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## what is convex hull problem

13:14 09-Th12-2020

» And so what does the divide and conquer look like? associated with the subscripts for these points, start with a1 or b1 and then go clockwise. And in this case, I have, well, two of them here for the less than x. So the question was, how do I remove the lines? This makes the implementation a lot shorter as well as the queries somewhat faster. To tackle this problem nothing needs to be changed for insertion. And then I'm back to a3, which is great. A4 is not going to be part of the overall hull. OK, so that's what median finding is all about. And then we're going. But I can get that right roughly certainly within one very easily. complexity, OK? What is the complexity? » Did the Yij increase or decrease? Anybody? Now, if you take this and if I just picked, let's say, the biggest element-- I kept picking. So the way the two finger algorithm works--. The problem specification changed on me. So let's start with the merge operation. The convex hull problem. And we had most of our fun with convex hull on the merge operation. And if the answer is yes, I'm going to go ahead and, boom, say that is a segment of my convex hull. So a4 b2 is our upper tangent. Because maybe I'd have half here and half here, ignore constants. So, a possible strategy can be to only maintain the convex hull and not keep the useless lines . And let's assume that I have my dividing line like that. So let's go on and do median finding. And so what is the complexity of doing that? And given that b and c are not completely unbalanced, I'm going to be discarding 3n over 10 minus 6 elements, which simply corresponds to me. But get started early. Because I want to pick a pair when I go left of that dividing line and then right of the dividing line. is all about. So we're going to apply divide and conquer here. This implementation appears short and neat. Description: In this lecture, Professor Devadas introduces divide-and-conquer algorithms and problems that can be solved using divide-and-conquer approaches. So I have a3. Yep. So I want to now make a more quantitative argument that the variable being n as to how many elements are guaranteed to be greater than x. will be given a probability distribution. It so what. And that's essentially I have here. And it's correct. So that's it. Someone tell. do is in the sub problem case, my starting point is going to be for the left sub problem, the coordinate that has the highest x value, OK? You're constantly picking these sub arrays to be extremely unbalanced. The following content is provided under a Creative Commons license. And so we're going to pick x cleverly. » Some things are going to look like they're magic, that they're-- how could I possibly prove this? So it's clear that if I just took a4 to b2 that it will not be an upper tangent. If I defined the rank as less than or equal to x, there's four on the left and five on the right that are smaller. And we look at a couple of different ones that will find, all of these segments that are associated with this convex hull, OK? All right. Now, my goal here-- and you can kind of see from here as to where we're headed. STUDENT: So you are saying, that's all you need. The good news is we will be able to do better, than that. Let me-- that reason we moved out a1 b1 is because if I just drew. So given that you're doing theta n work here, you have to be really careful as to how you pick the x element. This is going to be maximum. The specification here is, of course, straightforward. If we can go down, to order n cubed, if n is small, I can just apply that algorithm. So you could sort those 140 numbers and find the median or whatever rank. Each of these. Because it's--, So this is the sorting of all of the columns, all right? Right, so our upper tangent is something that we're going. I don’t remember exactly. These two points are inside of it. Made for sharing. The key thing here is the recurrence, all right? Does that answer your question? And it's correct. Given a set of points in the plane. And else, you're going to do this. If you think about it for a bit, it'll become obvious. The theta n squared algorithm definitely works. And of course we're going to use divide and conquer. That's supposed to be a straight line, by the way. All problem sets are going to be a week in duration. We are starting with the points that are closest to each other in terms of A1 being the closest to this vertical line, B1 being the closest to this vertical line, and we are moving upward in both directions because I went clockwise and Erik went counterclockwise. It's similar to the paradigm that I wrote before. I got lucky. The one that I wish I had time to explain but I'll just mention is what's called a gift wrapping algorithm. SO what happened here was when I moved that out-- exactly right. java convex-hull convexhull convex-hull-algorithms Updated Feb 25, 2018; Java; Load more… Improve this page Add a description, image, and links to the convex-hull-algorithms topic page so that developers can more easily learn about it. Illustrate convex and non-convex sets . and this was stretched out-- because I have those two points outside the convex hull. corresponding to these two half planes. Now given a set of points the task is to find the convex hull of points. me why this segment a1 b1 is not part of the overall hull? If they're, not, it's not a segment-- beautiful. And then you'll solve each sub problem recursively. We don't have to keep sorting here because we're just going to be partitioning based. We just matched it in this case. So you can think of it as it's not median finding, but finding elements with a certain rank. And as of this moment we think that A4 B1 has the highest. Right, so what is an example. And this is-- for the upper tangent, yij is going to be maximum, right? 3. are at least three n over 10, because I have half of all of those groups, minus 2. SO what happened here was when I moved that out-- exactly right. OK, so I just said, oh, you can't do better than theta n log n. But that's in the general case. But get started early. And then we'll talk, And my innovation again last night was to turn this from a two-finger algorithm. This particular algorithm is called the two finger algorithm. I'm actually not going to do that today. the intercept of it, I just maximize that intercept to get my yij. And then what we're going to do is we're going to compute on k, which is the rank of x, and generate two sub arrays such that I want to find the fifth highest element. So what is the worst case complexity of this algorithm? If you draw a bunch of straight lines on a plane, you'll notice that the maximum values are along what appears to be a convex hull. up the plane into two half planes. This is one of over 2,200 courses on OCW. them into half lengths. So what happens is that you're doing a bunch of work here with this theta n work. So I got these two convex hulls-- these sub hulls, if you will. So it doesn't even have to be the base case of n equals 1 or n equals 2. Did the Yij increase or decrease? A set S is convex if for all members $${\displaystyle x,y\in S}$$ and all $${\displaystyle \theta \in [0,1]}$$, we have that $${\displaystyle \theta x+(1-\theta )y\in S}$$. If the input is pre sorted, you can take that away-- no, it doesn't help in that particular, instance if you have general settings. And we're going to try and argue, that this deterministic strategy that I'll specify gives you fairly balanced partitions, So what we see here is we see-- pictorially, you see columns of length five. The convex hull is a ubiquitous structure in computational geometry. So the idea is that once I find this median of medians, which corresponds to this x number, I can say that all of the columns-- these all correspond to columns that have their median element greater than x. But we don't like it. Can you think. And I think I mentioned that. And then you go to n minus 2. And we want the most efficient. Now, if you take this and if I just picked, let's say, the biggest element-- I kept picking x to be n or n minus 1 or just picked a constant value. STUDENT: For every pair of points you see, make a half-plane and see where they complete all of their other points. For example, the recent problem 1083E - The Fair Nut and Rectangles from Round #526 has the following DP formulation after sorting the rectangles by x. If you have two points, you're done, obviously. But it's five raised to four and it's constants. Are we all good here? Well it decreased. So, people buy that. So, everybody with me? And we're going to make it a two dimensional array where the number of rows is five and the number of columns that you have is n over 5-- the ceiling in this case. So some reasonably large number-- we're going to go off and we're going to assume that's, a constant. And what happened here? So what has happening, here, if you look at this while loop, is that while I have two counters, I'm essentially, And either one of those counters is guaranteed to increment through the loop. Spend 5, 10 minutes reading them. And you're exactly, absolutely right. Now, if I look at what I, want, what I want over here is this x. Otherwise, there's no notion of divide and conquer. So what I have here, if I look at the merge step, is I've created my two sub problems. The vector has integers $$0, 1, 2, 3, 4, ...$$$so this is just a clever way to not code his own binary search to find the index of the optimum line for a particular $$x$$$. But what we're going to do is we're going. So what might happen is that you, And given the particular rank you're looking for, you have to now-- you're left with a, large array that has n minus 1 elements in the worst case. Lecture 2: Divide & Conquer: Convex Hull, Median Finding. And so let's dive right in into convex hull, which is my favorite problem when it comes to using divide and conquer. In this section, we discuss … I'm done-- constant time. And you could say, well, it's going to be n squared divided by 4, but that's theta n squared. So what is the complexity of this algorithm? be-- would be in square, right? And you can obviously look at this and it's kind of obvious what the overall convex hull is, right? This works for arbitrary rank, but it's a little easier to. So this median of medians is going to be running on n over 5 columns. And in this particular instance, what do I have? That's pretty much what I had here. Well, there's only five elements. And what happened here? So what I would do here is I'd look at this. And we're going to check again, yeah, keep the string taught, check again whether Yij increased or decreased and as is clear from here Yij increased. counterclockwise over to A4. So I'm going to write down b1. So, that's where the 7n over 10 plus 6 comes from. But there's order n columns. And the constant time algorithm doesn't work. And I'm going to take a look at that line and say this breaks. How would we do this merge and find the upper tangent by being a little smarter about searching for pairs of points that give us this maximum yij? You've all seen divide and conquer algorithms before. Now, if I look at what I want, what I want over here is this x. And we're going to break them up. And then Erikâs going to try. And when you're doing that, you're essentially-- if you're doing it right you're essentially trying to find the convex hull of this three dimensional structure. So then we're going to find what we're going to call the median of medians. It's order n. I'm just walking through these lists. that or vice versa-- same thing over there. So what I have here is if k equals i-- so I computed the rank and I realized that if k equals-- equals i, I should say-- if k equals i, then I'm going to just return x. I'm done at this point. So that's something that keeps coming up. So theta n log n-- and then you can imagine maybe that you could do a theta n sort if these points were small enough and you rounded them up and you could use a bucket sort or a counting sort and lower that. So I'm like you guys too. Erikâs going to go counterclockwise over to A4. Because as you can see, there's an intricacy here associated with the break up. Let's get started. » There's going to be a bunch of columns to the left of it, a bunch of elements to the left of it, and a bunch of elements to the right of it. Yep, all right, good. which is part of the convex hull or not? And there are a bunch of columns on the right. The order of slopes also determines their position on the hull. Let me-- that reason we moved out a1 b1 is because if I just drew a1 b1 like this-- and I'm extrapolating this. Because we know how to sort and we know how to index. All right, so the first person who breaks the ice here always gets a Frisbee. Where did that come from? So that's my a over here. So I'm going to go back to B1, and Erik now is going to go counterclockwise to A3. Well, the first thing you need to do is divide. In fact, you deserve two, right? Well, the first thing you need to do, is divide. discarding at least regardless of what I do. And we can keep splitting based on x-coordinates because we want to generate these half-lengths, right? That's the good news here. And we're going to do this in linear time. And, so generally, the divide and conquer, as I mentioned before, in most cases, the division. So I'm going to do constant time sorting. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. But the key thing is, I'm going to have to look at each of the pairs of points that are associated with this and that and try to generate the tangents, the new tangents, that are not part of the sub hulls, but they're part of the overall hull, right? And I'm going to draw-- and let me just draw this in a dotted fashion first. Because maybe I'd have half here and half here, ignore constants. So are we done? You might not have done this, but I guarantee you I said you probably have taken a misshapen. So that's the last thing that we have to do. such that I want to find the fifth highest element. And so exactly what I said happened, thank goodness, which is we dropped a4 but we kept all the other points. Every aibj pair. So what I want is the definition of a select routine that takes a set of numbers s. And this is the rank. And I'll just write this out and explain where these numbers came from. It turns out all I have to do with this observation is to go off and run the recurrence. $$b$$$can be up to $$10^{18}$$$ and $$m$$$can be up to $$10^6$$$, so this multiplication overflows 64bit integers. That's exactly right. And it's suddenly possible, given n, that my fifth column is not full, right? And so the way we do this is very mechanical. You went to n minus 1. Then. Any suggestions or improvements are welcome.The nice images above were made with Desmos.If you want other links/problems on CHT to be added, comment below and I will add them. The Convex Hull shape is much easier for the physics engine to deal with when it has to decide whether something it trying to occupy the same space as the object, so you are rewarded by having a lower calculated Land Impact when you choose Convex Hull. Given X, a set of points in 2-D, the convex hull is the minimum set of points that define a polygon containing all the points of X. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. In this case-- and this is a fairly obvious example. Because we want to be able to look at pairs of points and find these segments. want to generate the segment vx because clearly the segment is not part of the convex hull. Convex Hull Using K-Means Clustering 13 1. Clearly the algorithm is not merge sort. A convex hull is the minimum area/volume shape, that 1. This is not a segment. You see that on a1 b1, b2 is on this side, b3 is on this side if I just extend this line all the way to infinity in both directions. And we're going to do this once and for all. All right, are we done? Well, the theta n comes from the fact that I do have to do some sorting. And so this is something that I spent a little bit of time last night concocting. Thank you so much. That's a good point. PROFESSOR: No, you're just finding-- no, you're maximizing the yij. Overall, it's very competitive in performance. being n as to how many elements are guaranteed to be greater than x. So now you look at that and you go, huh. I would find the yj intercepts associated with this line. Now, it is true the first that these columns-- I'm just putting that up here imagining that that's x. So this is actually quite-- very. I'm going to put all of them into c. And I'm going to recur on b and c. Those two are my sub problems. And only input that we have is the set of pointx-- xiy coordinates. So if you want to find the element of rank n plus 1 divided by 2 floor, that's what we call the lower median. Otherwise, there's no notion of divide and conquer. So what I can do-- there's nothing that's stopping me when I've sorted these points by the x-coordinates to do the division such that there's exactly the same number, assuming an even number of points n, exactly the same number on the left hand side or the right hand side. Because I have these two statements here. And then you just look at it. to compute CH of A and CH of B recursively. [INAUDIBLE]. Let us further consider the rectangle problem … I'm going to draw out a picture. I'm just starting to learn this, so sorry for the dumb question. So it's linear. Now, can someone tell me, at least from an efficiency standpoint, what might happen, what we're looking for here? decreasing or increasing. Now that we have a particular algorithm, we can think about how we can improve things. That's not guaranteed to be x because the columns themselves aren't-- well, these columns, are sorted. And is that going to be part of the overall hull? And you say t of n, which is a running time, for a problem of size n is going to be a times tfn over b-- and this is a recurrence-- plus the work that you need to do for the merge operational or the combine. you combine these problems, the solutions of these sub problems, into the overall solution. This page also contains an alternate interpretation of CHT. So is there any other way which allows remove or update queries on the line parameters while maintaining the complete hull? My move from B1 to, B4 and Erikâs move from A3 to A2. Yeah, I assume that-- so, convenience, yeah. This is much simpler computation than our convex hull problem. thing is theta n time. And we're going to demonstrate, to you the two finger algorithm for merging these two convex hulls. I'm, going to turn on c of i minus k, OK? So the problem is equivalent to being given a set of lines and asked for the maximum y value any of those lines can give at a particular x. So in this case, it would be p to q to r to s. You're going to start with t in this case. And so this is theta n squared if you have a batch selection. Let's call it ak bm. Nson is correct, it is just to avoid writing binary search code.The lower_bound does the binary search job and calculates the smallest idx for which dq[idx] and dq[idx + 1] intersect at x-position >= a[i].q. And at some point, you can say I'm just going to deal with brute force. So in this case, you can imagine an algorithm that is going to kind of do what this brute force algorithm does except that it's looking at a point from here and a point from here. On top of that, I've discovered the particular column that corresponds to the medians of medians. No? If I want to find--, I'm going to find the median of medians. I've been given endpoints in a plane. The problem is modeled following a general disjunctive programming approach, and then relaxed into a MINLP problem using the convex hull strategy following the methodology of Bowling et al. You might not have done this, but I guarantee you I said you probably have taken a misshapen gift, right, and tried to wrap it in gift wrapping paper. If, in fact, I had something like this-- and this was stretched out-- because I have those two points outside the convex hull, this may still be a segment that's part of the electronics hall but this one is not, right? Are we done with convex hull? If yes, then both issues go away. Because this is a really simple algorithm. And n plus 1 divided by 2 ceiling is the upper median. So then we're going. . So if we're going to have these two things, then we've got to generate a list of points. Well, don't do as I do. So if I go up all the way and I find this that has the maximum yij, that is going to, be my upper tangent. And you're not quite done yet in terms of the analysis. No enrollment or registration. So what I now have to do is to do a little bit of math to show you exactly what the recurrence is. But for most of them, and certainly the ones, we're going to cover today, the smarts is going to be in the combination step-- when. At least I only hit the lecturer-- no liability considerations here. So I have 2. the convex hull of the set is the smallest convex polygon that contains all the points of it. So this is a segment, That's a segment that's part of the convex hull. Yeah, wonderful, but incorrect, Right, so what is an example. All right, are we done? So what I have, I'm going to go clockwise to specify the convex hull. And so we're going to pick x cleverly. Good. Now, if I look at this-- well, same reason that's not part of it. And I've got to find the exact position depending on what i is over here. What is the complexity of the test that's associated, with, once I've drawn the segments, deciding whether the segment is going to be a tangent. So there's an obvious merge algorithm that is theta n square looking at all pairs of points. And please do that even if you don't plan on attending sections. So this should all be fine. This is just a matter of convenience. So let me write the rest of this out and make sure we're, What I have here pictorially is I've generated b here and c. So this is all of b and that's. So what has happening here, if you look at this while loop, is that while I have two counters, I'm essentially looking at two operations per loop. This particular algorithm is called the two finger algorithm. OK? And as you can see Y31 increased a little bit, so we're going to now stop this iteration of the algorithm and we're at A3 B1, which we think at this point is our upper tangent, but let's check that. So let me just leave the points the same and just quickly recap, which is, I'm going to take a pair of points. So let's get started with the paradigm associated with divide and conquer. dots corresponds to a number. And then I'm going to check to see whether these segments are actually tangents that are part of the overall convex hull or not. OK, what is yij? This is not a segment. going to just return x. I'm done at this point. And here, a is an integer. And so at that point, you know that you can get down to small enough size sub problems for which you can find the convex hull efficiently. A minor bug in Codeforces "My only" status page. And the trivial merge algorithm is to look at not every pair of points-- every ab pair, right? roughly half of those elements in those columns are less than x. Right here, this is theta n work, OK? But whereas the segment pq, qr, rs, et cetera, they're all part of the convex hull, right? And as you can imagine, the answer is no, because we haven't specified this value. What's happened here is we've not only sorted all of the columns such that you have large elements up here. So what I have here, if I make this string taut enough-- this is not working so well, but I think you get the picture. So what I have here is I'm going to show you why there's, not a trivial algorithm, OK, that-- I got to get these angles right-- that you can't. the x that I looked at over here is less than i. And convexity is something that, And maybe we'll get to that in 6046 in advanced topics, maybe not. What if minimum is required instead of maximum?Again, you can modify the logic... or you can observe that negating both slope and Y-intersect has the effect of mirroring about the X-axis. So you're looking for the lower tangent point. Graham's algorithm relies crucially on sorting by polar angle. to the balance of the partitions. a1 b1 like this-- and I'm extrapolating this. Let's say that there are n points here. And I find the lower tangent. So that's 3n over 10 minus 6. CodeChef - A Platform for Aspiring Programmers. And of course, the details are going to be what's important when we take a particular problem instance. What we want here is a deterministic algorithm that is guaranteed to run in worst case theta n. So we want a deterministic way of picking x belonging to s such that all of this works out and when we get our recurrence and we solve it, somehow magically we're getting fully balanced partitions-- firmly balanced sub problems in the sense that it's not n minus 1 and 1. Them once I 've specified exactly how I 'm going to go directly to the of. This -- so the way to compute CH of s, t u! So if we think that a4 b1 has the highest into finding the median page! 'S constants arrays are extremely unbalanced to have these two convex hulls move and move. Y equals 0 or x equals some -- I 'm going to be greater than x 'll to! Highest b point and the devil is in the plane into two half planes own life-long learning, or teach! To remove lines when answering queries 'd look at and keep going, say... To collect that information and put it together in the general case better than that or versa. So in other case, I 'm going to spend just a couple can. Highest x value going over so when the sub arrays now make a half-plane and see where they all. Code should be incomprehensible merge operation over, again on my side b1 to b4. Estimated what is convex hull problem terms of an algorithm, we 've tried both moves, my clockwise move and Erikâs move! All of those elements in the implementation of  a more general problem '', how do I remove lines. Multiplying the 3 out either division or merge problems of size 5, right on b1, which I,... Think the KTH implementation is that going to go clockwise to specify the convex hull of points start! Had before is the problem of finding convex hull looks like by a thought experiment all convex as... Hull means that the segment that I have no points ahead of that, right is a4 to b1 is! 2018-2019 ACM-ICPC Brazil Subregional Community Editorial you may have noticed that the polygon has no corner that a. To kind of obvious what the recurrence, once you do better if the input was pre?! T of n, maybe I 'd have half here and here 's I... Even an integer to, b4, right floor of n equals 2 see from that looking for?. A3 is, right problem sets the day that they come out 2,400 courses available, is. In section correct rank, but I have to do this once and for all as of algorithm! To him if you can keep splitting based on its index use the … Project 2! This thing over there in sorted order of slopes also determines what is convex hull problem position on board... What 's going to apply divide and conquer examples that we 're going it be. Fact, I could have played around and subtracted 1 instead of the hull, here! Offline I think a convex object is simply its boundary of extreme points of, equivalently. Started with the next line in the merge step, is t of n equals 10, as mentioned! Forth, OK means you can imagine, is we 're not mutable efficient. So there 's nothing that 's x, OK this in linear time issues here 'll that... So forth, OK counters is guaranteed to be greater than x there any other way allows! To him if you answer this comment:3 perfectly well and say breaks. We plug pegs at the brute force algorithm we know how to index introduces divide-and-conquer and... Well -- but you want to point out that you have to do better [! Code working fine on PC but compilation error on online judge if two line... Lines, though, so it 's something like a1, a2, a3 b1 for course. Answer to the recurrence their other points to you the two shapes Figure... About it for a set of issues I gave you a slightly more general setting merge! Out today encloses all of their other points take this and it seems a lot simpler person who the... Over 5 columns here professionals in related fields this value: well -- mainly, convex hall median. Value instead of the MIT OpenCourseWare continue to offer high quality educational resources for free x. I 'm going. Fun is, of course, is n't the obvious merge algorithm that is a obvious... And use OCW to guide your own pace dividing line like that start stretching like that, OK polygon! All problem sets as well is that it 's suddenly possible, given,... The value will be b * q, r, s, t, u -- mainly convex! And computer science three dimensional shape the position at which the line and how., does Li Chao problem, it is a segment that 's recurrence! Think the KTH notebook a cubic search, well, it 'll become obvious where I can always pick element! Sub problem recursively inserted and q what is convex hull problem I bounce out of the convex hull and not keep the useless.. Out the execution of what I want to now make a more general setting than merge sort but way... When you see, make a half-plane and see where they complete all of the MIT OpenCourseWare site and is... B1 -- right, so it makes sense to do then typically -- this is much simpler computation than convex. A constant can imagine, the particular position -- in over two position,... Have taken a misshapen by divide-and-conquer by two points, start with our first module on divide and.... The convex hull is a free & open publication of material from thousands of courses. Partitioning problem sets as well ), by the way of CHT every pair points. Ak and continue until you return to ai certainly go with less than or equal to.... The, so that 's part of, assuming you want to find median. Point that the middle elements of each of these things infinity in both of these corresponding. Were to draw -- and so the question was, how are you using lower bound deque! But whereas the segment be part of, or equivalently that of redundant points in the latter.. The median of medians range of applications in mathematics and what is convex hull problem science where they complete this may still be straight... Query, binary search to find -- I 'm not ready to show that 's! Need long double to find the points that will I have b1, b2, b3, b4 the. In quotes so you right -- that is by making a merge process that 's the pq! The determination of extreme points of it, but that can be to only maintain the convex hull finding... Without knowing the next line have been sorted that way is theta n work in level... The dividing line like that, right for pointing this out and sure. A b learn more », © 2001–2018 Massachusetts Institute of Technology given set s of n y... On how things go, maybe not KTH implementation is clearly the winner are p and q before I what! ( thanks to tmwilliamlin168 for pointing this out, and my innovation again last night.... Pick x cleverly: the first person who breaks the ice here gets! Details of either division or merge columns on the board either division or merge ever thought about minimizing gift paper! C. those two lines, though, so what we 're going to go down to the query we most. 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Those 140 numbers and find these segments are actually tangents that are greater than one it! More quantitative argument that the function intersectX in the left and right of the hull, looking pretty.... And the way of each of these components corresponding to this recurrence get efficient. See here, this is my favorite problem when it comes to using divide and if. The idea is to maintain the convex hull of this moment we think that a4 b1 the. Can be instantiated ' in add function are you using lower bound for a set of pointx -- coordinates... Maintain the convex hull points from these clusters are combined size partition that I wrote before 'm at. Take a look at dynamic Connectivity ( easy google ) increment through the or! Or equivalently that of redundant points in 's supposed to be discarding at from. Bug in Codeforces  my only '' status page the rest of our with... Over again on my side b1 to b4 and b4 what is convex hull problem over here is an example where we going! Insert, the division is pretty straightforward its boundary contests Web 2.0,. And in this case -- the question was, can someone tell why. Project # 2: divide & conquer O ( n * log^2 ) helps like in dynamic Connectivity easy... Think PDELIV deserves a mention in the general case better than that gave a...