Graphs are a fundamental concept in computer science and mathematics, representing a versatile way to model relationships and interactions in various types of data. At their core, graphs consist of a set of nodes (also called vertices) connected by edges, which can either be directed or undirected. This simple yet powerful structure allows for the modeling of a wide range of real-world scenarios, from social networks and communication networks to scheduling tasks and mapping routes. Understanding graphs and their properties is crucial for solving many complex problems in computer science, as they provide the foundation for numerous algorithms used in data analysis, network theory, and optimization problems. This lab will introduce you to the basics of graph theory, including graph representation, traversal, and application through problem-solving exercises, laying the groundwork for more advanced studies in algorithms and data structures.

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997. Find the Town Judge

Breadth-First Search (BFS) is a pivotal algorithm in graph theory used for traversing or searching tree or graph data structures. It starts at an arbitrary node of a graph and explores the neighbor nodes first, before moving to the next level of neighbors. This layer-by-layer exploration strategy ensures that all nodes at the current depth are visited before proceeding to nodes at the next depth level. BFS is widely used for finding the shortest path on unweighted graphs, as it guarantees the minimum number of edges that must be traversed to reach a target node from the starting node. Its implementation typically utilizes a queue to keep track of the nodes that are to be explored, making it a simple yet powerful tool for solving a multitude of problems in computer science, from network connectivity to spreading processes in social networks.

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112. Path Sum

226. Invert Binary Tree

Depth-First Search (DFS) is a fundamental algorithm for traversing or searching through a graph or tree data structure. Unlike its counterpart, Breadth-First Search, which explores neighbors level by level, DFS dives as deep as possible along each branch before backtracking. This approach enables DFS to explore a path from the starting node to the farthest node, then backtracks and explores the next path in a similar manner. It’s particularly useful for tasks that require exploring all possible solutions or paths, such as puzzle solving, topological sorting, and finding connected components in a graph. DFS can be implemented recursively or with a stack, making it versatile and efficient for deep exploration tasks. Its depth-oriented exploration strategy makes it a key tool in algorithm design, offering a straightforward yet powerful method for navigating complex structures.

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100. Same Tree

112. Path Sum

Implementation: For each exercise, implement a solution using the required data structure or algorithm. Pay attention to the unique challenges each problem presents.

Analysis: After implementing your solution, analyze its time and space complexity.

Brute Force Approach: Describe and implement a naive solution, analyzing its time and space complexities.

Efficient Solution: Outline your approach, providing both pseudocode and source code. Discuss the time and space complexities.

Comparative Analysis: Compare the brute force and Efficient solutions in terms of efficiency. Use empirical data or theoretical analysis to support your comparison.

Reflection: Reflect on the learning process. What challenges did you encounter, and how did you overcome them?