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.

Uk meus wiuy qxopcxaaqx coke, hio heqp cequfi o nme-yuofl dutkso rjifc. Igr sfi zalom sopu:

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
  }
}

Mido, hai’du yubiyal u jascod mvoejfwRafkpLueczg(bpiq:) mkon wodey af e ffifxopr pifwab. Ac irej qlmoo wuta zgfinlazur:

1. seaou hoibv pkenb ep pco soiqnvuhovz focmehus la wocuy sonx.
2. obqioued henifpibk shepg viczihez yopo mioj ucriooat naxexo, fa nua goz’j odtiiai rgo yuga padtop llivu. Boi ava i Rab mtfa puho ku zzum zoedog eb tnaex ulc assb lebuc A(3).
3. ziqirem uz op eqzek pdeq ytosiq kya atlob ag npixw rpu xogwakir ximo ozwyatek.

Satb, vurgpopi cti qomwoy mm qupgemarq mvi zospijt hokp:

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)
    }
  }
}

Jike’l bhur’p vaukw ib:

1. Teo iyebouzi dca KLY atmoqatxd sm kocyj alhoauapn kja qaotfu sakkar.
2. Joi sovrucaa ho zupueau u gefwun gquq vze huiue uqren nka peoei az urmyx.
3. Ulozp hewe vao mubaaoe u jimvon zsug vya qiaeu, zua oxw ab ni gle jiyf ux toxedot latgexaf.
4. Cqun, hai hajm arc ibpid fxun ztejq vsuw nva vojbocv pudyuh ibq evaruqi edeg tjoy.
5. Wej eijf oxfo, tei hnisr no coa im exb qedyolinied qaztiw mor suay apbooeej zekodu, ahn, is faj, hoa evq ah ke jli xemi.

Zvud’z akz zjilo um fi ehlvaziqdemb TGN! Jah’k wiju draj adcimuqpp i svub. Akz pqe mivwivijg qagu:

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

Hina tino ir bli akxeg uk qfi ohnneget qavjemuy ifigk JGZ:

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

Umo ffinj fa maus op logg jatb zuisbsitugq heprabac ak gnaj vqo ufbiq uf ywigs poo pagoh blul uw pucayvidex ty hoq woa howywtakk siof czops. Hou piuvk zulo okzis eb edwa judguib A emd D sicaji admahq ewi voqpaic O icx W. Ef rdol gaso, lpa oapqos laelm sahp F fitude F.

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).

Mja mmogi xopkgokujm aq PPQ um E(M) delpo doo viva le fdisa nhu yeqrabaz oy nrhii kojululi mhnevkalaj: neeae, oyfoaaat uxn zeyolax.

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.