coursera – Design and Analysis of Algorithms I – 3.4 O(n log n) Algorithm for Closest Pair

coursera – Design and Analysis of Algorithms I – 3.4 O(n log n) Algorithm for Closest Pair


So in this video and the next, we’re gonna
study a very cool divide and conquer algorithm for the closest pair problem.
This is a problem where you’re given N points in the plane, and you wanna figure
out which pair of points are closest to each other. So this’ll be the first taste
we get of an application in computational geometry, which is the part of algorithms
which studies how to reason and manipulate geometric objects. So those algorithms are
important in, among other areas, robotics, computer vision, and computer graphics. So
this is relatively advanced material. It’s a bit more difficult. Than the other
applications of divide and conquer that we’ve seen. The algorithm is a little bit
tricky and it has a quite nontrivial proof of correctness. So just be ready for that.
And also be warned that because it’s more advanced. I’m gonna talk about the
material at a slightly faster phase than I do in most of the other videos. So let’s
begin now by defining the problem formally. So we’re given as input N points
in the plain. So each one just defined by its X coordinate and its Y coordinate. And
when we talk about the distance between two points in this problem, we’re gonna
focus on Euclidean distance. So, let me remind you what that is briefly. But we’ll
introduce some simple notation for that, which we’ll use for the rest of the
lecture. So we’re just gonna note the Euclidean distance between two points, PI
and PJ, by D of PI-PJ. So in terms of the X and Y coordinate of these two points, we
just look at the squared differences in each coordinate, sum them up, and take the
square root. Now as the name of the problem would suggest, the goal is to
identify among all pairs the points that pair which has the smallest distance
between them. Next, let’s start getting a feel for the problem by making some
preliminary observations. First, I wanna make an assumption, purely for
convenience, that there’s no ties. So that is, I’m gonna assume all N points have
distinct X coordinates. And also, all N points have distinct Y coordinates. It’s
not difficult to extend the algorithm to accommodate ties. I’ll leave it to you to
think about how to do that. So next, let’s draw some parallels with the problem of
counting inversions, which was an earlier application of divide and conquer that we
saw. The first parallel I wanna point out is that if we’re comfortable with a
quadratic time algorithm, then this is not a hard problem. We can simply solve this
by brute force search. And again, by brute force search, I just mean we set up a
double four loop, which iterates over all distinct pairs of points, to compute the
distance for each such pair. And we remember the smallest. That’s clearly a
correct algorithm. It has to iterate over a quadratic number of pairs, so it’s
running time in gonna be, theta of N squared. And, as always, the question is,
can we apply some algorithmic ingenuity to do better? Can we have a better algorithm
than this naive one which iterates over all pairs of points. You might have a, an
initial instinct that, because the problem asks about a quadratic number of different
objects, perhaps we fundamentally need to do quadratic work. But again, recall back
in counting inversions, using divide and conquer, we were able to get an N log N
algorithm, despite the fact that there might be as many as a quadratic number of
inversions in an array. So the question is, can we do something similar here for
the closest pair problem. Now, one of the keys to getting an N log N time algorithm
for counting inversions was to leverage a sorting subroutine. Recall that we
piggybacked on Merge Short to count the number of inversions in N log at a time.
So the question is, here, with the closest pair, perhaps sorting again can be useful
in some way. To beat the quadratic barrier. So to develop some evidence that
sorting will, indeed help us, compute the closest pair of points in better than
quadratic time, let’s look at a special case of the problem, really, an easier
version of the problem. Which is when the points are just in one dimension. So on
the line, rather than in two dimensions in the plane. So in the 1d version, all of
the points just lie on a line like this one and we’re given the points in some
arbitrary order, not necessarily in the sorted order. So a way to solve the
closest pair problem in one dimension is to simply sort the points and then of
course the closest pair better be adjacent in this ordering so you just iterate
through the n-1 consecutive pairs and see which one is closest to each other. So
more formally, here’s how you solve the one dimensional version of the problem.
You sort the points according to their only coordinate, because you’ve got to
remember, this is only one dimension. So as we’ve seen, using merge sort, we can
sort the points in [inaudible] time. And then we just do a scan through the points,
so this takes linear time. And for each consecutive pair, we compute their
distance and we remember the smallest of those consecutive pairs and we return
that. That’s got to be the closest pair. So in this picture on the right, I’m just
going to circle here in green, the closest pair of points. So this is something we
discover by sorting and then doing a linear scan. Now needless to say, this
isn’t directly useful. This is not the problem I started out with. We wanted to
find the closest pair among points in the plane, not points on the line. But I want
to point out that this, even on the line, there are a quadratic number of different
pairs, so brute force search is still a quadratic time algorithm, even in the 1D
case. So at least with one dimension, we can use sorting, piggyback on it to beat
the naive brute force search bound and solve the problem and N log it in time. So
our goal for this lecture is going to be to devise an equally good algorithm for
the two dimensional case. So we want to solve closest pair of points on the plane,
again, and log N time. So you will succeed in this goal, I’m gonna to show you an in,
log in, time out rhythm for 2D closer spirits. It’s gonna take us a couple
steps, we only begin with a high level approach. Right. So the first I’d even try
is to just copy what worked for us in the one dimensional case. So the one
dimensional case we first sorted the points by their coordinate, and that was
really useful. Now, in the 2D case, points have two coordinates, X coordinates and Y
coordinates, so there’s two ways to sort them. So let’s just sort them both ways.
That is the first tip of our algorithm, which you should really think of as a
preprocessing step. We’re gonna take the input points. We invoke Merge Short once
to sort them according to X coordinate, that’s one copy of the points. Then we
make a second copy of the points, which are sorted by Y coordinate. So we’re gonna
call those copies of points PX, that’s an array of the points sorted by X
coordinate. And PY for them sorted by Y coordinate. Now we know, Merge Short takes
N log N times for the preprocessing step. Only takes O of N log in time. And again,
given that we’re shooting for an algorithm with a running time Big O on N log N. Why
not sort the points? We don’t even know how we’re going to use this factor right
now. But it’s sort of harmless. It’s not going to affect our goal of getting a big
of O and log in time out algorithm. And indeed this illustrates a broader point
which is one of the themes of this course. So recall, I hope one of the things you
take away from this course if a sense for what are the four free parimatives? What
are manipulations or operations you can do on data which basically are costless.
Meaning that if your dataset fits in the main memory of your computer, you can
basically invoke the primitive and it’s just gonna run blazingly fast and you can
just do it even if you don’t know why. And again, sorting is the canonical for free
primitive, although we’ll see some more. Later in the course. And so here we’re
using exactly that principle. So we don’t even understand why yet. We might want the
points to be sorted. It just seems like it’s probably gonna be useful. Motivated
by the 1D case. So let’s go ahead and make assorted copies of the points my X and Y
up front. So reasoning by analogy with the [inaudible] case suggests that sorting the
points might be useful. But we can’t carry this analogy too far. So in particular,
we’re not gonna be able to get away with just a simple linear scan through these
arrays to identify the closest pair of points. So to see that, consider the
following example. So we’re gonna look at a point set which has six points. There’s
going to be two points, which I’ll put in blue. Which are very close in X
coordinate, but very far away in Y coordinate. And then, there’s gonna be
another pair of points, which I’ll do in green. Which are very close in Y
coordinate, but very far away in X coordinate. And then there is going to be
a red pair of points. Which are not too far away in either the x coordinate. Or
the Y coordinate. So in this set of six points, the closest pair is the pair of
red points. They’re not even gonna show up consecutively in either of the two arrays,
right? So in the array that’s sort of by X coordinate, this blue point here is going
to be wedged in between the two red points, they won’t be consecutive. And
similarly, in the, in PY, which is sorted by a Y coordinate, this green point’s
gonna be wedged between the two red points. So you won’t even notice these red
points if you just do a linear scan through PX and P, or PY, and look at the
consecutive pairs of points. So following our preprocessing step, where you just
invert, invoke Merge Short twice, we’re gonna do a quite non-trivial. Divide and
conquer algorithms that can beat the closest pair. So really, in this
algorithm, we’re applying the divide and conquer algorithm twice. First, internal
to the sorting subroutine. Assuming that we use the [inaudible] algorithm to sort,
divide and conquers being used there to get an N log in running time, in this
preprocessing step. And then we’re gonna use it again on the sorted arrays in a new
way. That’s what I’m gonna tell you about next. So let’s just briefly review the
divide and conquer algorithm design paradigm before we apply it to the closest
pair problem. So, as usual, the first step is to figure out a way to divide your
problem into smaller sub problems. Sometimes this has a reasonable amount of
ingenuity, but it’s not going to here in the closest pair problem. We’re going to
proceed exactly as we did in Merge Short in counting inversions problems, where we
took the array, and broke it into it’s left and right half. So here we’re gonna
take the input point set, and again, just recurse on the left half of the points,
and rescurse on the right half of the points, where hereby left and right, I
mean with respect to the points X coordinates. There’s pretty much never any
ingenuity in the conquer step. That just means you take the sub-problems you
identified in the first step and you solve them recursively. That’s what we’ll do
here. We’ll recursively compute the closest pair in the left half of the
points, and the closest pair in the right half of the points. So where all the
creativity in divide and conquer algorithms lie, is in the combined step.
Given solutions to your sub-problems, how do you somehow recover a solution to the
original problem, the one you actually care about? So for closest pair, the
question’s gonna be, given that you’ve computed the closest pair on the left half
of the points, the closest pair on the right half of the points, how do you then
quickly recover the closest pair from the whole. Point set. That’s a tricky problem.
That’s what we’re going to spend most of our time on. So let’s make this divide and
conquer approach for closest pair a little bit more precise. So let’s now actually
start spelling our, our closest pair algorithm. The input we’re given, it’s,
this follows the preprocessing steps. You’ll recall that we invoked merge sort,
twice, we get our two sorted copies of the point set PX, sorted by X coordinate, and
PY sorted by Y coordinate. So the first dividend is the division step. So given
that we have the copy of the points [inaudible] x of the x coordinate. It’s
easy to identify that left most part of the points goes with these. Those n over
two smallest x coordinates. And then the right half will be n over two largest x
coordinates. We’re gonna call those u and r [inaudible]. One thing I’m skipping over
is the base case, I’m not going to bother writing that down. So base case omitted.
But it’s what you would think it would be. So basically, once you have a small number
of points, say, two points or three points, then you can just solve the
problem in constant time by brute force search. You just look at all the pairs,
and you return the closest pair. So, think of it as being at least four points in the
input. Now, in order to [inaudible] to call [inaudible] again on the left and
right half?s, we need sorted versions of Q and R both by X coordinates and by Y
coordinates. So we’re just going to form those by doing suitable linear scans
through PX and PY. So one thing I encourage you to think through carefully
or maybe even code up. After the video is, how would you, form QX, QY, RX and RY,
given that you already have PX and PY. And if you think about it, because PX and PY
are already sorted, just producing these sorted sub lists takes linear. It’s, in
some sense, the opposite of the merge subroutine we used in Merge Short. Here,
we’re sort of splitting rather than merging. But again, this can be done in
linear time. That’s something you should think through carefully later. So that’s
the division step. Now we just conquer, meaning we recursively call closest pair
on each of the two sub-problems. So when we invoke, closest pair on the left half
of the points on Q, we’re going to get back what are indeed the closest pair of
points amongst those in Q. So we’re gonna call those P1 and PQ. So among all pairs
of points that both lie in Q, P1 and Q1 minimize the distance between them.
Similarly, we’re gonna call P2 Q2, the results of the second reversive call is P2
and Q2 are amongst all pairs of points that both lie in R, the pair that has the
minimum Euclidian distance. Now conceptually there’s two cases. There’s a
lucky case and there’s an unlucky case. In the original point set P, if we’re lucky,
the closet pair of points in all of P. Actually both of them lie in queue. Or
both of them lie in R, in this lucky case we’d already be done, if the closest pair
in the entire point set. They happen to both lie in q then this first [inaudible]
is going to recover them and we just have them in our hands: p1, q1. Similarly if
both, of the closest pair of points in all of p lies in the right side, in r. Then
they get handed to us on a silver platter by the second recursive call, that just
operates on R. So in the unlucky case, the closest pair of points in P happens to be
split. That is, one of the points lies in the left half in Q, and the other point
lies in the right half in R. Notice if the closest pair of points in all of P is
split, is half in Q and half in R. Neither recursive call is going to find it, okay.
The pair of points is not past to either of the two recursive calls, so there’s no
way it’s going to be returned to us, okay. So we have not identified the closest
pair. After these two recursive calls, if the closest pair happens to be split. This
is exactly analogous to what happened when we were counting inversions. The recursive
call on the left half of the array counted the left inversions, the recursive call on
the right half of the array counted the right inversions. But we still had to
count the split inversions. So, in this closest pair algorithm, we still need a
special purpose sub-routine that computes the closest pair for the case in which it
is split, and which is one point in q and one point in r. So, just like in counting
inversions, I’m gonna write down that subroutine, I’m gonna leave it
unimplemented for now. We’ll figure out how to implement it quickly. In the rest
of the lecture. Now, if we have a correct implementation of closest split pair. So
that takes, us, input the original point set, set up an X and Y coordinate, and
returns the smallest pair that’s split. Or one point’s in Q and one point’s in R,
then we’re done. So then the split, then the closest pair has to either be on the
left or on the right, or it has to be split. Steps two through four compute the
closest pair in each of those categories. So those are the only possible candidates
for the closest pair, and we just return the best of them. So that’s an argument
for Y, if we have a correct implementation of the closest split pair subroutine, then
that implies a correct implementation of closest. Now, what about the running time?
So the running time of the closest pair algorithm is going to be, in part,
determined by the running time of closest split pair. So on the next quiz, I want
you to think about what kind of running time we should be shooting for with the
closest split pair subroutine. So, the correct response to this quiz is the
second one. And, the reasoning is just by analogy with our previous algorithms for
Merge Short and for counting inversions. So what is all of the work that we would
do in this algorithm? Well, we do have this preprocessing step we call Merge
Short twice. We know that’s N log N. So we’re not gonna have a running time better
than N log N, ’cause we sort at the beginning. And then we have a recursive
algorithm with the following flavor. It makes two recursive calls. Each recursive
call is on a problem of exactly half the size, with half the points of the original
one. And outside of the recursive calls, by assumption, by, in the problem, we do a
linear amount of work in computing the closest split pair. So. With the exact
same [inaudible] tree which proves an N log inbound for merge short, proves an N
log inbound for how much work we do after the pre-processing step. So that gives us
an overall running time bound of N log N. Remember, that’s what we were shooting
for. We were working N log N already to solve the one dimensional version of
closest pair. And the goal of these lectures is to have an M-log in algorithm
for the 2D version so this would be great, in other words, the goal should be to have
a correct. Linear time implementation of the closest split pair, subroutine. If we
can do that, we’re home free. We get the desired in log and out algorithm. Now I am
gonna proceed in a little bit to show you how to implement closest with pair but
before I do that I want to point out one subtle but. Key idea which is going to
allow us to get this linear time correct implementation. So let me just put that on
the slide. So the key idea is that we don’t actually need a full blown correct
implementation of the closest split pair subroutine. So I’m not actually going to
show you a linear time subroutine that always correctly computes the closest
split pair of a point set. The reason I’m not going to do that, is, that’s actually
a strictly harder problem. Than what we need to have a correct recursive algorithm
we do not actually need a sub routine that for every point set always correctly
computes. The closest with a pair of points. Remember there’s a lucky case and
an unlucky case. The lucky case is where the closest pair in the whole points at P
happens, to lie entirely in the left half of the points Q. Or in the right half of
the point R. In that lucky case, we, one of our recursive calls will identify this
closest pair and hand it over to us on a silver platter. We could care less about
the split pairs in that case. We get the right answer without even looking at the
split pairs. Now there’s this unlucky case where the split pair happens to be the
closest pair of points. That is when we need this linear concept routine and only.
Then, only in the unlucky case where the closest pair of points happens to be
split. Now that’s in some sense a fairly trivial observation but there’s a lot of
ingenuity here in figuring out how to use that observation. The fact that we only
need to solve a strictly easier problem. And that will enable the linear time
implementation that I’m going to show you next. So now let’s rewrite the high level
recursive algorithm slightly to make use of this observation that the closest split
pair sub-routine only has to operate correctly in the regime of the unlucky
case when, in fact, the closest split pair is closer than the result of either
recursive call. So I’ve erased the previous steps four and five that, and,
but we’re going to rewrite them in a second. So before we evoke closest split
pair, what we’re going to do is we’re going to see how well did our recursive
calls do. That is, we’re going to define a parameter, little delta. Which is going to
be the closest pair that we found, or the distance to the closest pair we found by
either recursive call. So the minimum of the distance between P1 and Q1, the
closest pair that lies entirely on the left, and P2-Q2, the closest pair that
lies entirely on the right. Now, we’re going to pass this Delta information as a
parameter into our closest split pair subroutine. We’re gonna have to see why on
Earth that would be useful. I still owe you that information. But, for now, we’re
just gonna pass delta as a parameter for use in closest split pair. And then as
before, we just do a comparison between the three candidate closest pairs and
return the best of the, of the trio. So just we’re all clear on where things
stand. So what remains is to describe the [inaudible] of closest with pair. Before I
describe it. Let me be just crystal clear on what it is that we’re going to demand
of the [inaudible]. What do we mean to have a correct and [inaudible] login time
closest to [inaudible]. Well as you saw on the quiz, we want the running time to be O
of N always. And for correctness, what do we need? Again, we don’t need it to always
compute the closest split pair, but we need it to compute the closest split pair
in the event that there’s a split pair of distance strictly less than delta,
strictly better than the outcome of either recursive call. So now that we’re clear on
what we want, let’s go ahead and go through the pseudo-code for this closest
split pair subroutine. And I’m gonna tell you up front. It’s gonna be fairly
straightforward to figure out that this subroutine runs in linear time, O event
time. The correctness requirement of closest split pair will be highly non
obvious. In fact, after I show you this pseudo-code, you’re not gonna believe me.
You’re gonna look at the pseudo-code, and you’re gonna be, like, what are you
talking about? But in the second video on the closest [inaudible] lecture, we will
in fact show that this is, a correct subroutine. So how’s it work? Well, let’s
look at a point set. So the first thing we’re going to do is a filtering step.
We’re going to prune a bunch of the points away and sort of zoom in on a subset of
the points. And the subset of the points we’re going to look at is those that lie
in a vertical strip which is roughly centered in the middle of the point set.
So here’s what I mean. By centered I mean we’re going to look at the middle X
coordinate. So let X bar be the biggest X coordinate in the left half, so that is
the sorted version of the points by X coordinate. Look at the N over two
smallest coordinate. So in this example where we have six points, all this means
is we draw, we imagine drawing a line. Between the third points, so that’s gonna
be X bar. The X coordinate of the third point from the left. Now since we’re
passed as input a copy of the points sorted by X coordinate we can figure out
what X bar is in constant time just by accessing the relevant entry of the array
PX. Now, the way we’re going to use this parameter Delta that we pass. So, remember
what Delta is. So, before we invoke the closest split pair of subroutine and the
recursive algorithm. We make our two recursive calls. We find the closest pair
on the left, the closest pair on the right, and Delta is whatever the smaller
of those two distances are. So, Delta is the parameter that controls whether or not
we actually care about the closest split pair or not. We care if and only if there
is a split pair at distance less than Delta. So, how do we use Delta? Well that
is going to determine the width of our strip. So, the strip is going to have
width to Delta and it’s going to be centered around X. And the first thing
we’re going to do is, we’re going to ignore forevermore, points which do not
lie in this vertical strip. So the rest of the algorithm will operate only on the
subset of P, the subset of the points that lie in this strip. And we’re gonna keep
track of them sorted by Y coordinate. So the formal way to say that they lie in
this strip is if they have X coordinate in the interval, with lower end point xbar
minus delta, and upper end point Xbar plus Delta. Now how long does it take to
construct this set SY sorted by Y coordinate? Well fortunately we’ve been
passed as input a sorted version of the points PY. So to extract SY from PY, all
we need to do is a simple linear scan through PY checking for each point where
it’s X coordinate is. So this can be done in linear time. Now I haven’t yet shown
you why it’s useful to have this sorted set as Y, but if you take it on faith that
it’s useful to have the points in this vertical strip sorted by Y coordinate, you
now see why it was useful that we did this merge sort all the way at the beginning of
the algorithm before we even underwent any recursion. Remember, what is our running
time goal for closest split pair? We want this to run in linear time. That means we
cannot sort. Inside the closest split [inaudible] subroutine. That would take
too long. We want this to be linear time. Fortunately, since we sorted once and for
all at the beginning of the closest pair algorithm, extracting sorted sub lists
from those sorted list of points can be done in linear time, which is within our
goals here. Now it’s the rest of the sub-routine where you’re never gonna
believe me that it does anything useful. So, I claim that, essentially, with a
linear scan through SY, we’re gonna be able to identify the closest split pair of
points in the interesting, unlucky case where there is such a split pair, with
distance less than Delta. So here’s what I mean by that linear scan through SY. So,
as we do the scan we’re gonna keep track of the closest pair of points of a
particular type that we’ve seen so far. So let me introduce some variables to keep
track of the best candidate we’ve seen so far. Is going to be a variable best which
will initialize to be delta remember we’re uninterested in split pairs unless they
have distance stripping less than delta. And then we’re going to keep track of the
points themselves. So we’ll initialize the best pair to be null. Now here’s the
linear scan. So we go through the points of SY in order of Y coordinate. Okay, well
not quite all the points of SY. We stop at the eighth to last point and you’ll see
why in a second. And then for each position I of the array SY we investigate
the seven subsequent points of this same array SY. So for J going from one to
seven, we look at the Ith and I+Jth entry of SY. So if SY looks something like this
array here, at any given point in this double four loop, we’re generally looking
at. An index I, a point in this, in these I [inaudible] of the array, and then some
really quite nearby point in the array, I+J, because J here is going to be at most
seven Okay, so we’re constantly looking at pairs in this array but we’re not looking
at all pairs of all. We’re only looking at pairs that are very close to each other,
within seven positions of each other. And what do we do for each choice of I and J?
Well, we just look at those points. We compute the distance. We see if it’s
better than all of the pairs of points of this form that we’ve looked at in the
past, and if it is better, then we remember it. So, we just remember the
best, i.e., closest pair of points of this particular type for choices of I and J of
this form. So in more detail, if the distance between the current pair of
points P and Q is better than the best we’ve seen so far, we reset the best pair
of points to be equal to P and Q and we reset the best distance, the closest
distancing so far, to be the distance between P and Q, and that’s it. Then, once
this double four loop terminates, we just return at the best fare. So, one possible
execution of closest split pair is that it never finds a pair of points, p and q with
distance left in delta. In that case this is going to return null and, then, in the
outer call, in the closest pair, obviously you interpret a null pair of points to
having infinite distance. So, if you call closest with pair and it doesn’t return
any points, then the interpretation is that there’s no interesting split pair of
points and you just return the better of the results of the two recursive calls p1,
q1 or p2, q2. Now as far as the running time of this subroutine. What happens
here. What we do constant work, just initializing the variables, then notice
that the number of points in s-y, well, in the worst case you have all of the points
of p so it’s going to be in those end points and. So you do a linear number of
iterations in the outer fore loop, but here’s the key point. In the inner fore
loop, right, normally double fore loops give rise to [inaudible] running time, but
in this inner fore loop, you only look at a constant number of other positions. We
only look at seven other positions. And for each of those seven positions, we only
do a constant number at work, right. We just compare distance and make a couple of
other comparisons and reset some variables. So for each of the linear
number of outer iterations, we do a constant amount of work, so that gives us
a running time of. O event for this part of the algorithm. So as I promised,
analyzing the running time of this closest split pair [inaudible] was not
challenging. We just in a straight forward way looked at all the operations. And
again, because in the key linear scan we only do constant work per index the
overall running time is big O of N just as we want it. So this does mean that our
overall recursive algorithm will have running time O of N log N. What is totally
not obvious and perhaps even unbelievable is that this subroutine satisfies the
correctness requirements that we wanted. And remember what we needed. We needed
that whenever we’re in the unlucky case whenever in fact the. Pair of points in
the whole point set is split. This sub-routine. Better find it. So, but it
does, and that’s made precise in the following correctness claim. So let me
phrase the claim in terms of an arbitrary split pair which has distance less than
delta, not necessarily the closest such pair. Suppose there exists. A P. On the
left, a point on the left side, and a point on the right side, so that it’s a
split pair. And suppose the distance of this pair is less than Q. Now there may or
may not be such a pair of points P.Q. Don’t forget what this parameter delta
means. What delta is by definition, is the minimum, of D. Of P. One Q. One, where
P.1Q.1 is closest pair of points that lie entirely in the left half of the point set
Q., and D. Of P.2, Q.2. Where similarly P2Q2 is the closest pair of points that
lies entirely on the right, inside of R, so. If there’s a split pair with distance
less than Delta, this is exactly the unlucky case of the algorithm. This is
exactly where neither recursive call successfully identifies the closest pair
of points. Instead, that closest pair is a split pair. On the other hand, if we are
in the lucky case, then there will not be any split pairs with distance less than
Delta. Because the closest pair lies either all on the left or all on the
right, and it’s not split. But remember, so we’re interested in the case where
there is a split pair, that has distance less than Delta. Where there is a split
pair that is the closest pair. So the claim has two parts. The first part, part
A, says the following, it says that there’s a split pair P and Q of this type
then P and Q are members of S, Y. Now, let me just sorta redraw the cartoon. So
remember what SY is. Sy is that vertical strip. And again, the way we got that, is,
we drew a line through a medium x coordinate, and then we fattened it by
Delta on either side. And then we focused only one points that lie in the vertical
strip. Now, notice our counts, split pair subroutine. If [inaudible] returns a pair
of points, it’s gonna return a pair of points PQ that belong to SY. First, if it
filters down to SY, then it does a linear search through SY. So if we wanna believe
that our subroutine identifies S split pairs of points, then, in particular, such
split pairs of points better show up in SY, they better survive the filtering
step. So that’s precisely what part A of the claim is. Here’s part B of the claim.
And this is the more remarkable part of the plan, which is that P and Q are almost
next to each other in this sorted array SY. So they’re not necessarily adjacent
but they’re very close. They’re within seven positions away from each other. So
this is really the remarkable part of the algorithm. This is really what’s
surprising and what makes the whole algorithm work. So, just to make sure that
we’re all clear on everything, let’s show that if we prove this claim, then we’re
done. Then we have a correct fast implementation of a closest pair
algorithm. I certainly owe you the proof of the claim. That’s what the next video’s
gonna be all about. But let’s show that if the claim is true, then we’re home free.
So if this claim is true, then so is the following corollary, which I’ll call
corollary one. So corollary one says, if we’re in the unlucky case that we
discussed earlier. If we’re in the case where the closest point in the whole point
set P does not lie both on the left, does not lie both on the right. But rather, has
one point on the left and one on the right. That is, it’s a split pair. Then in
fact, the counts pair subroutine will correctly identify the closest split pair
and therefore the closest pair overall. Why is this true? Well what does count
split pair do? “Kay, so it has the double four through. And it, it thereby
explicitly examines a bunch of pairs of points, and it remembers the closest pair
of all the pairs of points that it examines. What does this, so what are the
criteria that are necessary for [inaudible] a pair to examine a pair of
points? Well, first of all the. Points p and q both have to survive the filtering
step. And make it into the [inaudible] as y. Right so cancel the pair. Only searches
over the ray as y. Secondly, it only searches over rays of point almost
adjacent to x y. That are only seven positions [inaudible]. But amongst the
pairs of points that satisfy those two criteria, counts, but [inaudible]
certainly compute the closest such pair, right? It just explicitly remembers the
best of them. Now, what’s the content of the claim? Well, the claim is guaranteeing
that every potentially interesting split pair of points, namely, every split pair
of points with distance less than Delta, meets. Both of the criteria which are
necessary to be examined by the count-split pair sub routine. So first of
all, and this is the content of part a, if you have an interesting split pair of
points with distance less than delta, then they’ll both survive the filtering step.
They’ll both make it into the array as Y. Part A says that. Part B says they’re
almost adjacent in SY. So you have an interesting split pair of points meaning
it has distance less than delta, then they will in fact be at most seven positions
apart. Therefore count-split pair will examine all such split pairs, all split
pairs with distances less than delta and, just by construction, it will compute the
closest pair of all of them. So again, in the unlucky case where the best pair of
points is a split pair, then this claim guarantees that the count-split pair will
compute the closest pair of points. Therefore, having handled correctness, we
can just combine that with our earlier observations about running time. And
corollary two just says, if he can prove the claim, then we have everything we
wanted. We have a correct O of N login, implementation for the closest pair of
points. So with further work and lot more ingenuity, we’ve replicated the guarantee
that we got, just by sorting for the one dimensional case. Now again this
corollaries hold only if the plane is in fact true. And I have given you no
justification of this plane. Even the statement of the plane, might be a little
bit shocking. For you, I would demand an explanation. For why this one is true and
that’s what I’m gonna give you in the next

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