An Efficient Algorithm to Count the Relations in a Range of Binary Relations Represented by a k²-Tree

Two sets
$A$
and
$B$
, whose elements fulfill a total order on operator
$\le $
, can have a binary relation
$R \subseteq A \times B$
represented by the k
²
-tree compact data structure, which greatly improves storage space. Currently,
$Count$
query is managed by either using
$Range$
query or to modify the structure to have aggregate information, implying additional time or space in order to perform the query. This article presents
Compact Count
, which exploits the k
²
-tree properties to reduce the paths to be scanned to count the numbers in a range
$r$
, thus ensuring an expected runtime of
$O(\log _{k} r \log _{k} n)$
and storage of
$O(\log _{k} r)$
with the k
²
-tree parameters
$n$
and
$k$
. Our algorithm was compared through a series of experiments that consider both synthetic data with different distributions and real data, with a solution based on the
Range
algorithm. Experimental results show that
Compact Count
is 250 to 1,000 times faster than
Range
on synthetic and real data, respectively, with a small additional storage cost, as expected by the theoretical analysis.