1dist

1dist is a shorthand used in some mathematical and computer science texts to refer to the L1 distance, also known as the Manhattan distance, between two vectors. It is defined for two equal-length vectors x and y in R^n as d1(x, y) = sum_i |x_i - y_i|. In machine learning and data analysis, 1dist is often described simply as the L1 norm of the difference, d1(x,y) = ||x - y||1.

As a metric on R^n, 1dist satisfies non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. It measures component-wise differences and is particularly sensitive to the number of dimensions in which two vectors differ, with each differing coordinate contributing a nonnegative amount to the total.

Applications include clustering methods such as k-medoids, nearest-neighbor search, and similarity measures for sparse, high-dimensional data (e.g., text data with bag-of-words representations). In optimization, L1 norms promote sparsity in certain problems; in distance computations, L1 is faster to compute than L2 because it avoids squaring and square roots.

Related distances include the L2 distance (Euclidean) and the L-infinity distance (maximum coordinate difference). For probability distributions over a finite set, the total variation distance equals half the L1 distance between the distributions.

The term 1dist is informal; the standard terminology in mathematics uses L1 distance or Manhattan distance. The concept is well established in normed vector spaces and underpins many algorithms in data analysis and machine learning.