PARTITIONING AND GEOMETRIC EMBBDDING ... - Semantic Scholar
PARTITIONING AND GEOMETRIC EMBBDDING OF RANGE SPACES OF FINITE VAPNIK-CHBRVONBNKIS DIMENSION
i . fntroduction
and statement
of results.
dimensional.
Permissionto copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permissionof the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 0 1987 ACMO-89791-231-4/87/GUG6/0331 754
331
~ucl,idean
space?
finite
A cr X it
for
A such
xcrnA)
produces
thet.,
can
for
be
a data
all
computed
r
in
in
lower
structure R,
time
I:(fl(Jo most
at
bound
kep,t
!
in
i,f
for
upper
results Taking
*!lAll. In
our
crerncd
results
with
I-nngc
we
the
sr:src:h
will
First
be
two
mainly
tions
con-
components
of
on
simpJy
a
problem.
a
Definition.
A range
(X,Rj,
where
tr i: r $. of’ S and S is
space
X is
X are
called
members
of
fSinjts.
and
H are
riwer
a pair
R E 2X.
:J.ements
or
called
(countable) . _----.--_-u
(rountnble).
S is
a set
Doifits
ranges
if
X is
of
which
of 3.
subsets with
finite
A’ A’
ble
lem
1.
Consider
the
(.W,Hz*,x,(No,+):,
of’
points
not
of
i.n
semigroup
real.
nonnegative
tion
and
the
problem
for
a given
p 0 i n t .7 in
the
plane
pciinis
of
h*
cient
Ly.
‘Phi.s
c,ommonly r:lnge
we
to
by
are
in
every
x
intcrest,ed a given
5% query
WC make
the
Definition. lel
A bc
hal-f-
X.
Then
RR(A)
sets of setting
is
set
of
o
a range
{Anr
: I-F.R).
that
A is
lhere
A C R2 that.
,Pol lowing
on
where
Obviously,
the
halfplanc.
the
assumptions
in
any
bp
docidod
(j
ij
for
ant!
A
x
senrrh
c X and in
any
const.ant.
!.ime
r
be
c H,
d
thon
in
cnnst,nnt-
l’l’he
rc!;ldc!r 11 6 (
x
i II lnally
c’awos.
I
for
r
C Ii,
implies G can
that
This
(i)
be
that ,
is
t.hc
f’oI.-
whc:rr
alhC.1,
set.
of’
the
set
i:hesr-:
in
u
of
infinj
S’i.s
of
1,hon
A of
by
exists,
is we
Ir:.
sub.-
intr+rllti !A)
z
WC! :;:iy
v-qnjk-simply d
X of’
shattered
01’
811
by i . c.
The
R.
s pace
f!lrrnrents
of’ r0 (or ___ largest
dimension
sion
a r n n gc
obtained in I?,
hy
is t.hc! d
(Vapni
l.hc
such
that
CII rcl i
R (if’ -1,
say
R is
and
the
i 1’ no
di.mc?n,.
0
k-Chervoncnki
se:rrc:h its
s)
~Ji~merr:aion
(x,K,o,~j
problem
of
A C. r?
Example
in
he
the
2. set.
underlying
not
no
problem
real
is
range
of is
assumyistic
(X* ,PREF(Z* i,t
our
332
t hc
~lphabrl.
OF
j.h~
spncc
for
not L!)
the
sot
))
2’* --nr~-fj~y
is
I y,
a range spa!.?..
the:
1~::. P’
nlph;rlai~l.
word
X).
For
the
set.
c.9 ii I I
a J.angunge
I wel.).
Obvious
anti
over
denotne w and,
= {pref(w) in
an
words empty
pref(w)
words
for
Y be all the
prefixes
performed
these
are
Let. of
P (including
(i)),
PRRP(L)
ergur!
a
(X,H).
c r?
t imc.
m’l.gh
r+xcopt.
to
= ZA,
sub set
maximal
PRGF(L) t .i 0
J.ower nnswC?r
[VC],
dimension ______.___^ S) is the
such
The
and in
;I gives
time,
decided
(which
1 addition
IlFI (A)
a
nality
w c: x*,
( iii
: which
( X , it ) be te
can bc a range If’
of ---
empty,
prohlem
constant
A C X and
n r 1: $7 can
S ::
shattered
dimension for
possi-
.
denotes
exists
3 range
(i)
of
romput.ntion;ll
Ix,rt,o,Gj:
c:an
all.
Y’ 5 Ii t.0
to
founrl,
a .fi.ni
A that A with
Chcarvonenkis .L--‘----.-.---dimension _--,-.--_-
problem
,Q, (2RZ,~)),
finiteset
Let
and
halfplane-
B Rz:.
2:”
dat.:i
for
number
leads
c:ont,c?xt)
number cffi-
reporting
if,
theorc:t.ic.
bc
further
reporting) as
an..’
t,his
a range
(see
can
some 6,
needed
naturally
di f Pcrrnnt
A of
a yuory
(1s the
(R” ,Hz*
for
of
that
b,i.g
to
k data
set
in
with
definitions
design
the
time
defi.nitions
problem.
3.s npccified &,(>i,i .: (x) herrA
Jie
halfplane-range
po-ints
to
Then
information
l’h:i.s
E
is
as
the
have
there
for
A of’ datn
t.i.mc
optimal,
be
lowi.ng
x
we
g. ,
query.
the
every
be determi.ned search problem
referred
‘Phc
1 for
is such
can range
counting
=
set
linear
semigroup
(c.
the
is
is
finite
A that.
5' Ha*
the
integers X(x)
s1.rticl’ure
plane
H:tl, ‘f%~ll) - a,; : I,-- J ) -i &Q.-J (n-,1)
n)O 01.1
for
n>O,
= 1 For
and
d,n)l.
PROPOSITlON 1.1.. of
,
in
hc* rl i-art&r!
int.r~gctrs
(Al,A>
U{Ai
if
S),
(X,K)
(
A is
E.
cal
A (with
a (v,m,
TA .
1 I
!Ai
Icd
respect.
E)-port
ition
P
n and
P. with
a nodo
Lrrar* i.n
t.imc.
a rca1
A partition-
for
such
A contains
i I ion-
dofinitionc
m ho
(v,m,&)--partition
The I’ ba
E be
r
trcr!. Let
v and
and
1
tion-tree
performed
;~.-iaurn[~-
r and A be sets. A if either A ‘: r
that
say
for
First
a parl:i
.I:h:lt.
rc-
performrttl on
query Let
the
di.mension
i)
c’on-
(Vo1.c
part
be
following
k 6 v, : l.(iO,
i s defined
thr!
recall
Lead
sures
TA,
dis.joint
time.
will
in
parts(p)
the
for we
1 inear
to
d.
Par
nor
C), in
Ai
oil
A for
that Definition.
in
parts
t-i 01)s ,
informaat
parts
I s; I-I:A~ a global
set
in
partjti.on
in
start
init.inl.ly
the
of
(due
the
bound
element
If
is
I-csuJ
qllotc
l,ow?r
query
dimension
r,
with
neutral
fo t.l.ow-up
sub1 inr!>tr the
which
12(d~ Ing’ fin
if
E{c?(x)
we
SUM,
dundant
of
compute!
a rnngc
nctually
problr:m
natur;tl
to
variable
tained
all
for
scclrch
Tho
whc.1 har
is,
CHII
sub I i near
achieved
dimension.
qcll‘:t t. i Iin
above,
be
order
let
taincd
fol.lowing from
proposition
calculations
can in
be
ob-,
[HW].
A the
PROPOSITION 1.2.
exac.-
search
that
space
333
S and
Let
with
problem
let
A be
P he
a
rongc
underlying .a set
r:tnc~r~ of
n
elements of S. If TA is a (v,n,&)partition tree of A with respect to (l
i . fntroduction
and statement
of results.
dimensional.
Permissionto copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permissionof the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 0 1987 ACMO-89791-231-4/87/GUG6/0331 754
331
~ucl,idean
space?
finite
A cr X it
for
A such
xcrnA)
produces
thet.,
can
for
be
a data
all
computed
r
in
in
lower
structure R,
time
I:(fl(Jo most
at
bound
kep,t
!
in
i,f
for
upper
results Taking
*!lAll. In
our
crerncd
results
with
I-nngc
we
the
sr:src:h
will
First
be
two
mainly
tions
con-
components
of
on
simpJy
a
problem.
a
Definition.
A range
(X,Rj,
where
tr i: r $. of’ S and S is
space
X is
X are
called
members
of
fSinjts.
and
H are
riwer
a pair
R E 2X.
:J.ements
or
called
(countable) . _----.--_-u
(rountnble).
S is
a set
Doifits
ranges
if
X is
of
which
of 3.
subsets with
finite
A’ A’
ble
lem
1.
Consider
the
(.W,Hz*,x,(No,+):,
of’
points
not
of
i.n
semigroup
real.
nonnegative
tion
and
the
problem
for
a given
p 0 i n t .7 in
the
plane
pciinis
of
h*
cient
Ly.
‘Phi.s
c,ommonly r:lnge
we
to
by
are
in
every
x
intcrest,ed a given
5% query
WC make
the
Definition. lel
A bc
hal-f-
X.
Then
RR(A)
sets of setting
is
set
of
o
a range
{Anr
: I-F.R).
that
A is
lhere
A C R2 that.
,Pol lowing
on
where
Obviously,
the
halfplanc.
the
assumptions
in
any
bp
docidod
(j
ij
for
ant!
A
x
senrrh
c X and in
any
const.ant.
!.ime
r
be
c H,
d
thon
in
cnnst,nnt-
l’l’he
rc!;ldc!r 11 6 (
x
i II lnally
c’awos.
I
for
r
C Ii,
implies G can
that
This
(i)
be
that ,
is
t.hc
f’oI.-
whc:rr
alhC.1,
set.
of’
the
set
i:hesr-:
in
u
of
infinj
S’i.s
of
1,hon
A of
by
exists,
is we
Ir:.
sub.-
intr+rllti !A)
z
WC! :;:iy
v-qnjk-simply d
X of’
shattered
01’
811
by i . c.
The
R.
s pace
f!lrrnrents
of’ r0 (or ___ largest
dimension
sion
a r n n gc
obtained in I?,
hy
is t.hc! d
(Vapni
l.hc
such
that
CII rcl i
R (if’ -1,
say
R is
and
the
i 1’ no
di.mc?n,.
0
k-Chervoncnki
se:rrc:h its
s)
~Ji~merr:aion
(x,K,o,~j
problem
of
A C. r?
Example
in
he
the
2. set.
underlying
not
no
problem
real
is
range
of is
assumyistic
(X* ,PREF(Z* i,t
our
332
t hc
~lphabrl.
OF
j.h~
spncc
for
not L!)
the
sot
))
2’* --nr~-fj~y
is
I y,
a range spa!.?..
the:
1~::. P’
nlph;rlai~l.
word
X).
For
the
set.
c.9 ii I I
a J.angunge
I wel.).
Obvious
anti
over
denotne w and,
= {pref(w) in
an
words empty
pref(w)
words
for
Y be all the
prefixes
performed
these
are
Let. of
P (including
(i)),
PRRP(L)
ergur!
a
(X,H).
c r?
t imc.
m’l.gh
r+xcopt.
to
= ZA,
sub set
maximal
PRGF(L) t .i 0
J.ower nnswC?r
[VC],
dimension ______.___^ S) is the
such
The
and in
;I gives
time,
decided
(which
1 addition
IlFI (A)
a
nality
w c: x*,
( iii
: which
( X , it ) be te
can bc a range If’
of ---
empty,
prohlem
constant
A C X and
n r 1: $7 can
S ::
shattered
dimension for
possi-
.
denotes
exists
3 range
(i)
of
romput.ntion;ll
Ix,rt,o,Gj:
c:an
all.
Y’ 5 Ii t.0
to
founrl,
a .fi.ni
A that A with
Chcarvonenkis .L--‘----.-.---dimension _--,-.--_-
problem
,Q, (2RZ,~)),
finiteset
Let
and
halfplane-
B Rz:.
2:”
dat.:i
for
number
leads
c:ont,c?xt)
number cffi-
reporting
if,
theorc:t.ic.
bc
further
reporting) as
an..’
t,his
a range
(see
can
some 6,
needed
naturally
di f Pcrrnnt
A of
a yuory
(1s the
(R” ,Hz*
for
of
that
b,i.g
to
k data
set
in
with
definitions
design
the
time
defi.nitions
problem.
3.s npccified &,(>i,i .: (x) herrA
Jie
halfplane-range
po-ints
to
Then
information
l’h:i.s
E
is
as
the
have
there
for
A of’ datn
t.i.mc
optimal,
be
lowi.ng
x
we
g. ,
query.
the
every
be determi.ned search problem
referred
‘Phc
1 for
is such
can range
counting
=
set
linear
semigroup
(c.
the
is
is
finite
A that.
5' Ha*
the
integers X(x)
s1.rticl’ure
plane
H:tl, ‘f%~ll) - a,; : I,-- J ) -i &Q.-J (n-,1)
n)O 01.1
for
n>O,
= 1 For
and
d,n)l.
PROPOSITlON 1.1.. of
,
in
hc* rl i-art&r!
int.r~gctrs
(Al,A>
U{Ai
if
S),
(X,K)
(
A is
E.
cal
A (with
a (v,m,
TA .
1 I
!Ai
Icd
respect.
E)-port
ition
P
n and
P. with
a nodo
Lrrar* i.n
t.imc.
a rca1
A partition-
for
such
A contains
i I ion-
dofinitionc
m ho
(v,m,&)--partition
The I’ ba
E be
r
trcr!. Let
v and
and
1
tion-tree
performed
;~.-iaurn[~-
r and A be sets. A if either A ‘: r
that
say
for
First
a parl:i
.I:h:lt.
rc-
performrttl on
query Let
the
di.mension
i)
c’on-
(Vo1.c
part
be
following
k 6 v, : l.(iO,
i s defined
thr!
recall
Lead
sures
TA,
dis.joint
time.
will
in
parts(p)
the
for we
1 inear
to
d.
Par
nor
C), in
Ai
oil
A for
that Definition.
in
parts
t-i 01)s ,
informaat
parts
I s; I-I:A~ a global
set
in
partjti.on
in
start
init.inl.ly
the
of
(due
the
bound
element
If
is
I-csuJ
qllotc
l,ow?r
query
dimension
r,
with
neutral
fo t.l.ow-up
sub1 inr!>tr the
which
12(d~ Ing’ fin
if
E{c?(x)
we
SUM,
dundant
of
compute!
a rnngc
nctually
problr:m
natur;tl
to
variable
tained
all
for
scclrch
Tho
whc.1 har
is,
CHII
sub I i near
achieved
dimension.
qcll‘:t t. i Iin
above,
be
order
let
taincd
fol.lowing from
proposition
calculations
can in
be
ob-,
[HW].
A the
PROPOSITION 1.2.
exac.-
search
that
space
333
S and
Let
with
problem
let
A be
P he
a
rongc
underlying .a set
r:tnc~r~ of
n
elements of S. If TA is a (v,n,&)partition tree of A with respect to (l
