Task Dependency Validator

You are given a project management system that needs to schedule numTasks tasks, labeled from 0 to numTasks - 1. Some tasks have prerequisite requirements that must be satisfied before they can begin execution.

You are provided with an array dependencies where each element dependencies[i] = [taskᵢ, requiredTaskᵢ] indicates that requiredTaskᵢ must be completed before taskᵢ can start. In other words, there is a directed dependency from requiredTaskᵢ to taskᵢ.

For example, the dependency [3, 1] means that task 1 must be finished before task 3 can begin.

Your goal is to determine whether it is possible to complete all tasks given these dependency constraints. If the dependencies form a valid execution order (no circular dependencies exist), return true. Otherwise, if there exists a cycle in the dependencies that makes it impossible to complete all tasks, return false.

This is fundamentally a cycle detection problem in a directed graph, where tasks are nodes and dependencies are directed edges. The problem can be solved using:

• Depth-First Search (DFS) with three-color marking to detect back edges
• Breadth-First Search (BFS) using Kahn's algorithm for topological sorting
• Topological Sort verification - if a valid topological order exists, there are no cycles

The key insight is that a valid task ordering exists if and only if the dependency graph is a Directed Acyclic Graph (DAG). Any cycle in the graph means there's a mutual dependency that cannot be resolved, making it impossible to complete all tasks.