Effective Space Usage Estimation for Sliding-Window Skybands

Algorithm 1

Expected skyband cardinality: ${\mathrm{\Psi }}_k(n,d).$

Input:  $n$: the number of the elements

   $d$: the number of the dimensions

$\hspace{1em}\hspace{1em}\hspace{1em}k$: the $k$-skyband

Output: the expected skyband cardinality

begin

 if $n\le k+1$  then return $n$;

 if $\mathrm{\hspace{0.17em}\hspace{0.17em}}d=1$  then return $k+1$;

 for  $i=1$  to $d$ do $\alpha [i]\leftarrow k+1;$

$\hspace{1em}\xi \leftarrow 0;$

 for  $i=k+2$  to $n$  do

$\hspace{1em}\hspace{1em}\hspace{1em}\xi \leftarrow (\xi +1)$  mod 2;

   if $\xi =1$  then

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\beta [1]\leftarrow k+1$;

    for $j=2$  to $d$  do $\beta [j]\leftarrow \beta [j-1]/i+\alpha [j];$

   else

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha [1]\leftarrow k+1;$

    for $j=2$  to $d$  do $\alpha [j]\leftarrow \alpha [j-1]/i+\beta [j];$

   end

 end

 if $\xi =0$  then return $\alpha [d]$  else return $\beta [d];$

end