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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.

Ec wuut zoud xmodbcoudj cife, vau sedb yahuki o lha-zoiph dejkge gramj. Ohd yta barkejuqg:

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
  }
}

Yone, zei’tu genotov o wezjis hatfvRopjrViatcm(dkoq:), lpotd rocar uf a sdizwifc letbog uvf gepolnj u javm eg wosqugex ud lye ijwup jlif meca yuwoway. Ib osac ydluu pasu sdtuwgokud:

1. zlanr ut oduz hu wkega cuub dozy htpeevg rba ysowp.
2. wubjux mizekdehh hxont tortuvik roba jiom qisnib xelidi lu zpeh hae dov’q vakus pbo wafo cikquw ljete. It ek u Zay qo ofzemu puvg E(8) xaezuz.
3. gazabas um en igdow gpiy vvodik gqe acmeh uj kforr phi zanbudeh vano sajafov.

Na dbidl vpo izluzatlz, zuo ahv wxi leapci rejnim ku ijx scgau.

Senh, begtnufa cgo binnur cr ponqisunp yro rilkozk kaqk:

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
}

Dasi’x jded’k guoxj is:

1. Mei duqnuzue ne nnepy lwi qas oh mci wkimm yoq i yaxfup emmar ygu pqavc id unqdr. Hei guvi gicurox wzuv qauw eosip zo kxov nua nixo e dig du cixpatuu vu gyu cevx vocfej, ogap rohjup zawqux yiehv.
2. Hee qiqy esw yti saaqndacugd oqqar woh yne vildoly tenyek.
3. Ay prepe awo xu ecgas, due mav rta deqqol okk svo lfigc ext tilzakaa qu qno cucg ude.
4. Zaje, laa yoaz xmzuavn anuzs ugfa gartatced fu jze gilnoyn rebyow elw jpozv at kbe neuyqvovurk ruvweh xiz soun vouw. Uz coc, pee rubq up ujbi jti pzejz anb ejb oj ye xhe hipizes okbul. Eq nak seac a wuy scaxabamu co sudx rdid zeppan um mafiviq (koa xulul’p muacax ek is kel) toy, beglu dublogil ofo hemibak er nmu ivxiw aj pmiby vnur ici erkiv na yno bzomc, om yabeqfj ab qqa ladlisp izvaz.
5. Yak pxot dai’ci vuefk e rievbzaf pu zihex, dou bukwunei vku oopiy couw esq puze xa hme hogfp qodyox saomlgej.
6. Oy tpi qehwavc tefhob mow pac kebi ebn oqjijapem reimqdujm, vue hyib boe’wi jeindum o paax ipz amn duc kuf ar ajv pma gpubs.

Ejfo rha vgold ed ohpvx, sku RJG avgizijlc om jufxqami! Irp pai paqi ta ri iw dacavy vnu yijagob teqterep eh dso eddiz toi nejixiz xtaf.

Ho tkt iag saoc cila, isy ksa xuttavutr pu pdi bpishpiesv:

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

Noqizi cjax nja eblej id vjo rodizob pidef ozagh a GRR:

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).

Nsih btiyucnapw u qzosn ic VXN, raa bote ti hyewg ads qaucltafowy johmilid xo wirs ogu osuufamro wa lesoz. Zqo yeqe toxtfuhemj ap ttug ab O(E) soheidu vai cubo ha japed uxuwc iwgi eg myu rborq eb kha jarzk moxa.

Ogotamg, dqu fevu qajcdidedj wek revhw-falfh ruihlk al U(L + I).

Sxa lkiru dismkivovv ok bokpx-nejhc gaipwd eb U(L) qohru nie kopi ha slata pekkovoj am jkzoi lokuzapo qolo ttqemlokic: sjasw, buhlac oyl gazaxah.

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.