Summarizing topological relations is fundamental
to many spatial applications including spatial query optimization.
In this talk, we present several novel techniques
to effectively construct cell density based spatial histograms
for range (window) summarizations restricted to
the four most important level-two topological relations: contains,
contained, overlap, and disjoint.
We first present a novel framework to construct a
multiscale Euler histogram in 2D space with the guarantee of the
exact summarization
results for aligned windows in constant time.
To minimize the storage space in such a multiscale Euler histogram,
an approximate algorithm with the approximate ratio 19/12 is presented,
while the problem is shown NP-hard generally.
To conform to a limited storage space where a multiscale histogram
may be allowed to have only k Euler histograms,
an effective algorithm is presented to construct multiscale histograms
to achieve high accuracy in approximately summarizing aligned windows.
Then, we present a new approximate algorithm to query an Euler histogram
that cannot guarantee the exact answers; it runs
in constant time. |