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 in n

log n 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 in

the plane, again, in n log n, n time. So we will succeed in this goal. I'm going

to show you an n log n time algo rithm for 2D closest pair. It's going to take us a

couple steps. So let me begin with a high level approach. Alright. So the first I

need to try is just to 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 step of our algorithm, which you should really think of as a

preprocessing step. We're going to take the input points. We invoke merge sort

once to sort them according to x coordinate, that's one copy of the points.

And then we make a second copy of the points where they're sorted by y

coordinates. So we're going to 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, so this preprocessing step only takes o of n

log n time. And again, given that we're shooting for an algorithm with running

time big O of n log n, why not sort the points? We don't even know how we're going

to use this fact right now, but it's sort of harmless. It's not going to effect our

goal of getting a big of O n log n time 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 is a sense for what are the four free

primitives, what are manipulations or operations you can do on data which

basically are costless. Meaning that if your data set fits in the main memory of

your computer, you can basically invoke the primitive and it's just going to 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 wa nt the points to be sorted. It

just seems like it's probably going to be useful, motivated by the 1D case, so let's

go ahead and make assorted copies of the points by x and y coordinate upfront. So

reasoning by analogy with the 1D suggests that sorting the points might be useful,

but we can't carry this analogy too far. So in particular, we're not going to 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 going to 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 going to 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's 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 going to show up

consecutively on either of the two arrays, right? So in the array that's sorted 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 sort of

by y coordinate, this green coordinate is going to be wedged between the two red

points. So you won't even notice these red point if you just do a linear scan if your

px and py, or py look at the consecutive pairs of points. So, following our

preprocessing step where we just invert, invoke merge sort twice we're going to do

a quite nontrivial divide and conquer algorithm to compute 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 merge sort

algorithm to sort. Divide and conquer is being used there to get an n log n running

time in this preprocessing step, and the n, we're going to use it again on sorted

arrays in a new way and that's what I'm going to 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 subproblems. 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 the merge sort and counting

inversions problems, where we took the array and broke it into its left and right

half. So here, we're going to take the input point set, and again, just recurse

on the left half of the points, and recurse on the right half of the points.

So here, by 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 complete 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 is in the

combined step. Given the solutions to your sub problems, how do you somehow recover a

solution to the original problem? The one that you actually care about. So for

closest pair, the questionis going to be, given that you've computed the closest

pair on the left half of the points, and the closest pair on the right half of the

points, how do you then quickly recover the closest pair for 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 out our closest pair algorithm.

The input we're given, it's, this follows the preprocessing steps or recall that we

invoke, merge sort, we get our two sorted copies of the poin t 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 a copy of the points px sorted by x coordinate, it's

easy to identify the leftmost half of the points, those with the, those n over two

smallest x coordinates and in the right half, those were the n over two largest x

coordinates. We're going to call those Q and R respectively. 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 point, say two points or three points, then you can just solve the

problem in constant time by a brute-force search. You just look at all the pairs and

you return the closest pair. So think of it being at least four points in the

input. Now, in order to recurse, to call clo pair again, in the left and right

halves, we need sorted version of Q and R, both by x coordinate and by y coordinate,

so we're just going to form those by doing suitable linear scans through px and py.

And 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 sublists takes linear time. It's in some sense the

opposite of the merge subroutine used in merge sort. 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 line on each of the two

subproblems, 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 going to 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 going to

call Q2Q2 the results of the second recursive call, that is, P2 and Q2 are

amongst all pairs of points that both lie in R, the pair that has the minimum

Euclidean 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

closest pair of points in all of P, actually, both of them lie in Q 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 recursive call is

going to recover them and we just have them in our hands P1Q1. Similarly, if both

of the closest pair of points in all of P lies on the right side in R, then they get

handed to us on a silver platter by the second recursive call that just operate on

R. So in the unlucky case, the closest pair of point 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 passed 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 analagous 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 subroutine that computes the closest pair for the

case in which it is split, in which there is one point in Q and one point in R. So

just like in counting inversions, I'm going to write down that subroutine and

I'm going to leave it unimplemented for now, we'll figur e 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 sort of the x and y

coordinate, and returns the smallest pair that's split or one points in Q and one

points in R, then we're done. So then, the split, then the closest pair has to either

be on the lef or onm 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 returned the best of them. So that's

an argument for y, if we have a correct implementation of the closest split para

subroutine, then that implies a correct implementation of closest pair. Now, what

about the running time? So the running time of the closest para algorithm is

going to be in part determined by the running time of closest split pair. So in

the next quiz, I want you to think about what kind of running time we should be

shooting for with a closest split pair subroutine. So the correct response of

this quiz is the second one, and the reasoning is just by analogy with our

previous algorithms for merge sort and for counting inversions. So, what is all of

the work that we would do in this algorithm or we do have this preprocessing

step we call merge sort twice, we know that's n log n, so we're not going to 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 we, the exact same recursion tree which proves

an n log n bound for merge sort, proves an n log n bound for how much work we do

after the preprocessing step, so that gives us an overall running time bound of

n log n. Remem ber, 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 n log n algorithm for the 2D versions. So this

would be great. So 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 n log algorithm. Now, I'm going to proceed in a

little bit to show you how to implement closest split pair, but before I do that,

I want to point out one subtle, the 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 closets split pair subroutine. So, I'm not actually going to show you a

linear time subroutine that always correctly computes the closets split pair

of a point set. The reason I'm 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 subroutine that, for every point sets, always correctly computes the

closest split pair of points. Remember, there's a lucky case and there's an

unlucky case. The lucky case is where the closest pair in the whole point set P

happens to lie entirely in the left half of the points Q or in the right half of

the points 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 pair, pairs. Now, there's this unlucky case where the split pairs happens

to be the closest pair of points. That is when we need this linear time subroutine,

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 i n 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 subroutine 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 4 and 5, that, but we're

going to rewrite them in a second. So, before we invoke close 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 of 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 P2Q2, 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 going to have to see why on earth that

would be useful and I still owe you that information, but, for now, we're just

going to pass delta as a parameter for use in the 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. And so, just so we're all clear on, on where things

stand, so what remains is to describe the implementation of closest split pair, and

before I describe it, let me just be crystal clear on what it is that we're

going to demand of the subroutine. What do we need to have a correct in o of n log n

time closest pair algorithm. 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 events that there is 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 pseudocode for this closest

split pair subroutine. And I'm going to tell you upfront, iIt's going to be fairly

straightforward to figure out that the subroutine runs in linear time, o of n

time. The correctness requirement of closest split pair will be highly

non-obvious. In fact, after I show you this pseudo you're not going to believe

me. You're going to look at the pseudocode and you'd be like, what are you talking

about? But in the second video, on the closest pair lecture, we will in fact show

that this is a correct sub-routine. So, how does 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 so to 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

center dot, 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 in the sorted version of the

points by x coordinate, we look at the n over two smallest ex-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 going to 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're passed, so remember what delta is. So

before we invoke the closest split pair subroutine in 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's going to determine the width

of our strip, so the strip's going to have width 2 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 line 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 on the strip, and we're going to keep track of them sorted by y

coordinate. So the formal way to say that they line the strip, is that they have x

coordinate in the interval with lower endpoint x bar minus delta and upper

endpoint x bar 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 p y checking for each point where its 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 recurssion. 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 pair subroutine. That would take too long. We want this to be in

linear time. Fortunately, since we sorted once and for all at the beginning of the

closest pair algorithm, extracting sorted sublists from those sorted lists of points

can be done, done in linear time, which is within our goals here. Now, it's the rest

of t he subroutine where you're never going to believe me that it does anything

useful. So, I claim that essentially with a linear scan through Sy, we're going to

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, we're going to 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. There's going to be a vary, variable best which will initialize to be

delta. Remember, we're uninterested in split pairs unless they have distance

strictly less than delta. So, and then we're going to keep track of the points

themselves, so we'll initialize the best pair to be null. Now, here is the linear

scan. So we go through the points of Sy in order 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 the same array Sy. So for j going from one to seven, we look at

the Ith, and I plus jth entry of Sy. So if sy looks something like this array here,

in any given point in this double for loop, we're generally looking at an index

I, a point in this, in this of the array, and then some really quite nearby point in

the array I plus j, because j here's 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, ie c losest 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 of 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 distance seemed so

far to be the distance between p and q and that's it. Then, once this double for loop

terminates, we just return it the best pair. So one possible execution of closest

split pair is that it never finds a pair of points, p and q, at distance less than

delta. In that case, this is going to return null and then in the outer call. In

the closet pair, obviously, you interpret a null pair of points to have an infinite

distance. So if you call closest split 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 p1Q1 or P2Q2. Now,

as far as the running time of the subroutine, what happens here? Well, we do

constant work just initializing the variables. Then notice that the number of

points in Sy, well in the worst case, you have all of the points of P. So, it's

going to be the most endpoints, and so, you do a linear number of iterations in

the outer for loop. But here is the key point, in the inner for loop, right,

normally double for loops give rise to quadratic running time, but in this inner

for loop we 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 of work. Right? We just, we want to compare distance and make a couple 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

of n for this part of the algorithm. So as I promised, analyzing the running time of

this closest split pair subroutine was not challenging. We just , in a

straightforward way, looked at all the operations. 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 wanted. 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 satifies the correctness requirements that

we wanted. Remember, what we needed, we needed that whenever we're in the unlucky

case, whenever, in fact, the closest pair of points in the whole point set is split,

this subroutine better find it. So, but it does, and that's being precise in the

following correctness claim. So let me rephrase the claim in terms of an

arbitrary split pair, which has distance less than delta, not necessarily the

closest such pair. So suppose, there exists, a p on the left, a point on the

left side and a point on the right side so that is 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, PQ.. Don't forget what this parameter delta means. What delta is, by

definition, is the minimum of d of p1q1, for p1q1 is the closest pair of points

that lie entirely in the left half of the point set Q and d of p2q2, or similarly,

p2Q2 is the closest pair of points that 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 on the right, and it's not split. But remember, we're interested in

the case where there is a split pair that has a 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 if there's a split pair p and, and q

of this type, then p and q are members of Sy. And let me just sort of 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 median x coordinate and then we fattened

it by delta on either side, and then, we focused only on points that lie in the

vertical strip. Now, notice our counts split pair subroutine, if it ever returns

a pair of points, it's going to return a pair of points pq that belong to Sy.

First, it filters down to Sy, then it does a linear search through Sy. So if we want

to believe that our subroutine identifies best 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 claim, 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

is going to 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 corollaryl 1. So corollary 1 says, if we're in the unlucky case that we

discussed earlier, if we're in the case where the closest point and the whole

points of 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 but as it's a split pair, th en in

fact, the count split 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? Okay, so it has this double for loop, and thereby, explicitly examines

a bunch of pairs of points and it remembers the closest pair of all of the

pairs of points that it examines. What does this, so what are the criteria that

are necessary for count split pair to examine a pair point? Well, first of all,

the points p and q both have to survive the filtering step and make it into the

array Sy. Right? So count split pair only searches over the array Sy. Secondly, it

only searches over pairs of points that are almost adjacent in Sy, that are only

seven positions apart, but amongst pairs of points that satisfy those two criteria,

counts but pair will 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 and 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

subroutine. 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 Sy., part A

says that. Part B says they're almost adjacent in Sy. So if 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 distance 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 h andled correctness, we can just combine that with our earlier

observations about running time and corollary 2 just says, if we can prove the

claim, then we have everything we wanted. We have a correct O of n log n

implementation for the closest pair of points. So with further work and a lot

more ingenuity, we've replicated the guarantee that we got just by sorting for

the one-dimensional case. Now again, these corrollaries hold only if this claim is,

in fact, true and I have given you no justification for this claim. And even the

statement of the claim, I think, is a little bit shocking. So if I were you I

would demand an explanation for why this claim is true, and that's what I'm going

to give you in the next video.