Manhattan Distance: A Key Metric for High-Dimensional Nearest Neighbor Search and Applications

Manhattan Distance, also known as L1 distance or taxicab distance, is a metric used to calculate the distance between two points in a grid-like space by summing the absolute differences of their coordinates. It has gained importance in machine learning, particularly in high-dimensional nearest neighbor search, due to its effectiveness compared to the Euclidean distance.

In the realm of machine learning, Manhattan Distance has been applied to various problems, including the Quadratic Assignment Problem (QAP), where it has been used to obtain new lower bounds for specific cases. Additionally, researchers have explored the properties of circular paths on integer lattices using Manhattan Distance, leading to interesting findings related to the constant π in discrete settings.

Recent research has focused on developing sublinear time algorithms for Nearest Neighbor Search (NNS) over generalized weighted Manhattan distances. For instance, two novel hashing schemes, ($d_w^{l_1},l_2$)-ALSH and ($d_w^{l_1},\theta$)-ALSH, have been proposed to achieve this goal. These advancements have the potential to make high-dimensional NNS more practical and efficient.

Manhattan Distance has also found applications in various fields, such as:

1. Infrastructure planning and transportation networks: The shortest path distance in Manhattan Poisson Line Cox Process has been studied to aid in the design and optimization of urban infrastructure and transportation systems.

2. Machine learning for chemistry: Positive definite Manhattan kernels, such as the Laplace kernel, have been widely used in machine learning applications related to chemistry.

3. Code theory: Bounds for codes in the Manhattan distance metric have been investigated, providing insights into the properties of codes in non-symmetric channels and ternary channels.

One company leveraging Manhattan Distance is XYZ (hypothetical company), which uses the metric to optimize its delivery routes in urban environments. By employing Manhattan Distance, XYZ can efficiently calculate the shortest paths between delivery points, reducing travel time and fuel consumption.

In conclusion, Manhattan Distance has proven to be a valuable metric in various machine learning applications, particularly in high-dimensional nearest neighbor search. Its effectiveness in these contexts, along with its applicability in diverse fields, highlights the importance of Manhattan Distance as a versatile and powerful tool in both theoretical and practical settings.

# Manhattan Distance

## Manhattan Distance Further Reading

1.A Remark on the Manhattan Distance Matrix of a Rectangular Grid http://arxiv.org/abs/1208.5150v1 A. Y. Alfakih2.Sublinear Time Nearest Neighbor Search over Generalized Weighted Manhattan Distance http://arxiv.org/abs/2104.04902v2 Huan Hu, Jianzhong Li3.Pi Visits Manhattan http://arxiv.org/abs/1708.00766v1 Michelle Rudolph-Lilith4.Product Constructions for Perfect Lee Codes http://arxiv.org/abs/1103.3933v2 Tuvi Etzion5.Polylogarithmic Approximation for Generalized Minimum Manhattan Networks http://arxiv.org/abs/1203.6481v2 Aparna Das, Krzysztof Fleszar, Stephen Kobourov, Joachim Spoerhase, Sankar Veeramoni, Alexander Wolff6.Statistical Physics of the Travelling Salesman Problem http://arxiv.org/abs/cond-mat/0001069v1 Anirban Chakraborti, Bikas K. Chakrabarti7.Metric Transforms and Low Rank Matrices via Representation Theory of the Real Hyperrectangle http://arxiv.org/abs/2011.11503v2 Josh Alman, Timothy Chu, Gary Miller, Shyam Narayanan, Mark Sellke, Zhao Song8.On Grid Codes http://arxiv.org/abs/2202.10005v4 E. J. García-Claro, I. S. Gutiérrez9.Shortest Path Distance in Manhattan Poisson Line Cox Process http://arxiv.org/abs/1811.11332v3 Vishnu Vardhan Chetlur, Harpreet S. Dhillon, Carl P. Dettmann10.Bounds for codes for a non-symmetric ternary channel http://arxiv.org/abs/1004.1511v1 Ludo Tolhuizen## Manhattan Distance Frequently Asked Questions

## What is Manhattan distance formula?

Manhattan distance, also known as L1 distance or taxicab distance, is a metric used to calculate the distance between two points in a grid-like space. The formula for Manhattan distance between two points (x1, y1) and (x2, y2) is: `Manhattan Distance = |x1 - x2| + |y1 - y2|` This formula can be extended to higher dimensions by summing the absolute differences of each coordinate.

## What is Manhattan distance in machine learning?

In machine learning, Manhattan distance is used as a similarity measure between data points, particularly in high-dimensional nearest neighbor search. It is effective in these contexts because it is less sensitive to outliers and can better handle high-dimensional data compared to Euclidean distance. Manhattan distance has been applied to various problems, such as the Quadratic Assignment Problem (QAP) and Nearest Neighbor Search (NNS) over generalized weighted Manhattan distances.

## What is the Manhattan distance example?

Consider two points A(2, 3) and B(5, 7) in a 2D grid. To calculate the Manhattan distance between these points, we use the formula: `Manhattan Distance = |x1 - x2| + |y1 - y2|` So, the Manhattan distance between A and B is: `Manhattan Distance = |2 - 5| + |3 - 7| = 3 + 4 = 7`

## Why is Manhattan distance called so?

Manhattan distance is called so because it resembles the distance a taxi would have to travel in a grid-like city layout, such as Manhattan in New York City. In such a layout, a taxi can only move along the grid lines (streets) and cannot travel diagonally. The Manhattan distance measures the total number of horizontal and vertical moves required to travel between two points, which is similar to the actual distance a taxi would cover.

## How does Manhattan distance differ from Euclidean distance?

Manhattan distance and Euclidean distance are both metrics used to calculate the distance between two points. The key difference between them lies in how they measure this distance. Manhattan distance calculates the sum of the absolute differences of the coordinates, while Euclidean distance calculates the square root of the sum of the squared differences of the coordinates. In a grid-like space, Manhattan distance is more appropriate for measuring distances along the grid lines, whereas Euclidean distance is suitable for measuring straight-line distances.

## In which applications is Manhattan distance commonly used?

Manhattan distance has found applications in various fields, such as: 1. Infrastructure planning and transportation networks: It is used to aid in the design and optimization of urban infrastructure and transportation systems. 2. Machine learning for chemistry: Positive definite Manhattan kernels, such as the Laplace kernel, are widely used in machine learning applications related to chemistry. 3. Code theory: Bounds for codes in the Manhattan distance metric have been investigated, providing insights into the properties of codes in non-symmetric channels and ternary channels. 4. Route optimization: Companies like XYZ (hypothetical company) use Manhattan distance to optimize their delivery routes in urban environments, reducing travel time and fuel consumption.

## What are the advantages of using Manhattan distance in high-dimensional nearest neighbor search?

Manhattan distance is particularly effective in high-dimensional nearest neighbor search due to its ability to handle high-dimensional data and its robustness to outliers. In high-dimensional spaces, Euclidean distance can be affected by the 'curse of dimensionality,' which makes it difficult to distinguish between close and distant points. Manhattan distance, on the other hand, is less sensitive to this issue and can provide more accurate results in high-dimensional settings. Additionally, Manhattan distance is less influenced by outliers, making it a more reliable metric for similarity measurement in machine learning applications.

## Explore More Machine Learning Terms & Concepts