Box Trees and R trees with Near Optimal Query Time
y z x x
Pankaj K Agarwal Mark de Berg Joachim Gudmundsson Mikael Hammar
z
Herman J Haverkort
Abstract graphics spatial databases GIS and rob otics In
order to exp edite and simplify the data structure a
A b ox tree is a b ounding b ox hierarchy that uses
window query is answered in two steps In the rst
axis aligned b oxes as b ounding volumes The stab
step called the ltering step each ob ject is replaced
bing number of a b ox tree with resp ect to a given
by the smallest b ox containing the ob ject and the
type of query is the maximum number of no des vis
query pro cedure computes the b ounding b oxes that
ited when answering such a query We describ e sev
intersect the query window Instead of b oxes other
eral new algorithms to construct b ox trees with small
simple shap es such as spheres ellipsoids cylinders
stabbing number with resp ect to queries with axis
have also b een used The second step called the
parallel b oxes and with p oints We also prove lower
re nement step extracts the actual ob jects among
b ounds on the worst case stabbing number for b ox
these b ounding b oxes that intersect the query win
trees which show that our results are optimal or close
dow A few recent results show that under
to optimal Finally we present algorithms to convert
certain reasonable assumptions on the input ob jects
b ox trees to R trees resulting in R trees with al
the number of b ounding b oxes intersecting a query
most optimal stabbing number
window is not much larger than the number of ob
jects intersecting the window which makes this ap
Introduction
proach quite attractive see the pap er by Zhou and
Suri and the references therein There has b een
Motivation and problem statement Prepro
d
much work on the ltering step and we also fo cus
cessing a set S of geometric ob jects in R for an
on this step More precisely we wish to prepro cess
swering window queries rep ort all ob jects of S that
d
a set S of n d rectangles in R so that all rectan
intersect a query d rectangle is central to many
gles of S intersecting a query d rectangle can b e re
applications and has b een widely studied in several
p orted e ciently We will refer to this query as the
areas including computational geometry computer
rectangle intersection query A related query is the
The work by P A is supp orted by Army Research O ce
rectangle containment query in which we want to re
MURI grant DAAH by a Sloan fellowship by
p ort all rectangles in S that contain a query p oint
NSF grants EIA EIA and CCR
and by grant from the U S Israeli Binational Science Foun
dation The work by H H is supp orted by the Netherlands A number of data structures with go o d provable
Organization for Scienti c Research NWO
b ounds have b een prop osed for answering rectangle
y
Center for Geometric Computing Department of Com
intersection queries Unfortunately they are of lim
puter Science Box Duke University Durham NC
ited practical use b ecause the amount of storage they
USA
z
use is rather high O n log n storage and even O n
Institute of Information and Computing Sciences Utrecht
University PO Box TB Utrecht the Netherlands
storage with a large hidden constant are often unac
x
Department of Computer Science Lund University Box
ceptable Therefore in practice one usually uses sim
Lund Sweden
1
pler data structures A commonly used structure for
A d rectangle also called a d dimensional hyperrectangle
is the pro duct of d intervals a b a b answering rectangle intersection queries rectangle
1 1
d d
containment queries and in fact many other types of b ounding b ox hierarchies b oth in main and external
queries is the bounding box hierarchy sometimes also memory that have low stabbing number
called AABB tree this is a tree T in which each
Previous results As noted ab ove several e
leaf is asso ciated with a rectangle of S and each in
cient data structures have b een prop osed for an
terior no de is asso ciated with the smallest b ox B
swering a rectangle intersection query For example
enclosing all the rectangles stored at the leaves of the
Chazelle showed that a compressed range tree
subtree ro oted at All the rectangles of S intersect
can b e used to answer a d dimensional rectangle
d
ing a query rectangle R are rep orted by traversing T
intersection query in time O log n k using
d
in a top down manner Supp ose the query pro cedure
O n log n log log n space This data structure
is visiting a no de If B R there is nothing to
is to o complex to b e practical even in R Since a d
do If B R then it rep orts all rectangles stored
d d
rectangle in R can b e mapp ed to a p oint in R
in subtree ro oted at Finally if B R but
we can use a k d tree to construct a data structure of
B R it recursively visits the children of We
O n size that can answer a d dimensional rectangle
say that R crosses a no de if B R and B R
d
intersection query in time O n k A num
If the fan out of T is b ounded then the query time
b er of heuristics based on k d trees have also b een pro
is prop ortional to the number of no des of T that R
p osed to answer rectangle intersection queries
crosses plus the number of rectangles rep orted We
Several pap ers describ e how to construct
call the stabbing number of T denoted S T to
b ounding b ox hierarchies or other b ounding volume
b e the maximum number of its no des crossed by a
hierarchies for example using spheres or k DOPs as
rectangle It is therefore desirable to construct a
b ounding volumes but they do not obtain b ounds
b ounding b ox hierarchy with small stabbing number
on the worst case query complexity
In many applications esp ecially in the database
Some of the most widely used external memory
applications the set S is to o large to t in the main
b ounding b ox hierarchies are the R tree and its
memory therefore it is stored on disk In that case
variants An R tree originally introduced by
the main goal is to minimize the number of disk ac
Guttmann is a B tree each of whose leaves is
cesses needed to answer a window query and the
asso ciated with an input rectangle All leaves of an
p erformance of an algorithm is analyzed under the
R tree are at the same level the degree of all inter
standard external memory mo del This mo del as
nal no des except of the ro ot is b etween t and t for
sumes that each disk access transmits a contiguous
a given parameter t and the degree of the ro ot varies
blo ck of t units of data in a single input output op
b etween and t We will refer to t as the mini
eration or I O The e ciency of a data structure
mum degree of the tree Although several metho ds
is measured in terms of the amount of disk space it
have b een prop osed for ordering the
uses measured in units of disk blo cks the number
input rectangles along the leaves varying from sim
of I Os required to answer a query and the number
ple heuristics to space lling curves to minimize the
of I Os need to construct the data structure In the
stabbing number none of them guarantee the worst
context of b ounding b ox hierarchies several schemes
case p erformance In the worst case a linear number
have b een prop osed that construct a tree as ab ove but
of b ounding b oxes might intersect a query rectan
in which the fanout of each no de dep ends on t Some
gle even if it intersects only O input rectangles
notable examples of external memory b ounding b ox
The only analytical result is by Faloutsos et al
hierarchies are various variants of R trees see the
but they prove b ounds on the query time only in
survey pap er We can still de ne the crossing
the dimensional case when the input intervals are
no des and the stabbing number as earlier and one
2
Barequet et al gave an algorithm to construct a
can argue that the number of I Os needed to answer
2
b ounding b ox hierarchy in R and they claimed that if the
a query is prop ortional to the stabbing number plus
rectangles in S are pairwise disjoint then any p oint lies in
the output size
O log n b ounding b oxes But the argument presented in the
In this pap er we study the problem of constructing pap er has a technical problem
sults to our b oxtrees we improve the result of de uniformly distributed and have at most two di erent
Berg et al our query complexity do es not de lengths Recently de Berg et al describ ed an al
p end on the parameter w which makes their query gorithm for construcing an R tree on rectangles in R
complexity linear in the worst case and it is linear so that all k rectangles containing a query p oint b e re
p
in p orted in O log log n log t I Os Here is the instead of We also introduce the concept of
ratio of the maximum and the minimum x lengths of semi R trees these are similar to ordinary R trees
the input rectangles and is the point stabbing num the degree of each internal no de except for the ro ot
ber of S that is is the maximum number of input is b etween t and t for some given parameter t
rectangles containing any p oint in the plane They except that the leaves do not have to b e at the same
also describ ed another algorithm for constructing an level We give a general algorithm to convert a b ox
R tree that can answer a rectangle intersection query tree with small stabbing number into a semi R trees
in O log w k log n I Os where and with small stabbing number the stabbing number
t
are as ab ove and w is the ratio of the x length of the obtained here is b etter than that for R trees This
query rectangle to the smallest x length of an input leads to semi R trees with almost optimal stabbing
rectangle number
Lower Bounds
Our results In this pap er we rst describ e several
algorithms for constructing b ox trees and we prove
In this section we give lower b ounds on the stabbing
lower b ounds on the worst case stabbing number of
number of semi R trees of minimum degree t in var
b ox trees Our rst algorithm constructs a b ox tree
ious settings Since semi R trees are more general
on a set of p ossibly intersecting d rectangles in
than R trees the same b ounds hold for R trees By
d
R so that a rectangle intersection query can b e an
choosing t we obtain lower b ounds for b ox trees
d
swered in O n k time Our lower b ound shows
We start with a simple generalization of the
that this is optimal
dimensional lower b ound given by de Berg et al
In the plane we show that if the input is disjoint
It is obtained by putting n unit hypercub es in an
d d
more generally if the p oint stabbing number of
n n grid the argument showing that
the input is small how to construct a b ox tree that
this gives the stated b ound is given in the full pap er
still has almost optimal query time for rectangle
Theorem For any n and d > there is a set of
intersection queries but much b etter query times for
d
n disjoint unit hypercub es in R with the following
p oint queries More precisely the time for rectangle
p p
prop erty for any semi R tree T of minimum degree t
n log n log n k intersection queries is O
p
there is a query b ox not intersecting any b ox from S
and the time for p oint queries is O log n k We
d
p
such that a query with that b ox visits n t
also develop a b ox tree with O log n k
no des in T
query time when the asp ect ratio of the query rect
angles is small One would hop e that similar improve
Next we describ e a construction that proves lower
ments are p ossible in higher dimensions One of our
b ounds on rectangle containment queries and that
lower bound results shows that this is not p ossible
will also b e useful for a number of other cases For
d
in dimensions d > the n k lower b ound
any we call a d rectangle an hypercube if the
on the query complexity holds even for hypercub es as
length of each edge is b etween and We x a pa
query ranges and any b ounding b ox hierarchy that
rameter and construct a set S fb b g
n
d
achieves this query time cannot have a b etter worst
of n hypercub es in R We also construct two sets
case query time for p oint queries even when the input
of query p oints Q and Q called primary and sec
consists of disjoint almost unit hypercub es
ondary p oint sets that lie in the common exterior of
Finally we give general metho ds to convert b ox rectangles in S and have the following crucial prop
trees with small stabbing number into R trees with erty For any semi R tree T on S with minimum de
small stabbing number When we apply these re gree t either a p oint of Q lies in at least b ounding
d
b oxes of T or a p oint of Q lies in n t can shift them a little to make them disjoint from all
b ounding b oxes of T We rst decsrib e the set S and input b oxes
then construct the p oint sets
Lemma Let T b e any semi R tree of minimum
Let n n b e the outward normals of a d
d
degree t on the set S constructed ab ove Then ei
rectangle We can pair these normals into d pairs
ther there is a primary query p oint contained in
n n n n so that no pair contains op
d d
b ounding b oxes stored in T or one of the secondary
p osite normals that is n n for i d
d
i i
query p oints is contained in n t b ound
Let h b e the plane spanned by the vectors n and
i i
ing b oxes stored in T
n and containing the origin Let b b e a d rectangle
i
Pro of We prove the lemma for b ox trees the gen
containing the origin Since n n the facets
i i
eralization to semi R trees is describ ed in the full
f f of b normal to n and n resp ectively share
i i i i
pap er Supp ose that all primary query p oints are
a d face f which is orthogonal to the plane
i
contained in less than b ounding b oxes stored
h The intersection of f and h is a p oint c Con
i i i i
in the interior no des in T Then the number of
versely by sp ecifying a p oint c on each h i d
i i
incidences b etween these p oints and interior no des
we can represent a unique d rectangle in which c lies
i
b ounding b oxes is at most n Since there
on the facets normal to n and n We will therefore
i i
d
are n interior no des in T they store at least
de ne each rectangle b S by a d tuple c c
j
j j
n b ounding b oxes that contain less than
where the facets of b whose outward normals are n
j i
i
primary query p oints Observe that a b ounding b ox
and n pass through c We next describ e how to
i
j
for input b oxes b b S contains jj j j primary
i
j
j
choose the p oints c for i d and j n
j
query p oints b ecause there are that many concave
For each plane h we choose a line of slop e
i i
corners in the staircase b etween corners c and c
0
j
j
the exact equation of will sp eci ed b elow We
i
We conclude that there are at least n b ound
will refer to h as the primary plane and every h
i
0
and p erhaps some ing b oxes that store b oxes b b
j j
with i will b e called a secondary plane Set
more b oxes with jj j j But if jj j j
d
on p We place n p oints p
n
then j j mo d so b b This im
j
j
for every j n p sorted along and set c
j j
plies that there is at least one i with 6 i 6 d
i i
on p For each j we place p oints p
i i
such that c c Hence the b ounding b ox storing
0
j
j
i
and assign c to these p oints as follows Let
i
j
0
will contain one of the secondary query p oints b b
j j
j j j b e the representation of
d
So in total we have at least n incidences b e
P
d
j
j mo d in radix that is j mo d
j
tween secondary query p oints and b ounding b oxes
j
i i
d
Note that n p oints have the We set c p
so one of the d b O secondary
j
j
i
d
i
query p oints is contained in n b ounding
same value of c Finally we choose and the p oints
i
j
b oxes 2
on so that each b is an hypercub e The easy de
i j
tails are omitted from this abstract
We can use this lemma to prove lower b ounds for sev
Finally we choose a set Q of n p oints on the
eral settings We start with a lower b ound for p oint
primary plane h and a set Q of d p oints
queries
on the secondary planes as follows Supp ose h is the
Theorem For any n d > and there
x x plane For each j n we choose the
d
is a set S of n hypercub es in R with the following
p oint q x p x p and add it to Q In
j
j j
prop erty for any semi R tree T of minimum degree
other words if we regard the p oints on as a stair
t there is a p oint not contained in any b ox from S
case Q is the set of concave corners of the stair
d
such that a query with that p oint visits n t
case In order to construct Q we rep eat the same
no des in T
step for each of the secondary planes thus obtaining
p oints on each of them These p oints will b e Next we mo dify the ab ove construction so that the
on the b oundary of some of the input b oxes but we same b ound can b e achieved in d even if the
b ox tree in workspace pro ceed as follows Replace input consists of a set of n disjoint hypercub es and
the representative tuple in each leaf by the corre the queries are hypercub es The idea is to use the
sp onding input b ox Then going b ottom up store in construction ab ove in d dimensional space and
each internal no de the b ounding b ox of its children use the remaining dimension to turn the intersecting
We call the resulting b ox tree a con guration space d dimensional b oxes into disjoint d dimensional
boxtree or cs box tree for short b oxes details are in the full pap er
For the analysis of the range stabbing number in
Theorem For any n d > and there is
the resulting cs b ox tree we need the following fact
d
a set S of n disjoint hypercub es in R with the fol
ab out kd trees given here without pro of
lowing prop erty for any semi R tree T of minimum
degree t there is a hypercub e not intersecting any b ox
Lemma The number of cells at depth i in a d
from S such that a query with that hypercub e visits
dimensional kd tree that intersect an axis parallel f
d
if d
n t no des in T
at 6 f 6 d is O
Finally we observe that the pro of of the preceding
A kd tree and hence our b ox tree has the following
theorem actually shows that in higher dimensions any
prop erty the number of ob jects stored in the two
semi R tree with small say p olylogarithmic query
subtrees of any given no de di er by at most one We
complexity for p oints must have large near linear
call such trees perfectly balanced That our b ox tree
query complexity for ranges More precisely it shows
is p erfectly balanced will b e advantageous when we
the following result
will convert it to an R tree We can now analyse the
range stabbing number in a cs b ox tree
Theorem For any n d > and there
d
is a set S of n disjoint hypercub es in R with the
Lemma Let S b e a set of n p ossibly intersecting
following prop erty for any semi R tree T of mini
b oxes in the plane There is a p erfectly balanced b ox
mum degree t if the number of no des visited by any
tree for S such that the number of no des at level i
p oint query is then there is a hypercub e not inter
that are visited by a range query with an axis aligned
i d
secting any b ox from S such that a query with that
b ox is O k where k is the number of b oxes
d
hypercub e visits n t no des in T
in S intersecting the query range
Q
d
From kd trees to Box Trees
x Q x Q b e a query Pro of Let Q
i i
i
range We can restrict our attention to the interior
In this section we describ e and analyze several meth
no des visited since the number of visited leaves is at
o ds to construct b ox trees using kd trees For conve
most one more We distinguish two types of visited
nience we will allow no des of degree up to d it
interior no des The rst type is where at least one
is easy to convert these trees to binary trees with
of the input b oxes stored in the subtree of inter
out a ecting the asymptotic b ounds on the stabbing
sects Q Obviously there are only O k such no des
numbers
at a given level i The second type is where all input
The con guration space approach
b oxes in the subtree of are disjoint from Q Any
input b ox disjoint from Q must b e separated from Q
The basic metho d Let S b e a set of n
by a hyperplane through a facet of Q Clearly not
arbitrary p ossibly overlapping d rectangles
d
all input b oxes can b e separated from Q by the same
in R which we call the workspace As
hyperplane otherwise the b ounding b ox of would
noted in Introduction a d rectangle b
Q
d
not intersect Q and would not b e visited Hence
x b x b can b e represented by a p oint
i i
i
there are at least two such hyperplanes separating Q
x b x b x b x b x b x b in
d d
d
from an input b ox in the subtree of
R which we call the con guration space We
Assume w l o g that x x Q is one of these build a d dimensional kd tree on these tuples To
i
i
separating hyperplanes and let b b e the input b ox transform the kd tree in con guration space into a
normals of the facets of B take the b ox from S that it separates from Q Then we must have x b
i
extends furthest in the direction of that normal This x Q But there must also b e a b ox b with x b
i i
results in a set S of at most d b oxes Each b ox Q otherwise the b ounding b ox of would not x
i
in S is put in a so called priority leaf which is an intersect Q We can conclude that the p oints rep
immediate child of If the set S n S of remaining resenting b and b in the con guration space lie on
b oxes contains less than two b oxes then this b ox Q Now x opp osite sides of the hyperplane x
i i
if it exists is put as a leaf child of If two or Q inter x this implies that the hyperplane x
i i
more b oxes remain we split the set of b oxes into sects the cell in con guration space of the no de in the
two almost equal sized subsets with a hyperplane kd tree corresp onding to
in con guration space This hyperplane is orthogonal
We can apply the same argument to the second hy
to the cutting direction of the level we are working
p erplane separating Q from an input b ox the hyper
at as b efore the cutting directions of the levels cycle
plane x x Q for example to show that there
j
j
through the d directions in con guration space
is a hyperplane in con guration space with p oints on
The subset of b oxes whose representative p oints
opp osite sides x x Q in the example
j j
lie to one side of the cutting hyperplane are stored
We can conclude the following Supp ose Q visits
recursively in one subtree of The subset of b oxes
a no de of the second type Then in con guration
whose representative p oints lie to the other side of the
space there is a pair of hyperplanes b oth of the form
cutting hyperplane are stored recursively in another
x x Q or x x Q and b oth intersecting
i i i i
subtree of
the cell in con guration space of the no de in the kd
Next we analyze the query complexity of the tree
tree corresp onding to But then the cell must also
resulting from this construction which we call a cs
b e intersected by the d at that is the intersec
priority box tree In our analysis we b ound the num
tion of these two hyperplanes By Lemma there
i d d i d
b er of visited no des of a given weight where the
are only O O such no des at
weight of a no de is de ned as the number of input
level i 2
b oxes stored in its subtree This will b e useful when
This leads directly to the following theorem
we convert this b ox tree into a semi R tree
Theorem Let S b e a set of n p ossibly intersect
Lemma The number of no des of weight at
ing b oxes in the plane There is a p erfectly balanced
least w visited by a query with a query b ox Q is
d
b ox tree for S such that the number of no des vis
O n w k
ited by a range query with an axis aligned b ox is
Q
d
d
Pro of Let Q x Q x Q We can re
i i
i
O n k log n where k is the number of b oxes
strict our attention to the visited no des of weight at
in S intersecting the query range
least d as the total number of visited no des is at
Improving the query time We now show how
most a constant times larger than this number Let
to reduce the O k log n term in the query complexity
b e such a visited no de of weight at least d There
to O k The idea is the same as in a priority search
are two cases
tree input elements b oxes in our case that have
The rst case is where one of the priority leaves
a high chance of b eing an answer are pushed to high
directly b elow stores a b ox intersecting Q Clearly
levels in the tree In our case this means that we store
there are at most k such no des The second case is
b oxes that extend the farthest in some direction high
when all priority leaves directly b elow store b oxes
in the tree More precisely the construction of the
disjoint from Q Thus each such b ox is separated
d
tree T for a set S of b oxes in R is as follows
from Q by a hyperplane through a facet of Q We
If jS j then T consists of a single leaf no de claim that not all b oxes can b e separated by the
storing the input b ox in S Otherwise we make a same hyperplane Supp ose for a contradiction that
no de storing the b ounding b ox B of all b oxes in there is a facet f whose containing hyperplane sep
S and we pro ceed as follows For each of the d inner arates all b oxes of the priority leaves from Q Then
in particular it would separate the b ox that extends queries increases only slightly This approach only
furthest in the direction of the inner normal of the works in two dimensions Theorem states that a
facet f contradicting that Q intersects the b ounding similar result in more than two dimensions cannot b e
b ox stored at So we have two distinct hyperplanes obtained
through facets of Q separating a b ox in the subtree The basic idea b ehind kd interval trees is again to
of from Q use a kd tree but this time in the workspace which
The b ox tree that we have constructed basically is now the plane Since the ob jects in the workspace
corresp onds to a kd tree in con guration space as are rectangles not p oints many of them may inter
b efore The priority leaves make that the tree in con sect the cutting line These b oxes are taken out and
guration space is strictly sp eaking not a kd tree but handled separately like in an interval tree To make
it is easy to see that Lemma still holds More kd interval trees more e cient we introduce priority
over there is still a one to one corresp ondence b e leaves like in the previous section
tween no des of the b ox tree and no des of the kd tree
The dimensional case First we describ e how
in con guration space Hence we can use the fact
a set S of b oxes that all intersect a given line are
that there are two distinct hyperplanes through facets
handled With a slight abuse of terminology we call
of Q separating a b ox in the subtree of from Q in
a tree for this case a dimensional kd interval tree
the same way as in the pro of of Lemma it implies
If jS j then T consists of a single leaf no de storing
that there is a d at in con guration space de
the input rectangle in S Otherwise we make a no de
ned by a pair of facets of Q intersecting the cell in
storing the b ounding b ox B of all rectangles in S
the kd tree corresp onding to It follows that the
and we pro ceed as follows For each of the inner
total number of no des to which the second case
normals of the edges of B take the rectangle from
i d
applies at a given level i is O
S that extends furthest in the direction of that nor
To nish the pro of observe that no des at the low
mal This results in a set S of at most rectangles
ermost blog w d c levels have weight less than w
Each rectangle in S is put in a so called priority leaf
Adding the b ounds for the second case on the remain
Consider the set of intersections of the edges of the
P
dlog ne blog w d c
i d
ing levels we get O
i remaining rectangles with Let p b e the median of
d
O n w 2
these intersection p oints The rectangles in S n S
containing p are stored in a subtree of that is a
The following theorem follows directly
dimensional cs priority box tree as describ ed in the
Theorem Let S b e a set of n p ossibly intersect
previous section The rectangles in S n S completely
d
ing b oxes in R There is a b ox tree for S such that
to one side of p are stored recursively in one subtree
the number of no des that are visited by a range query
of The rectangles in S n S completely to the other
d
with an axis aligned b ox is O n k where k
side of p are stored recursively in another subtree of
is the number of b oxes in S intersecting the query
range
We call the no des in the main dimensional kd
interval tree D nodes Such a no de corresp onds to
The kd interval tree approach
an interval on the de ning line We call the no des
The cs b ox tree of the previous section has optimal
of the dimensional cs priority box trees cs nodes
query complexity for p oint queries and for range
We start by analyzing the query complexity when
queries if the input consists of arbitrary intersecting
we query with a segment on the line The pro of of
b oxes Unfortunately if the input b oxes are disjoint
the following lemma is omitted
then the query complexity for p oint queries do es not
improve In this section we develop a di erent b ox Lemma If we query a dimensional kd interval
tree the kd interval tree whose query complexity is tree storing a set S of n rectangles with a line segment
much b etter if the p oint stabbing number of the on the de ning line then we visit at most O log n
input set S is small The query complexity for range k no des where k is the number of rectangles to b e
Now consider a dimensional cs priority box tree
that is visited Supp ose the interval of the D no de
that is the parent of this subtree is completely con
tained in Q Then we can argue again using the
priority leaves that we can charge all the visited
Q
no des to rectangles intersecting Q If the interval
of the D no de that is the parent of this subtree
is not completely contained in the pro jection of Q
we argue as follows First we observe that the in
terval must then contain an endp oint of the pro jec
tion of Q so there are only O log n such parent
no des In the dimensional con guration space b ox
tree b elow such a parent we can use Lemma to
Figure Querying a dimensional kd interval tree with
b ound the number of visited no des of weight w by
p
a b ox Q
O n w k where n is the number of b oxes
stored in the con guration space b ox tree and k is
the number of answers rep orted in this subtree Note
that n O since the con guration space b ox
rep orted
trees are used only to store sets of b oxes that share
a single p oint Hence the overal number of cs no des
Next we analyze the query complexity when we query
p
w log n k nishing the pro of of visited is O
with a b ox
part i of the lemma
ii The pro of of the second part of the lemma can
Lemma i If we query a dimensional kd
b e found in the full version of this pap er 2
interval tree storing a set S of n rectangles with a
p
w log n k query b ox Q then we visit at most O
no des of weight at least w where k is the number of
The dimensional case Our kd interval tree for
rectangles to b e rep orted
a general set S of rectangles in the plane is de ned
ii If the pro jection of Q onto the line that stabs the
as follows If jS j then T consists of a single leaf
rectangles in S contains the intersections of all rect
no de storing the input b ox in S Otherwise we make
angles with then the query time reduces to O k
a no de storing the b ounding b ox B of all b oxes in
Pro of i See Figure If Q intersects then the S construct priority no des like in the dimensional
query is equivalent to querying with Q so the case and split the remaining set of b oxes by a ver
result follows from the previous lemma Otherwise tical or horizontal line dep ending on the level in
assume w l o g that is vertical and that Q lies to the the tree This splitting line is chosen such that the
right of Consider a D no de that is visited when number of rectangles lying completely to one side of
we query with Q When the interval corresp onding di ers at most one from the number lying completely
to this no de is completely contained in the pro jection to the other one side The rectangles lying completely
of Q onto then the rectangle in the subtree extend to one side of are stored recursively in one subtree
ing furthest to the right must b e intersected This of The rectangles to the other side are stored re
rectangle is stored in a priority leaf immediately b e cursively in another subtree of The rectangles in
low to which we can charge the visit of Hence tersecting are stored in a dimensional kd interval
there cannot b e more than k such no des When the tree as explained ab ove We call the no des of the
interval is not completely contained in the pro jection main tree which corresp ond to dimensional cells
of Q then it contains an endp oint of the pro jection D nodes Next we analyze the p erformance of the
of Q and there are only O log n such no des kd interval tree
Lemma The number of no des of weight at least so ciated dimensional kd interval trees we have to
w that are visited by a range query with an axis multiply this by the b ound of Lemma leading to
p p p
aligned b ox is O a total of O n w log n w log n k w log n k
where k is the number of rep orted answers The There are no other types of no des whose b ounding
number of such no des visited by a p oint query is b oxes intersect Q Adding up the number of no des
p
w log n k O for all four cases gives the desired b ound 2
This leads to the following theorem
Pro of Consider a D no de that is visited when we
query with an axis aligned rectangle Q We distin
Theorem Let S b e a set of n p ossibly inter
guish four di erent types of such no des We b ound
secting b oxes in the plane such that no single p oint
their number and the number of no des visited in
is contained in more than b oxes There is a
dimensional kd interval subtrees separately
b ox tree for S such that the number of no des vis
Inner nodes These are D no des whose b ounding
ited by a range query with an axis aligned b ox is
p p
b oxes lie completely inside Q The number of inner
n log n log n k where k is the number of O
no des is easy to b ound since all rectangles in the
b oxes in S intersecting the query range The number
p
subtree of such a no de intersect Q Hence the total
log n k of no des visited by a p oint query is O
number of such no des or no des in their dimensional
The longest side rst approach
asso ciated kd interval trees is O k
Side nodes These are D no des whose b ounding
Recall that a kd interval tree is basically a mo di ed
b oxes cut exactly one edge of Q In this case the
kd tree where each no de is split by a line The ori
rectangle that extends furthest into the direction of
entations of these lines dep end on the level in the
the inner normal of this edge must intersect Q This
tree in such a way that orientations take turns in a
rectangle is stored in a priority leaf immediately b e
round robin fashion on any path from the ro ot down
low the no de The same reasoning applies to their
into the tree An interesting variation of the kd
dimensional asso ciated kd interval trees Hence the
interval tree arises when we replace the round robin
total number of side no des or no des in their asso ci
splitting strategy by the longest side splitting rule as
ated kd interval trees is O k
suggested by Dickerson et al In such a longest
Piercing nodes These are D no des that cut two op
side rst kd tree the number of no des whose corre
p osing edges of Q but do not contain any corners of
sp onding cell is pierced by a query rectangle is small
Q From Lemma and the fact that no des at the
if the query rectangle is fat We can use this to prove
lowermost blog w d c levels of the tree must have
the following result
weight less than w we conclude that the number of
Theorem Let S b e a set of n b oxes in the
D no des of weight at least w that intersect any edge
P
dlog ne blog w d c
i
plane with stabbing number There is a b ox tree
O of Q is b ounded by
i
p
for S such that the number of no des that are vis
n w Now there are two cases the splitting line O
ited by a range query with a rectangular range of
used at such a no de is orthogonal to the intersected
p
asp ect ratio is O log k where k is the
edges or it is parallel to them In the former case we
number of b oxes in S intersecting the query range
can apply Lemma to obtain a O log n k b ound
The number of such no des visited by a p oint query
on the number of no des visited in the dimensional
p
log n k is O
kd interval tree asso ciated with where k is the
number of rep orted answers In the latter case we
From b ox trees to semi R trees
can apply Lemma ii to get a b ound of O k
p
In the previous section we describ ed several algo Hence we get a grand total of O n w log n k
rithms to construct b ox trees with go o d query com corner nodes These are D no des that contain one
plexity In this section we give general theorems to or more corners of Q There are O log n such no des
convert them to semi R trees To obtain the total number of visited no des in the as
We start with a general theorem that converts any settle for semi R trees instead of real R trees Recall
b ox tree to an R tree Recall that the weight of a that the di erence b etween a semi R tree and an R
b ox tree no de is the number of input b oxes stored in tree is that in the former we do not require all leaves
its subtree to b e at the same depth
Theorem Let T b e a b ox tree for a set of n
Theorem Let T b e a b ox tree for a set of n
d
d
b oxes in R such that any query with a range of a
b oxes in R such that any query with a range of a
given type visits at most f w no des of weight w or
given type visits at most f w no des of weight w or
more Then T can b e converted to an R tree of min
more Then T can b e converted to a semi R tree of
imum degree t where every query with a range of the
minimum degree t where every query with a range of
same type visits at most O f t log n log t no des
the same type visits at most O f t no des
Pro of We simply read out the leaves from T in
By applying the conversion algorithms of the theo
order and then construct an R tree where the b oxes
rems ab ove to the structures from the previous sec
o ccur in the same order in the leaves for example
tion we obtain the following results
using an algorithm to construct B trees
d
Consider a b ounding b ox B stored in the R tree It
Corollary Let S b e a set of n b oxes in R with
is the b ounding b ox for some input b oxes that were
stabbing number
stored in consecutive leaves in the b ox tree T Let
i There is an R tree for S of minimum degree t such
B b e the lowest common ancestor of these leaves
that the number of no des visited by any b ox query is
d
Since the minimum degree in the R tree is t the
O n t k log n log t where k is the number
weight of B is t or more Furthermore the no des
of rep orted answers
B for the b ounding b oxes B stored at a xed level
ii There is an semi R tree for S of minimum degree
in the R tree must b e distinct b ecause their de ning
t such that the number of no des visited by any b ox
d
sets form a partition of the leaves in T into consec
query is O n t k
utive sequences Hence we can charge the visited
iii When d there is a semi R tree for S of min
no des of the R tree to visited no des of weight t or
imum degree t such that the number of no des visited
p p
more in T in such a way that a no de in T do es not
n t log n t log n k by any b ox query is O
get charged more than once from no des at a xed
and the the number of no des visited by any p oint
p
level in the R tree Since the depth of the R tree is
t log n k In b oth b ounds k is query is O
O log n log t the b ound follows 2
the number of rep orted answers
iv When d there is a semi R tree for S of min
The construction of Theorem results in losing a
imum degree t such that the number of no des visited
logarithmic factor in the query complexity In the
by any query with a rectangle of asp ect ratio is
p
full pap er we show how to improve this result for
O t log n k where k is the number of
p erfectly balanced b ox trees Recall that a b ox tree
rep orted answers
is called p erfectly balanced if for any no de the weight
v For the cases mentioned under iii and iv there
of its left and right child di er by at most one
is also an R tree of minimum degree t for which the
Theorem Let T b e a p erfectly balanced b ox
number of visited no des is O log n log t times the
d
tree for a set of n b oxes in R such that any
number of visited no des in the semi R tree
query with a range of a given type visits at most
f i no des at level i in T Then T can b e con
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