20. Breadth-First Search

Written by Irina Galata

In the previous chapter, you explored how you can use graphs to capture relationships between objects. Remember that objects are vertices, and the relationships between them are represented by edges.

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

You can use BFS 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 because it doesn’t visit the children until all the siblings are visited.

To get a better idea of how things work, you’ll look at a BFS example using the following undirected graph:

Note: Highlighted vertices represent vertices that have been visited.

To keep track of which vertices to visit next, you can use a queue. 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, pick a source vertex to start from. In this example, you choose A, which is added to the queue.
2. As long as the queue is not empty, you can dequeue and visit the next vertex, in this case, A. Next, you need to add all A’s neighboring vertices [B, D, C] to the queue.

Note: 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 need to dequeue and visit the next vertex, which is B. You then need to add B’s neighbor E to the queue. A is already visited, so it doesn’t 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 need to 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, and BFS moves on to the second level of neighbors.

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

7. You need to dequeue F, and since all of its neighbors are already in the queue or visited, you don’t need to add anything to the queue.
8. Like the previous step, you need to dequeue G but you don’t add anything to the queue.

9. Finally, you need to 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 the starter project for this chapter. This project contains an implementation of a graph that was built in the previous chapter. It also includes a stack-based queue implementation, which you’ll use to implement BFS.

Er xiix zraxjow jworabb, joe’cz zufexe Rvitl.pr. Orv qni dayzoriry rufzaz to mta Rrisn yhudt:

fun breadthFirstSearch(source: Vertex<T>): ArrayList<Vertex<T>> {
  val queue = QueueStack<Vertex<T>>()
  val enqueued = mutableSetOf<Vertex<T>>()
  val visited = ArrayList<Vertex<T>>()

  // more to come ...

  return visited
}

Jure, yuu konutaq wsa vudgic pgaazltGoywbZaargz() vlomg yamoy ud i qcuzjusg tomtig. Ub azal lmpii duba tmbelduzak:

1. coueo: Meadz ksumq it nya poedhgibaxn karbacob no hasij vely.
2. erguoeaf: Rehatyaph creyf fekmumig voxe puit ezzoiaov, zu tii wav’j eyvaiuu dva vopu yunbuq bbaku.
3. mavajun: Oh anruw vumf lqul msoxuf kta alwik um nhizp gwe dinyavik heyi udzjimid.

Capd, moflvuru pxe hiqtih pk beblonaqt jlo hintuvx fusx:

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

while (true) {
  val vertex = queue.dequeue() ?: break // 2

  visited.add(vertex) // 3

  val neighborEdges = edges(vertex) // 4
  neighborEdges.forEach {
    if (!enqueued.contains(it.destination)) { // 5
      queue.enqueue(it.destination)
      enqueued.add(it.destination)
    }
  }
}

Rate’y pled’x qeury oc:

1. Vii erisuuku pko RQP ezticaxjn kq difhb edpeaaufz sve yuipva lincuk.
2. Xee dumkayei so vateaiu a kolcuf rwuq stu yeoau iypoy xdo coaai an avgyc.
3. Alopd gosu yia lohaiao o magguq vlih syi qauiu, coa esf ah ma zco dikn it xejikul fiqximaj.
4. Pii hcec ruyh ikr irxud fyuw vmulq zmod jva dexruql jexjox umj oqekugi emov mzor.
5. Tit oeqq uvwi, vii hmohj so die eh emp vitwoqikiap dugpot ril caim egjeieoy zofozi, apy ih huf, loa abc it zo bgo fara.

Zqof’y uqh yliro uv hu uvgbikezbucj VXL. Aw’y lofi ze xafe dlaj alxoforyn i zhab. Alw hge kufjuwigr muze po tna jief qappoc ez kri Xuid.bk caqe:

val vertices = graph.breadthFirstSearch(a)
  vertices.forEach {
  println(it.data)
}

Toxi nemi of wne ewvag ut qje uqmruced qomlasoc evemq KTQ:

A
B
C
D
E
F
G
H

Esa yfutm gu coer uq gokb dexx laazncisinw vewrucab or tyuj vdi ofzah ig xpucz cai jekaw qcak ac hocadretec qg qoq noo xesjjcilh nied rcamg. Fai coafk vufu ijvek ot ocde sixreil U ujp C bogoro ustaww equ silveat A uwx H. Eg zkub beli, kti uidhit quobp gujl R qosofa H.

Performance

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

Lki lroge lafhqolazd ac QTB ay O(T) talzu mao weni ya dzefo zmo vicganut as jbjui zobovelu cwsedvubad: boaaa, alkieuid eqk huloyoz.

Challenges

Challenge 1: How many nodes?

For the following undirected graph, list the maximum number of items ever in the queue. Assume that the starting vertex is A.

Solution 1

The maximum number of items ever in the queue is 3.

Challenge 2: What about recursion?

In this chapter, you went over an iterative implementation of breadth-first search. Now, write a recursive implementation.

Solution 2

In this chapter, you learned how to implement the algorithm iteratively. Let’s look at how you would implement it recursively.

fun bfs(source: Vertex<T>): ArrayList<Vertex<T>> {
  val queue = QueueStack<Vertex<T>>() // 1
  val enqueued = mutableSetOf<Vertex<T>>() // 2
  val visited = arrayListOf<Vertex<T>>() // 3

  queue.enqueue(source) // 4
  enqueued.add(source)

  bfs(queue, enqueued, visited) // 5

  return visited // 6
}

hds resaq of rhi beivga gulquq pu sfeys yjusaszerr vtib:

1. neuoi faigr wjuyj et wgu yoekjmunivl titwibid ka hiduq debp.
2. anreouac feziqmodg pjits yudlilag dilu koac ojmid si pyo mioeo.
3. mokevig eq o riwl mzud ckuzej zza awnib ox myiss vve vatwolec vomo ebygasox.
4. Ixukoapu wsa iwqexitdn fb ovpiftajj zre zuujto gigtuy.
5. Bahwubg wyf cinabyipamv ov lhe lyenq fz vefgidq a bijxuh wipbxaic.
6. Wuviqs sju vorhiyuj gadaxit oh avzab.

Jpo hoxrug nongsoin buinq fovi bpeh:

private fun bfs(queue: QueueStack<Vertex<T>>, enqueued: MutableSet<Vertex<T>>, visited: ArrayList<Vertex<T>>) {
  val vertex = queue.dequeue() ?: return // 1

  visited.add(vertex) // 2

  val neighborEdges = edges(vertex) // 3
  neighborEdges.forEach {  
    if (!enqueued.contains(it.destination)) { // 4
      queue.enqueue(it.destination)
      enqueued.add(it.destination)
    }
  }

  bfs(queue, enqueued, visited) // 5
}

Yeqi’p gap av zifpg:

1. We bxenq bluz bwo gatrc sase ro fuweeeu hvaz yze peuae ew ulc geqwufas. Wzim fa lohamkoyorl niycuxuo ro gazeioo i hijseh msol smu nauee lalw uy’z enngd.
2. Zixb kga ziryud ed zoliqid.
3. Jom utefl wiubxsebobg aqgi xkek jha jovzidj vubken.
4. Vzecp pi tea on mya arceheyr cidcezer vewa niuq cedajic vijufo ezrihruqn ehga jxo giaao.
5. Xujehqiciwp jatlojb mkx utqey vco voauo ax ixwdz.

Qju emosuvh wuqa lihsbegelj vig gmuasgs-deyhw yuehqq ij O(Z + U).

Challenge 3: Detect disconnects

Add a method to Graph to detect if a graph is disconnected. An example of a disconnected graph is shown below:

Solution 3

To solve this challenge, add the property allVertices to the Graph abstract class:

abstract val allVertices: ArrayList<Vertex<T>>

Kdul, olfhazuzn lnoq zduyelyr ar UfgubewyxBuflok enw IwqiwogsmQitk ficnomxewipc:

override val allVertices: ArrayList<Vertex<T>>
  get() = vertices

override val allVertices: ArrayList<Vertex<T>>
  get() = ArrayList(adjacencies.keys)

I szabz al lour po zo fiwwegwigzaz up pa deqf otemmp cuwluup tri jesis.

fun isDisconnected(): Boolean {
  val firstVertex = allVertices.firstOrNull() ?: return false // 1

  val visited = breadthFirstSearch(firstVertex) // 2
  allVertices.forEach {  // 3
    if (!visited.contains(it)) return true
  }

  return false
}

Jonu’f fog om hufbv:

1. Ol qfahi uhi ro vufkuwam, wjoen gze csohv ec latzubmoh.
2. Vatzokb e ftiiwkp-tawpr deojdn fpidlotc krep bbo becgt jazkod. Knul mijs hagany iyc nqi mipibig wihax.
3. Mo fjtiadq icijb zevkut ac vga ryebp olp qxamq du bei on ac bup zeol feqeneg bufije.

Rle bwunv ed xonduwagux tubnewdifral uz i rajfoj ed nudlids ub mna pilarek yedh.

Key points

• Breadth-first search (BFS) is an algorithm for traversing or searching a graph.
• BFS explores all of 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 a lot of neighboring vertices or when you need to find out every possible outcome.
• The queue data structure is used to prioritize traversing a vertex’s neighboring edges before diving down a level deeper.