### gepardo's blog

By gepardo, history, 7 years ago, translation,

This is an editorial for the problems of the today's contest. I tried my best to describe the solutions for the problems as detailed as possible, but if something is not understandable for you, you can write in comments! :)

785A - Антон и многогранники

Hint
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785B - Антон и занятия

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785C - Антон и сказка

This problem had weak pretests. It was made intentionally to cause more hacks. And I have warned you about it in the announcement :)

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785D - Антон и школа - 2

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785E - Антон и перестановка

I'm sorry that this problem was not original. I will try my best to prevent this from happening again.

If you have TL or ML in this problem, don't think that the time/memory limits are too strict. The model solution works in 1.2 seconds and consumes 9 MB memory.

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Alternative solution from netman:

Java code
• +186

| Write comment?
 » 7 years ago, # |   0 Auto comment: topic has been updated by gepardo (previous revision, new revision, compare).
 » 7 years ago, # |   0 Also notice that k can be found using a formula, but such solutions could fail by accuracy, because the formula is using the square root function. Is there any way to get the solution by formula either in c++ or python while still retaining accuracy?
•  » » 7 years ago, # ^ |   +5 use long double throughout
•  » » » 7 years ago, # ^ |   +4 Can you tell why double gives error and not long double ?? I didn't get convincing proof about it.
•  » » » » 7 years ago, # ^ |   0 it's because of how the IEEE 754 standard encodes floating point numbers. double is a 64-bit floating point type which means it has less than 64 bits for the mantissa (meaning it has less than 64 bits of precision; I think it has 53 bits of precision)long double however is typically an 80-bit type which is AFAIK made specifically to have 64 bits for the actual data we care about.even though the problem can be solved by this method, alternative methods that don't require fidgeting with extra precision floating point types are less likely to produce errors so I can't really recommend doing this always.
•  » » 7 years ago, # ^ |   0 I did exactly that :)25530268
•  » » 7 years ago, # ^ |   0
•  » » 7 years ago, # ^ |   0 I thick you can get this ... solution
 » 7 years ago, # |   +2 785A - Anton and Polyhedrons Hint: 404 Not Found  quit interesting:)
 » 7 years ago, # |   +25 Thanks! I like the way the editorial is written :D
 » 7 years ago, # |   +8 Wow, Great Editorial. Equality of all languages.
 » 7 years ago, # |   +4 This editorial with hints and opening blocks! Something new and beautiful in this edutorial! I've never senn something like this!
 » 7 years ago, # |   +9 the iterative solution of C passes in time limit ... should've taken care about that..
•  » » 7 years ago, # ^ |   +6 It's hard to hack such solutions because of the following: 109 passes the time limit. The constraints cannot be increased because 1018 is an almost maximal range for the constraints.
•  » » » 7 years ago, # ^ | ← Rev. 3 →   +11 Test cases could have been added so that these kind of solutions get TLE. Anyways the contest was really good, so is the editorial. Thanks! :)
•  » » » » 7 years ago, # ^ |   0 Do you really know the test that can cause these solutions to fail? As mentioned above, 10^9*(some really simple operations) fits the TL.
•  » » » » » 7 years ago, # ^ |   0 I was talking about multiple test cases in a single judge input file. One single test will always pass..
•  » » » 7 years ago, # ^ |   0 Or you can lowering the time limit :)
•  » » » 7 years ago, # ^ |   0 gepardo why you searched in 2*10^9 ? How you guess that for 10^18 1 ans will must between them ? What is the proof ?
•  » » » » 7 years ago, # ^ | ← Rev. 2 →   0 If you see a formula given in the editorial and try k = 2·109, you'll see that it's quite enough to be the right binary search range.
 » 7 years ago, # |   +1 It is also possible to solve C using linear search: 25518550
 » 7 years ago, # |   0 salute to alex256.. this is the best editorial i've ever seen..i've always wanted to have those "hints" kind of thing
 » 7 years ago, # | ← Rev. 4 →   +8 problem E: — TLE — OK 1310 ms.(what i'm doing wrong?)
•  » » 7 years ago, # ^ | ← Rev. 2 →   0 I also passed with . Are you sure your solution with square root decomposition isn't actually because if it is it actually makes sense that your TLEd.UPD: Your solution really is .
•  » » » 7 years ago, # ^ |   +5 yeah, you are right
•  » » 7 years ago, # ^ |   0 I passed in under 2000ms when i changed to BIT from segment tree with Order statistics tree in
•  » » » 7 years ago, # ^ |   0 How would you use BiT instead?
•  » » » » 7 years ago, # ^ |   0 You need two information to perform the swaps in every query: At any index i, how many a[j] > a[i] such that j [1, i-1]. Let this value be x At any index i, how many a[j] < a[i] such that j [i+1, n]. Let this be y I only maintain the first information using BIT. The second information can be extracted as y = (a[i] - 1) - ((i - 1) - x)25530551
•  » » » » » 7 years ago, # ^ |   0 Does the ordered tree act as multiset? Can we erase single value from ordered tree as we can delete a single value from multiset using iterator.
•  » » 7 years ago, # ^ |   -33 logx has a higher growth rate than sqrt(x) for larger inputs
•  » » » 7 years ago, # ^ |   +8 nooo... its just some hidden constant in pb_ds and segment tree
•  » » 7 years ago, # ^ |   +15 I have a completely different algorithm that runs in time O(n log n + q sqrt(n)). It's here: 25590906 It is much simpler than the one in the editorial.Draw the permutation as a 2-D plot. The key things that you need to be able to do are to (1) count the number of points below and to the left of a query point and (2) insert and delete points into the plot.So we use a sqrt(n) decomposition of the plot into sqrt(n) by sqrt(n) squares of size sqrt(n). For each square we keep the number of points in it. We also keep a hierarchy of squares doubled in size. So there are .5 log n of these levels.To solve a query we first note that there is an ell-shaped part that is that is outside of the squares described above. These can be counted in O(sqrt(n)) time. We maintain the permutation and its inverse. So the horizontal part of the ell (and the vertical too) can be counted row by row. Now to count the number of points in the squares we peel off possibly the top row and column and just count them. Now we move into the next level of boxes that are twice as big. The total time for this count is therefore O(sqrt(n)).Inserting and deleting from the boxes is easy. Just make a pass through the levels. This takes O(log n) time.
 » 7 years ago, # |   +2 This is truly one of the best editorials on CF till date. Giving hints and then the complete idea to the question is a good way to help us improve ourselves. Also, giving reference code in three different languages is a very appreciable effort. Congratulations to the writers of the contest for this excellent editorial and we hope to look more such editorials in the future.
 » 7 years ago, # | ← Rev. 2 →   0 Can anyone help me get rid of that 1e018 in 9th test case in problem C. 25537798 There are other submissions where the same formula is used but have an AC.
 » 7 years ago, # |   0 In problem C why do we have to binary search up to 2e9 not 1e9(it give WA)?Look at my submission with 1e9(25537413) and 2e9(25538410)???
•  » » 7 years ago, # ^ | ← Rev. 2 →   +1 for test case n = 1e18 m =1 ans is greater than 1e9 (approx. sqrt(2)*1e9)) put n,m values in the equation you will get maximum possible ans for the problem n — m <= k * (k + 1)/ 2
 » 7 years ago, # |   0 Can someone help me with the proof of — 785D — Anton and School — 2 , I am not getting the explanation given in tutorial . Thanks in advance
•  » » 7 years ago, # ^ | ← Rev. 2 →   +2 Where didn't you undestand it? I'll try to explain it better.
•  » » » 7 years ago, # ^ |   0 this part, this formula also means the number of ways to match the string with the sequence of zeros and ones of the same length
•  » » » » 7 years ago, # ^ |   +1 Consider the number of ways to make a sequence of length x + y with x ones and y zeros. Won't it be ?Later, it's explained how to match an RSBS to every such sequence.
•  » » » » » 7 years ago, # ^ |   +5 Is empty RSBS included in the answer?, e.g. for n=3,m=3 and string ((( ))), we include empty RSBS corresponding to 111000.
•  » » » » » » 7 years ago, # ^ |   0 Yes, in the simplified version an empty string is included to the answer. Later in the solution, there won't be empty RSBS, because it will contain the iterated opening bracket.
•  » » » » 7 years ago, # ^ |   0 It's kind of equivalent to stars and bars also.
•  » » » » » 7 years ago, # ^ |   +3 I understand the significance of (x+y)C(x) but I don't know how the author originally came up with it. I had the formula given below in mind, but I don't know how to reduce it. xC1*yC1 + xC2*yC2 + xC3*yC3 +....+ xC(min(x,y))*yC(min(x,y)) I would appreciate it if you could please explain the formula using stars and bars. Thank you!
•  » » » » » » 7 years ago, # ^ |   0 I couldn't think of any intuitive way of explaining the problem from a stars and bars perspective. I'm just noting that its equivalent.
•  » » » » » » 7 years ago, # ^ |   0 write xCi as xC(x-i).Observe that sigma(xCi*yCi) is coefficient of a^x in (1+a)^x * (1+a)^y.Convince yourself that x+yCx -1 is your reduced from.IDK why in editorial -1 is absent.
•  » » » » » » » 7 years ago, # ^ |   +8 In editorial it is mentioned But we also have an additional condition: we must necessarily take the last opening bracketThus we are interested in sigma{ xCi * (y-1)C(i-1) ) } = (x+y-1)Cx[from i=1 to i=min(x,y)]
•  » » » 7 years ago, # ^ |   0 How do you justify the statement " Also opening brackets appear earlier than closing brackers " ??
•  » » » » 7 years ago, # ^ |   +1 I think it's obvious because in the simplified version we deal with the sequence like "((((...((()))))))))...))))" and every its subsequence will contain opening brackets earlier than closing brackets.
•  » » » » » 7 years ago, # ^ |   0 Thanks a lot !! I forgot the earlier explanation of simplified version .
 » 7 years ago, # |   0 Has anyone got AC in E using segment tree with multiset(supporting order stats)? I am getting TLE on #7.
•  » » 7 years ago, # ^ |   0 TheTerminalGuy did, I guess. Here's his code. 25524777
•  » » » 7 years ago, # ^ |   0 3978ms seems quite a risk. The same solution may not pass again.
•  » » » » 7 years ago, # ^ |   +8 My code, 3337 ms.
•  » » » » 7 years ago, # ^ |   0 cin / cout are to blame mostly.
•  » » » » » 7 years ago, # ^ |   +5 I too used cin/cout. I guess the time is due to the find function being called every time before erasing some value. As we have inserted tge value in set we can be sure that it will be present in set and remove it without worrying about runtime error.
 » 7 years ago, # | ← Rev. 2 →   0 if problem E2 is data structures, it may be not hard to solve =))
 » 7 years ago, # |   +15 My funny task A solution. long sum = 0; for (int n = nextInt(); n > 0; n--) { long x = -next().codePointAt(0); sum -= (((13 * x + 4870) * x + 662555) * x + 39072170) * x + 846175272; } println(sum / 33660); Also fan fact, that all task contains test case with "404" answer.
•  » » 7 years ago, # ^ |   +13 Haha! That is awesome, I had to stare at it for quite a long time to figure out how it worked. Excellent practice, thanks. May I ask how you fit that polynomial function?I know your program was meant to be cool and not short, but anyway it made me think of writing something where the main logic was in a one-line formula, like yours. So here's my "shortest code" submission, 127 non-whitespace characters (http://codeforces.com/contest/785/submission/25567260): #include int t, n; main() { scanf("%d\n", &n); while (n--) { char s[99]; gets(s); t += "4!8D6<"[*s%7]-48; } printf("%d\n", t); } Would love to see shorter solutions .. in any language!
•  » » » 7 years ago, # ^ |   0 25547848 — just one line in python225745756 — just minified your code (without any thinking, only deleting unnessesary code)
•  » » » » 7 years ago, # ^ |   +8 Knew people would make it shorter but not THAT MUCH shorter. :-O :-)
 » 7 years ago, # |   +5 Presentation of the editorial is really amazing... I think I'd never seen that before. Thanks a lot.
 » 7 years ago, # | ← Rev. 5 →   0 for question E Can we have a 2D BIT where BIT[x][y] where x will denote block no and y is actual number depending on given input.Then each query is solved in sqrt(N)+log(block)*log(N) and updates also in sqrt(N)+ log(block)*log(N). Correct me if I am wrong..
 » 7 years ago, # |   +5 Very detailed & useful tutorial. Thank you very much!
 » 7 years ago, # |   +19 First time in my whole journey (which is albeit short), I have seen an editorialist with so much creativity and patience taking so much interest in really explaining the solutions in a comment. It's sad how whistle blogs about complaining all about this contest. More power to you, gepardo.
•  » » 7 years ago, # ^ |   +1 Thank you :)
 » 7 years ago, # |   0 D is a very nice especially if one derives it without using the identity directly! Thank you for this problem.
 » 7 years ago, # |   0 I tried to solve problemE with segment tree plus ordered multisets but I am getting TLE on test 7.The complexity of build function in my code in O(N * LOGN * LOGN * C). where C is the constant factor in segment tree.(C ≈ 4).http://codeforces.com/contest/785/submission/25568923Is there any way to optimize the build function.
•  » » 7 years ago, # ^ |   +3 What do the query1 and query2 methods do?
•  » » » 7 years ago, # ^ |   0 query1 calculates no of elements greater than val in ranger l to r and query2 calculates no of elements less than val in ranger l to r.
•  » » » » 7 years ago, # ^ |   0 These 2 values are connected.Suppose you have the element x in the position i in the array and you know the answer to one of the queries, e.g. there are ans elements greater than x in [1,i-1]. This necessarily means there are i-1-ans elements smaller than x in [1, i-1]. Since the total amount of elements smaller than x is x-1, there x-1-(i-1-ans) = x-i+ans elements smaller than x in [i+1, N]. Thus, the total amount of inversions is x-i+2*ANS.This thing should reduce the constant of your solution. Also, don't use long long everywhere — this slows down the solution, which already runs pretty close to the TL.
•  » » » » » 7 years ago, # ^ |   0 Thanks for this idea but I want to reduce the complexity of build function which works in O(n * log(n)2 * c) because for 7th test case , even the build function is not able to execute in 4s.
•  » » » » » » 7 years ago, # ^ |   0 Does it? I just ran your build() function of CF for 250000 and it seems to take about 1.3s.
 » 7 years ago, # |   +1 Is there any divMod explanations? I don't understand how it works. Thanks
•  » » 7 years ago, # ^ |   +1 divMod is a function that computes a / b modulo m I don't understand how it works. At first instead of dividing it computes (a * inv(b)) % m where inv(b) is an inverse element for b. Inverse element can be computed using an extended Euclidian algorithm. You can learn more about this here
•  » » » 7 years ago, # ^ |   +5 Thank you! That article is very easy to understand.
 » 7 years ago, # | ← Rev. 2 →   0 why is it low+m and not the midvalue+m? in 785C
•  » » 7 years ago, # ^ |   0 Because I write the binary search in this way. I exit from the binsearch when l=r. So the answer will be stored obviously in l
 » 7 years ago, # | ← Rev. 2 →   0 Can Someone explain me why number of total possible valid sub sequence is x+yCx as explained in Editorial in 4th Question ? I am not able to understand that
•  » » 7 years ago, # ^ | ← Rev. 2 →   0 Which part of the editorial you don't understand? I'll try to explain it better.
•  » » » 7 years ago, # ^ |   0 "At first, let's simplify the problem: let our string consists of x + y characters, begins with x characters "(" and ends with y characters ")". How to find the number of RSBS in such string?Let's prove that this number is equal to x+y C x.This is what I am not able to understand. If we take a string that consists of x + y characters, begins with x characters "(" and ends with y characters ")", then how total number of RSBS is (fact(x+y) / fact(x)*fact(y) ) ?
•  » » » » 7 years ago, # ^ |   0 There's a proof in the editorial. What exactly you don't understand in it?
•  » » » » » 7 years ago, # ^ |   0 I didn't understand how did you decide to represent which bracket with 0 and which with 1. and so I am not able to understand the proof. Sorry if this is silly but I seriously do not get it
•  » » » » » » 7 years ago, # ^ |   +6 OK. Let's take ANY sequence of x ones and y zeros. Now write it under the bracket sequence in the following way:Now let's take opening brackets under which we have zero and closing brackets under which we have one (green brackets on the picture).Now, we must prove that number of opening brackets will be equal to number of closing brackets. How to do it? Suppose we have z zeros under opening brackets and, therefore, x - z ones (because we have x opening brackets)under opening brackets. So the rest of ones are under closing brackets. Their count will be x - (x - z) = z (because we have x ones. So we have equal count of opening and closing brackets. So we took an RSBS in this way.Now more understandable?
•  » » » » » » » 7 years ago, # ^ |   +8 Okay.. Now I completely got it.. Now it is much easier to visualize for me for the further part of your tutorial. So thank you so much for tolerating and quickly answering to my silly doubt. Truly appreciate your quick reply to each and every comment. Thank you so much.
•  » » » » » » » 7 years ago, # ^ |   0 Does the difficulty of this task correspond to Div2?
•  » » » » » » » » 7 years ago, # ^ |   0 I think yes, because 94 people in Div. 2 solved it. But I agree that it's harder than Div. 2 D level.
•  » » » » » » » 7 years ago, # ^ |   0 How do we assign 0s and 1s?
•  » » » » » » » » 7 years ago, # ^ |   0 You can choose any of the sequences that contain x ones and y zeros.
 » 7 years ago, # |   0 in the problem:B we have to calculate the max distance between the periods that is the time between end of first period and start of second period but in the editorial why we are calculating int minR1 = infinity, maxL1 = -infinity; int minR2 = infinity, maxL2 = -infinity; for (int i = 0; i < n; i++) { maxL1 = max(maxL1, a[i].first); minR1 = min(minR1, a[i].second); } for (int i = 0; i < m; i++) { maxL2 = max(maxL2, b[i].first); minR2 = min(minR2, b[i].second); }
•  » » 7 years ago, # ^ |   0 maxL1, maxL2 — maximum left ranges in chess variants and programming variantsminR1, minR2 — minimum right ranges in chess variants and programming variantsThen we use them for both cases. In first case we need the distance between minimum right chess range and maximum left programming range. So we simply use maxL2 - minR1. The same in the second case.
 » 7 years ago, # |   0 In problem 785D Anton and School 2 for the case when we have x opening brackets and then y closing, total number of RSBS according to the editorial Should we subtract 1 from it? (case when we have x ones in the beginning and then y zeros as I understand is empty sequence which is not RSBS according to the statement).I was thinking in a bit different way about number of RSBS when we have x opening brackets and then y closing, if we sum over the length of first half we can find that number of RSBS is: But I don't know how to simplify this one. Maybe someone was doing similar things and achieved result?
•  » » 7 years ago, # ^ |   +3 Should we subtract 1 from it? Yes, because in this formula we count an empty string as an RSBS. But in the further problem we don't have to subtract 1, because we always take the last opening bracket, so RSBS won't be empty.
•  » » » 7 years ago, # ^ |   +8 Do you know is there a way to simplify this sum: ?
•  » » » » 7 years ago, # ^ |   0 No, unfortunately I know a normal way how to simplify this to . I came to this simple formula bruteforcing the answer and then found that beautiful proof. So I didn't try to simplifity such expression.
•  » » » » 7 years ago, # ^ |   0 Hi did you find any way to simplify this equation, Even the approach I am thinking boils to this equation
•  » » » » » 7 years ago, # ^ |   0 This may help — Vandermonde's Identity. We can count this as taking i objects from a pile of x objects, and y - i objects from a pile of y objects, where i varies from 0 to Min(x, y). This is basically equivalent to taking a total of y objects total, from a collection of x + y objects. Hence, our summation above can be rewritten as:
•  » » » » 7 years ago, # ^ |   +3 Multiply (1+x)^n1 and (x+1)^n2 See its binomial expansion. You will find the required ans to be coefficient of n1+1 in (1+x)^(n1+n2). Again use binomial to find coefficient. You will get formula given in editorial. Here n1=x, n2=y
 » 7 years ago, # |   +5 I just solved problem E today. I can see somebody here has same approach with me: using Balanced BST (I choose Treap) on segment tree. My second submission run in O((N + Q) × log2(N)) time complexity and I got TLE. Then I reduce my solution to O(N × log(N) + Q × log2(N)) time complexity and got AC by reducing the build tree step to O(N × log(N)) time. More over, each node is a treap can be built in O(N) time. Hope this help someone implement this problem easier.
 » 7 years ago, # | ← Rev. 4 →   -8 Isn't it the algorithm for Problem D?But I am getting Memory limit exceeded on test 6 verdict :( My Code#include #include #include using namespace std; typedef long long int ll; #define M 1000000007 vector dp[200001];// 2D vector to store nCr ll combinatorics_of(int n,int r) //return nCr { if(dp[n][r]!=0) return dp[n][r]; if(n==r) return 1; if(r==1) return n; return dp[n][r]=(combinatorics_of(n-1,r-1)%M+combinatorics_of(n-1,r)%M)%M; } int main() { string s; cin>>s; int l=s.length(); for(int i=0;i
•  » » 7 years ago, # ^ | ← Rev. 3 →   0 But I am getting Memory limit exceeded on test 6 verdict :( Consider how much memory the line vector dp[200001]; consumes. It'll be bytes, which is about 20 GB. It won't fit any time limit. Isn't it the algorithm for Problem D? Yes, it is, but you need a way to find the number of combinations faster. To do this, you can look at my code, which is in the editorial.To do this, you can use a naive formula . To calculate this fast, you can do the following: Precalculate all the factorials under 106 by modulo. Learn how to divide by modulo. Dividing by modulo means multiplying by an inverse element. How to find the inverse element, see the article P. S I recommend you not to place you code directly in the comment. You may place a link to submission instead, or place your code to ideone or pastebin and provide a link to it. Or you can use spoilers, like this: C++ code#include using namespace std; int main() { cout << "Spoilers are good" << endl; return 0; } 
•  » » » 7 years ago, # ^ | ← Rev. 2 →   +8 Thank you so much for the link :)P.S:The comment has been edited as it is recommended.
•  » » » 7 years ago, # ^ |   +8 Thank you so much for the great editorial.For modular multiplicative inverse by extended Euclidean algorithm, we have a statement x = (x % m + m) % m. This will give an element in the range [0,m), right? My doubt here is can the modular inverse be negative. Else we could have written x = (x % m) directly. Sorry if this doesn't make any sense.
•  » » » » 7 years ago, # ^ |   0 No, the modular inverse cannot be negative. When we deal with arithmetics by modulo, we always take integers in the [0;m) range.
 » 7 years ago, # |   0 Hello, can some tell me what is wrong with this solution with C problem? 25806694Thanks in adavanced.
•  » » 7 years ago, # ^ | ← Rev. 2 →   0 In the 7th test case your l > 2e9, so l > r and you don't enter the binary search and just print m+1 (that is l).
 » 7 years ago, # | ← Rev. 2 →   0 why to use the logn time complexity ? This can be done in O(1). We dont need to use the binary search. At the end of mth day the no of grains in the barn is n — m; from the next day it will be like (m — (m+1) , (m — (m+2)), ... , (m — (m+k)) which can be written as k * (k+1)/2. lets formulate the equation n = (k *(k+1))/2. hence k^2 + k — 2*n = 0; the root is (-1 + (sqrt(1 + 4*2*n))/2. let this be s. if(s*s + s < 2*n) then add a day. day = m + s is the answer.
 » 7 years ago, # | ← Rev. 2 →   +16 785D — Anton and School — 2I think this sentence is not correct in solution : number of RSBS in string that begins with x characters "(" and ends with y characters ")" equal to ( x + y)! / ( x! * y! ) it must be : ( ( x + y)! / ( x! * y ! ) ) — 1Why we subtract 1 ?In solution written that : formula also means the number of ways to match the string with the sequence of zeros and ones of the same length, which contains exactly x onesLets look picture in example : ( ( ( ) ) ) ) one combination is 1 1 1 0 0 0 0 If we create string with following algorithm which is written in solution : all the opening brackets that match zeros and all the closing brackets that match ones Resulting string will be empty string. And empty string is not RSBS. And this means we must subtract 1 . But Code work fine. Because in code we write ( x + y-1)! / ( x! * (y-1)! )_ ( explained in solution ). This prevent taking empty string
 » 6 years ago, # |   0 Can someone help me with my solution of Problem E ? Link: https://ideone.com/500vNF I am using segment tree with each node having a set .
 » 5 years ago, # |   0 In the solution of problem C,why is the upper limit of binary search 2e9??
•  » » 5 years ago, # ^ |   0 Because the maximum possible answer is around $1.414\cdot10^9$. But I just took a larger boundary for binary search.
•  » » » 5 years ago, # ^ |   +10 thanks....