LeetCode Meditations — Chapter 11: Graphs

Table of contents

• Introduction
• Representing graphs
  • Edge List
  • Adjacency Matrix
  • Adjacency List
• Traversals
  • Breadth-First Search (BFS)
  • Depth-First Search (DFS)
• Resources

---

A graph is probably the data structure that everyone is familiar with, regardless of their profession or interests.

Graph theory is a very broad topic, but we'll simply look at some of the main ingredients of what makes a graph and how to represent it, as well as basic graph traversals.

---

In a graph, there are two main components: vertices (or nodes) and edges that connect those vertices.

Note
Here, we're going to use "vertex" and "node" interchangeably. The terms "adjacent vertices" and "neighbors" are used interchangeably as well.

A graph can be directed or undirected. With a directed edge, we have an origin and a destination vertex. On the other hand, an undirected edge is bidirectional, origin and destination are not fixed.

Note
There might also be mixed graphs that have both directed and undirected edges.

A graph can also be weighted or unweighted, each edge can have different weights, usually representing the cost of going from one vertex to the other.

We can define a graph like this:

$$G = (V, \ E)$$

$V$ is a set of vertices, and $E$ is a set of edges.

For example, if we have a directed graph like this:

Then, we have the vertices:

$$V = \lbrace A, \ B, \ C, \ D \rbrace$$

And, the edges are:

$$E = \lbrace(A, \ B), \ (A, \ C), \ (C, \ B), \ (C, \ D)\, \ (D, \ C)\rbrace$$

If we have an undirected graph such as this one:

We have the same vertices:

$$V = \lbrace A, \ B, \ C, \ D \rbrace$$

But our edges can look like this:

$$E = \lbrace \lbrace B, A \rbrace, \lbrace A, C \rbrace, \lbrace C, B \rbrace, \lbrace D, C \rbrace \rbrace$$

Note
We use parentheses when it comes to directed edges but curly braces with undirected edges as there is no direction from one vertex to the other.

When two vertices share an edge, they are adjacent to each other. The degree of a vertex is the number of adjacent vertices to it. We can also define the degree as the number of edges coming out of the vertex.
In our example above, the vertex A has the degree of 2.

---

A simple path is the one that we don't repeat any vertices while traversing the graph.

An example might look like this:

A cycle is a simple path, except that we end up at the vertex we started with:

---

When it comes to representing graphs, there are several ways to do it, and we'll look at three of them: an edge list, an adjacency matrix, and an adjacency list.

Edge List

We can simply put all the edges in an array:

[ [A, B], [A, C], [B, C], [C, D] ]

However, to find an edge in an edge list, we'll have to iterate through them, so it will have $O(E)$ time complexity, where in the worst case, we'll search the whole list to find an edge. Similarly, it needs $O(E)$ amount of space to represent all the edges.

Adjacency Matrix

The adjacency matrix for our example might look like this:

$$\left\lceil \begin{matrix} & A & B & C & D \newline A & 0 & 1 & 1 & 0 \newline B & 1 & 0 & 1 & 0 \newline C & 1 & 1 & 0 & 1 \newline D & 0 & 0 & 1 & 0 \end{matrix} \right\rceil$$

Each row is for a vertex, and the matching column shows the relationship between those vertices. For example, the vertex A doesn't have an edge pointing to D, so the cell that matches A and D is 0. On the other hand, A is connected to B and C, so those cells have the value 1.

Note
If the graph is weighted, we can simply put the weight instead of 1, and when there is no edge, the value can stay 0.

Note
An adjacency matrix will have 0s in the "main diagonal" if there are no self-loops.

Let's try implementing it in TypeScript.

We'll start with a minimal graph vertex:

class GraphVertex {
  public value: string | number;

  constructor(value: string | number) {
    this.value = value;
  }
}

Now we can define our graph. We'll make it really simple with three properties to hold: matrix to represent the graph as an adjacency matrix, vertices to hold vertices, and isDirected to indicate whether our graph is directed:

class Graph {
  public matrix: number[][];
  public vertices: GraphVertex[];
  public isDirected: boolean;

  constructor(vertices: GraphVertex[], isDirected = true) {
    this.vertices = vertices;
    this.isDirected = isDirected;
    ...
  }

  ...
}

Initializing our adjacency matrix might look like this:

this.matrix = Array.from({ length: vertices.length }, () => {
  return Array.from({ length: vertices.length }, () => 0)
});

We'll have an array with the length of vertices, each item in the array is an array with the length of vertices as well, but filled with zeroes.

In our example with four vertices, the initial adjacency matrix looks like this:

[ [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0] ]

Then, adding an edge is just marking the corresponding value as 1, so that we can represent a connection between two vertices:

this.matrix[this.vertices.indexOf(v1)][this.vertices.indexOf(v2)] = 1;

Note
This implementation assumes that all vertices are distinct.

If we have an undirected graph, we can have it both ways:

if (!this.isDirected) {
  this.matrix[this.vertices.indexOf(v2)][this.vertices.indexOf(v1)] = 1;
}

Removing an edge, in this case, will be just resetting the value to 0:

this.matrix[this.vertices.indexOf(v1)][this.vertices.indexOf(v2)] = 0;

And, checking for the existence of an edge is simply checking whether the corresponding value is 0 or not:

this.matrix[this.vertices.indexOf(v1)][this.vertices.indexOf(v2)] !== 0;

And, here is the whole example:

class Graph {
  public matrix: number[][];
  public vertices: GraphVertex[];
  public isDirected: boolean;

  constructor(vertices: GraphVertex[], isDirected = true) {
    this.vertices = vertices;
    this.matrix = Array.from({ length: vertices.length }, () => {
      return Array.from({ length: vertices.length }, () => 0)
    });
    this.isDirected = isDirected;
  }

  addEdge(v1: GraphVertex, v2: GraphVertex) {
    this._checkVertexIsInGraph(v1);
    this._checkVertexIsInGraph(v2);

    this.matrix[this.vertices.indexOf(v1)][this.vertices.indexOf(v2)] = 1;

    if (!this.isDirected) {
      this.matrix[this.vertices.indexOf(v2)][this.vertices.indexOf(v1)] = 1;
    }
  }

  /* 
  For a weighted graph:

  addEdge(v1: GraphVertex, v2: GraphVertex, weight: number) {
    this._checkVertexIsInGraph(v1);
    this._checkVertexIsInGraph(v2);

    this.matrix[this.vertices.indexOf(v1)][this.vertices.indexOf(v2)] = weight;
    if (!this.isDirected) {
      this.matrix[this.vertices.indexOf(v2)][this.vertices.indexOf(v1)] = weight;
    }
  }
  */

  removeEdge(v1: GraphVertex, v2: GraphVertex) {
    this._checkVertexIsInGraph(v1);
    this._checkVertexIsInGraph(v2);

    this.matrix[this.vertices.indexOf(v1)][this.vertices.indexOf(v2)] = 0;

    if (!this.isDirected) {
      this.matrix[this.vertices.indexOf(v2)][this.vertices.indexOf(v1)] = 0;
    }
  }

  hasEdge(v1: GraphVertex, v2: GraphVertex) {
    this._checkVertexIsInGraph(v1);
    this._checkVertexIsInGraph(v2);

    return this.matrix[this.vertices.indexOf(v1)][this.vertices.indexOf(v2)] !== 0;
  }

  getAdjacencyMatrix() {
    return this.matrix;
  }

  _checkVertexIsInGraph(v: GraphVertex) {
    if (!this.vertices.includes(v)) {
      throw new Error('Vertex doesn\'t exist');
    }
  }
}


let a = new GraphVertex('A');
let b = new GraphVertex('B');
let c = new GraphVertex('C');
let d = new GraphVertex('D');

let graph = new Graph([a, b, c, d], false);

graph.addEdge(a, b);
graph.addEdge(a, c);
graph.addEdge(b, c);
graph.addEdge(c, d);

console.log(graph.getAdjacencyMatrix());
// -> [ [0, 1, 1, 0], [1, 0, 1, 0], [1, 1, 0, 1], [0, 0, 1, 0] ]

Operations on an adjacency matrix has $O(1)$ time complexity; however, our storage needs will be $O(V^2)$ where $V$ is the number of vertices.

Adjacency List

In an adjacency list, usually a hashmap or an array of linked lists is used. For example:

let graph = {
  'A': ['B', 'C'],
  'B': ['A', 'C'],
  'C': ['A', 'B', 'D'],
  'D': ['C']
}

Let's see how we can modify our code above to use an adjacency list instead.

Instead of having a matrix which is an array of arrays, we can have a Map that maps the vertices to an array of their neighbors.

We can initialize it as a map that has the vertices as keys, each of which has a value of an empty array for now:

this.list = new Map<GraphVertex, GraphVertex[]>();
for (const v of vertices) {
  this.list.set(v, []);
}

Adding an edge will be just pushing to the array of corresponding vertex:

this.list.get(v1)!.push(v2);

If our graph is undirected, we can do it both ways here as well:

if (!this.isDirected) {
  this.list.get(v2)!.push(v1);
}

Removing an edge will be deleting that vertex from the array:

this.list.set(v1, this.list.get(v1)!.filter(v => v !== v2));

Checking if an edge exists is just checking the existence of that vertex in the array:

this.list.get(v1)!.includes(v2);

Note
We're using a non-null assertion operator. As we'll see below, we're first checking if the vertex is in the graph. And since we're adding all the vertices in the graph as keys to this.list, we're sure that getting that vertex from the list is not undefined. However, TypeScript will warn us because if a key is not found in a Map object, it could potentially return undefined.

Here is our graph:

class Graph {
  public list: Map<GraphVertex, GraphVertex[]>;
  public vertices: GraphVertex[];
  public isDirected: boolean;

  constructor(vertices: GraphVertex[], isDirected = true) {
    this.vertices = vertices;
    this.list = new Map();
    for (const v of vertices) {
      this.list.set(v, []);
    }
    this.isDirected = isDirected;
  }

  addEdge(v1: GraphVertex, v2: GraphVertex) {
    this._checkVertexIsInGraph(v1);
    this._checkVertexIsInGraph(v2);

    this.list.get(v1)!.push(v2);

    if (!this.isDirected) {
      this.list.get(v2)!.push(v1);
    }
  }

  removeEdge(v1: GraphVertex, v2: GraphVertex) {
    this._checkVertexIsInGraph(v1);
    this._checkVertexIsInGraph(v2);

    this.list.set(v1, this.list.get(v1)!.filter(v => v !== v2));

    if (!this.isDirected) {
      this.list.set(v2, this.list.get(v2)!.filter(v => v !== v1));
    }
  }

  hasEdge(v1: GraphVertex, v2: GraphVertex) {
    this._checkVertexIsInGraph(v1);
    this._checkVertexIsInGraph(v2);

    return this.list.get(v1)!.includes(v2);
  }

  getAdjacencyList() {
    return this.list;
  }

  _checkVertexIsInGraph(v: GraphVertex) {
    if (!this.vertices.includes(v)) {
      throw new Error('Vertex doesn\'t exist');
    }
  }
}


let a = new GraphVertex('A');
let b = new GraphVertex('B');
let c = new GraphVertex('C');
let d = new GraphVertex('D');

let graph = new Graph([a, b, c, d], false);

graph.addEdge(a, b);
graph.addEdge(a, c);
graph.addEdge(b, c);
graph.addEdge(c, d);

console.log(graph.getAdjacencyList());

/* Output:

Map (4) {
  GraphVertex: { "value": "A" } => [
    GraphVertex: { "value": "B" }, 
    GraphVertex: { "value": "C" }
  ], 
  GraphVertex: { "value": "B" } => [
    GraphVertex: { "value": "A" }, 
    GraphVertex: { "value": "C" }
  ], 
  GraphVertex: { "value": "C" } => [
    GraphVertex: { "value": "A" }, 
    GraphVertex: { "value": "B" }, 
    GraphVertex: { "value": "D" }
  ], 
  GraphVertex: { "value": "D" } => [
    GraphVertex: { "value": "C" }
  ]
} 

*/

Getting the neighbors of a vertex is $O(1)$ because we're just looking up a key in a map. However, finding a particular edge can be $O(d)$ where $d$ is the number of degrees of the vertex, because we might need to traverse all the neighbors to find it. And, it could be $V - 1$ where $V$ is the number of vertices in the graph. It's the case when that vertex has all the other vertices as its neighbor.

The space complexity can be $O(V + E)$ where $V$ is the number of vertices and $E$ is the number of edges.

---

Continuing with the adjacency list representation, let's now take a look at two (very familiar!) ways to traverse a graph: breadth-first search and depth-first search.

But first, we'll modify our graph a little bit. We'll add a new vertex 'E' and update some edges:

let a = new GraphVertex('A');
let b = new GraphVertex('B');
let c = new GraphVertex('C');
let d = new GraphVertex('D');
let e = new GraphVertex('E');


let graph = new Graph([a, b, c, d, e], false);

graph.addEdge(a, b);
graph.addEdge(a, c);
graph.addEdge(b, d);
graph.addEdge(c, e);

The important idea to remember is that there is no hierarchy of vertices, so we don't have a root node.

For a breadth-first or depth-first search, we can use an arbitrary node as a starting point.

Breadth-First Search

With our new graph, a breadth-first search traversal looks like this:

When it comes to breadth-first search, usually a queue is used, and the idea is simple: given a current node, we'll add the adjacent nodes first, marking them as visited as we go.

Inside the Graph class, we can implement a bfs method that does just that:

bfs(startNode: GraphVertex) {
  const visited = new Set();
  const queue = [startNode];
  visited.add(startNode);

  while (queue.length > 0) {
    const currentNode = queue.shift();
    // console.log(currentNode);
    this.list.get(currentNode as GraphVertex)!.forEach((node) => {
      if (!visited.has(node)) {
        visited.add(node);
        queue.push(node);
      }
    });
  }
}

If we log currentNode to console each time we go, it's as we expected:

GraphVertex { value: 'A' }
GraphVertex { value: 'B' }
GraphVertex { value: 'C' }
GraphVertex { value: 'D' }
GraphVertex { value: 'E' } 

With the adjacency list, using a BFS has $O(V + E)$ time complexity (sum of the vertices and edges) as we're traversing the whole graph.

Depth-First Search

With the same modified graph, a depth-first search looks like this:

With depth-first search there is usually recursion involved as we're traversing through a path until we have visited all the nodes in that path. Once we hit a dead end, we'll backtrack and continue exploring until we have visited all the vertices in the graph.

dfs(startNode: GraphVertex, visited = new Set()) {
  visited.add(startNode);
  // console.log(startNode);
  this.list.get(startNode)!.forEach((node) => {
    if (!visited.has(node)) {
      this.dfs(node, visited);
    }
  });
}

Starting with a node, we check how deep we can go from there. Once we reach a dead end (when the dfs inside forEach returns), we continue checking other neighbors (with forEach) until none is left. We essentially do the same thing until all the vertices are visited.

Logging the output matches our expectation:

GraphVertex { value: 'A' }
GraphVertex { value: 'B' }
GraphVertex { value: 'D' }
GraphVertex { value: 'C' }
GraphVertex { value: 'E' }

The time complexity for a depth-first search traversal of a graph is the similar to BFS, $O(V + E)$ .

---

The first problem we'll look at in this chapter is Number of Islands. Until then, happy coding.

Resources

• Graphs - brilliant.org
• Time and Space Complexity of Adjacency Matrix and List - baeldung.com
• "Going Broad In A Graph: BFS Traversal" - Vaidehi Joshi
• "Deep Dive Through A Graph: DFS Traversal" - Vaidehi Joshi

Comments

[Thaísa Vieira]

Woah, what a great article! I remember just a few things about graphs from when I was an undergraduate Physics student.