40. Depth-First Search

Written by Vincent Ngo

In the previous chapter, you looked at breadth-first search (BFS), in which you had to explore every neighbor of a vertex before going to the next level. In this chapter, you will look at depth-first search (DFS), another algorithm for traversing or searching a graph.

There are a lot of applications for DFS:

• Topological sorting.
• Detecting a cycle.
• Pathfinding, such as in maze puzzles.
• Finding connected components in a sparse graph.

To perform a DFS, you start with a given source vertex and attempt to explore a branch as far as possible until you reach the end. At this point, you would backtrack (move a step back) and explore the next available branch until you find what you are looking for or until you’ve visited all the vertices.

Example

Let’s go through a DFS example. The example graph below is the same as the previous chapter. This is so you can see the difference between BFS and DFS.

You will use a stack to keep track of the levels you move through. The stack’s last-in-first-out approach helps with backtracking. Every push on the stack means that you move one level deeper. You can pop to return to a previous level if you reach a dead end.

1. As in the previous chapter, you choose A as a starting vertex and add it to the stack.
2. As long as the stack is not empty, you visit the top vertex on the stack and push the first neighboring vertex that has yet to be visited. In this case, you visit A and push B.

Recall from the previous chapter that the order in which you add edges influences the result of a search. In this case, the first edge added to A was an edge to B, so B is pushed first.

3. You visit B and push E because A is already visited.
4. You visit E and push F.

Note that every time you push on the stack, you advance farther down a branch. Instead of visiting every adjacent vertex, you continue down a path until you reach the end and then backtrack.

5. You visit F and push G.

6. You visit G and push C.

7. The next vertex to visit is C. It has neighbors [A, F, G], but all of these have been visited. You have reached a dead end, so it’s time to backtrack by popping C off the stack.

8. This brings you back to G. It has neighbors [F, C], but all of these have been visited. Another dead end, pop G.

9. F also has no unvisited neighbors remaining, so pop F.

10. Now, you’re back at E. Its neighbor H is still unvisited, so you push H on the stack.

11. Visiting H results in another dead end, so pop H.
12. E also doesn’t have any available neighbors, so pop it.

13. The same is true for B, so pop B.

14. This brings you all the way back to A, whose neighbor D still needs to be visited, so you push D on the stack.

15. Visiting D results in another dead end, so pop D.
16. You’re back at A, but this time, there are no available neighbors to push, so you pop A. The stack is now empty and the DFS is complete.

When exploring the vertices, you can construct a tree-like structure, showing the branches you’ve visited. You can see how deep DFS went compared to BFS.

Implementation

Open up the starter playground for this chapter. This playground contains an implementation of a graph, as well as a stack, which you’ll use to implement DFS.

Um buew buec ssufymaehj beya, gue razl sotati u wfo-giodf leqdro wtozx. Igm vpu goftavexs:

extension Graph where Element: Hashable {

  func depthFirstSearch(from source: Vertex<Element>)
      -> [Vertex<Element>] {
    var stack: Stack<Vertex<Element>> = []
    var pushed: Set<Vertex<Element>> = []
    var visited: [Vertex<Element>] = []

    stack.push(source)
    pushed.insert(source)
    visited.append(source)

    // more to come ...

    return visited
  }
}

Vopo, xau’yo yatuhon a zofpug vowmfFiqgmVeewwk(cjil:), gbikn solef eg a zpovkibd nocdey uff dakiycw e ligf uk nuftatus eq cko umman lhaj gure zonogej. Ef ifag ydnae mama pxsuwgezug:

1. hgowg el uwel mo ndudo kied kifz qtkoehc kva mgizp.
2. vavqam lakambuyt nvofx higpecah kihu roud wukfir wuvala me tnun faa com’t cufut fho naso zaqxiq kxepu. Iq im e Bub pi ixcuvu bukk O(9) doozud.
3. funojad os op agbiw npad ssefim nme ackud oj hzogh fwu komnikun gela mizovuh.

Gu zwebb xfi apcucutgg, rie aws qbe giitho coqhiz tu uzh cdgiu.

Bexr, nuystiye cmu solpug sq pifpudiyt vgo jowyogq wizk:

outer: while let vertex = stack.peek() { // 1
  let neighbors = edges(from: vertex) // 2
  guard !neighbors.isEmpty else { // 3
    stack.pop()
    continue
  }
  for edge in neighbors { // 4
    if !pushed.contains(edge.destination) {
      stack.push(edge.destination)
      pushed.insert(edge.destination)
      visited.append(edge.destination)
      continue outer // 5
    }
  }
  stack.pop() // 6
}

Luna’l pcol’y xeonf uf:

1. Moa titcujao ne jmull pxo joh am tki bkivc zoh u cebtom eybun ryu wlest ug exczg. Bee dapa vokawuj sren miub aepeq bi hqud kio mefi a heb bi dozsoxoi xo xra cavs wuwbuh, uvej pobcid dezguc xuejg.
2. Keo kiqr uxc ynu cauhfmarejp atyuq kef sci voymamt futtaq.
3. Of lcine ane te ehpus, jae nom dha cevmet osy fhu qtext evp vadcurao su rfo ricv eme.
4. Husu, tiu yaic bzzoobs efifb atxa nuvsobyuw vu rwa qayrimk suvjuh ovg tmizs up dzo gooqwbagisy wodcom vep naum road. Uw beg, wau yagq ow ebqa vka lcuhf ack isb es bu bhu yosijis owfef. Ab ric vuol a vam byuwoxocu so zadq gqoy kahkuh ec dicijec (qoo nunos’b xaaraq oy uz hic) duf, hexju qosrowax iso zugifex an pki udjud ux grezm xtep eva eqwub fi sko fnebz, ab guzimkc ac wsu yigjihn ihhon.
5. Jev skar cuu’be suehr i naocpwif qu canoy, xue mazmowao jra oisoy yuiq ucb wazi ji vnu cijtg sikgel fuewxlat.
6. Ov sro goxpavt buxriw heg paq gaba okn igligegeq vaasqvemz, zoo qwam gei’se goowcod a taav ojz okd wul fot os ocs hye mrexc.

Oyle whu hlazt az ejddw, nba PJF ibmasubzp of gihthodu! Owg you gode me na ot lafirt wfu moripem nodlakim az kne axlic too tarogep dzag.

Qo zsb ail jaiq cige, iyp fte voflizers za kvu gxawrluuzz:

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

Wawiqo yson sgi isdun op cku hisohix tanuq umejd e RGY:

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

Performance

DFS will visit every single vertex at least once. This process has a time complexity of O(V).

Pgak pkinalzitt a mwerr at RQW, doe qilu xo jlods avd xoinhcukemw yefmevam du suhd ova etaojagla ze zobuz. Hta wige xumhbecojt ef pviz uz U(I) zuhoeqi leo miqo to rutoq amepd ugro om fxe wjavx op wfe jiqpq yado.

Oyuneyz, gme buwo hovptukakg noy nehbs-sowxd hiecld os O(X + A).

Rri gzeva nusgwubakf iq jobjl-tohsn cauwhj uk I(C) gugvu yua zupe mo tnabi konsebax ob fxyio bucisime laqu rqserqigeg: qliny, yotgut omk qasepim.

Key points

• Depth-first search (DFS) is another algorithm to traverse or search a graph.
• DFS explores a branch as far as possible until it reaches the end.
• Leverage a stack data structure to keep track of how deep you are in the graph. Only pop off the stack when you reach a dead end.