This paper provides an overview of efficient algorithms for approximate and exact nearest neighbor problems. The goal is to pre-process a set of objects (such as images) and quickly return the object in the dataset that is most similar to a new query object. This problem is of great interest in various fields, including data compression, databases, information retrieval, and machine learning. The paper is divided into two main sections. The first section discusses a series of algorithms based on locality-sensitive hashing (LSH), which have already been established for practical applications. The second section focuses on LSH algorithms for d-dimensional Euclidean space, which are believed to be among the best in terms of performance. The introduction defines the nearest neighbor problem and highlights its significance in various research areas. It notes that while efficient algorithms exist for small dimensions (up to 10 or 20), they often suffer from the "curse of dimensionality" when d is large. To address this, approximate algorithms that return a point within a constant factor of the true nearest neighbor have been proposed. These algorithms are particularly useful when the exact distance calculation is not critical. The paper then delves into LSH algorithms, which use a set of hash functions to map points to buckets in a way that nearby points are more likely to collide. The core idea is to choose hash functions that ensure collisions between nearby points are more probable than those between far points. This property allows for efficient nearest neighbor search by querying the bucket containing the query point. LSH algorithms have been applied to various computational problems, including web clustering, computational biology, computer vision, and computational linguistics. The paper also discusses the theoretical and practical aspects of LSH, including the design of LSH functions and the amplification process to achieve the desired collision probabilities. Finally, the paper outlines the basic LSH algorithm, which involves preprocessing the dataset by mapping each point to a set of buckets and then searching the buckets for the query point. Two search strategies are discussed: one that returns the first \( L' \) points found and another that continues until all points are found. The paper provides detailed analyses of these strategies, including their time complexities and performance guarantees. Overall, the paper aims to provide a comprehensive understanding of LSH algorithms and their applications in high-dimensional nearest neighbor search.This paper provides an overview of efficient algorithms for approximate and exact nearest neighbor problems. The goal is to pre-process a set of objects (such as images) and quickly return the object in the dataset that is most similar to a new query object. This problem is of great interest in various fields, including data compression, databases, information retrieval, and machine learning. The paper is divided into two main sections. The first section discusses a series of algorithms based on locality-sensitive hashing (LSH), which have already been established for practical applications. The second section focuses on LSH algorithms for d-dimensional Euclidean space, which are believed to be among the best in terms of performance. The introduction defines the nearest neighbor problem and highlights its significance in various research areas. It notes that while efficient algorithms exist for small dimensions (up to 10 or 20), they often suffer from the "curse of dimensionality" when d is large. To address this, approximate algorithms that return a point within a constant factor of the true nearest neighbor have been proposed. These algorithms are particularly useful when the exact distance calculation is not critical. The paper then delves into LSH algorithms, which use a set of hash functions to map points to buckets in a way that nearby points are more likely to collide. The core idea is to choose hash functions that ensure collisions between nearby points are more probable than those between far points. This property allows for efficient nearest neighbor search by querying the bucket containing the query point. LSH algorithms have been applied to various computational problems, including web clustering, computational biology, computer vision, and computational linguistics. The paper also discusses the theoretical and practical aspects of LSH, including the design of LSH functions and the amplification process to achieve the desired collision probabilities. Finally, the paper outlines the basic LSH algorithm, which involves preprocessing the dataset by mapping each point to a set of buckets and then searching the buckets for the query point. Two search strategies are discussed: one that returns the first \( L' \) points found and another that continues until all points are found. The paper provides detailed analyses of these strategies, including their time complexities and performance guarantees. Overall, the paper aims to provide a comprehensive understanding of LSH algorithms and their applications in high-dimensional nearest neighbor search.