Approximating the Depth via Sampling and Emptiness - PowerPoint Presentation

Slide1

Approximating the Depth via Sampling and Emptiness

Shirly

YakubovSlide2

Motivation

Given a set

S

of n objects we

want to store them in a data-structure that could answer range queries. For a range r we have:range-searching counting queries – number of objects from S intersected by r, denoted by .range-searching emptiness queries – does r intersect any of the objects in S.Counting queries are harder than emptiness ones.We want to approximate using emptiness queries.

 Slide3

MotivationSlide4

Outline

Constructing the data-structure

Applications: halfplane/halfspace, disks, pseudo-disksRelative approximation via samplingconstructing a data-structure that for a prespecified

and

decides

whether

with allowed mistake if

Using

polylogarithmic

emptiness queries, output a number

such that

(the answers are correct with high probability)

 Slide5

The data-structure

Let

be M independent random samples of

S

formed by picking every element with probability , where

Build M separate emptiness-query data-structures

respectively, and let

 Slide6

The data-structure

Answering a query

of a range

r

:Let ,

Compute

using emptiness

queries

in each

Compute

If

return

, else return

Since each object is picked with

probability, if

we expect to catch more objects from r to our samples

 Slide7

The data-structure

Correctness

we will show that with high

probability the data-structure returns the correct answer outside of the range

, that is: and

are small.

 Slide8

The data-structure

The idea

By repeating the experiment M times we get an estimation of

by .  

As bigger the gap is between

and

, the

fewer experiments required

for a reliable estimation.

We can see that the gap is

.

 Slide9

The data-structure

The idea

We want to bound the probability that the difference between

/M

) and our estimation of exceeds this gap.

is a

sum of

independent

Bernoulli random variables.

Using

Chernoff's

inequality

we can bound this probability. Slide10

The data-structure

Observation:

For

, we have

Lemma:

, where

can be made arbitrarily large by a choice of large enough

.

Proof:

We will from now on assume

 Slide11

The data-structure

,

-

By the

observation

 Slide12

The data-structure

Deploying

Chernoff

inequality we have:

 Slide13

The data-structure

Given a set

S

of n objects, a parameter

, and , one can construct a data-structure which, given a range r, returns either L or R. if it return L then

, else

. It might return either answer if

.

(All results are correct with high probability)

CorollarySlide14

The data-structure

D

consists of

emptiness data-structures. The space and preprocessing time needed to build them are

,

where S(m) is the space (and time) needed for a single emptiness data-structure storing m objects.

The query time is

,

where Q(m) is the query time of one emptiness data-structure.

(All

bounds hold with

high probability)

 CorollarySlide15

The data-structure

A data structure

that uses

logarithmic number of emptiness data structures as a black box. decides if (L) or

(R)

, and the answer is correct

with high

probability.

So far we have:

R

LSlide16

Approximating the depth

The goal:

Given a set

S

of n objects and a parameter , build a data-structure that for a range r outputs a number , with .The idea: build several data-structures as described before and perform a binary search on them to find

.

 Slide17

Approximating the depth

The data-structure

For small values of

(

we construct data structures .  Slide18

Approximating the depth

The data-structure

Next, consider the values

for

, where . Note that

:

We

build a data-structure

for each

.

 Slide19

Approximating the depth

Answering a query range r

First, we can determine if

is at least

or smaller then (using ).if smaller, we can perform binary search on

to find the exact (+-1) value of

.

. . . .

 Slide20

Approximating the depth

Answering a query range r

If bigger, with high probability, if we were to query all

, we would get a sequence of

Rs, followed by a sequence of Ls:  

We can perform a binary search to find this changeover.

Notice that for every j:

 Slide21

Approximating the depth

Answering a query range r

Let

be the last data-structure returning R, we have:

and

,

let

 Slide22

We have:

and

Approximating the depth

Answering a query range r

Now,

yields the required approximation:

.

 Slide23

Approximating the depth

Complexity :

We have

)

data-structures. Each of them consists of emptiness data-structures.We used

queries on

.

Therefore, The overall query time is

,

were

is the emptiness-query time of the emptiness data-structures we use. We get the following theorem:

 Slide24

Approximating the depth

Theorem :

Given a set

S

of n objects and a parameter , assume that one can construct, using space, in time, a data-structure that answers emptiness queries in time. Then, one can construct, using

space, in

)

time, a data-structure that, given a range

r

, outputs a number

, with

, in

time.(All results and bounds hold with high probability)

 Slide25

Applications

We will show several examples of emptiness data-structures that answer emptiness queries in

logarithmic

time, using

linear space. Applying these data-structures to our theorem will give us efficient data-structures for approximating the depth. Slide26

S = points in

/

, r =

halfplane

/ halfspace Dobkin and Kirkpatrick hierarchySlide27

Answering emptiness queries for

halfplane

using

Dobkin

and Kirkpatrick hierarchyThe data structure:Given n points in the plane, compute their convex hull P (can be computed in time, using space).  Build the DK hierarchy for P (constructing time is , the space required is ).

Convex hull:

S = points in

/

, r =

halfplane

/

halfspace

 Slide28

Answering emptiness queries for

halfplane

using

Dobkin

and Kirkpatrick hierarchyAnswering a query:S = points in /, r = halfplane/ halfspace Slide29

Answering emptiness queries for

halfplane

using

Dobkin

and Kirkpatrick hierarchyAnswering a query:S = points in /, r = halfplane/ halfspace Slide30

Answering emptiness queries for

halfplane

using

Dobkin

and Kirkpatrick hierarchyAnswering a query:We have , and .

S = points in

/

, r =

halfplane

/

halfspace

 Slide31

Corollary:

Given a set

P

of

n points in two/three dimensions and a parameter , one can construct in time a data-structure of size , such that given a halfplane/halfspace (reps.) r, it outputs a number , such that

in

time.

*

For exact counting quires, the best result that uses

space gives query time

for two dimensions, and

for three dimensions. 

(Corollary

results and bounds hold with high probability)

S = points in

/

, r =

halfplane

/

halfspace Slide32

Answering emptiness queries for disks using

Dobkin

and Kirkpatrick hierarchy

S = points in

, r = disks

 Slide33

Corollary:

Given a set

P

of

n points in the plane and a parameter, one can construct in time a data-structure of size , such that given a disk r, it outputs a number , such that

in

time.

(All results and bounds hold with high probability)

S = points in

, r = disks

 Slide34

Pseudo-disks

a set S of n simply connected regions in the plane is called a set of pseudo-disks if the boundaries of any two distinct regions cross at most twice.

S = pseudo-disks, r = a pointSlide35

Pseudo-disks

We have

,

and . 

S = pseudo-disks, r = a pointSlide36

Corollary:

Given a set

S

of

n pseudo-disks in the plane and a parameter , one can construct in time a data-structure of size , such that given a point q, it outputs a number , such that

in

time.

(All results and bounds hold with high probability)

S = pseudo-disks, r = a pointSlide37

Relative approximation via sampling

What can be done if we want to use a single sample instead of many ones?

We sample each

object with probability

p in a random sample R.If the depth of a range r is sufficiently large (roughly ), then it can be estimated reliably by The interesting fact is that as deeper r is the better this estimate is. Slide38

Relative approximation via sampling

Lemma:

Let

S

be a set of n objects, , and r a range of depth in S. Let R be a random sample of S, such that every element is picked with probability .Let be the depth of r

in

R

. then

with probability

.

 Slide39

Relative approximation via sampling

Reminder: (

)

In previous lectures we saw how to find a sample R, that with high probability, is an

, that is:In our case, the lemma gives us the following: 

 Slide40

Relative approximation via sampling

Proof:

W

e have that

. By (*) we have:

(*):

Chernoff

:

for

.

Theorem:

.

 Slide41

Relative approximation via sampling

Conclusion

Note that if depth(

r,S

) is (say) , then

,

which is (relatively) a small number.

Via sampling, we turned the task of estimating the depth of heavy ranges into the task of estimating the depth of a shallow range.