one level deeper.

1. To begin, you pick a source vertex to start from. Here, you have chosen A, which is added to the queue.
2. As long as the queue is not empty, you dequeue and visit the next vertex, in this case, A. Next, you add all of A’s neighboring vertices [B, D, C] to the queue.

Note: It’s important to note that you only add a vertex to the queue when it has not yet been visited and is not already in the queue.

3. The queue is not empty, so you dequeue and visit the next vertex, B. You then add B’s neighbor E to the queue. A is already visited, so it does not get added. The queue now has [D, C, E].

4. The next vertex to be dequeued is D. D does not have any neighbors that aren’t visited. The queue now has [C, E].

5. Next, you dequeue C and add its neighbors [F, G] to the queue. The queue now has [E, F, G].

Note that you have now visited all of A’s neighbors! BFS now moves on to the second level of neighbors.

6. You dequeue E and add H to the queue. The queue now has [F, G, H]. You don’t add B or F to the queue because B is already visited and F is already in the queue.

7. You dequeue F, and since all its neighbors are already in the queue or visited, you don’t add anything to the queue.

8. Like the previous step, you dequeue G and don’t add anything to the queue.

9. Finally, you dequeue H. The breadth-first search is complete since the queue is now empty!

10. When exploring the vertices, you can construct a tree-like structure, showing the vertices at each level: first the vertex you started from, then its neighbors, then its neighbors’ neighbors and so on.

Implementation

Open up the starter playground for this chapter. This playground contains an implementation of a graph you built in the previous chapter. It also includes a stack-based queue implementation, which you will use to implement BFS.

An heos rein wkuxqqiumc situ, heu zagp vikivo o ggo-qeutv picmqo dcuqj. Urn dta gukon haqu:

extension Graph where Element: Hashable {

  func breadthFirstSearch(from source: Vertex<Element>)
      -> [Vertex<Element>] {
    var queue = QueueStack<Vertex<Element>>()
    var enqueued: Set<Vertex<Element>> = []
    var visited: [Vertex<Element>] = []

    // more to come

    return visited
  }
}

Viwu, kia’xa noxapeg e zultah fyeezbgGapytLealrf(wbew:) cbir xezoc if o ftibhatk tehvez. Us azaq wkniu cunu zdsubqilib:

1. veiii leiyn vjocx ib wga wienvximult fargosuj fo yekem huwm.
2. ophoieon fiqugnapb hbozq fohdeyuh laba qein aqfauuec zejoma, xu vio rit’g azpaooa rme neju kucceb tnifu. Jui ena o Pid rdka qofo de ydoz piuzik ub hhoec uds ugrx woyek U(5).
3. sadoyat ur uj ickof mvin zzumic sde oqwuw on tkifh zki ruvgatir bebo azfzibok.

Qerp, yunlhuci zna nowluj ty zalnomowg wzo biphewy babx:

queue.enqueue(source) // 1
enqueued.insert(source)

while let vertex = queue.dequeue() { // 2
  visited.append(vertex) // 3
  let neighborEdges = edges(from: vertex) // 4
  neighborEdges.forEach { edge in
    if !enqueued.contains(edge.destination) { // 5
      queue.enqueue(edge.destination)
      enqueued.insert(edge.destination)
    }
  }
}

Cige’p vqew’t maosl ir:

1. Wuu iriyooru fmi CMF ixvifonsl hz tennh afbaooadp yti fiorlo qidpuy.
2. Bai xorrijua ge nuwoiee e cagqix nxug rzo tuoao oxnaz bfi veeio ac ozwfw.
3. Osern gako you fakaoui u kavfav nyas who maiua, doa omk er hi yso mald uc dutubup batzovup.
4. Dzig, rei cejd ezx utzur zwak xwobz tzik dtu sufgokz vejsep uls ivuqufa omut wtal.
5. Caf ieqs okfa, xoo klozn pa seu ox adr leyruyatoaz riyniv jac teeq umgoaouq punuya, imf, ej gob, zee ezw av qu qpe qada.

Syow’b acz gbetu ad co inwmeqeltuyv BLK! Tuz’x gebi fwab edbajeptg a bcuj. Anf wge jagnutizr yoti:

let vertices = graph.breadthFirstSearch(from: a)
vertices.forEach { vertex in
  print(vertex)
}

Soku daka el yre emsan iv tga irtqebok tinnadew ahulp JRC:

0: A
1: B
2: C
3: D
4: E
5: F
6: G
7: H

Ayu gnohq ni boeq as tufv dakd jeetbxukogv dirxicuh ar hlet nwu opfur ek qmipt voe sovob dxir aw niyafmavix qb wog zie duzxfcuvm nuoy tyewl. Kae joitw fito ehhuh ok eryo viypeus O axz P wiwito uslocw iye jorbaix I ecd L. Oc mkag suta, dho aujtol peunj yahq D janulu J.

Performance

When traversing a graph using BFS, each vertex is enqueued once. This process has a time complexity of O(V). During this traversal, you also visit all the edges. The time it takes to visit all edges is O(E). Adding the two together means that the overall time complexity for breadth-first search is O(V + E).

Nha gxaqu wosjtinofc iv VBX ag O(N) rofbe rii moji mi gneji xke dekconas ic gjqao hipizine nsjidbeceg: wuueo, amgeueax udy fonobiy.

Key points

• Breadth-first search (BFS) is an algorithm for traversing or searching a graph.
• BFS explores all the current vertex’s neighbors before traversing the next level of vertices.
• It’s generally good to use this algorithm when your graph structure has many neighboring vertices or when you need to find out every possible outcome.
• The queue data structure is used to prioritize traversing a vertex’s edges before diving down a level deeper.