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# Efficiently implementing adjacency list in directed graph

In computer science, graphs are a fundamental data structure used to represent relationships between different entities. A directed graph, also known as a digraph, is a type of graph where edges have a specific direction from one node to another. An adjacency list is a common way to represent a graph, where each node is associated with a list of its neighboring nodes.

## Problem Statement

The problem at hand involves efficiently implementing an adjacency list for a directed graph. This means creating a data structure that stores the relationships between nodes in a way that allows for quick access to neighboring nodes.

## Description

Consider a scenario where you want to represent a network of connections, such as a social media platform where users are connected to each other. This network can be modeled as a directed graph, where users are nodes and connections between users are directed edges.

For instance, let's take a simplified social network with 10 users. The following connections exist:

• User 0 is connected to users 1, 2, and 7.
• User 1 is connected to users 5 and 7.
• User 2 is connected to users 3 and 7.
• User 3 is connected to users 4 and 8.
• User 4 is connected to user 9.
• User 5 is connected to users 6 and 8.
• User 6 is connected to users 8 and 9.
• User 7 is connected to user 8.

## Solution Idea

To efficiently implement an adjacency list for this directed graph, we use a vector of vectors. In Java, the `Vector` class is used to create dynamic arrays, similar to lists in Python. The outer vector represents the nodes, and each inner vector represents the neighbors of a node.

## Pseudocode

``````class Graph
Initialize class variables:
vertices

Initialize constructor:
Accept the number of vertices

Check if u and v are valid vertices

Function printGraph():
Iterate through vertices:
Print vertex
Print neighbors
``````

## Algorithm Explanation

1. Initialize a `Graph` class with a constructor that accepts the number of vertices. Inside the constructor, create an empty `adgeList` vector with as many elements as there are vertices.

2. Implement the `addEdge(u, v)` function to add an edge from vertex `u` to vertex `v`. Before adding the edge, make sure that both `u` and `v` are valid vertex indices.

3. Implement the `printGraph()` function to iterate through each vertex and its corresponding `adgeList`. Print the vertex's index and then iterate through the `adgeList`, printing the neighbors.

## Time Complexity

• Creating the graph and adding edges both have a time complexity of O(1), as they involve simple operations.
• Printing the graph takes O(V + E) time, where V is the number of vertices and E is the total number of edges. This is because each vertex and its neighboring edges are visited once.

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