Thanks at this thanks.

Yes So this is partly a bass and
Joe work with invited whims and if I have

time again I can tell you a little bit
about some more of a paper by my own.

But your goal is to stop him with
questions I don't get to my own paper.

So what do I need to
harness the summation P.

Here is one what David example
it's called the maximum

inner product you have two
sets of vectors in a zero one.

And you want to put them next to A and B.

The maximize be.

Simple and you know this has
a bunch of other names you can like

Max intersection and so on and
there we have other results for

a long close pair with longer scum and
subsequence approximate regular Bubba but

before I continue telling you about
continuing to tell you about.

How amazing the results
are let me tell you about.

Our mistake so.

There are two variants of this problem

there is the by chromatic Vane which I'll
talk about we have two sets A and B.

and you want to find the maximum.

You can also also consider
the monochromatic variant.

Where you just have one set anyone to
find any two vectors from that one set

all of them want to come out a very and
sounds more natural the amount is actually

a more interesting form because it
the lower bounds imply lower bounds for

data structures I'll talk
more about that later but

in the Fox recent Fox was sitting so
we claim there are techniques also applied

to the monochromatic version that was
a mistake and we're sorry for that

and so this monochromatic case
is still open even for exact.

Nothing is known even if you
don't care with approximation.

And.

You know one one sad thing about this so
we make we made this

mistake it's our fault but now for
ever in the proceedings of Fox

twenty seventeen there will be a false
theorem and for those if you are sitting

there will be sitting in the steering
committee of the P.C. chairs and so

on I want to pose this problem of fighting
we still have printed proceedings.

I.

Know but you can't change that.

You cannot change the proceedings and

I try to ask about arrived in the said
post an archive or something but

there is nothing you can do that's what I
was told by that you know publisher so.

I'm sure that I'm still trying but.

OK.

And I said that's wrong but.

Back it back to what what is correct so
you know by communiqué so

we understand pretty well.

And I should tell you the parameters so we
care about the you think of the states and

me as having said large number
of factors that capital and and.

The dimension is short so
smaller than any point I'm going in

now there is a trivial algorithm and
times the time.

You just enumerate all of the Paris and.

Compute the inner product and the real
question is can you do truly some predict

time so enter the two minute steps on for
some concepts alone and.

So.

So so this sort of results sort of extend
to some of those like clothes clothes for

over long is coming subsequence or
it isn't so

this is similar to clothes spare
overlong is common subsequence

I'll get back to maybe a touch in
a distance a little bit and maybe.

So I.

Know they're not special cases so it's
not obvious that either poem is harder

you know you can consider many metrics
this is just an example to get the flavor

of things OK and so

and have the paper that sort of
started Soul area run with them so

that if you want to solve this problem
exactly then you cannot do this in.

And squared my steps on into
the to my stepson for any and so.

Assuming this strong special time
a process which all defines and.

And then last year there was a paper.

But it pérez a sign and so
the mystery that prove that even for some.

Even for very good approximation
you still cannot do better than

much better than quadratic time.

But you know one point zero or
one can do it's open to approximation

hundred books a nation
log an approximation.

A log of these are all open but
if you want to get like.

You know put them in a harness that's

impossible because the dimension
is less than point Amul and

getting one over Does she really just try
to get one coordinate where you match so

one overseas approximation is trivial so
less than point normal but

where there is a question like where
is what's the right how hard is this.

On the support summation line.

And where we can prove is that.

You can get almost attitude to the if
you had to the log in there would be.

The point that we cannot get so we have
to log in over a simple logs doctors.

So almost everything and

like I said that we can do even
a little bit better so really.

Tied up the constant exponent but
your goal is to ask many questions so

I don't talk about that part all I'm
really excited about this new result.

So.

Yeah I know there are the same and
there might be different.

Yeah but as you can see there are almost.

There is up to some constant
there you know it's.

But up to some concept
there are centrally.

OK let me tell you about.

Another problem.

Instead of not similar problem
talk about clothes spare over

to magick and
this is a very similar problem right like.

But now.

There is a click talked about
close peer over and hadn't.

Yeah.

Yes the other without If you had a harness
for the monochromatic version it's easy to

get harness for the by part of
vision with the other way it.

Is not known.

Yet and in general it's.

It's also I mean assuming it's stronger.

If you look at the really small dimensions
so dimension like a log log in.

And over over the reals then.

The monochromatic version
is strictly easier.

So there are algorithms for
the monochromatic version that for

money come out if you can solve
them if the monochromatic was hard

if the by magic was it was easy then
you could just pick the signs randomly.

And then.

If if you have an algorithm for
the by chromatic case and

you want to solve the monochromatic
case then you can compete you can pick

a color so mentally color
the vectors at random red and blue.

Yeah and then we probably have.

But yeah.

OK so we can tell we can
have this like a close pair.

And here the what's known
as a little bit different.

We have again so again the trivial
algorithm is squared times the inner

one of the faster and

again run with as a result for
exact You cannot do faster but here

there are some ation algorithm so if you
use the local ecology sensitive hashing.

You can get it in time into the step son
you can get a one plus Ted off epsilon.

Approximation So

this is already ways Indian I mean
everything that we see in this line is

already like assuming a flick of what was
a small segment in the previous picture so

this is much better than any of the
hardest results and there are even better

algorithms surveillance I'll
give you can get one plus.

Or a focus on squared approximation and
then there is a point on method.

Ten and Williams.

Give one plus a proc simply epsilon cubed.

And you can ask if there isn't one
plus Epsilon the fourth I don't know

another thing can ask is
can you actually get.

Something that approaches one
without dependence on Epsilon so

in into the one point nine time
something that is arbitrarily close to.

And here is all that we have know so
there is some function of

Epsilon the depends is not so nice but
there is some function of epsilon where.

If you want to run into the steps on
time you cannot do better than one plus

function of epsilon oxidation So

let me put this another way for
every epsilon there exists a delta so

that if you want to build up like
summation you need into the steps on time.

And what I referred so
particular you don't have a.

Sort of like a fixed parameter so could
not approximation like it after the delta

of Delta times until one point in time.

And what if I mentioned earlier but.

In the data structure plan is if
you have if you use a poor normal.

People at the same time you cannot do
better than into the one of the steps on

time for query for answering
queries questions about the result

of A So let's go back to the simpler
this maxim inner product problem and

let me show you how to get this.

That exact harness
the original exactness So

the strong special time passes
states that if you have K.

set for a large enough constant K.

you need approximately to the nth time and
here's the reduction

we start with our case if
feet yeah there is a depends

it's to the Oneness epsilon or
too many steps on time scale to the N.

where epsilon.

Thanks.

So we have some star from that.

That's OK We start from
a case in this field and

variables and we partition the variables.

We look at all the half assignments to
the first half of the variables and

on the half of times the second
half of the variables.

Yeah so I want to show that.

Proving this rate so
this is this is hard OK So

solving the problem from exactly is
hard so I'll construct a victory for

each half assignment because
activator an A for each F.

assignment.

To the first of the very
most constructive Victor and

before each have some of the second
half of the variables and

then the use of gadgets to
say that if a dog large and.

To have Simon together jointly satisfy it.

And the important thing to notice is that
the side of the reduction is approximately

two to the end over two.

Or Kaplan is approximate
to the number two and

now by the song exponential time
opposes any the proximity to the N.

which is like and squared.

So those is it now my goal for most of the
stock will be to show you how to get it to

harness and hopefully we can get it into
the two to the longer a little off long.

OK So the way we typically get harnessed
the summation is this P.C.P. through.

So the speaker has two from relations

one the original one is
a politically checkable proof.

So you have a three set Phoenix
satisfying assignment X.

you want to generate a proof by a fax

that the verifier can read
only a few bits from plastics.

In a very five X.

actually satisfies.

Another way these are equivalent but
in another way you can look at the P.C.P.

through his harness of a harness and
implication harness of summation.

So you have a three set
fee again in variables and

you can suck to three
set three prime of N.

prime variables.

And now approximately solving the problem
is as hard as solving the original five

and then we focus on the second definition
and in particular on the depends when N.

variables to and prime variables so
this is what I call the P.C. people up.

In the original proof.

You need to approximate the N.

squared.

To be approximate and squared.

And got it to very close to linear and
the Norse

peaceably get gets even closer to lean
years its End Times poll law given and

it's a major open ball whether you can get
actually near dependence I think on some.

Now let's assume that we had this
fantasy P.C.P. with only ten in blow up.

And try to use it to get harness
of approximation from Sith.

So if you do this is a reduction
the reduction size is.

Approximate to the end crime over
two which is to the five and

even if you had this fantasy P.C.P..

And that doesn't refute
anything interesting.

So let me tell you how to overcome this.

This is the first half of the talk
title but before I actually tell you

what seemed to P.C.P.'s are let me warm up
with distributed error correcting code.

I should mention by the way not
if you know what the soup P.C.P.

is Arca that's something that I made up so
don't don't believe I don't know what.

So I have a message I want to encode and
I give half the message to Alice and

I give half a message to Bob Allison
codes here have been codes his half.

And then I write them coding next to each
other then talk between them at all but

this is a pretty good encoding
of the original message X..

Right so maybe I lose a constant factor in
the rate or the distance of the code but

it's a pretty good or greater good and I
want to do the same thing for P.C. piece.

Let me remind you the.

First affiliation of species so

I'm trying to construct a proof
I have a satisfying Simon X.

and I want to prove that X.

actually satisfies the C.

and if.

OK I don't have it here let's see so
everyone knows that.

This was Alice.

Everyone knows the three set right and

I give half the assignment I
give half the Simon to Bob and

now Alice writes like a local P.C.P.
And Bob writes a local P.C.P..

Yeah and I put the.

Piece of music each other and I let the
very fiery bits from whichever party he

wants and hopefully this is a pretty good
P.C.P. for Alpha come a better satisfied.

And the reason I'm telling you about this.

Notion of the sea with P.C.P.

is that it completely circumvents
this piece of people up there.

Know so so so the P.C. P.C.P. takes.

X.
write the assignment and

construct a fix and
the very frightened look at

now Alice only gets half a fix and
she constructs some powerful for.

Someone something she could write
something she write something.

And then.

Takes it on and write something

and you know at the very far it
has access to everything and

he reads very far it's a few bits
from whatever side that he wants and

then he has to decide whether it's a.

Good question

so it's not like this there's this notion
of like multi prove or this is different.

And you can eat can even not really worry
about malicious some of it can for the for

our purposes you can even
assume there are sort of.

Saying with a door something but.

The the point is that Alice doesn't
see the entire assignment and

Bob you know in the mood to prove or.

To prove a proof systems the Privacy
is the previous see everything and

then and then they can and
then you put them in separate rooms

here Alice didn't see better at
all she doesn't know part of.

The same and
she only sees her half that Simon and

she writes something she has some
function from half assignments to.

Instantly Bob sees his half and
write something.

And we're hoping that together what we've
got a good proof that Alpha comma better.

So it will this is strictly her right so
I mean the best thing you can hope for

is then time spoiler again but let me show
you that you know even if you had put

no real length any polio or even some
experimental they'll be great OK So

yes so we're completely circumventing
the species people appreciate.

OK so so here is the new reduction.

So again they put is the formula.

And now we construct a vector
instead of constructing a vector for

each half assignment to this longer three
set we construct a vector for each.

Possible for
each half of Simon's original three set.

You construct a victor but

now this sector is corresponds to
the half proof two path Alpha.

And then use some guy just
to make sure the B.'s

approximately the probability
the very far except.

It to be very flexible and
puppet us right and

the crucial thing to notice is the number
of gadgets remains the same so it doesn't

I don't care at all what the blow up is
the number of factors remain the same for

the doctors a little bit longer but like I
said it's going to be less in point but.

Could be.

Slightly longer vectors and still be
interesting OK so I'll leave this.

Like Senator observe that this is this is
going to be harder than it classically

should be but maybe if we allow more
even more blow up maybe we can do this

does anyone have a guess
about how much we would need

can we do really like and
spoiling in a point.

OK so.

We actually know we're actually looking
for good sections of such a stupid piece

of piece so I've been walking
around with this distributed B.C.P.

conjecture for
a while until I talked to gold and

he gave me one proof that they're
actually impossible OK So

this is the proof it's actually
not even if he said to set that.

It's an easy problem right but
this is the formula so it's done and over.

For every I I want either not Alpha I or
not better I.

Wonder when or better satisfying this

formula under what conditions I'll
feel better is a set of things I mean

so the answer is already in the screen and
they satisfy if and

only if alphabet or joint.

I mean.

I need to for every I either
Alf eyes on the set or better.

Yet.

How distributed P.C.P.

then I can do I can use it to solve
said join us so alpha so Alice is local

He writes her local piece to be involved
locally writes his local P.C.P..

And then they don't have to
communicate the entire local P.C.P.

They just simulate the very far equerries.

So if I had a.

P.C.P. with less than.

Anything less an exponential
blowup than I would get the.

Communication protocol for
sedition it's OK No cities doing this.

It's hard for randomized protocols it's
hard even for non to me support of course

so we're pretty stuck here but fortunately
there is a million Arthur protocol for

said just doing this the runts
in your communication so

Millan is the wizard The knows everything
but will restrict merely into a.

Limited amount of communication so

here's a protocol there and
sort of it doesn't make sense Alice.

Approximately route and
it really knows everything including that

Alice assignments and Bob Simons and
everything OK then Alice and

Bob toss a small number of coins

then Bob's cents Alice
approximately written bits.

And Alice reads millions message
the coins and Bob's message.

And she decides whether to accept or
reject so and we only use the proximal

written bits and

so we need this magical advice for
Merlin but

fortunately in the context of reduction
from Strong N.T. age you know we can

enumerate over all the two of the written
messages from million but it's still.

Exponential time so it doesn't it's
nothing compared to the end deficit.

Let me just show you what.

How this listening non-deterministic
you seem to be sleepy.

And I'll show you more details so.

Again this non-domestic the soon to be
sleepy Alice above have half assignments.

And.

Marilyn has this.

Marilyn Milian says gives
an advice of think of it like.

It's like the witness and

people all right but now real that with
randomness after we see millions message.

So Bob writes.

His local proof and
Alice writes her local proof but

it can also depend on the short hint for
Marilyn.

And how this guy can tell you his name but
with a V.

So this is a very fire.

He reads one symbol there are a small
number of bits from each of.

Alice's proof and Bob's proof and

on the outs with accept or reject So this
is this is what we're going to construct

in this show is that you know
it's hard to approximate

so they could only come in
because there's a to the N.

over to right even the quadratic
of the correct time

yeah it's still the two then over to.

Root and so it's negligible.

This is a good time for more questions.

Excellent question so

three players actually the whole
reason I mean OK so three players.

It's actually not known that said just
doing this requires linear communication.

And there is a paperweight Williams
where they say if you could do that.

The number of forehead communication for
sedition is for three players in less than

a linear communication then you can use
that to refute the strong at the age.

No You can also ask.

Yeah.

There are.

So I'm not sure I'm not to say this and

I'm saying I think it and
we don't know but.

A friend Solomon told me that
there are a very nice paper

are using this methodology for.

What they consider OK.

Parties You know I can maybe
I can tell you was like OK.

Yeah so there are problems where
you might be the limit but mostly.

I mean you can talk about like.

You know a magazine or park with three
victories something like that but

the problem is you're thinking
of like Oprah shortest path.

So for those problems

we don't know any reduction from
a strong age with approximation or not.

And I'll go back to that
hopefully the last line.

OK.

OK this is so if you have more high
level questions you can ask If not I'll

want to try to like instantiate this
is like a real protocol might work.

And you know I was looking for

Bart Simpson pictures and
so on over six like this.

OK you know how to.

Turn this off.

And.

OK so.

Let me tell you this.

Let me instantiate this.

Million Arthur communication protocol
first it's so simple I feel it's a shame

not to show you the proof and
then we can try to improve so.

This is going to be Alice and

she's going to take care she has
an end Victor with ones and zeroes and

she's going to write it in a square so
this is zero one and it's going to be.

And.

Bob is going to see it

and then they're going to say we
want something like harness the for

examination approximation they're
correcting codes right so

they're going to extend
this as a no degree extend.

Load of people in on us.

Over some field you know you
can think of fuse largely.

OK And.

Let's call this room so

this room has some loading people and
amulets call it alpha T..

And this road is going to be.

Better.

And now I have to tell
you what milling does.

Summerlin considers.

And he truly has just like a.

Muti is just going to be.

Is just a product of the.

Alpha teen ability to

know what said just doing this
means is that we want that for each

in the square we want all the interest to
be the product of all that just to be zero

and now we define you to be
this isn't the sum of the.

Tease and

this is the first of the first step in
the protocol is merely in sense this.

Sense some of the music.

You know that you're into an entire Victor
I mean it's a lot of people who will

write because of something to some of so

these are look at your product and
take yourself a lot of people only some.

So not really and
since it's low to be a porno male.

And notice that if it's if
the this is here this is

we want this to be all zeros and
we want this wrote this song to be.

Known as correspond to things do it and
now the second step in the protocol is.

Alice involved.

Are going to draw.

An entry in the field around them so
my nephew.

And then Bob since.

It T.

a fight.

For a life.

So for
Altie I ran them in the forest Alice.

Very five is that.

What send what Bob sent match so
I want to verify that mean I.

Is the sun

because you're you're a great
question because we want it we don't

want to some of.

Them to some of the T.'s want
the some of the product right.

Can you still see here.

So this gives.

This gives like a two factor.

So this gives like a Souness
of one half in the particle or

a two factor harness approximation.

YEAH SO SO SO SO SO SO IF.

If so how can a mailing chip

really can try to send the wrong
the wrong low to the point amil and

if the degree of the point I was at
most a half the size of the field.

Then this is probably at least
one half of catching him really.

Now how do we amplify the.

Soundness of a particle Rican
repeated many times rate so

repeat it how many times
can we afford to repeat it.

We want the communication to
still be the tall I can merely

Arthur communication still be.

And this is approximately
this is something.

To repeat approximately ten times.

And this gives the.

Two of them.

So this gives something
like that goes like.

O'Toole or food and so that's like
root log and log capital and but

I promise you almost like a Blimey so

then the next thing me you notice is.

You know we have a choice as you
say that Rudin is tight from

really north of there is a lot on food and
communication.

But.

So the way that.

When you repeat the protocol we don't
have to repeat merely steps Malians.

Message because all the ins happens here
so there is no reason to repeat the.

Message really goes in one message
we can just repeat these steps.

And no we won't so we can afford to make
millions message longer the expense of

making Bob's message shorter and

in general what we can do is in then.

We can make this.

And.

Yeah and yeah exactly and I know I can.

Either I can make good repeat almost or
Delfin over three times.

But I can make a log.

And I can repeat so.

You have to be a little bit careful
right because the communication is

the number of bits that we communicate
also depends on the size of the field so

there is a log in factor on each end but
this gives.

So you can repeat proximately.

And over probably logon and
this gives like the.

Two of the problems you log in over.

A long square long.

Factor.

So this is.

You know maybe you know you
want to get rid of this log log

I don't think that's so
interesting but if you want to go to.

Applications like.

L two closed for so
far this leaves nothing and

the reason is that we lose this.

Factor of.

And the.

How to say.

No So this was we get together to
fuck to factor harness right and

then we got even better we get.

A repeating and then we reap we can repeat
even more and say I still have this

annoying log log factor if you
know I don't care about it so

much here but for this all this thing or
any distance doesn't give anything.

But I want to get more so.

You know that's the reason that we lose
log in is because of this field size which

happens in the reeds on coding.

And you know if you look at.

Just.

The.

Point of elsewhere and

rate if you look at it as a binary cord
as a binary code the rate is at most

one of the logon So that's like
kills the logon factor immediately

OK We can also look at what other
codes that we know we know.

So it's again a lot of people in M.L.S.
but not now there are.

More to vary it but if you look at the.

Rate as a binary code the rate
gets even even worse.

We're How much worse it depends on how
much of what Hamilton very there is vs.

And you know we know what we know a lot of
constructions that have a constant rate

but most of them don't satisfy
this nice Probably you can

multiply code words and so on.

But.

Finally we have something
called a big jump.

So these are still the degree but
now they're not going to be.

Very it but

now they're not exactly pornos
there are rational functions.

And so
we have if you have a queue of this.

You know.

So yes so

these achieve exactly sort of what do
they give us they give us a constant rate

in the other thing that's important is
that something is called the point home

and put in a mill closure or.

You can multiply them like and
that's what.

They did here there is a crucial step
where we are deploying code words and

we're taking sums of court words and
that's something that if you act like a.

You know your favorite consummate
code like a man or something.

You know how the property OK And

they also have this property
the call called us systematic

This means that you know
if you look at the.

Letter you put in most you can pick
the first substance to be whatever.

Whatever values you want and
you can just extend it.

And they sort of have
this property this a G.

codes.

That actual construction of a G.

codes is a very complicated as I told you
over there although to be multivariate

rational functions that's nice but
if you would try to write out the values

everywhere you again you would not get
concentrate so there are just evaluated

the small really some like specific
points solutions of Philip to care of and

so on so I can't tell you much
about that but it's a whole theory.

Great and
I should say these have been used through.

Design.

Like linear sized P.C.P.'s before that
always get the sound as the P.C.P. So.

There is.

Where they get.

Linear So it's P.C.P. So that would be
like the fantasy I told you about but

there are complexities into the salon and
there is.

Another been since on all paper

from last year and they get linear.

I.O.P. So this interactive.

Oracle proof I think so
you see send the very front of P.C.P.

in the piece at the very for reads a few
bits and then queries and then sends

your message back and you can send another
piece of being if it's a few bits but and

you can they can do this is the earlier
size and consonant are friends but

actually getting kind of
the cell phone is it and

those I should mention those don't
give any harness approximation.

Yes So now we want to just erase
all the logon factors everywhere.

And then we can get this but you have to
be careful because the reason we get.

The reason we get in your size is that.

We know work over a constant field.

But when you're working over a concept
field it's likelihood of sum up

the point of else so we use the fact that
when you when we saw a bunch of zeros and

ones you forgot zero we knew
it's everything was zero but

that's not true anymore
over a constant field.

You can try to do that so
that this you can do something similar and

that gives an A.M.A. protocol.

OK I'll get back to that a second but
the for you know our purposes the.

Eventually have to keep union
over a exponential number of.

If you have any support pairs so
that's like two capital and

square pairs have an exponential
sized thing it exponentially.

That's hopeless.

What do you can do is so

we still lose a factor here but the fact
of the really is it's just the log.

And in our application.

Very small because right
remember when I want to make T.

small and over T.

large so millions messages long so we lose
a lot of the factor of the constant so.

It's OK we get all the hardest and.

So I the last few minutes
were handily but.

The last that you get for having.

A larger field size you
lose a log the factor.

Is.

OK that the find the final practical
political looks like this instead of flow

to be extensions.

You have a.

Coding.

And OK you have to be careful so.

You have some feel that if you squared.

The load in the ages a G.

code the constructions work you
have to be you have to be some Q.

squared but and you lose a factor of.

Luck you have to have to pick Q.

large enough.

So you have to pick one Q That's a
cardinality you want to be larger than T.

sorry.

But an obligation to mostly lose a log Q.

which is log defect which is

so I have a couple of slides of.

Open questions I want to put up but

the first one is going to
be how do you actually get.

A root and communication.

Merely north of protocol so I can do
this like I am a particle but even for

a game there are no lower bounds so
whereas for M.

A That would be tight and you know.

So this this log the skew get
into the hack work here but.

In another question I'll put up on

the slides in a second I just want
what I have this stuff is there is.

In your paper by writing them
something in a second what it does but

the way does it is in our construction.

In which And I'm sorry and have time to.

Look at all the.

Entries in queue you pick.

From the random and
then you are going to have an average

Alice has a good chance of catching
the male and if she's the.

Other thing you can do is if you're
working over a large field you can pick

a point way out there which is so
large that you're going to it's not a zero

and you have this you're just doing this
particle you're looking just at one point

and that's something else I
don't know how you doing.

With those age because where if I
want to work over a small field.

Almost certainly just.

Flashed a few slides of questions.

OK this is a reduction I don't
have time to show you see it's.

I was in for

but you see it fits in one slide very
easily so it's very simple deduction.

Here are some questions easy I don't
know how to solve any of them but

the book it looks like if you have those
aged cards are just around the corner but

I don't know how to do them so I told you
I'm a communication complex I don't know

if it's written or a routine logon.

And the other thing can do is you can
generalize this into a multi around.

Communication protocol or
IP communication complexity where

you know Millington's a message Alice in
some message really since a message and

then instead of having this rectangle
you have tens or in high dimensions

and if you work out
the details you get to log and

times log log and
communication complexity and

the reason is that you
have log in rounds and

any Tron you want to one that has one of
the logon failure probability so to get

one of us one of the logon filler problems
you need field size at least logon

now maybe with age codes you can
somehow get the state logon but.

I don't know and then the other thing of
the News newspaper those telling you about

you can ask if you want to
solve exactly close pair.

With less than mention so
I told you ds like a logon or

a slightly less important mail but
what if you want to really.

You know constant or very small dimension
so by looking at this one point in.

You know it for a large enough entry.

Iran gets a lot longer.

And you pose this open
question of can you get.

Constant hits I was so excited when I
saw the paper here I notice all this but

it turns out that even with eighty
codes I don't know how to solve this.

Now I'm just one more sleight
of real prompts and then I'll.

Let you go to lunch.

So.

That to question some
of us earlier before.

You know before a little too close pair
we don't get the exact dependence on

Epsilon But the fact that we run
the running time is slightly sub

quadratic in the epsilon depends on
the proxy mation is morally right.

For Ed it is since we don't even know
a log and log the approximation for the.

Distance close pair but
the only harness of summation I know for

a distance is by taking the L.

to harness and working even more
to get the distance harness.

And you know be really nice to get
anything in this for the distance flows.

Another question is that you can talk
about I guess about any sensor long and

longest common subsequence with just
two long strings so it's not a bunch of

factors of just one too long strings in
the way if you want exact harness the way

the reduction works is you take all
the gadgets you constructed and you

just compare the need them with some more
gadgets and then you try to match them and

if there is one satisfying assignment
you get one pair of gadgets the match

is a little bit better and gives you a
slightly better longest common subsequence

but if you think about
longest come subsequence.

All the bad gadgets also contribute to
your score because you can always pick

one symbol and just take one over
the alphabet size right so the.

Instead of.

The approximate it doesn't only correspond
to the number of clauses it also

corresponds to the fraction of satisfying
assignments in an additive sense and

approximating the number of satisfying
assignments is easy you can just

draw a random sample draw random
assignments and you can approximate

an ad to summation them or felt random a
sentence number of satisfying assignments.

And what we can show is under some.

Under some.

Assumptions.

You can show that there are no
deterministic approximation algorithms and

if you get a better This uses this IP
protocol I told you about unless I

need to get a better IP protocol you
can get slightly weaker substance here.

And OK you mentioned you know there
are problems like dynamic Maxima matching

where there are no reductions for
from assumptions

like three some triangle attention and
all nine matrix vector multiplication.

And so you know none of this this is not
from seven ounces applies to this and

it would be nice to get like a P.C.P.
like harness the folks in NATION.

For this in the general
I think the new P.C.P.

like models that solve any of the above
problems would be excellent thank you.

Thank you.

Thanks.