生成一个矩阵，其中包含取自 n 个向量的元素的所有组合 [英] Generate a matrix containing all combinations of elements taken from n vectors

问题描述

这个问题经常以一种或另一种形式出现(参见例如此处或此处).所以我想我会以一般形式呈现它，并提供一个答案以供将来参考.

This question pops up quite often in one form or another (see for example here or here). So I thought I'd present it in a general form, and provide an answer which might serve for future reference.

给定任意数量的 n 个可能不同大小的向量，生成一个 n 列矩阵，其行描述从这些向量中提取的元素的所有组合(笛卡尔积).

Given an arbitrary number n of vectors of possibly different sizes, generate an n-column matrix whose rows describe all combinations of elements taken from those vectors (Cartesian product) .

例如

vectors = { [1 2], [3 6 9], [10 20] }

应该给

combs = [ 1     3    10
          1     3    20
          1     6    10
          1     6    20
          1     9    10
          1     9    20
          2     3    10
          2     3    20
          2     6    10
          2     6    20
          2     9    10
          2     9    20 ]

推荐答案

ndgrid 函数几乎给出了答案，但有一个警告:必须明确定义 n 输出变量才能调用它.由于 n 是任意的，最好的方法是使用 逗号分隔列表(从带有 n 个单元格的单元格数组生成)作为输出.生成的 n 矩阵然后连接成所需的 n 列矩阵:

The ndgrid function almost gives the answer, but has one caveat: n output variables must be explicitly defined to call it. Since n is arbitrary, the best way is to use a comma-separated list (generated from a cell array with ncells) to serve as output. The resulting n matrices are then concatenated into the desired n-column matrix:

vectors = { [1 2], [3 6 9], [10 20] }; %// input data: cell array of vectors

n = numel(vectors); %// number of vectors
combs = cell(1,n); %// pre-define to generate comma-separated list
[combs{end:-1:1}] = ndgrid(vectors{end:-1:1}); %// the reverse order in these two
%// comma-separated lists is needed to produce the rows of the result matrix in
%// lexicographical order 
combs = cat(n+1, combs{:}); %// concat the n n-dim arrays along dimension n+1
combs = reshape(combs,[],n); %// reshape to obtain desired matrix

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