### Kuroni's blog

By Kuroni, history, 20 months ago,

Hello everyone, this is the editorial for Codeforces Round #558 (Div. 2). I hope you enjoy the problem as well as I did!

1163A - Eating Soup

Author: ArguteOnAir

Tutorial
Implementation

1163B2 - Cat Party (Hard Edition)

Author: Shirone

Tutorial
Implementation

1163C2 - Power Transmission (Hard Edition)

Author: GreymaneSilverfang

Tutorial
Implementation

1163D - Mysterious Code

Author: Kuroni

Tutorial
Implementation

1163E - Magical Permutation

Author: Kuroni

Tutorial
Implementation
Implementation with DFS

1163F - Indecisive Taxi Fee

Author: Kuroni

Tutorial
Implementation

• +75

 » 20 months ago, # |   +16 Auto comment: topic has been updated by Kuroni (previous revision, new revision, compare).
 » 20 months ago, # |   +13 Fast editorial :)
•  » » 20 months ago, # ^ |   +6 Thanks!
 » 20 months ago, # |   +57 The way Grey code was included in E, so natural, I can only say it was extremely elegant. Another kudo for you guys. ;)
•  » » 20 months ago, # ^ |   +7 can you please explain the solution for E a little bit more? I didn't get it :(
 » 20 months ago, # |   +18 I really liked problem B.
 » 20 months ago, # |   +22 Problem E is just beautiful.
•  » » 20 months ago, # ^ |   0 You can say it was magical :D
•  » » » 20 months ago, # ^ |   0 in your if block of add function, no need to sort. again, great problem!
•  » » 20 months ago, # ^ | ← Rev. 2 →   +5 "such that the xor value of any two consecutive elements belongs to the basis; or in other words, the corresponding bitmask of any two consecutive elements in the magical permutation differs by exactly 1 bit." Can you explain this part?
•  » » » 20 months ago, # ^ |   +8 Suppose the elements needed to create basis are $v_0, v_1, ... v_{x - 1}$. Suppose x = 2; Grey code is, 00 01 11 10 Now you can create magical permutation by taking xor's of all $v_i$'s for all set bits i. In this case magical permutation is$0$, $v_0$, $v_1\oplus v_0$, $v_1$
 » 20 months ago, # |   +3 really good problem set, I need more clarification about the counting techniques in problem B, any help would be appreciated
•  » » 20 months ago, # ^ |   0 Same Here
•  » » 20 months ago, # ^ |   0 It just finding maximum prefix that sustainable for this requirement. The requirement is you must removing one integer from this prefix and then the number from each integer(in the prefix) is same.
•  » » 20 months ago, # ^ |   +2 You use 2 counting arrays, one for counting the occurrence of colors, and one for counting the number of colors which have the same occurrences. You can have a look at the implementation for more details.
•  » » » 20 months ago, # ^ |   0 About idea maintain a "max" variable: frequency that appears most often. I see this brilliant. Maintain count[] and freq[] are also very beautiful. How can you come up with this solution ? Do we have any similar problems using this technique ? Thanks so much.
•  » » » 20 months ago, # ^ |   0 could you please explain in a bit more detail. Didnt understand how to implement the same, even after going through the implementation codeThanks
 » 20 months ago, # | ← Rev. 2 →   0 Can someone explain the solution of the problem Cat Party in a little bit more Detail
•  » » 20 months ago, # ^ | ← Rev. 2 →   0 It just finding maximum prefix that sustainable for this requirement. The requirement is you must removing one integer from this prefix and then the number from each integer(in the prefix) is same.
 » 20 months ago, # |   +4 Why long double fails for C whereas double passes ?
•  » » 20 months ago, # ^ |   +17 When you use floating point numbers there are always rounding errors. Using double just happens to discard the rounding errors < $10^{15}$ so something actually wrong (like a == b) just happens to work. Do NOT use floating point numbers in problems like this. They'll catch you and knock down you, maybe in an important contest.
•  » » » 20 months ago, # ^ |   0 Is there any possible test case that would fail for double as well?
•  » » » » 19 months ago, # ^ | ← Rev. 2 →   0 Rounding off of double leads to incorrect key for mappings.
•  » » » » 19 months ago, # ^ |   +1 On 32-bit x86 machines (for example, Codeforces runners) map is essentially broken because x87 registers use 80-bit representation while the result stored into the memory is truncated into 64-bit. It only "works" because of blind luck.Maybe map will work (since volatile forces the value to be stored into the memory and eliminates the "existence" of 80-bit value).Even if SSE is used (for e.g on x86-64) or we somehow workaround this x87 problem, a1/b1 and a2/b2 may be different but truncated to the same value. I'm not sure if it could happen under the constraint of this problem.
•  » » » » » 19 months ago, # ^ |   0 Thank You.
 » 20 months ago, # |   +19 My Solution of Problem BIterating through the given array of colors from 1 to n I figure out if given size satisfies the condition or not. If it does, I store it in variable "answer", since we need the maximum one.So the interesting part, How Do I know, if given subarray from 1 to say k satisfies the condition? I have two additional arrays and a set. The set just stores different colors I have seen so far.1) vector count_colors stores number of colors I have seen so far. For example, If I had seen color "9" 3 times in the given array, count_colors[9] = 3 2) vector count_numbers stores number of groups with i values in each. For example, I had seen color "9" 3 times in the given array, color "10" 3 times and color "7" 1 time, then count_numbers[1] = 1 for "7" and count_numbers[3] = 2 for "9" and "10". Also I keep track of maximum value in count_numbers, can do it on each change since we keep seeing colors and numbers never decrease.So, at first I have not seen any color. When seeing new color, I add it to set of seen colors. Then I decrease value of previous value of count_numbers[count_colors[colors[i]]]--; There are 1 less groups consisting of 7 elements, for example, and 1 more group consisting of 8 ones, that is the logic. I also increase count_colors[colors[i]]++; by definition. max_num = max (max_num, count_colors[colors[i]]); gets updated here too.Now for the checking part. There are 3 cases that I check for. 1) All values met once -> answer is yes, we can remove any. I think I did unnecessary comparison in the second part in my code though, that should be enough. We can remove any hat and be left with none or again each hat met once, that satisfies the answer 2) One value seen once and others seen x times, x > 1. I check count_numbers[1] to be equal to 1, some color group consists of only one element. Then I check if count_numbers[max_num] == num_seen.size() — 1. x should be equal to max_num, otherwise there is third value between 1 and x. And number of groups of size max_num should be num_seen.size(), number of seen minus 1 that is one separate value. 3) One value bigger than all others by one, so that when we remove such element, it would again satisfy condition. I check for count_numbers[max_num] == 1, one element group is maximum. And count_numbers[max_num — 1] == num_seen.size() — 1, others are in amounts of maximum — 1. .
•  » » 20 months ago, # ^ |   0 Can you tell me how can you come up with this data structure ? Or we just need to practice more ? I see the idea of maintain a "max" variable: frequency that has maximum color. How can we come up with this idea ?
•  » » » 20 months ago, # ^ |   0 No data structure needed, just using vectors. I just thought about the variables that we need to get the answer and maintained them.
 » 20 months ago, # |   0 Since am extremely smart when it comes to math problems I was able to solve problem A with a bruteforce approach. I feel ashamed with my code https://codeforces.com/contest/1163/submission/53912810
 » 20 months ago, # |   0 Problem C."that whenever two different electric wires intersect, they may interfere with each other and cause damage. So he wonders, how many pairs are intersecting?". Maybe someone can explain when is the intersect wire counted.Thank you.
 » 20 months ago, # |   +7 Can someone suggest similar problems to D since many people said it is standard?
 » 20 months ago, # |   +16 Are you sure about time complexity of D? should it not be c*|s|*|t|*26?
•  » » 20 months ago, # ^ |   +8 The exact complexity is indeed $O(26 * |c| * |s| * |t|)$, but I removed the constant to simplify it.
 » 20 months ago, # |   0 Any links to the tutorial for solving such problems using Gaussian elimination ? Also any links for learning general Gaussian elimination?
•  » » 20 months ago, # ^ |   +5 For learning Gaussian elimination, you may refer this: https://cp-algorithms.com/linear_algebra/linear-system-gauss.html
 » 20 months ago, # | ← Rev. 2 →   0 For C, is this really true...? These are the pairs with the same slope (i.e. same value of a and b), I wonder how about such cases like 4x + 6y = 5 and 2x + 3y = 4, which are parallel(My code in the contest time had this same bug)
•  » » 20 months ago, # ^ |   0 Ah, now I understand Such a case like 4x + 6y = 5 does not occur, for this has no integer solutions: gcd(a, b) and gcd(a, b, c) always coincide
 » 20 months ago, # | ← Rev. 2 →   0 My code got accepted for C1. But it is giving wrong answer in C2. Can someone please help
•  » » 20 months ago, # ^ |   0 Assuming your logic is correct throughout the code, the only thing that I think could cause this is if you are not using longs as you generate the final answer in C2.
•  » » » 20 months ago, # ^ | ← Rev. 2 →   0 Thanks, missed the fact that set.size() returns int value.
 » 20 months ago, # | ← Rev. 2 →   +19 Problem F is almost the same as 《故乡的梦》 on bzoj（which is a Chinese OJ). Anyway,Problem E is very interesting.(Although some part of this problem appeared in Atcoder AGC 31 C) :D
•  » » 20 months ago, # ^ |   0 AGC031 C actually inspired this problem :)
 » 20 months ago, # |   0 can someone help me with my code for D?I am getting RE on test 18.
 » 20 months ago, # |   +19 We can optimize problem D even further. Suppose at a dp state, we have $ks \leq kt$, then we don't even need to save $ks$ as a state, because we can directly calculate $ks$ using $kt$. Hence, the number of dp states are now only $O(|c| \cdot \max(|s|, |t|))$.
 » 20 months ago, # | ← Rev. 2 →   +3 Problem D can be solved in a better way.We build an ACAM(Aho-Corasick Automaton) for $s$ and $t$ and then we do dynamic programming on the ACAM.Let $dp_{i,j}$ be the largest answer we can get in a walk on the ACAM which consists of $j$ steps and ends at node $i$. Initially $dp_{rt,0}=0$ and the others are $-\infty$, where $rt$ denotes the root of the trie of the ACAM. Finally the answer is the maximum value of all those $dp_{i,|c|}$s where i denotes a node on the ACAM. Do not forget to add all $i$'s fail-nodes' value to $i$'s value.The time complexity is $O(26\times |c|\times (|s|+|t|))$.An ACAM can deal with similar problems with 2 strings or more. How nice! :DCode: 53944139
 » 20 months ago, # | ← Rev. 2 →   0 Problem A- I think the solution provided for 1163A-Eating soup is a little bit wrong between the statement "Otherwise, if m+1≥⌊n2⌋, each independent cat to leave decreases the number of groups so the answer is n−m", in this if m+1=lower(n/2) the solution concide with m-1
 » 20 months ago, # |   0 can someone tell me what's wrong with my solution?I am not sure why am I getting wrong answer for bigger values of N. https://codeforces.com/contest/1163/submission/53945287
•  » » 20 months ago, # ^ |   0 The precision of float is not enough, try double.
•  » » » 20 months ago, # ^ |   0 thanks, that got me an AC, finally...
 » 20 months ago, # |   0 For D, What's the complexity that get the nxt array? for (int i = 0; i <= n; i++) for (char c = 'a'; c <= 'z'; c++) { int cur = i; while (cur > 0 && s[cur + 1] != c) cur = kmp[cur]; if (s[cur + 1] == c) ++cur; nxt[i][c - 'a'] = cur; } 
•  » » 20 months ago, # ^ |   +3 $O(|s|^{2} + |t|^{2})$
•  » » » 20 months ago, # ^ |   0 Thanks.
 » 20 months ago, # |   0 Use Aho-Corasick automation,Problem D's Complexity can be O(|c|⋅(|s|+|t|)).
 » 20 months ago, # |   +44 I think the tests of the problem E is not strong enough.Here is my solution: https://codeforces.com/contest/1163/submission/53950641Here is the hacking test: 2 4 5 But my solution gets: 1 0 0 It is obviously wrong. Because my solution thinks if it is possible to get all $2^i\ (0 \leq i \leq x)$ from the basis, $x$ is going to be valid. Maybe not only me wrote like that.
 » 20 months ago, # |   0 In problem C2 tutorial, can anyone explain why c=y1x2−y2x1 ?
•  » » 20 months ago, # ^ |   0 Equation of a line is (y-y1)=m(x-x1) now m=(y2-y1)/(x2-x1) now try to change this formula in form of equation of line y=mx+c You will get c=y1x2-y2x1.
 » 20 months ago, # | ← Rev. 2 →   +5 E can be solved in $O(n+MAX)$ by using counting sort and Gray code.53956574
 » 20 months ago, # | ← Rev. 2 →   0 Can you explain "We now iterate over these "blocks" of parallel lines and count the number of pairs each block contributes — a block of size s gives s(s−1)2 pairs." ?? And in problem B, why cnt[mx — 1] * (mx — 1) == i — mx && cnt[mx] == 1 is one color has the occurence 1 more than any other color ?? I don't understand. Thank you very much <3 <3 <3
 » 20 months ago, # | ← Rev. 2 →   +8 Thanks for the round!
 » 20 months ago, # |   +3 Can someone explain a bit more in detail solution for D? What exactly is the array nxt[N][26], and how it is constructed? I know about KMP as algorithm for finding pattern in text in O(N) time complexity.
•  » » 20 months ago, # ^ |   0 you can view nxt[i][c] as, i (before) matched length, c new char, nxt will be (after) matched length. similar in KMP search method, [26] just pre-process those possibility.
 » 20 months ago, # |   0 Need help for problem C2. Why my submission giving WA. Even it is working fine at IDE.
•  » » 20 months ago, # ^ |   0 Because of double comparison (use on map and set) I think.
•  » » 19 months ago, # ^ | ← Rev. 3 →   0 Rounding off of double leads to incorrect key for mappings.
 » 20 months ago, # |   +8 I'm having trouble understanding one detail for problem F. To handle the case that an edge $e$ on the main path increases in cost, we need a way to find the shortest path from $1$ to $N$ not using $e$. The proposed approach generates a set of candidate paths, each of which avoids some interval of edges on the main path, and finds the shortest among those avoiding $e$. How can we prove that for every $e$, the shortest possible distance will certainly appear within that set of candidates?
•  » » 20 months ago, # ^ |   +8 Now if for each $u$, $l_u$ is tightly bounded, i.e. every shortest path from $1$ to $u$ must uses the first $l_u$ edges of the main path (same analogy for tight $r_u$), then the the set shortest possible distance not using the edge $e$ is maximized.Now, what if $l_u$ and $r_u$ is not tightly bounded? I will call the tight bound of $l_u$ as $tl_u$, and tight bound of $r_u$ as $tr_u$. I will skip over this part a little fast, but basically take the shortest path using the tight bound $tl_u$ and $tr_v$ of a candidate edge $(u, v)$, and let's call the set of edges on this shortest path from $tl_u$ to $tr_v$ as $S_e$. Then I can prove that the union of the ranges $l_{u_e}$, $r_{v_e}$ of all the edges belong to this $S_e$ set to be exactly $tl_u$ to $tr_v$, without any interruption. So, in one way or another, the value of the shortest path that must pass through the candidate edge $(u, v)$ will appear in the range $tl_u$ to $tr_v$, regardless of the values of $l_u$ and $r_v$.
 » 20 months ago, # |   0 My way of solving B is maintaining a multiset of the counts of each color. At each step, it is possible if (1st element is 1 and 2nd element = greatest element) or (1st element = next-to-last element and next-to-last element = last element - 1).
 » 20 months ago, # |   0 Can anyone help me? I am unable to find where I am going wrong. This is my solution for problem C2(Power Transmission), I am getting wrong answer on test 10. link: https://codeforces.com/contest/1163/submission/53963521 Thank You.
•  » » 20 months ago, # ^ |   +3 fil.size() return unsigned int (4bytes) on codeforces which will overflow when multiply.
•  » » » 20 months ago, # ^ |   0 Thanks
 » 20 months ago, # |   0 In C2 code implementation , I didn't understand this part . please help me --- // simplify equation int d = gcd(a, b); a /= d, b /= d; if (a < 0 || (a == 0 && b < 0)) { a = -a; b = -b; }
•  » » 20 months ago, # ^ | ← Rev. 2 →   0 When using gcd function with negative numbers, so whenever the numerator is negative, the result becomes negative. So after a and b are divided by their gcd i.e. a /= d, b /= d , their sign may get reversed. This happens only when if numerator 'a' is negative or if 'a' is zero and denominator 'b' is negative.try with an example it will be more clear.
 » 20 months ago, # |   0 Why does https://codeforces.com/contest/1163/submission/53983303 get accepted but https://codeforces.com/contest/1163/submission/53982305 give wrong answer?
 » 20 months ago, # |   0 I am getting wrong answer on test case 10 in Power transmission( Hard version ) Is there any corner case in problem?
 » 20 months ago, # |   0 Can someone please elaborate editorial's idea on solving D, in more detail?
•  » » 20 months ago, # ^ |   0 I will try explaining how i solved it :) lets startwell ... solid understanding of kmp and dynamics would do to solve this one // let f(id,ks , kt) be the recursive function ... we will call f(0,0,0 ) to get our answer ... // here id,ks ,kt represents current index of mysterios code, string s and string t ... respectively and base case will be when id >= length of ( mysterios code ) recursive partso we will try to maximize ans as follows int f (id, ks , kt) { if(id>=sz(mysteriosString) ) return 0; real_ks = ks; real_kt = kt ; // adjusting ks , kt ... if( ks>= sz(s) )ks--; if( kt>= sz(t) )kt--; ans = -infinity ; for(i=0; i<26 ; i++ ) { char ch = i+'a' ; if( mysteriosString[id] == ch or mysteriosString[id] == '*' ) { ks = real_ks ; kt = real_kt ; ks = nxt_s [ks][i] ; kt = nxt_t [kt][i] ; // updating ks , kt for next call ... if(s[ks] == ch ) ks++ ; if(t[kt] == ch) kt++ ; //if ks == sz(s) it will contribute +1 //if kt == sz(t) it will contribute -1 to ans ans = max(f(id+1, ks, kt) + (ks==sz(s) ) -(kt==sz(t) ) ,ans); } } return ans ; }  handling ks kt ...i will explain for ks ... for kt its similarnxt_s[ks ][ i ] it represents where should we shift ks for i th char of mysteriosString or length of maximum matching prefix of s till now example .... mysteriosString : xabaabaa string s : abaa so string s occurs 2 times in mysteriosString so say ... i = 4, ks = 3 , then nxt_s[ks][4] = 3.. // i = 5 (which is character b ) , ks = 3 ; we will adjust ks to 3 here because we want maximum matching prefix which is also a suffix ... so nxt_s[ks][5] = 1 ... hope u have understood till now and code bottom up yourself :)
•  » » 20 months ago, # ^ |   0 I discussed a solution here
 » 20 months ago, # |   0 I tried submitting problem D's solution in practice without precomputing the next arrays, which the Author's solution does and it passed in almost the same time as the solution with the next arrays precomputed. Here's the code: 53966779The solution has a worst case complexity of $O(26*|c|*|s|*|t|*max(|s|,|t|))$. So how can it pass? Is it because of weak test cases or is there a reason for this?
 » 20 months ago, # | ← Rev. 6 →   -9 Anyone has come across any similar questions like D? Plz put their link.
 » 20 months ago, # |   +3 Could anyone provide a clear explanation for problem D pls ?
 » 20 months ago, # | ← Rev. 2 →   0 In the problem E, why does the dfs-solution (the second solution attached on the tutorial) works? Gray-Code works obviously, but I can not prove we can replace it with a simple dfs. (or am I misreading something?)
•  » » 20 months ago, # ^ | ← Rev. 2 →   +3 That's what I was surprised with to be honest, at the beginning I wrote this DFS to try and hack it, but it ended up as a completely viable solution :/Since when I DFS like in the implementation, the first usable bit is always used, the sequence of the bit changed is actually the same as the Gray code, i.e. the bit changed are $0$, $1$, $0$, $2$, $0$, $1$, $0$, $3$, $0$, $1$, ... So, the DFS will trace a straight path of $2^x - 1$ numbers right away, since it is the same as Gray code.
•  » » » 20 months ago, # ^ | ← Rev. 2 →   +8 Wow, so we can construct gray-code simply by using succinct DFS. I didn't know that.
 » 20 months ago, # | ← Rev. 3 →   +11 Problem D : (Just Discussing a solution)1. Given strings S and A , both contain lowercase English char only , we have to count how many occurrences of A in S. ...what we will do?..KMP right? so , we have LPS[] for string A. ( LPS[i] = length of " longest prefix which is also suffix " of sub-string A[0...i]) . Let's make an recursive implementation of KMP.2. Now assume S can have some '*' which can be replaced by any char in between 'a'-'z' . How many occurrences now ?....KMP again..but we need LPS[i][c] now. Where LPS[i][c] = length of " longest prefix which is also suffix " of string( A[0...i] + (char)(c+'a') ); what change will come in our above recursive solution?... something like this, right?int lo = 0 , hi = 25; if(s[pos] != '*') lo = hi = s[pos]-'a'; for(int c = lo; c<=hi ; c++){ res = (LPS[pos_A][c] == A.size() ) + func(pos+1 , LPS[pos_A][c]); } 3. Now come to our actual problem D. We need two different lps arrays LPS_A[][] and LPS_B[][] . how the func will change now? Like this?for(int c = lo; c<=hi ; c++){ int la = LPS_A[pos_A][c]; int lb = LPS_B[pos_B][c]; now = (la == A.size() ) - (lb == B.size()) + func(pos+1 ,la,lb); ret = max(ret , now); } 4. we're actually done!..just need to pre-calculate two LPS arrays using brute force and trivial dp memoization .Code
 » 20 months ago, # |   0 Hi, this solution: http://codeforces.com/contest/1163/submission/54373882 returns WA for both C1 and C2, but in my IDE it returns the correct answer. Is there a problem with the compiler? Thanks.
•  » » 20 months ago, # ^ |   0 I didn't watch your code carefully, but as someone earlier said — using floating point numbers in such problems is a big NO NO. Try to remake your algorithm so that you operate only on integers.
•  » » » 20 months ago, # ^ |   0 Alright, thanks a lot.
•  » » 19 months ago, # ^ |   0 db m = (a[i].second — a[j].second)/ ((a[i].first — a[j].first) * 1.0); The value of slope m lies in the range (-INF, +INF). db cannot store slope when it exceeds the capacity of db.
•  » » » 19 months ago, # ^ | ← Rev. 2 →   0 Ok, thanks, but still I don't get why I get the correct answer on my computer. Could someone test it yourselves locally? Thanks a lot.
 » 20 months ago, # |   0 why does ceil function doesn't work in my solution??solution link-> https://codeforces.com/contest/1163/submission/54708474
 » 20 months ago, # |   0 what is wrong with my code for problem C1 Solution
 » 20 months ago, # |   0 for problem C what happen if two line are as given2x — 3y = 4;and4x — 6y = 8;because c is not divided by gcd of a and b.
 » 20 months ago, # |   0 Help in C: Giving WA for large test cases... My logic for C is: Calculate number of distinct lines . then find total number of intersections.And then from total, subtract all the parallel lines which won't be intersecting. Suppose for slope m , then are k parallel lines. Then subtract k*(k-1)/2 from total.Do this for all the slopes.Its giving WA for large test cases. Submission link: Submission Link Question Link: Question LinkThanks in advance:)
 » 14 months ago, # |   0 Firstly I was thinking that Problem B's solution is wrong but i realise that here clearly mentioned that we have to consider "First x days" not any contiguous x days.. Thank God ...!!
 » 9 months ago, # | ← Rev. 2 →   0 KMP state for the replaced sub-code to be ks and kt Can anyone explain this line from the editorial of problem D What is KMP state??
 » 2 months ago, # |   0 Could someone explain problem D to me? I understand that the kmp array is the prefix function, but what is the nxt array supposed to be?