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Data Structures in JavaScript: Depth-First Search Graph Traversal

February 05, 2021

jarednielsen data structure graph bfs

At some point in your career (today?!) you will want to learn data structures. It’s not just to ace the technical interview and land your dream job. Learning data structures will help you understand how software works and improve your problem-solving skills. In this tutorial, you will learn depth-first search graph traversal in JavaScript.

If you’re just joining us, you may want to start with Learn JavaScript Graph Data Structure.

Retrieval Practice

Retrieval practice is the surest way to solidify any new learning. Attempt to answer the following questions before proceeding:

  • What is a Graph?

  • What problem(s) does a Graph solve?

  • What problem(s) do data structures solve?

What is a Graph?

A graph consists of a set of nodes, or vertices, connected by edges. An edge consists of a pair of vertices. For example, if we establish a pair between two vertices, A and B, we refer t0 this related pairing as an edge. Because they are connected by an edge, A and B are adjacent.

What Problem(s) Does a Graph Solve?

  • Optimization: We can use the graph data structure in conjunction with an optimization algorithm for determining an optimal path, such as GPS

  • Network topology: We can use the graph data structure when modeling network topology, such as the internet or your friends on Facebook!

What Problem(s) Do Data Structures Solve?

According to Wikipedia:

Different types of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks. Data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Usually, efficient data structures are key to designing efficient algorithms.

Let’s Get Meta

  • What is Depth-First Search?

  • What is the difference between Breadth-First Search and Depth-First Search?

  • What problem(s) does Depth-First Search solve?

Graph Traversal: BFS vs. DFS

There are two algorithms for graph traversal:

  • breadth-first search (BFS)

  • depth-first search (DFS)

How do we search a graph?

Unlike a binary tree, we don’t necessarily have a root and we definitely don’t have a predetermined structure of branches. What do we have?

  • vertices

  • edges

What do we know about these things?

  • vertices are connected to each other with edges

  • vertices can be connected to any number of adjacent vertices (including zero!)

Let’s draw a simple graph so we can visualize this:

jarednielsen data structure graph bfs a g

If we can’t start at a root, where do we begin?

Anywhere!

We just pick a vertex and start searching.

For the sake of simplicity, let’s choose vertex A as our starting point, or root, and G as our goal.

As we can see, A is connected to vertices B, C, and D.

Now we need to make a decision.

Do we first search the vertices connected to A? Or do we choose one of the vertices connected to A and then search the vertices connected to it?

🤔

This is the difference between BFS and DFS.

With Breadth-First Search, we search all of the edges connected to a vertex before moving on to search the edges of the connected vertices.

With Depth-First Search, we follow the paths of the edges connected to our starting vertex, or search key, one at a time, until we reach the end, then we backtrack and search the alternate paths, until we find the vertex we are looking for or we arrive back where we started.

Depth-First Search (DFS) in JavaScript

Let’s declare our Graph data structure.

class Graph {
  constructor() {
      this.vertices = [];
      this.adjacent = {};
      this.edges = 0;
  }

  addVertex(v) {
      this.vertices.push(v);
      this.adjacent[v] = [];
  }

  addEdge(v, w) {
      this.adjacent[v].push(w);
      this.adjacent[w].push(v);
      this.edges++;
  }
}

Next, let’s initialize a new Graph and add vertices and edges.

const g = new Graph();

g.addVertex("A");
g.addVertex("B");
g.addVertex("C");
g.addVertex("D");
g.addVertex("E");
g.addVertex("F");
g.addVertex("G");

g.addEdge("A","B");
g.addEdge("A","C");
g.addEdge("A","D");
g.addEdge("B","C");
g.addEdge("B","D");
g.addEdge("C","D");
g.addEdge("C","E");
g.addEdge("D","F");
g.addEdge("F","G");

Let’s restate the goal of our dfs method:

Given a graph, a starting vertex, and a goal, start at the root and search path by path

To our Graph class, let’s add a dfs method. We need to declare two parameters, goal and v.

    dfs(goal, v) {

        return false;
    }

📝 We will return false for the time being because we haven’t found our goal yet.

We can take our declaration one step further using JavaScript’s default parameters.

    dfs(goal, v = this.vertices[0]) {

        return false;
    }

📝 You will see many different implementations of DFS. Some do not specify a root while others do not specify a goal. The goal here (no pun intended) is to demonstrate an approach that covers DFS variations in-depth (pun intended).

Now what?

Let’s outline a rough approach in pseudocode:

  • Check the root.

  • If the root is equal to the goal, return true.

  • If the root is not equal to our goal, check the first vertex adjacent to our root.

  • If it is equal to our goal, return true.

  • If the first adjacent vertex is not equal to our goal, check the first adjacent vertex to our first adjacent vertex…

✋ Hey!

What is this starting to look like?

🤔

Recursion!

What do we know about recursion?

Like proof by induction, we need to establish two things:

  • base case

  • recursive, or iterative, case

So what is our base case?

Our base case is whether or not we discovered our goal.

    dfs(goal, v = this.vertices[0]) {
        const discovered = [];
        discovered[v] = true;

        return discovered[goal] || false;
    }

We declare an array, discovered and create a key with the value of v. We short circuit the return, if goal is in discovered, and return true, otherwise we return false.

If we call our dfs with an argument of “G”, g.dfs("G"), it will return false.

But if we call our dfs with an argument of “A”, g.dfs("A"), it will return true.

Base case established. Now what?

We need to establish the recursive, or iterative, case. Let’s make things easier on ourselves, and let this.adjacent equal adj.

    dfs(goal, v = this.vertices[0]) {
        let adj = this.adjacent;

        const discovered = [];
        discovered[v] = true;

        return discovered[goal] || false;
    }

Now we need to iterate over adj, our array of vertices. For each vertex adjacent to v, we want to check if it discovered. If it’s not discovered, we make our recursive call.

    dfs(goal, v = this.vertices[0]) {
        let adj = this.adjacent;

        const discovered = [];
        discovered[v] = true;

        for (let i = 0; i < adj[v].length; i++){
            let w = adj[v][i];

            if (!discovered[w]) {
                this.dfs(goal, w);
            }

        }

        return discovered[goal] || false;
    }

What happens when we run our dfs?

RangeError: Maximum call stack size exceeded

🤔

WTF?

Where’s the problem?

With each recursive call of dfs, we are declaring a new discovered array. Each time we enter our for loop, we “discover” that w is not yet discovered and we make another recursive call to dfs.

What’s the solution?

Hint: It’s dynamic!

We need to pass our discovered array as an argument to our recursive dfs calls.

    dfs(goal, v = this.vertices[0]) {
        let adj = this.adjacent;

        const discovered = [];
        discovered[v] = true;

        for (let i = 0; i < adj[v].length; i++){
            let w = adj[v][i];

            if (!discovered[w]) {
                this.dfs(goal, w, discovered);
            }
        }

        return discovered[goal] || false;
    }

But this won’t work. Why?

Because discovered is not a parameter of our dfs declaration. Let’s fix that.

    dfs(goal, v = this.vertices[0], discovered = []) {
        let adj = this.adjacent;

        discovered[v] = true;

        for (let i = 0; i < adj[v].length; i++){
            let w = adj[v][i];

            if (!discovered[w]) {
                this.dfs(goal, w, discovered);
            }
        }

        return discovered[goal] || false;
    }

This is memoization, one of two approaches referred to as dyanmic programming . With memoization, we make a “memo” of the values returned by the preceding recursive calls. Speaking of memos…

📝 We removed the const declaration of discovered.

Now, if we call our dfs with an argument of “G”, g.dfs("G"), it will return true.

And if we call our dfs with an argument of “H”, g.dfs("H"), it will return false.

Here’s our complete Graph class:

class Graph {
    constructor() {
        this.vertices = [];
        this.adjacent = {};
        this.edges = 0;
    }

    addVertex(v) {
        this.vertices.push(v);
        this.adjacent[v] = [];
    }

    addEdge(v, w) {
        this.adjacent[v].push(w);
        this.adjacent[w].push(v);
        this.edges++;
    }


    bfs(goal, root = this.vertices[0]) {
        let adj = this.adjacent;

        const queue = [];
        queue.push(root);

        const discovered = [];
        discovered[root] = true;

        const edges = [];
        edges[root] = 0;

        const predecessors = [];
        predecessors[root] = null;

        const buildPath = (goal, root, predecessors) => {
            const stack = [];
            stack.push(goal);

            let u = predecessors[goal];

            while(u != root) {
                stack.push(u);
                u = predecessors[u];
            }

            stack.push(root);

            let path = stack.reverse().join('-');

            return path;
        }
    

        while(queue.length) {
            let v = queue.shift();

            if (v === goal) {
                return { 
                    distance: edges[goal],
                    path: buildPath(goal, root, predecessors)
                };
            }

            for (let i = 0; i < adj[v].length; i++) {
                if (!discovered[adj[v][i]]) {
                    discovered[adj[v][i]] = true;
                    queue.push(adj[v][i]);
                    edges[adj[v][i]] = edges[v] + 1;
                    predecessors[adj[v][i]] = v;
                }
            }
        }

        return false;
    }

    dfs(goal, v = this.vertices[0], discovered = []) {
        let adj = this.adjacent;

        discovered[v] = true;

        for (let i = 0; i < adj[v].length; i++){
            let w = adj[v][i];

            if (!discovered[w]) {
                this.dfs(goal, w, discovered);
            }
        }

        return discovered[goal] || false;
    }
}

That’s it!

That’s Depth-First Search!`

“What good is this inefficient algorithm?”

I’m glad you asked.

In the next tutorial, we’ll modify our dfs method to perform topological sorting! Stay tuned!

Reflection

  • What is Depth-First Search?

  • What is the difference between Breadth-First Search and Depth-First Search?

  • What problem(s) does Depth-First Search solve?

What is Depth-First Search?

Breadth-First Search is an algorithm that searches a graph for a specific goal by checking all of the vertices connected on a path before moving on to check the adjacent vertices.

What is the Difference Between Breadth-First Search and Depth-First Search?

Breadth-First Search checks all of the vertices adjacent to a given vertex before checking the vertices adjacent to those vertices. Depth-First Search, on the other hand, checks all of the vertices on a path and then backtracks.

What Problem(s) Does Depth-First Search Solve?

There are a number of specific use cases, such as finding the bridges in a graph, finding the solution to a maze, or defining the topology of a graph.

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