多重和の解法

【たじゅうわのかいほう (solution of multiple summation)】

一般に, 多重和問題

$\displaystyle {\sum \{g(x_{1},x_{2},x_{3},\ldots ,x_{N+1})}\displaystyle {\mid (x_{2},x_{3},\ldots ,x_{N+1})\in X^{N}\}}\,$

は次の後向き再帰式で解ける:

${\begin{array}{l}\displaystyle {w_{N+1}(x^{N+1})=g(x^{N+1}),\quad x^{N+1}\in X^{N+1}}\\\displaystyle {w_{n}(x^{n})=\sum _{y\in X}w_{n+1}(x^{n},y),}\ \ \ \ \displaystyle {x^{n}\in X^{n},~1\leq n\leq N.}\end{array}}\,$

ただし, $x^{n}=(x_{1},x_{2},\ldots ,x_{n}).\,$