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USC Computer Science Technical Reports, no. 701 (1999)

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USC Computer Science Technical Reports, no. 701 (1999)

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Content D T op ological and Direction Relations in the W orld of
Minim um Bounding Circles
Ma ytham Safar Cyrus Shahabi
In tegrated Media Systems Cen ter
Departmen t of Computer Science
Univ ersit y of Southern California
Los Angeles California Email msafar shahabiuscedu
Jan uary Abstract
The represen tation and pro cessing of spatial queries is imp ortantinn umerous application domains in the
areas of cartograph y computeraided design computer vision spatial reasoning cognitiv e science image and
m ultimedia databases and geographic applicatio ns A sample query t yp e in these applications is to nd ob jects
that satisfy a sp ecic order in space with resp ect to a giv en ob ject termed direction relations or to nd ob jects
that are in the neigh b orho o d inciden t or included in a giv en ob ject termed top ological relations Due to the
large size of these databases and the complexit y of geometric algorithms appropriate indexing tec hniques and
ob ject appro ximations are crucial The problem is that the relationships b et w een ob ject appro ximations are
usually a sup erset of the relationships b et w een their corresp onding ob jects resulting in a n um ber of false hits
when supp orting spatial queries using ob ject appro ximation s A c hallenge hence is to reduce the n um ber of
false hits in order to decrease the n um b er of times wein v ok e the complex geometric algorithms on the actual
represen tations of the ob jects
In this pap er w e describ e the supp ort of top ological and direction queries using spatial data structures based
on the minim um b ounding circle appro ximation MBC By iden tifying sp ecial cases in MBC relations w e
prop ose extra ltering steps to reduce the n um b er of false hits in determining the actual relationship s among
the ob jects F or top ological queries these sp ecial cases are deriv ed based on the relation b et w een the center s
and r adii of the ob jects MBC s and the w aythese MBCscon tain their ob jects F or direction relations using
cone dir ections metho d the sp ecial cases are based on the partitions of MBCsco v ered b y their corresp onding
ob jects
Key W ords m ultimedia database managemen t spatial relations top ological and direction relations digital
libraries
General T opics access metho ds and data structures m ultimedia systems and digital libraries spatial and
temp oral databases query pro cessing and optimization
In tro duction
Man y applications in the areas of cartograph y computeraided design computer vision spatial reasoning cogni
tiv e science image and m ultim edia databases and geographic applications require to represen t and pro cess spatial
This researchw as supp orted in part b y NASAJPL con tract nr and unrestricted cashequipme n t gifts from In tel and NSF
gran ts EEC IMSC ER C and MRI

relations b et w een ob jects A ma jor data t yp e stored and managed b y these applications is represen tation of t w o
dimensional D ob jects These ob ject databases are then queried and searc hed for dieren t purp oses F or example
am ultim edia application suc h as structured video animation and MPEG standard dene sp ecic ob jects
that constitute dieren t scenes of a con tin uous presen tation These scenes and their ob jects can b e stored in a
database for future queries Tw o imp ortantt yp es of spatial queries are top ological and direction queries With
top ological queries w e are in terested in nding ob jects that are in the neigh borhood inciden t or included in the
query ob ject While with direction queries w eare in terested in nding the ob jects that satisfy a certain lo cation
in space eg nor th south with resp ect to the query ob ject F or the example of m ultimedia applications here a
sample query mightbe nd al l the sc enes that an obje ct A is on the top or ne ar an obje ct B Therefore one of
the imp ortan t functionalities required b y all of these applications is the capabilit y to nd ob jects in a database that
satises a spatial relation with a giv en ob ject
In a spatial database system the ob jects are organized and accessed b y spatial access metho ds SAM s Ho w
ev er since SAM s are not able to organize complex p olygon ob jects directly a common strategy is to store ob ject
appro ximations and use these appro ximations to index the data space Appro ximations main tain the most imp ortan t
features of the ob jects p osition and extension and therefore they can b e used to ecien tly estimate the result of
a spatial queryThe adv an tage of using appro ximations is that the exact represen tation of the ob ject is not often
required to b e loaded in to main memory and b e examined b y exp ensiv e computational geometry algorithms
Almost all the previous studies on spatial databases use the minimum b ounding r e ctangle MBR of an ob ject as
its appro ximation Hence the main access metho d used b y these and other studies is based on
the R tr ee index structure or its v ariations They describ e top ological relations using MBRs and also describ e
direction relations based on either the pr oj ection based metho d or cone dir ections metho d In this
pap er ho w ev er w e fo cus on the minimum b ounding cir cle MBC s of an ob ject as its appro ximatio n Our primary
motiv ation for this decision is that in a previous study w esho w ed the in teresting prop erties of MBC to impro v e
the eciency of similarit y queries for the retriev al of D ob jects b y shap e Therefore in order to utilize the same
appro ximation for other spatial queries ie top ological and direction queries w eneed toin v estigate the usefulness
and feasibilityof MBC s for suc h queries In this pap er hence w eshowho w a subset of MBC features could also
b e utilized to answ er spatial queries As a result there is no need to main tain t woappro ximations ie MBR and
MBC p er ob ject for ecien t supp ort of dieren t queries ie similarit y and spatial In addition w e argue that
dep ending on the application using MBC appro ximations can b e more b enecial as compared to using MBRs First
circles are insensitiveto orien tation and hence an ob jects MBC is unique and in v arian t to translation and rotation
This prop ert y suits the requiremen ts of the top ological relations in whic h MBCsstayin v arian t under top ological

transformations suc h as translation rotation and scaling Second in an indexing tec hnique S pher e tr ee w as prop osed for circles that could b e used to ecien tly supp ort MBC based spatial queries see Sec for further
details Third in applications where queries of t yp e nding neighb ors within a distanc e are frequen tly submitted
and the query regions are irregular MBC s are more suited than MBRs Finally circles o ccup y less storage space
as compared to rectangles
The geometric algorithms emplo y ed to examine spatial relationships b et w een t w o ob jects are usually computa
tionally exp ensiv e and complex Instead relationships b et w een the appro ximations of the ob jects MBCsor MBRs
can b e examined quite ecien tly Therefore a t ypical tec hnique to impro v e the p erformance of spatial queries is
to examine the ob jects appro ximations instead of the actual represen tations of the ob jects t ypically represen ted b y
pol y g ons The problem ho w ev er is that b y using ob jects appro ximations wein tro duce f al se hits in whic h the
relations b et w een the appro ximations is a sup erset of the relations b et w een the actual ob jects Th us a successful
appro ximation is the one that reduces the n um b er of false hits In this pap er w eshowthat MBC could b e utilized
to supp ort b oth top ological and direction relations Moreo v er w eshowthat MBC is a successful appro ximation
for top ological o v erlap queries imp ortan t for in tersection joins W esho w that the p erformance of o v erlap queries
can b e impro v ed b y distinguishing among dieren tt yp es of o v erlaps b et w een the MBC s Then w e determine
the cases in whic h the relations b et w een the actual ob jects can b e determined ecien tly from the relations b et w een
their corresp onding MBC s F urthermore w e distinguish b et w een hal f filled and f ully f illed M B C s whic h
helps in iden tifying sp ecial cases for o v erlap and equal relations that reduces the n um b er of false hits ev en further
F or direction relations b y emplo ying the concept of cone dir ections w ein tro duce t w o metho ds to reduce the
n um b er of false hits W eac hievethat b y partitioning the MBCsin to regions and nd the direction b et w een ob jects
based on the regions they co v er
The remainder of this pap er is organized as follo ws Sec pro vides some bac kground material on spatial rela
tionships and ob ject represen tations In Sec w e dene the top ological relations b et w een ob jects based on their
MBC s Sec pro vides the description of the direction relations b et w een ob jects based also on their MBC s Sec discuss some similarities b et w een MBR and MBC appro ximations In addition it sho ws ho w D ob jects could b e
organized and indexed using the S pher e tr ee index structure and ho w to optimize pro cessing of complex queries
Finally Sec concludes the pap er and pro vides an o v erview on our future plans

Spatial Queries with MBC
In a spatial database system ob jects are organized and accessed using spatial access metho ds SAM s Since SAM s
are not able to organize complex p olygon ob jects directly a common strategy is to store ob ject appro ximations
Consequen tly those appro ximations are used to access the spatial data structures Appro ximations main tain the
most imp ortan t features of the ob jects Therefore spatial queries can b e ecien tly pro cessed on the basis of
appro ximations Ho w ev er spatial relations b et w een appro ximations do es not necessarily coincide with the spatial
relations b et w een the actual ob jects In most cases if the appro ximations of the actual ob jects satisfy a giv en
spatial relation the actual ob jects ma y satisfy a n um b er of p ossible spatial relations see Fig Therefore if
the spatial relations of in terest cannot b e determined b y using the appro ximations then the original ob jects w ould
ha v e to b e loaded in to memory Subsequen tly complex geometric algorithms w ould b e emplo y ed to determine the
exact relations F or example supp ose the top ological relation of in terest is whether t w o ob jects meet or not If the
appro ximations of these t w o ob jects do not share common pointsw e can conclude that the ob jects do not meetOn
the other hand if the appro ximations of these t w o ob jects ha v e common points no conclusion can b e immediately
dra wn ab out the top ological relation b et w een the actual ob jects ie the actual ob jects migh t meet o v erlap etc
In general the appro ximationbased spatial query pro cessing is p erformed in t w o steps First a f il ter step iden
ties based on appro ximations a set of the ob jects that could satisfy the query candidate set The candidate set
ma y con tain ob jects that do not satisfy the query ie false hits Therefore a r ef inement step is required to in
sp ect the exact represen tation of each objectofthe candidate set and eliminate the f al se hits In this step complex
and CPUtime in tensiv e geometric algorithms are used to detect and eliminate the false hits The strategy could
also b e extended to include more lter steps with ner appro ximatio ns in order to exclude more false hits from the
candidate set If the relations b et w een the ob jects are unam biguously determined b y examining the spatial relations
bet w een the appro ximations of the ob jects the renemen t step could b e completely a v oided
T op ological direction and appro ximate distance relations b et w een t w o ob jects are examples of spatial relations
A sample spatial comp osition incorp orating top ological direction and distance features is sho wn in Fig It depicts
t w o ob jects p and q that are disj oint q is Weak North E ast of p and the minim um Euclidean distance b et w een
them is b ounded b y dF or this pap er distance relations will not b e discussed an y further
The p erformance of appro ximation based query pro cessing dep ends on the t yp e of appro ximation used A suitable
appro ximation is crucial for reducing the size of the candidate set ie reducing the n um b er of false hits MBR
as appro ximation has b een studied extensiv ely in the literature Sev eral other approac hes ha v e b een suggested to
main tain nonrectangular appro ximations b y adequate spatial access metho ds eg circles in the S pher e tr eeor

rp
rq
d
Cp
Cq
p
q
Figure T op ological direction and distance features
con v ex p olygons in the Cell tr ee P ol y hedr a tr ee or P tr ee Here w e fo cus on MBC as appro ximation and in
Secs and w e explain ho w MBC s can b e emplo y ed to appro ximate top ological and direction relations b et w een
t w o ob jects W eiden tify the top ological and direction information that MBCscon v ey ab out the actual ob jects they
enclose with resp ect to the inter section mo del In addition w e discuss ho w S pher e tr ees can b e utilized
to index MBC s to exp edite spatial queries The spatial relations b et w een ob jects based on the minim um b ounding
rectangles MBRs appro ximation are describ ed in and summarized in App endix A
T op ological Relations
T op ological relations are describ ed based on the in tersection of the ob jects top ological in teriors b oundaries and
exteriors They are crucial for describing the concepts of neigh b orho o d inclusion and incidence
Although appro ximations main tain the most imp ortan t features of the ob jects top ological relations b et w een
appro ximations do not necessarily coincide with the top ological relations b et w een the actual ob jects F or example
if t w o MBC s meet then the relation b et w een their actual ob jects migh t b e either meet or disj oint see Fig b
Moreo v er dieren t appro ximations of ob jects w ould lead to dieren t sets of top ological relations that could b e
deducted b et w een the ob jects When appro ximating ob jects b y MBRs the top ological relations b et w een MBRs
are determined b y the relation b et w een the lo w er and upp er p oin ts of the b ounding rectangles of the ob jects On
the other hand when appro ximating ob jects b y MBC the top ological relations b et w een ob jects are determined
b y the relation b et w een the radius and the cen ter of the b ounding circles of the ob jects F or example with MBR
appro ximation if MBR of p eq ual s the MBR of q then the relation b et w een p and q couldbe eq ual cov er ed by ov er l ap cov er s meetor disj oint While with MBC appro ximation the relation b et w een p and q could b e the
same as for MBR except that it cannot b e disj oint In general w e could classify the relations b et w een MBRsas
follo ws

In tersect at no p oint ie represen ts the top ological relations inside or disjoin t
In tersect at one p oin t represen ts meet
In tersect at t wopoin ts represen ts o v erlap
In tersect at a line represen ts meet or co v ers
Equal represen ts equal
and with MBC w e get dieren t relations as follo ws
In tersect at no p oin t represen ts inside or disjoin t
In tersect at one p oin t represen ts meet or co v ers
In tersect at t wopoin ts represen ts o v erlap
Equal represen ts equal
Therefore w e need to describ e those relations that MBC s could con v ey ab out their corresp onding ob jects see
Fig Those relations can also b e expressed with the follo wing rules where p and q are MBC s of p and q resp ectiv ely
disj oint p q disj oint p q meet p q disj oint p q meet p q ov er l aps p q disj oint p q meet p q ov er l ap p q contains p q cov er s p q inside p q cov er ed by p q cov er s p q disj oint p q meet p q ov er l ap p q contains p q cov er s p q cov er ed by p q disj oint p q meet p q ov er l ap p q inside p q cov er ed by p q contains p q disj oint p q meet p q ov er l ap p q contains p q cov er s p q inside p q disj oint p q meet p q ov er l ap p q inside p q cov er ed by p q eq ual p q meet p q ov er l ap p q cov er ed by p q cov er s p q eq ual p q This section studies the top ological information that MBCscon v ey ab out the actual ob jects they enclose with
resp ect to the inter section mo del W e also in v estigate the p ossible top ological relations b et w een MBC s

and the corresp onding top ological relations b et w een the actual ob jects that they appro ximate When the top ological
relation b et w een appro ximations is ov er l ap see Fig c the n um b er of p ossible top ological relations b et w een the
actual ob jects is large hence leads to a large n um b er of false hits Therefore w e dene some sp ecial cases that
dep ends on the top ological relations b et w een MBCs eg o v erlap relations F or the case where the MBCsofthe
ob jects ov erlapw ecan ha v e more information ab out the actual ob jects bykno wing the relation b et w een the cen ters
and radii of the MBC s F urthermore w e dene a feature that could b e extracted from MBC of an ob ject to classify
it as hal f filled or fully filled Consequen tly those sp ecial cases and features are used to reduce the n um ber
of f al se hits of the lter step of appro ximatebased top ological relation query pro cessing
Denitions
W e start b y dening some features that could b e extracted from MBC and used later to reduce the n um b er of false
hit in the lter step Hence reducing the renemen t step pro cessing time b y reducing the n um ber of in v o cations of
complex geometric algorithms
Denitio n Sep ar atorDiameter of a circle is the diameter that separates the circle in to t w o halv es one is empt y
and the other half con taining the ob ject see Fig a
Denitio n Halfl le dcir cle is a circle that has a separ ator diameter see Fig a
Denitio n F ul lyl le dcir cle is a circle in whichanydra wn diameter separates it in to t wohalv es eac hhalf
con taining a part of the con tained ob ject see Fig b
Lemma If a cir cle is hal f filledthen at le ast two vertic es of the actual obje ct lays on the separ ator diameter
se e Fig c
Pro of Supp ose weha v e an ob ject suchthatits MBC is hal f filled and has only one p oin t on the separ ator diameter Consequen tlyw e could nd another circle with a smaller diameter to b e the MBC of that ob ject This
con tradicts our assumption that the rst circle w as the MBC of the ob ject
W e refer to the minim um b ounding circle of ob ject q as q and use RE LAT I O N p q to dene that p is related
to q according to the relation RE LAT I O N The query ob ject MBC is referred to as the pr imar y ob ject MBC
p p The ob ject in the database to whic h the pr imar y ob ject MBC is compared is referred to as the r ef er ence
ob ject MBC q q In the examples hereafter the r ef er ence ob ject q is the grey ob ject while the pr imar y ob ject
p is the transparen t ob ject

q
disjoint
p
q
disjoint
p
q
meet
p
a MBC of p disjoin t with MBC of q b MBC of p meet with MBC of q
p
disjoint
q p
meet
q p
overlap
qp
contains
p
covers
q
q q
inside
q
covered-by
p p
c MBC of p o v erlaps MBC of q
p
disjoint
q p
meet
q p
overlap
qp
contains
p
covers
q
q
d MBC of p co v ers MBC of q
q
disjoint
p q
meet
p q
overlap
pq
inside
q
covered-by
p
p
e MBC of p co v ered b y MBC of q
p
disjoint
qp
meet
q p
overlap
q p
contains
q
p
covers
q
f MBC of p con tains MBC of q
q
disjoint
pq
meet
pq
overlap
p q
inside
p
q
covered-by
p
g MBC of p inside MBC of q
meet overlap covered-by covers equal
q p
p
q
q
p
q
p
p
q
h MBC of p equal MBC of q
Figure P ossible relations b et w een actual ob ject giv en the relation b et w een their MBC s

Separator-diameter
Empty-half
Actual object
Separator-diameter
Actual object
Separator-diameter
Empty-half
Actual object
a halflled circle b fullylled circle
c halflled circle with least n um ber of
touc hing v ertices
Figure Half and F ully lled circles
p q
p'
q'
p con tains q and p is inside q
Figure Relations cannot b e satised b et w een actual ob jects dep ending MBC relations
T op ological Relations Using MB C
In order to retriev e ob jects that satisfy a particular relation with qw e need to kno w what relation exist b et w een the
MBC of q and the MBC of those ob jects F or instance in order to answ er the query nd al l obje cts that c ontains
obje ct q the candidate M B C s are those that satisfy the relations contains cov er s or ov er l aps with the MBC of
qT able lists all the top ological queries and the p ossible relations b et w een their corresp onding candidate M B C s
In Fig w epresen t all p ossible relations b et w een actual ob jects when their MBC s are related b y one of the
p ossible relations b et w een MBC s Some relations b et w een MBCsmayiden tify few relations b et w een the actual
ob jects while other relations b et w een MBCsmayiden tify more relations b et w een the actual ob jects F or example
if t w o MBC s meet then the relation b et w een the actual ob jects is either meet or disj oint see Fig b On the
other hand if t w o MBC s are equal the the relation b et w een the actual ob jects could b e meet ov erlap cov er ed by cov er sor eq ual T o answ er a query ab out ob jects that ha v e a particular relation with the query ob ject w e need to nd the
set of candidate M B C s This set con tains MBC s that satisfy a particular relation with the MBC of the query
ob ject see cases in Fig Only those MBC s could include ob jects that could satisfy the queryF or instance
in order to answ er the query nd al l obje cts that c ontain obje ct q the candidate M B Csw ould b e those that
satisfy the relations contains cov er s or ov er l aps the MBC of the q C andidate M B Csto answ er a top ological
relation queryma y satisfy other relations F or example to answ er the query nd al l obje ctspe qual to obje ct q

T able Queries and the candidate M B Cstoansw er them
The query to b e answ ered The relations that the retriev ed MBCsm ust satisfy
nd all ob jects p eq ual to ob ject q eq ual p q nd all ob jects p that contains ob ject q contains p q cov er s p q ov er l aps p q nd all ob jects p inside ob ject q inside p q cov er ed by p q ov er l aps p q nd all ob jects p that cov er s ob ject q contains p q cov er s p q eq ual p q ov er l aps p q nd all ob jects p cov er ed by ob ject q inside p q cov er ed by p q eq ual p q ov er l aps p q nd all ob jects p that ov er l aps ob ject q cov er ed by p q contains p q cov er s p q inside p q eq ual p q ov er l aps p q nd all ob jects p that meet ob ject q cov er ed by p q contains p q cov er s p q inside p q meet p q eq ual p q ov er l aps p q nd all ob jects p disj oint with ob ject q cov er ed by p q contains p q cov er s p q inside p q meet p q disj oint p q ov er l aps p q the only candidate M B C s are those that satisfy the relation eq ual On the other hand those candidate M B Csma y
also enclose ob jects that satisfy the relations meet ov er l ap cov er ed by or covers with resp ect to q see Fig h
Therefore a renemen t step is required if the candidate M B Csare not disj oint Sometimes bykno wing the relation b et w een the MBC s of ob jects one could deduce what relations b et w een the
actual ob jects could not b e satised F or example Fig depicts MBC p contains M B C q Consequen tly one
can deduce that p cannot b e inside q b ecause if p is inside qthen p cannot b e the MBC of p Sp ecial Cases of T op ological Relations
There are some sp ecial cases in whic hw e could utilize the top ological relations b et w een MBC s to reduce the n um ber
of false hits from the lter step W e dene t wosuc h cases where in the rst case w e use the information pro vided
bykno wing what class of ov erlap the relation b et w een MBC s b elongs to see Fig for classes of o v erlap F or
instance if MBC s p and q are as in Fig a then the actual ob jects p and q cannot satisfy the relations contains inside cov er s and covered by see T able for other congurations F or the second case if the t w o MBCsare
equal w e examine if MBC s of the ob jects are hal f filled or fully f illed see Defs and in Sec F or instance in the case where the MBC s of the ob jects are eq ual and giv en that MBC p is hal f f illed and
MBC q is f ully f illed then ob ject p cannot satisfy the relations cov er s p q or eq ual p q see T able for other
congurations The t w o cases are discussed in details as follo ws
MBCs of ob jects o v erlap Here w e assume that the relation b et w een MBCsis ov er l ap W e can ha v e more
information ab out the actual ob jects if w e knew the relation b et w een the cen ters of the MBC s Suc h relation

o v erlapa o v erlapb o v erlapc
o v erlapd o v erlape o v erlapf
o v erlapg o v erlaph o v erlapi
Figure Dieren to v erlap cases according to the cen ters of the MBC s
w ould b e whether eac h MBC con tains the cen ter of the other MBC or not Fig depicts the dieren t cases
in whicht w o ob jects MBC s could o v erlap they dier in whether their cen ters are con tained in the MBC s
of eac h other or not F or instance if MBC s p and q ov er l ap but p do es not con tain the cen ter of q and
vice v ersa see Fig o v erlapa then the actual ob jects p and q cannot satisfy the relations contains inside cov er s and cov er ed by T able illustrates the congurations for whic h the renemen t step is not required when the MBC s of the
ob jects ov er l ap Eac hro w of the table presen ts a dieren tt yp e of ov erlap bet w een the MBC s of the ob jects
Eac h X in a ro w means that the relation b et w een the actual ob jects cannot b e satised according to the heading
of the column F or example in ro w the relation b et w een MBCsis ov er l ap c Hence the relations b et w een
the actual ob jects cannot b e contains but could b e inside Therefore if the relation b et w een MBCsis ov er l aps
then a renemen t step is required in some cases and not in other cases Fig sho ws some examples where a
renemen t step is required to iden tify the relation b et w een the t w o actual ob jects These examples corresp ond
to the cases where the en tries in T able are blanks F or example Fig a demonstrates cases where t w o
MBC s ov er l ap are of t yp e ov erlap g ov er l ap hor ov er l ap i as dened in Fig but w e do not kno w if the
actual ob ject p contains q or not In Fig w esho w some examples where a renemen t step is not required to
dene the relation b et w een the t w o actual ob jects These examples corresp ond to the cases where the en tries

T able Congurations for whic h a renemen t step is not required when MBCs of the ob jects o v erlap
con tainspq insidep q co v erspq co v ered b ypq
Ov erlapa X X X X
Ov erlapb X X X X
Ov erlapc X X
Ov erlapd X X X X
Ov erlape X X X X
Ov erlapf X X
Ov erlapg X X
Ov erlaph X X
in T able are mark ed byan X eg the case where t w o MBC s ov er l ap of t yp e ov er l ap dand w ew antto
kno w if the actual ob ject p contains q or not
Lemma If p overlaps overlapd with q then the obje ct c ontainedby p c annot c ontain the obje ct c ontaine d
by q Pro of p contains q implies that q lls only the dark shaded area in Fig a whic h leads to a con tradiction
that q is the MBC of qW e could ha v e a smaller circle q to b e the MBC of q notice that q has a smaller
radius than q Lemma If p overlaps overlape with q then the obje ct c ontainedby p c annot b e inside the obje ct
c ontainedby q Pro of In Fig b w e assume that p is a hal f filled circle with the separ ator diameter sho wn in the
gure F or p to b e inside q w ould imply that p is completely con tained in the dark shaded area That is only
one v ertex of ob ject p la ys on the separ ator diameter whichcon tradicts the fact that a hal f f illed circle
has at least t wov ertices of the actual ob ject la ying on the separ ator diameter see Lemma MBCs of ob jects are equal Here w e describ e another w a y to reduce the n um b er of false hits from the lter
step Supp ose the relation b et w een MBCsis eq ualw e can obtain more information ab out the actual ob jects
if w e kno w whether their MBCsare hal f filled or fully f illed see Defs and T able illustrates
the congurations for whic h the renemen t step can b e en tirely eliminated when the MBC s of the ob jects are
eq ual Eac hro w of the table presen ts a dieren tt yp e of MBC s Eac h X in a ro w means that the relation
bet w een the actual ob jects cannot b e satised according to the heading of the column F or instance in the
case where the MBC s of the ob jects are eq ual and giv en that the MBC p is hal f filled and the MBC q is

a Examples where p con tains or not con tains q
con tainspq
q
p
q
p
q
p
o v erlapg o v erlaph o v erlapi
not con tainspq
q
p
q
p
q
p
o v erlapg o v erlaph o v erlapi
b Examples where p inside or not inside q
insidepq
p
q
p
q
p
q
o v erlapc o v erlapf o v erlapi
not insidepq
p
q
p
q
p
q
o v erlapc o v erlapf o v erlapi
c Examples where p co v ers or co v ered byq
co v erspq
q
p
q
p
q
p
o v erlapg o v erlaph o v erlapi
co v ered b ypq
p
q
p
q
p
q
o v erlapc o v erlapf o v erlapi
Figure Examples in whic h the t w o MBCs are o v erlapp ed and a renemen t step is required

p'
q'
q''
p'
q'
Separator-diameter
o v erlapd o v erlape
a Example where p cannot con tain q b Example where p cannot b e inside q
Figure Examples in whic hthe t w o MBCs are o v erlapp ed and a renemen t step is not required
T able Congurations for whic h a renemen t step is not required when MBCs of the ob jects are equal
co v ered b ypq co v erspq equalpq
p halflle d q fullyll ed X X
p fullyll ed q halfll ed X X
f ully f illed ro w then the ob ject p cannot satisfy the relations cov er s p q or eq ual p q column and

Direction Relations
Direction relations describ e the order of ob jects in space eg nor th south and are crucial for establishing spatial
lo cation and path nding With MBR appro ximatio n a pr oj ect based metho d is t ypically used to dene
direction relations b et w een ob jects With this metho d a plane is partitioned in to some subpartitions and the direction
relation b et w een t w o ob jects is dened b y the subpartitions they o ccup y This metho d ho w ev er is not suited when
appro ximating ob jects with MBC Therefore w e utilize another metho d named cone dir ections that could b e
easily adapted to MBC s With cone based a plane is divided in to partitions that denes the primitiv e direction
relations see Fig d No w it is necessary to in v estigate what direction relations MBC s could con v ey ab out their
actual ob jects
Section presen ts the direction relations b et w een ob jects based on the concept of cone dir ections in whic h
the direction domain is divided in to a n um ber of cones or triangles with a similar resolution size W eshowho w
the cone dir ection is w ell suited for MBC s As the case with top ological relation query pro cessing since w euse
MBC as ob ject appro ximation to answ er the queries a lter and a renemen t steps are required F urthermore w e
in tro duce t w o metho ds to reduce the n um b er of false hits of the lter step b y breaking the MBC s of the ob jects
in to partitions and further partitioning the same level regions of the ob jects Subsequen tly the renemen tstepis

North
South
East West
p
q
100
o
North
South
East West
q
p
North
South
East West
q
p
North
South
East
West
q
Same Level
a Primitiv epoin ts b A p oin t and an ob ject c Tw o ob jects d Using MBCs
Figure Primitiv e direction relations using coneshap ed metho d North
South
East West
q
p1
p2
p3
p4
North
South
East West
q
p
p
r
a Ob jects spanning more than one partition b Incorrect inferences
Figure Sp ecial Cases
needed to eliminate all the false hits pro duced b y the lter step
Conebased Direction Relations
This section presen ts the direction relation b et w een ob jects based on the concept of cone dir ections With this
concept the direction domain is divided in to a n um ber of cones or triangles with a similar resolution size and a
single p oin t q
i
is used to representthe ref erence ob ject q The cones are consequen tly used to dene the direction
relation relativ e to the cone that con tains the p oin t p
j
whic h represen ts the pr imar y ob ject p see Fig a In
addition the direction relation b et w een t w o ob jects can b e sp ecied more accurately using the angle b et w een a line
connecting the ob jects and a xed direction in space F rom Fig a the direction relation of ob ject p relativeto
ob ject q is represen ted as p is north of ob ject q with an angle v alue of deg The relation b et w een t w o p oin ts
could also b e extended to dene direction relations b et w een actual ob jects W e illustrate that bythe t w o examples
sho wn in Fig b and Fig c in whic hw e dene the direction relation b et w een a p oin t and an ob ject and b et w een
t w o ob jects

1
3
42
North
South
East West
q
p'
1
0
1 0
r
a P artitions on MBCs b Direction relations on partitions
Figure Minim um b ounding circle partitioning
The cone based metho d could b e easily adapted to our represen tation of ob jects as MBC s The direction relation
bet w een anyt w o ob jects can b e computed b y using their MBC s as ob ject represen tativ e W e dene these relations
according to the direction relations in Fig d where the plane is represen ted b y direction partitions The circle
represen ts the MBC of the r ef er ence ob ject the parts of the primary ob ject that la y in the MBC of the r ef er ence
ob ject are considered to b e at the same lev el with the reference ob ject
Some ob jects could span more than one partition of the direction partitions see Fig a for example an
ob ject could ha v e parts in the North partition and other parts in the E ast partition In order to dene suc h relations
correctlyw e dene the direction relation b et w een t w o ob jects as a p erm utation from the set fN north S south E east W w est SL same lev elg Therefore for the ob jects p p p and pin Figaw e could dene their
direction relations with resp ect to the r ef er ence ob ject q as fNg fNE g fWNE gand fNESW g resp ectiv ely Using MBC s as represen tativ e for ob jects when nding the direction relations b et w een ob jects ma y lead to
incorrect inferences ie false hits F or example in Fig b the direction relation b et w een ob jects p and p with
resp ect to q isamem ber of fNorthEast g but the actual ob jects do not satisfy these relations ie p satises fEastg
and r satises fNorthg A naiv e solution to this problem could b e a renemen t step whic h requires the retriev al of
the original ob jects Our prop osed solution ho w ev er is to lter out more false hits in the lter step W eac hiev e
that b y dividing the MBC s and the same level regions in to partitions see Fig a F or the partitions where
the actual ob ject la ys in w e assign a otherwise w e assign a Subsequen tly the direction relation b et w een t w o
ob jects could b e dened b y using only the partitions mark ed F or example in Fig b the direction relation
bet w een ob ject q and p is dened as fNg only and not fNE g The same concept could b e used for the case of
ob jects at the same lev el F or instance in Fig c the direction relation b et w een p and q could b e dened more
accurately as fNorthg instead of fNorthEastW estg The denition of direction relations could b e extended to dene direction relations b et w een D ob jects In D
w orld w e appro ximate eachobject b y its minim um b ounding sphere Giv enapointinDplane w e dene eigh t

North
South
East West
p
q
1
1 1
0
0
1
1 1
North
South
West
q
p
q
a partitions of the same lev el direction b partition of ob ject p and q c the direction relation b et w een p and q
Figure Same lev el partitioning
direction relation partitions with the reference p oin t as its cen ter see Fig F our of the partitions are considered
to b e abov e the r ef er ence poin t and the other four are considered to b e bel ow the r ef er ence poin t The follo wing
is a list of all direction partitions
par tition all p oin ts in this partition and its extension to the outside are considered to b e Below Front Lef t
of the r ef er ence poin t
par tition Below F r ont Rig ht
par tition Below Back Rig ht
par tition Below Back Lef t
par tition Abov e F r ont Lef t
par tition Abov e F r ont Rig ht
par tition Abov e B ack Rig ht
par tition Abov e B ack Lef t
All the discussion for the D ob jects direction relations could b e easily extended for D ob jects The only
dierence is that w e divide eac h MBS to partitions instead of partitions of MBC s as for D ob jects The
MBS of the r ef er ence ob ject is considered as par tition W e divide the rest of the space in to dieren t subspaces
where eac h one of them is the extension of the partitions in the MBS of the ref erence ob jects see Fig MBC Similari ti es to MBR
T o exp edite the pro cessing of spatial queries ob jects could b e organized and indexed using the Sphere tr ee index
structure S pher e tr ee is a spatial access metho d used for storing and retrieving circles and spheres as opp osed

Figure Primitiv e direction relations in D space
Figure Primitiv e direction relations partitions in D
to the R tr ee whic h is used for rectangles Therefore Sphere tr ee is a go o d candidate for indexing MBC s
Ho w ev er in order to utilize Sphere tr ee to index MBCsw e need to address the mismatchbet w een appro ximation
relations and actual relations for in termediate no des of the tree In tuitiv ely since MBC appro ximations of ob jects
are dieren t than their MBR appro ximations w e exp ected that this w ould c hange the rules of propagation in the
in termediate no des of S pher e tr ee compared to those of R tr ee On the con traryour in v estigations sho w ed
that the relations that ma y b e satised b et w een an in termediate no de P and q so that the no de b e selected for
propagation in the Sphere tr ee are similar to those relations obtained for the case where Rtree w as used in The same results w ere also obtained when in v estigating the empt y results query optimization ie where the result
of the query is kno wn to b e empt y without running the query The t w o cases are discussed in details as follo ws
Sphere Index for MBCs In order to retriev e the spatial relations using Sphere tr eesw e need to dene more
general relations for propagation of the in termediate no des ma ycon tain sev eral MBC s of the tree structure

P
q'
p'
P
q'
p'
a con tainsP q b co v ersP q Figure In termediate no des propagation
By kno wing the spatial relation b et w een the MBC of the in termediate no de P and that of the r ef er ence
ob ject q one could conclude the p ossible spatial relations b et w een q and all the pr imar y ob jects con tained
in PF or instance the in termediate no des that could enclose p that cov er s q ma y satisfy the more general
constrain t contains P q cov er s P q Otherwise the in termediate no de w ould not con tain an y p that
satises the relation cov er s p q Th us all the MBCsinside thisin termediate no de ma y b e ignored from
further searc h F ollo wing this strategy the searc h space is pruned b y excluding the in termediate no des P that
do not satisfy the previous constrain t Fig pro vides examples of in termediate no des P that ma y con tain
MBC s that cov er s the MBC of q In App endix A w e included T able that presen ts the relations that ma y b e satised b et w een an in termediate
no de P and q of the r ef er ence ob ject so that the no de b e selected for propagation in the tree structure as
in Notice that a similar relation b et w een in termediate no des and the r ef er ence ob jects MBC exists for all
the lev els of the tree structure F or instance for the example sho wn in Fig an in termediate no de P whic h
encloses the in termediate no de P that satises the general constrain t contains P q covers P q ma y
also satisfy a similar constrain t This can b e easily concluded from T able and is applicable to all top ological
relations of Fig The same concept could b e applied to direction relations In order to nd all the ob jects
to the North of a r ef er ence ob ject qw e need to retriev e all ob jects whic h has nonempt y partitions mark ed
in the nor th partition of the MBC of q A similar partitioning could b e applied to the in termediate no des
of the S pher e tr ees and could b e used to prune the searc hspace b y excluding the in termediate no des P that
do not ha v e nonempt y partitions in the nor th partition of the MBC of q Complex Queries Sometimes top ological relations could b e dened as disjunctions of the relations of mtlo w er
qualitativ e resolution Consider the query nd al l the p ostal oc es in a given c ounty in a map The
result should b e all the oces that are inside or cov er ed by that coun t yTh us the in terpretation of in is
inside covered by In general the MBC s to b e retriev ed are the union of the MBC s to b e retriev ed b y
eac h of the relations that b elongs to the disjunction

q1
q2
p
Figure A query with t w o reference ob jects and empt y results
In some cases eg empt y results query the results of a top ological relation query could b e returned without
running the query seman tic query optimization The empt y result is deduced from the relations required in
the query b et w een the ref erence ob ject and the pr imar y ob jects An example of queries of this form is nd
al l obje cts that ar e cov er ed by q and ar e inside q W e can determine that the result of this query is
emptyif w e knew that q meets q As Fig illustrates if p is cov er ed by qand q meets q it cannot
b e the case that p is inside q In App endix A w e included T able that illustrates the relations b et w een
ref erence ob ject and pr imar y ob jects for whichanempt y result is returned without running the query Conclusion and F uture W ork
In this pap er w e describ ed some tec hniques to supp ort spatial queries on D ob jects using their MBC appro xima
tions First w e fo cused on the supp ort of top ological relations as dened b y the in tersection mo del W e prop osed
some ltering tec hniques based on sp ecial cases for MBC in order to reduce the n um b er of false hits Those cases
dep end on the top ological relations b et w een MBC s suc h as kind of o v erlap relations or halflled and fullylled
MBC s Next w e prop ose an extra ltering step to exp edite direction relations b et w een ob jects utilizing their MBCs
based on cone dir ections metho d Wein tend to extend this w ork to supp ort three dimensional ob jects Our preliminary in v estigations showthat
analogous to the sp ecial cases based on MBC of D ob jects w e can dene sp ecial cases for D ob jects b y using
their minim um b ounding spheres F urthermore w e plan to design a top ologicaldirection mo del for the sp ecication
of the spatiotemp oral relationships among ob jects eg ob jects in a video sequence
References
S Ghandeharizadeh StreamBased V ersus Structured Video Ob jects Issues Solutions and Challenges In S Jajo dia and
V Subr ahmanian e ds Multime dia DB Systems Issues and R es Dir e ct SpringerV erla g T Brinkho HP Kriegel and R Sc hneider Comparison of Appro ximations of Complex Ob jects Used for Appro ximation
based Query Pro cessing in Spatial Database Systems In the Pr o c e e dings of the th Internationa l Confer enc e on Data
Engine ering ICDE
R Bec kmann and HP Kriegel The Rtree An Ecien t and Robust Access Metho d for P oin ts and Rectangles Pr o c
A CM SIGMOD Int Conf on Management of Data A tlantic City NJ pp A Guttman Rtrees A Dynamic Index Structure for Spatial Searc hing Pr o c A CM SIGMOD Int Conf on Management
of Data pp P V O osterom and E Claassen Orien tation Insensitiv e Indexing Metho ds for Geometric Ob jects th Internationa l
Symp osium on Sp atial Data Hand ling Zurich Switzerland p H Samet Spatial Data Structures App e ars in Mo dern Datab ase Systems The Obje ct Mo del Inter op er abili ty and Beyond
W Kim ed A ddison WesleyA CM Pr ess pp MJ Egenhofer Reasoning ab out Binary T op ological Relations In the Pr o c e e dings of the Se c ond Symp osium on the
Design and Implementation of L ar ge Sp atial Datab ases SpringerV erla g LNCS MJ Egenhofer The DirectionRelati on Matrix A Represen tation for Direction Relations b et w een Extended Spatial
Ob jects httpwwwsp atial maine e d u maxmax ht ml K Zimmermann and C F reksa Qualitativ e Spatial Reasoning Using Orien tation Distance and P ath Kno wledge IJCAI
Workshop on Sp atial and T emp or al R e asoning Chamb ery August D Greene An Implemen tation and P erformance Analysis of Spatial Data Access Metho ds In the Pr o c e e dings of the th
International Confer enc e on Data Engine ering ICDE D P apadias and T Sellis The Seman tics of Relations in D Space Using Represen tativeP oin ts Spatial Indexes In
F r ank AU Camp ari I e ds Pr o c e e dings of the Eur op e an Confer enceon Sp atial Information The ory COSIT Springer
V erlag D P apadias Y Theo doridis and T Sellis The Retriev al of Direction Relations using Rtrees In the Pr o c e e dings of the
th Internationa l Confer enceon Datab ases and Exp ert Systems Applic ationsDEXA Springer V erlag LNCS Y Theo doridis D P apadias and E Stefanakis Supp orting Direction Relations in Spatial Database Systems T e chnic al
R ep ort KDBSLABTR National T e chnic al university of A thens Gr e e c e D P apadias and Y Theo doridis Spatial Relations Minim um Bounding Rectangles and Spatial Data Structures T e chnic al
R ep ort KDBSLABTR National T e chnic al University of A then Gr e e c e D P apadias Y Theo doridis T Sellis and MJ Egenhofer T op ological Relations in the W orld of Minim um Bounding
Rectangles A study with Rtrees Pr o c e e dings of A CM SIGMOD International Confer ence onManagementofData
Y Theo doridis and T Sellis On the P erformance Analysis of Multidimensi on al Rtreebased Data Structures T e chnic al
R ep ort KDBSLABTR National T e chnic al university of A thens Gr e e c e MJ Egenhofer On the Robustness of Qualitativ e Distance and DirectionReasoni ng In the Pr o c e e dings of A utoCarto
in Charlotte North Car olina pp C F reksa Using Orien tation Information for Qualitativ e Spatial Reasoning App earsin AUF r ank I Camp ari U
F ormentini e ds The ories and Metho ds of Sp atioT emp or al R e asoning in Ge o gr aphic Sp ac e LNCS SpringerV erla g
Berlin AU F rank Qualitativ e Spatial Reasoning with Cardinal Directions Pr o c e e dings of the Seventh A ustrian Confer enceon
A rticial Intel ligenc e Wien Springer Berlin pp AU F rank Qualitativ e Spatial Reasoning ab out Distances and Directions in Geographic Space Journal of Visual
L anguages and Computing pp Emo W elzl Smallest Enclosing Disks Balls and Ellipsoid s New R esults and new T r ends in Computer Scienc e L e ctur e
Notes in Computer Scienc e RH Guting An In tro duction to Spatial Database Systems Invite d Contribution to a Sp e cial Issue on Sp atial Datab ase
Systems of the VLDB JournalV ol No Octob er CH P apadimitrio u D Suciu and V Vian uT op ological Queries in Spatial Databases A CM PODS Montr e al Queb e c
Canada ChangWhan Sul KeeChang Lee and Kw angyun W ohn Virtual Stage A Lo cationBased Karok e System IEEE Multi
me dia pp C Shahabi M Safar and A Hezhi Multiple Index Structures for Ecien tRetriev al of D Ob jects T o app e ar in the
Pr o c e e ding of IEEE th Internation al Confer enc e on Data Engine ering Sydney A ustr alia D P equet and Z CiXiang An Algorithm to Determine the Directional Relationshi p b et w een Arbitrarily Shap ed P olygons
in the Plane Pattern R e c o gnitionV ol No pp A Spatial Relations Using Minim um Bounding Rectangles
This section describ es the primitiv e top ological and direction relation sets as dened in
T able Relations for the in termediate no des
Relation b et w een MBC s Relations b et w een in termediate no de P that ma yenclose
MBC p and reference MBC
eq ual p q eq ual P q cov er s P q contains P q contains p q contains P q inside p q ov er laps P q eq ual P q inside P q cov er s P q covered by P q contains P q cov er s p q cov er s P q contains P q cov er ed by p q ov er laps P q eq ual P q cov er s P q cov er ed by P q contains P q disj oint p q ov er laps P q meet P q covers P q contains P q disj oint P q meet p q contains P q cov er s P q meet P q ov er laps P q ov er lap p q contains P q cov er s P q ov er laps P q A T op ological Relations Using MB Rs
W e rst start b ysho wing all the p ossible top ological relations b et w een MBRs as dened b ythe inter section mo del These
relations are illustrated in Fig The corresp onding top ological relations for ob ject appro ximated bytheir MBC are sho wn
in Fig p q
p q p q
q p
disjoin tpq meetpq o v erlappq co v erspq
q p qp qp pq
con tainspq co v eredb ypq insidepq equalpq
Figure P ossible top ological relations b et w een MBRs
p
q
p
q
p
q
p
q
disjoin tpq meetpq o v erlappq co v erspq
p
q
q
p
p
q
pq
con tainspq co v eredb ypq insidepq equalpq
Figure P ossible top ological relations b et w een MBC s
T op ological relation set mt consists of the relations disj oint and not disj oint This set is used if the only top ological
relation of in terest is to c heckift w o ob jects share some common p oin ts or not A second set of top ological relations is
named mt This set consists of the top ological relations describ ed in Fig disj oint meet ov erlap cov er s contains cov er ed by inside and eq ual T able presen ts the relations that ma y b e satised b et weenanin termediate no de P and the q of the ref erence ob ject
so that the no de b e selected for propagation in the tree structure as in T able illustrates the relations b et w een ref erence ob ject and pr imar y ob jects for whic h an empt y result is returned
without running the query as in When a query in v olving t w o ref erence ob jects qand q is giv en if the top ological
relation b et w een the r ef er ence ob jects exists in T able the output of the query is empt yF or the ab o v e query in addition

T able MBCs relations and empt y results queries
disjoin tpq meetpq equalpq insidepq cvrdb ypq con tainpq co v erspq o v erlappq
disjoin tpq e ct cv
m e i
cb ct
cv o
e ct cv
e ct cv
m e i
cb ct
cv o
m e i
cb ct
cv o
e ct cv
meetpq e i cb i ct d e i cb ct cv o
d m e ct cv
d e ct
cv
m e i cb ct cv o
e i cb ct cv o
e ct cv
equalpq m e i cb ct cv o
d e i cb ct cv o
d m i cb ct cv o
d m e cb ct cv o
d m e i ct cv o
d m e i cb cv o
d m e i cb ct
d m e i cb ct cv
insidepq e i cb d m e i cb
d m e i cb
cv o
d m d m e i cb o
d m e i cb cv o
d m e i cb cv
d m e i cb
cvrdb ypq e i cb d e i cb d m e i cb ct o
d m e
ct cv
d m i ct
d m e i cb cv o
d m e
i cb o
d m e i cb
con tainspq m e i cb ct cv o
m e i cb ct cv o
d m e cb ct cv o
d m e cb ct cv o
d m e cb ct cv o
e ct cv e ct cv
co v erspq m e i cb ct cv o
e i cb ct cv o
d m e i ct cv o
d m e cb ct cv o
d m e ct cv o
e i cb i ct e ct cv
o v erlappq e i cb e i cb d m e i cb ct cv
d m e ct cv
d m e ct cv
e i cb e i cb Where d disj ointm meete eq uali insidecb cov er ed byct containscv covers and o ov er lap
to meet if qand q are related b y eq ual contains disj oint or cov er ed bythe result isalsoempt y Eac hen try at ro w
ripq and column rjpq where ri and rj are relations of mt is the comp osition of the relations ripq and rjpq
with resp ect to mt

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USC Computer Science Technical Reports, no. 948 (2014)

Description

Maytham Safar and Cyrus Shahabi. "2D topological and direction relations in the world of minimum bounding circles." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 701 (1999).

Asset Metadata

Creator Safar, Maytham (author), Shahabi, Cyrus (author)

Core Title USC Computer Science Technical Reports, no. 701 (1999)

Alternative Title 2D topological and direction relations in the world of minimum bounding circles (title)

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)

Tag OAI-PMH Harvest

Format 23 pages (extent), technical reports (aat)

Language English

Unique identifier UC16269795

Identifier 99-701 2D Topological and Direction Relations in the World of Minimum Bounding Circles (filename)

Legacy Identifier usc-cstr-99-701

Format 23 pages (extent),technical reports (aat)

Rights Department of Computer Science (University of Southern California) and the author(s).

Internet Media Type application/pdf

Source 20180426-rozan-cstechreports-shoaf (batch), Computer Science Technical Report Archive (collection), University of Southern California. Department of Computer Science. Technical Reports (series)

Access Conditions The author(s) retain rights to their work according to U.S. copyright law. Electronic access is being provided by the USC Libraries, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Repository Email csdept@usc.edu

Inherited Values

Title Computer Science Technical Report Archive

Description Archive of computer science technical reports published by the USC Department of Computer Science from 1991 - 2017.

Coverage Temporal 1991/2017

Repository Email csdept@usc.edu

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)