#57 Graph traversal algorithms BFS and DFS
Intro
Graphs for beginners seem really hard, but the truth is that they are not so difficult when you learn the basics and you really understand the main mechanisms. Let’s start with some basic knowledge.
What is a graph?

Consists of nodes which are connected by edges

Each edge can have a direction (directed graph) or not (undirected graph)

Additionally, graph can be weighted or not. Weighted means that edge has some value. It can for example represent the distance between two nodes.

Example of the graph you can see here (Undirected Cyclic Graph):
Graph representation
Typically we represent a graph by adjacency list (it is the most popular option). Every node has a list pointing to other nodes to which it has a connection. For an implementation point of view we can use for that a dictionary.
For example, we can construct an adjacency list like this:
var adjList: [Int: [Int]] = [:]
adjList[1] = [2, 3]
adjList[2] = [1, 4]
adjList[3] = [1, 4]
adjList[2] = [5]
The second option to represent a graph is a matrix:
0  1  1  0  0 

1  0  0  0  1 
1  0  0  1  0 
0  1  1  1  0 
0  1  0  0  0 
This matrix shows that element node 1 (columns 1) is connected with node 2 and 3 (so that’s why row 2 and row 4 has ones in column 1).
Then node 2 (column 2) is connected with node 1, 4, 5 (that’s why in row 1, 4 and 5 there is “1” in column 2), etc.
To be honest I have never used matrix representation in any graph related problem, but for sure it is good to know that this kind of representation exists.
Very popular graph terms
 Cycle: if within a graph we travel and meet the visited node once again: it means that we have a cycle in the graph. Finding a path of a cycle can be complicated and we can use an algorithm to do that (BFS or DFS  more about them in further parts of the article).
 Connected component (cc): we can imagine a connected component as a lonely island which is not connected to other lands. There exist algorithms for finding connected components in the graph. I hope in the future I will write more about it in a separate post.
The most popular graph traversal algorithm
Let’s focus on the main subject of this post: Graph traversal algorithms. We can distinguish two:
 DFS (Depth First Search)
 BFS (Breadth First Search)
It is really important to understand how they work and when they are most useful (in which use case). Below I tried to highlight the main features of both. I presented it in the table for a better understanding of the difference between them.
DFS  BFS  

Main usability  Is commonly used for visiting (traversing) all nodes in the graph (or to be more concrete: in connected component )  Is used when we want to find the shortest path in the unweighted graph 
Implementation  Using recursion (iterative as well but definitely less popular and used)  Iterative 
Data structure used for iterative implementation  Stack  Queue 
Visiting nodes order  Branch by branch (so it goes by depth as much as possible, then it switches to the next branch)  It visits the nodes level by level. First, it starts with the nearest neighborhood of starting node and then it goes to the next level etc. 
Cycle detection  Commonly used for cycle detection  Not very efficient for finding a cycle in the graph. 
Complexity  O(V+ E)  O(V+ E) 
Practice   https://leetcode.com/problems/shortestpathinagridwithobstacleselimination/description/  https://leetcode.com/problems/minimumgeneticmutation/solution/ https://leetcode.com/problems/nearestexitfromentranceinmaze/ 
Visualization
DFS
As we said, the DFS is going branch by branch, is going deep as much as possible than it turns back and starts investigating a new branch. Having this in mind let’s look at what could be the order of visiting nodes if we started from the node with an id equal to 3:
Algorithm starts with the selection of some first branch and goes to the end (end means that the algorithm encounters a node with no other children, or all the children are already visited):
Node 5 has no other children so we come back to the place when it can check another branch (in this case it will be node 3)
And then algorithm starts to vist the new unexplored branch, so it goes to node one and then it finishes the job.
BFS
The visualization shows that we go level by level and I think this is the best way how we can imaging the algorithm .
If we start with node id = 3 (first layer), then we visit 1 and 4 (second layer) then 2 (third layer) and last will be 5 (fourth layer).
Implementation
Having this knowledge let’s implement DFS and BFS in Swift. Additionally let’s take a look at some additional topics which are very interesting as well:
 Finding the shortest path with unweighted graph
 Finding a cycle in an undirected graph
 Finding a cycle in a directed graph
DFS
// This class will be used later in every algorithm
class Node {
let id: Int
var children: [Node]
init(id: Int, children: [Node] = []) {
self.id = id
self.children = children
}
}
var visited: Set<Int> = []
func dfs(_ root: Node?) {
guard let root = root else {
return
}
visited.insert(root.id)
for child in root.children {
if !visited.contains(child.id) {
dfs(child)
}
}
}
Download playground from here.
BFS
func bfs(_ start: Node) {
var queue: [Node] = []
var visited: Set<Int> = []
visited.insert(start.id)
queue.append(start)
while !queue.isEmpty {
for item in queue {
let first = queue.removeFirst()
for child in first.children {
if !visited.contains(child.id) {
visited.insert(child.id)
queue.append(child)
}
}
}
}
}
Download playground from here.
BFS + shortest path
Implementation with additional finding distance from the start node and printing a path (an important note here is that here we are discussing unweighted graphs. If we want to find the shortest distance for a weighted graph, we need to consider another algorithm like Dijktstra’s algorithm).
var parent: [Int: Int] = [:]
func bfs(_ start: Node) {
var queue: [Node] = []
var visited: Set<Int> = []
var distance: [Int: Int] = [:]
visited.insert(start.id)
queue.append(start)
while !queue.isEmpty {
for item in queue {
let first = queue.removeFirst()
for child in first.children {
if !visited.contains(child.id) {
distance[child.id] = distance[item.id, default: 0] + 1
parent[child.id, default: 0] = item.id
visited.insert(child.id)
queue.append(child)
}
}
}
}
}
func getPath(_ target: Node, _ start: Node) {
var x = target.id
var path: [Int] = []
path.append(target.id)
while x != start.id {
x = parent[x, default: 0]
path.append(x)
}
path.reverse()
}
Download playground from here.
Detecting cycle in an undirected graph
var visited: Set<Int> = []
var parent: [Int: Int] = [:]
func dfs(_ root: Node?, _ previous: Node?) {
guard let root = root else {
return
}
visited.insert(root.id)
let children = root.children
for child in children {
if !visited.contains(child.id) {
parent[child.id] = root.id
dfs(child, root)
} else if child.id != previous?.id {
// here we can be sure that we don't have a cycle which consists of only two elements
// and then we can print stack starting from child
getPath(root, child)
return
}
}
}
Download playground from here.
Detecting cycle in a directed graph
Undirected graphs are much easier to handle, because we don’t care about direction of each edge. In this section I would like to show you how to detect a cycle in a directed graph and as well how to find a path for the detected cycle.
There are two options to find a cycle:
 DFS with node coloring
enum NodeState {
case unvisited
case visiting
case visited
}
var parent: [Int: Int?] = [:]
var visited: [Int: NodeState] = [:]
func dfs(_ root: Node?) {
guard let root = root else {
return
}
visited[root.id] = .visiting
let children = root.children
for child in children {
parent[child.id] = root.id
if visited[child.id] == .unvisited {
dfs(child)
} else if visited[child.id] == .visiting {
findCyclePath(root, child)
return
}
}
visited[root.id] = .visited
}
private func findCyclePath(_ start: Node?, _ end: Node) {
var path: [Int?] = []
var parentElement: Int? = start?.id
path.append(end.id)
while parentElement != nil && parentElement != end.id {
path.append(parentElement)
parentElement = parent[parentElement!, default: 0]
}
}
Download playground from here.
 Alternate way for finding cycle with DFS is the the usage of two arrays:
var isVisited: [Bool]
var inStack: [Bool]
 used to track the items on which we go deep and then if we visited all branches then we need to clear it or it say that we found a cycle
var visited: Set<Int> = []
var inStack: Set<Int> = []
func dfs(_ root: Node?, _ stack: [Node]) {
guard let root = root else {
return
}
inStack.insert(root.id)
visited.insert(root.id)
let children = root.children
for child in children {
if !visited.contains(child.id) {
var newStack = stack
newStack.append(child)
dfs(child, newStack)
} else if inStack.contains(child.id) {
var path = stack
path.append(child)
print(path.map { $0.id })
}
}
inStack.remove(root.id)
}
Download playground from here.
Summary
Thank you for reading it! I hope it somehow was helpful to you. In terms of any questions please email me: bartlomiej.woronin@gmail.com ;)