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.. default-domain:: cpp

.. _alg_minkowski_distance:

==================
Minkowski distance
==================

.. include::  ../../../includes/pairwise-distances/minkowski-distance-introduction.rst

------------------------
Mathematical formulation
------------------------

.. _minkowski_distance_c_math:

Computing
---------

Given a set :math:`U` of :math:`n` feature vectors :math:`u_1 = (u_{11}, \ldots, u_{1k}),
\ldots, u_n = (u_{n1}, \ldots, u_{nk})` of dimension :math:`k` and a set :math:`V` of :math:`m`
feature vectors :math:`v_1 = (v_{11}, \ldots, v_{1k}), \ldots, v_m = (v_{m1}, \ldots, v_{mk})`
of dimension :math:`k`, the problem is to compute the Minkowski distance
:math:`||u_i, v_j||_{p}` for any pair of input vectors:

.. math::
    ||u_i, v_j||_{p} = \sum_{l=1}^{k} {({|u_{il} - v_{jl}|}^p)}^{1/p},

where :math:`\quad 1 \leq i \leq n, \quad 1 \leq j \leq m, \quad p > 0`.

.. _minkowski_distance_c_math_dense:

Computation method: *dense*
---------------------------
The method defines Minkowski distance metric, which is used in other algorithms
for the distance computation. There are no separate computation mode to compute distance
manually.

---------------------
Programming Interface
---------------------

Refer to :ref:`API Reference: Minkowski distance <api_minkowski_distance>`.