Consider a multi$-$dimensional, metric space with eight points $($p;$-$pg$)$ and two multi$-$dimensional distance

range queries $($q:$,,)$ and $($qo$,2),$ as shown in the Figure below $($left part$).$ Recall that a data

point p is a result of a multi$-$dimensional distance range query $($q$,),$ if dist$($g$,$p$)<,$ where dist

denotes the Euclidean distance. Based on the two queries, we can divide the space into $4$ parts: one

part containing points which are results to both queries $($ps$),$ one part containing points which are

results to g; but not g $($p$2,$p$7),$ one part containing points which are results to g$2$ but not q$1($pa$,$ Pe$),$

and one part containing points which are results to neither query $(1,3,$ ps$).$ Such a division, can be

done hierarchically to form a pivot tree $($P$-$tree$)$ as shown in the figure. We can use $($q$,1)$ as root

to divide the space to points inside and outside distance range query $($q:$,1)$ and then use $($go$,2)$ to

further divide each partition into two parts based on whether they are inside range $($q$2,2).$ Data

points are then allocated to the leaves of the tree.