Chebyshevdistance

The Chebyshev distance, also known as the L∞ norm or maximum metric, is a distance metric defined between two points in a vector space. It is the maximum of the absolute differences of their coordinates. For two points p = (p1, p2, ..., pn) and q = (q1, q2, ..., qn) in an n-dimensional space, the Chebyshev distance is given by:

D_Chebyshev(p, q) = max_i(|p_i - q_i|)

This means that the distance is determined by the largest difference along any single dimension. Imagine a king moving on a chessboard; the minimum number of moves a king needs to go from one square to another is the Chebyshev distance between those squares. This is because the king can move one square in any direction, including diagonally, effectively covering the maximum difference in both the row and column simultaneously in a single move.

In contrast to Euclidean distance, which measures the straight-line distance between two points, or Manhattan distance, which measures the distance by summing the absolute differences of coordinates, the Chebyshev distance focuses on the greatest divergence in any one dimension. This metric is particularly useful in scenarios where movement or difference is limited by the maximum deviation in any single component, such as in certain game AI algorithms or in image processing. It is one of the p-norms, specifically the case where p approaches infinity.