Educational Codeforces Round 21 — Editorial

8 years ago, # ^ |

I have the same doubt with you! I also what to know how to use ternary search sloving this problem.

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Steaunk

8 years ago, # ^ |

Have you solve this? If not, you can read my code, I use ternary search to solve this problem.

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8 years ago, # ^ |

I am sorry for that I saw your reply too late.But does your code is use ternary search? I consider ternary search is trie but your code is use binary search. Anyhow, thanks your new idea.

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krijgertje

8 years ago, # ^ |

Consider the case n=4, m=7 with items A=(1,3), B1=(2,4), B2=(2,4) and C=(3,6).

Then the (only) optimal solution for dp[4]=A+C, for dp[5]=A+B1+B2, dp[6]=A+Bx+C and dp[7]=B1+B2+C, but there is no way to create the solution for dp[7] by adding items to any of the three solutions before.

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barnabis

8 years ago, # ^ |

It apparently doesn't work for weights 1, 2 and 3. but can you give a proof that it works for weights 1 and 2 only? I can't figure it out.

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krijgertje

8 years ago, # ^ |

When you have only two weights, you can solve the problem greedily: first take the minimal number of necessary items so there is a multiple of the lcm remaining (in this case this means taking a single item of weight 1 when the wanted sum is odd) and after that greedily take the best groups of a single item of total weight equal to the lcm until you have the wanted sum (in this case this means either taking a single item of weigth 2 or two items of weight 1).

It is then relatively easy to prove that the dp solution gives the same results as the greedy solution above (for example by induction).

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8 years ago, # ^ |

Why can't we make the same argument for three numbers 1, 2, 3 with lcm 6?

At a given state there could be 6x + a number of 1's, 3y + b number of 2's and 2z + c number of 3's. So if we consider all values of a, b and c for dp transitions, it should work right?

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ghostkkk

6 years ago, # ^ |

How to take items to let the remaining weight be a multiple of lcm ?

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8 years ago, # ^ |

thanks your sample, it very useful for me because my poor dp skill.

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akela1101b

8 years ago, # ^ |

Many thanks! Your example was extremely useful. I've managed to solve it with quadruple dp. The point was in trying to change dp[m-1] or dp[m-2]. If they have 1-element, it can be replaced with more expensive 2-element or 3-element respectively.

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calabash_boy

8 years ago, # ^ |

I tried this method, and got several WA, but finally received an AC. If you take 1&2&3-element in consideration at the same time, then there are four possible way to renew dp[i]:first:dp[i-1]+ 1-element; second: dp[i-2]+ 2-element; third: dp[i-3]+ 3-element; ※fourth: dp[i-2]- 1-element+ 3-element.What's more, it can be proved that all other way is equal to the four ways above. This is my AC code: http://paste.ubuntu.com/24652108/

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8 years ago, # ^ |

thanks a lot, what my poor dp skill

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matheusdacach

4 years ago, # ^ |

great comment, thank you very much!

managed to get AC considering this new fourth case

very instructive

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u33how

4 years ago, # ^ |

Could you kindly share the proof or hints for the fourth case?

808E - Selling Souvenirs

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Serh.kry

8 years ago, # |

any suggestions how this solutions for D was hacked? http://mirror.codeforces.com/contest/808/submission/27138172

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calabash_boy

8 years ago, # ^ |

When you check the position of target delta value

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teja349

8 years ago, # |

what does unexpected verdict during hacking mean?

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marcospaulo

8 years ago, # |

← Rev. 12 →

My solution of E is a kind of greedy and Dp solution. In this problem we have a knapsack problem. The knapsack has a size much bigger than the size of its components. So we can make a greedy approach until the size of knapsack is X, then make a knapsack DP on the rest of components on a knapsack of size X. I fix X = 300 (3*100 ) arbitrarily. A doubt: How we can find the minimum X that we guarantee a optimal solution in this case? :p My solution:http://mirror.codeforces.com/contest/808/submission/27144481

Edit: It is Wrong! Sorry!

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8 years ago, # ^ |

← Rev. 2 →

+12

It is wrong because if you have, for example, one souvenir with cost 5 and weight 3 and then two hundred souvenirs with cost 3 and weight 2 and you are able to carry 400 weight, you would greedily choose the 3-cost souvenir right at the start, which is not optimal.

Now, what if you make sure you leave enough unpicked items of each weight before the end of the greedy phase? That is, you refuse to pick the item with the best cost-to-weight ratio if it makes your list with that specific weight too short. Then, the dynamic programming should have enough of each weight to optmize the remainders of the knapsack.

My intuition says this approach should work, but I can't come up with a proof or even the right limits for when to start the DP and for how many items of each weight we should keep.

EDIT: Nevermind, it's just wrong. Unless there are specific and artifical limits on the costs of the items, one greedy choice in enough to make the algorithm incorrect.

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UnknownNooby

8 years ago, # |

← Rev. 2 →

+18

Ignore

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UnknownNooby

8 years ago, # ^ |

+21

Flow network must be oriented, that isn't said in the editorial. Hope that this will help someone

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BledDest

8 years ago, # ^ |

Thank you for pointing it out. It is added to the editorial now.

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AndreBtt

8 years ago, # |

I am not able to understand letter E, can someone explain me more clearly please ?

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Deng2X

8 years ago, # |

Can anybody explain why we just need to optimize the 'cost' in problem E solution without regarding to the 'cnt1' and 'cnt2' ?

Much appreciated!

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mig

8 years ago, # ^ |

if(dp[i - 1].cnt1 < t1 && dp[i - 2].cnt2 < t2) 
    if(dp[i - 1].cost + c1[dp[i - 1].cnt1] > dp[i - 2].cost + c2[dp[i - 2].cnt2]) {
    *****//renew dp[i] //********
    }

my code is a implementation of E in editorial, cnt1 and cnt2 can be used to determine how to renew dp[i](like above). Here it is 27157016

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Deng2X

8 years ago, # ^ |

← Rev. 2 →

+28

I mean, why we ONLY need to optimize(maximize) the COST.

The cnt1 and cnt2 definitely means something. But why we can ignore it.

I want a proof or something like that.

my code here 27149084

and another code using quadruple get WA which I cannot figure out why. 27163738

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rahul_1234

8 years ago, # ^ |

Hey, can this be cleared out. If Deng2X got it or anybody, please tell why there is WA with 1, 2, 3 and cost together but not with 1, 2 and cost?

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angelg

8 years ago, # |

Can someone explain the ternary search solution to problem E?

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Daniar

8 years ago, # ^ |

keep the best m elements of weight 1

keep the best m/2 elements of weight 2

keep the best m/3 elements of weight 3

sort all of them in non-increasing order

sum[i][j] is the sum of the first j elements of weight i

/* from the editorial above

We can iterate on the number of 3-elements we will take (in this editorial k-element is a souvenir with weight k). When fixing the number of 3-elements (let it be A), we want to know the best possible answer for the weight m - 3A, while taking into account only 1-elements and 2-elements.

let y be m — 3A , we need to take B 2-elemensts and C 1-elements with total weight equal to y

y = B*2 + C*1

we will take the first B elements of weight 2 and the first C elements of weight 1

the ternary search is on B with this function F(i) = sum[2][i] + sum[1][y-2*i]

it is correct because F(0) <= F(1) <= .... <= F(B) >= F(B+1) ...>= F(m/2)

and we are searching for the best B

the answer will be : sum[3][A] + sum[2][B] + sum[1][C]

take a look at my code if you need : code

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DoanPhuDuc

8 years ago, # ^ |

← Rev. 2 →

+11

Can you give me your evidence to prove prove that F(0) <= F(1) <= .... <= F(B) >= F(B+1) ...>= F(m/2)

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8 years ago, # ^ |

I also dont get this. What if last 1-element ( that has least price) has bigger cost than first 2-element (has highest cost of all 2-elements) ? then F(0) >= F(1)

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Daniar

8 years ago, # ^ |

← Rev. 2 →

B isn't literally bigger than 0 and less than m/2

it might be 0 or m/2

the idea is that the graph of the function F is increasing then at some point is decreasing

it's correct because we sorted the 3 types of elements in decreasing order

when we take 1 more 2-element with sum greater than the last 2 1-elements we took before the answer is increasing

but at some point , taking 1 more 2-element with sum less than the last 2 1-elements we took before the answer will start to decrease

you need to try an example to understand it well

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DoanPhuDuc

8 years ago, # ^ |

Thank you your help. I unserstood

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8 years ago, # ^ |

← Rev. 3 →

Why do u take first m/2 elements of weight 2 and m/3 elements of weight 3? Maybe in optimal solution we will take worst element of weight 3? Please explain. Also, any good reference on ternary search? Edit: wow, cant believe I asked this..thanks for reply

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nikich340

8 years ago, # ^ |

Cos it maximal number for 3-w elements could be only m/3 and for 2-w m/2.

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HandsomeHow

8 years ago, # ^ |

Does ternary search still work if we iterate on the number of 1-elements?

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memeset

8 years ago, # ^ |

I thought ternary search is only possible when the function is strictly increasing then decreasing (or vice versa). How come this still works if you have the <= instead of < in this case?

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krijgertje

8 years ago, # ^ |

It works because in this case F(0) < F(1) < .... < F(B1) = F(B1+1) = ... = F(B2) > F(B2+1) ...> F(m/2). If there also are equal elements in other places, you are right and ternary search is not guaranteed to work.

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cronenberg

8 years ago, # |

I don't understand D?

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hrushikesht

8 years ago, # ^ |

Here's one approach.

Maintain two multisets, one for prefix elements and one for suffix elements. Traverse through the array in the given order and keep prefix and suffix sums as you traverse. Also maintain the two multisets, adding the current element in prefix multiset and erasing it from suffix multiset.

If prefix sum is greater than suffix sum, search in the prefix multiset for their difference halved. If such element exists, output YES.

If suffix sum is greater than prefix sum, search in suffix multiset. If you can't find such element, output NO.

Corner Cases: When n=1, answer is always NO.

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-wolfie-

8 years ago, # ^ |

← Rev. 2 →

I traversed the array both ways keeping the sum of all elements encountered. Then when ever the sum is exceeding the half of the total sum I'm checking whether the difference has been encountered. This means I'll be able to spot an element which i have to remove from the prefix and add to the suffix in order to get the answer. 27175419

Please let me know if I have miss understood the question.

EDIT: This is the working solution 27176375...

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Daniar

8 years ago, # |

+22

Problem F is something good to learn about mincut , thanks

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haleyk100198

8 years ago, # |

I think problem F is solvable with binary search + edmond blossom with a complexity O(N^3logN), not sure if it would pass.

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code: http://mirror.codeforces.com/contest/808/submission/27160206

8 years ago, # |

Can anyone explain binary(not ternary!) search solution for problem E (if there is one)? I tried to solve it this way with binary search but it fails on test case 9: separate elements in 3 arrays, every weight goes in one of them. Sort arrays Lets fix number of 3-elements we will use. then I do 2 steps: 1. step — first binary search for number of 2-elements, and while doing that, binary search for number of 1-elements (so binary search in binary search :D ) 2. step — same but oppposite: first binary search for 1-elements, then binary search for 2-elements inside. Why we do bs in bs twice? Because maybe optimal solution is to take 0 2-elements and 100 1-elements, so if we just binary search for 2-elements first, then we may skip this solution.

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Kettlebell

8 years ago, # |

Does problem D mean that we get to swap the elements? For instance, example #3 shows that we need to swap the elements 3 and 4 to make the sums equal. (2 2 3 4 5) -> (2 2 4 3 5). I'm asking this because it did not work, obviously :(.

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8 years ago, # ^ |

Not swap elements, just PUT one element in different position. In example above, you took number 3 and put it right from number 4. Number 4 didnt move anywhere, but it seemed as it did because you moved 3 from its position.

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8 years ago, # ^ |

In general, no, you cannot swap two arbitrary elements, you are only allowed to remove one element and place it somewhere else. Notice, hoewever, that if you take the 3 of (2, 2, 3, 4, 5) and place it one position to the left, the final result is equivalent to that of swapping it with the 4.

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Kettlebell

8 years ago, # ^ |

So, from what I understand from your kind answers, there are two cases

Removing an element from prefix(or suffix) and add it to suffix(or prefix), which means <sumPrefix — element, sumSuffix + element>
Moving an element at the boundary of prefix / suffix in which case, the swap happens <sumprefix — prefixElem + suffixElem, sumsuffix — sufixElem + prefixElem>

Please correct me if I'm wrong

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8 years ago, # ^ |

If I understand what you are saying, you are correct, but there is no need to divide the problem into two cases, the second one is just a particular occurence of the first.

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jk_qq

8 years ago, # |

← Rev. 3 →

Another approach for E: Sort all souvenirs in decreasing order by cost / weight (if equal it's better to choose souvenir with least weight).

Now pick greedily up to m - X weight (we'll choose X later). After that we have res + X weight to fill (1 < = res < = 3 — residue after greedy stage of algorithm).

Now we'll use naive approach (either standard knapsack dp or even iterate on all possible counts of 1-souvenir, 2-souvenir, 3-souvenir). It turns out as far as we have really small weights, we can choose X to be relatively small (intuitively we can choose X to be 3 * 3 + 2 * 3 + 1 * 3 = 6 * 3 = 18).

My submission http://mirror.codeforces.com/contest/808/submission/27144235 so feel free to hack it;)

P.S. For ones who might want to understand crappy code above. In submission I have 3 vectors (for 1-souvenirs, 2-souvenirs, 3-souvenirs respectively) and 3 pointers to handle greedy picking souvenirs and loops at the end.

Edited:

Thanks @Lewin for the hack. Yeah, interesting. I was pretty sure this approach works:) Gotta find out if it's a bug or incorrect approach

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8 years ago, # ^ |

This comment proposed a similar solution, and you can see my counterexample as an answer.

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jk_qq

8 years ago, # ^ |

← Rev. 2 →

In fact I still believe this approach works with the following tune: After greedy step we have to remove worst 3 1-souvenirs, worst 3 2-souvenirs, worst 3 3-souvenirs and perform naive approach (looping and knapsack dp).

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8 years ago, # ^ |

Nice idea. I got AC with it.

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mkagenius

7 years ago, # ^ |

I happen to think this should work, let me modify my code.

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NotImplemented

8 years ago, # |

+18

What if we have several pairs of cnt1, cnt2 for optimal cost dp[w]?

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8 years ago, # ^ |

Exactly..Why no one has the answer yet?

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Vasiljko

8 years ago, # |

← Rev. 2 →

What is the time complexity of D using sets of search.

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unused

8 years ago, # |

← Rev. 2 →

+58

Here is another approach to problem E. Suppose we have only two kinds of weight, 2 and 3. Then we can easily solve the problem with sorting and (two pointers or binary search).

The main idea is that we can reduce the main problem to this easy one. Solve the problem twice: choosing odd number of items of weight 1, or even. If we decide to choose even, we can pair the weight-1 items and convert them to weight-2 items. And if we decide to choose odd, first pick the biggest item of weight 1. The remaining part is same as the even version.

27164728

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8 years ago, # ^ |

Such a beautiful solution. While trying to solve problem, I also came on idea to pair weight-1 items to get weight-2 or even to take triples of weight 1 items. But I thought that it wont be correct because it may be optimal to choose only one weight-1 item. Now I see — either you pick greatest weight-1 item and pair others, or you pair all of them... Thank you.

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sas

8 years ago, # ^ |

Why can this subtask be solved using greedy approach unlike thr initial?

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unused

8 years ago, # ^ |

Initial problem can also be solved with greedy approach(selecting the most valuable items from each group). But the time complexity will be O(N^2) as we have to deal with 3 groups. After reducing the number of groups from 3 to 2, the problem can be solved in O(N log N).

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OO_I_LOVE_YZQ__OO

8 years ago, # |

http://mirror.codeforces.com/contest/808/submission/27164939 About the problem E ? I don't think is wrong. Who can help me?

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Al.Cash

8 years ago, # |

+14

In G you don't need to consider all 26 characters. From position j you either move forward by t_j or you start from position P(j), where P is prefix function. My solution for reference: 27165597

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8 years ago, # |

I thought I understand DP solution for E, but then I realised that I dont. Here is my problem; lets say that dp[i].cost = x. then you say we update dp[i+1], dp[i+2],dp[i+3] with this value. But what if there is way to get weight i, with cost y, such that y < x, but in that case we used less elements of weight 1 lets say. So, maybe dp[i+1] will be better if updated with cost y + cost of next weight 1. Am I right?

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sas

8 years ago, # ^ |

← Rev. 2 →

I don't think there is a difference due to the fact we use best k-elements available. For instance, to get x you use 1-elements e1[1] + ... + e1[m] (in sorted order) and to get y e1[1] + ... + e1[m -1]. You claim that y < x and y + e[m] > x + e[m + 1] (if e[m + 1] exists) which is obviously wrong, in worst case (if e[m + 1] doesn't exist) y + e[m] = x, so y can't be strictly bigger than x, I suppose.

Edit: y + e[m] can be bigger than x in worst case, of course, if we take into account 2 and 3-elements , so the question remains unanswered.

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8 years ago, # ^ |

Yeah, I wanted to say you that you didnt count 2 elements and 3 elements, and then saw that you updated comment. So, anyone with answer?

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nikich340

8 years ago, # ^ |

← Rev. 2 →

You can not update [i+3] state choosing only from 1s and 2s. And we don't care about free souvenirs, we will use the most expensive of them anyway. Example: 1-w: [2 3] 2-w: [2 5] m = 2. dp[0] = {0, 0, 0}, dp[1] = {3, 1, 0}. dp[2] choosing from (0 + 5) and (3 + 2). They are equal, so two variants of new tuple: {5, 0, 1} and {5, 2, 0}. But we are interested only in the first value, and it must be the biggest. if (cost of 1-w + 1-w) == (cost of 2-w) we could use any of them (weights are equal) and use another later if have extra weight reserve.

Sorry if I didn't unserstand you question and wrote something strange :D

My realization if needed: http://mirror.codeforces.com/contest/808/submission/27421196

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LoveIsSoFar

8 years ago, # |

why F we can't use more than one card of magic number is 1?

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jk_qq

8 years ago, # ^ |

← Rev. 2 →

Cause in this case we have sum equal to 2(1+1) which is prime.

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silent_lamp

8 years ago, # |

Could someone please help me understand as to why greedy approach of G not correct. As i am replacing the given string in the specified string from back and then count the total number of the given string .Any explaination of this would be really helpful .

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segwit

8 years ago, # ^ |

+10

Dude, u better change your dp before asking any question.

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8 years ago, # ^ |

Consider s1 = "a????bcab", s2 = "abca????b" and t = abcab.

For s1 optimal answer is "aabcabcab" and for s2 it is "abcabcabb", and t occurs 2 times in both of these.

If you do greedy from front, you'd get "abcabbcab" as answer for s1, which has 1 occurrence of t.

If you do greedy from back, you'd get "abcaabcab" as answer for s2, which has 1 occurrence of t.

Hope this helps!

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silent_lamp

8 years ago, # ^ |

Thanks Dude but could you help me understanding the dp approach ?

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8 years ago, # ^ |

Do you know the KMP algorithm and understand it well?

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silent_lamp

8 years ago, # ^ |

i know KMP algo but dont have a deep understanding of it just basics !

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8 years ago, # ^ |

My solution is here.

There are two main functions pre and go.

In pre, first I create the table b where b[i+1] stores length of longest proper suffix (which is also its prefix) of the string p[0:i+1] (substring of p starting at 0 and of length i+1). Then I calculate n[i][j] which is same as the next array defined in the editorial.

My go function now uses the above two arrays to calculate the dp and cnt values. dp[i,j] tells if there exists a placing of characters such that last j characters of s[0:i+1] are same as first j characters of p.

Use this information and try to understand the code, and why it is correct.

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capta1n

8 years ago, # |

← Rev. 2 →

I got TLE in this code for problem B can someone please point out the error??

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quotitquot

8 years ago, # ^ |

you have used "cin,cout", because of this, you got TLE, if you change "cin,cout" to "scanf,printf), it will work much faster

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Xahandd

8 years ago, # ^ |

if you submit it with c++14 (http://mirror.codeforces.com/contest/808/submission/27183006) it will get AC. cin,cout are faster in c++14 than c++11 if you use boost(ios::syn..).

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Vasiljko

8 years ago, # |

What is the time-complexity for D?

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BledDest

8 years ago, # ^ |

O(n log(n)) if you use something like std::set to maintain the elements on the prefix.

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Vasiljko

8 years ago, # ^ |

I thought of the same complexity, but i thought it was wrong. Thank you :)

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memeset

8 years ago, # |

How do we do ternary search (or its reduction to the binary search version) for E if it isn't necessarily true that the function is strictly increasing/decreasing?

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8 years ago, # ^ |

See my comment: http://mirror.codeforces.com/blog/entry/52010?#comment-360483 Here I explained my TRY of binary search inside binary search. It passed 9 test cases and I didnt really try much to debug it. But from my solution you can see that you can do bs. If you separate items (weight i in group i — so 3 gruops totally) and then sort, you have increasing function.

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memeset

8 years ago, # ^ |

Isn't it only nondecreasing, not increasing?

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8 years ago, # ^ |

Well for binary search non decreasing is ok. And I cant help you with ternary cause I also have problems with that solution.somewhere here in comments there is ternary search solution written but I dont understand why function is first increasing and then decreasing (user didnt prove it, he just said it).

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pr3pony

8 years ago, # |

How did this randomized solution 27148367 pass 808F - Card Game ?
Are the test data too weak or the probability of success is really high?

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Omar_Morsi

8 years ago, # |

In Problem F, I get WA test 11.

Can someone help me please? My solution ==> 27199504

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pr3pony

8 years ago, # ^ |

You forgot to add reverse edges in dinic.adjList[]

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Omar_Morsi

8 years ago, # ^ |

You are right, Thank you very much xD

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akumar1503

8 years ago, # |

In problem G: what are other ways to represent the states? thanks

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ChandyShot

8 years ago, # |

My code http://mirror.codeforces.com/contest/808/submission/27212742 for Problem A is running wrong in test case 2, for input 201, when checked in submission it shows output 96, whereas in my compiler and other online compilers it is showing output 99, don't know what to do.. help if possible...

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cyand1317

8 years ago, # ^ |

Rounding inside pow() differs slightly among implementations (details here), so surrounding pow() with another round() fixes the issue. It's usually preferred to use exponentiation by squaring when exponents are integers, though.

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Ishan.nitj

8 years ago, # |

← Rev. 2 →

I am getting WA on test 21 for problem D ,although I checked it with lot of hack cases. Any help would be appreciated. http://mirror.codeforces.com/contest/808/submission/27217259

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GreenGrape

8 years ago, # ^ |

Changing int pre[] to ll pre[] will do.

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Ishan.nitj

8 years ago, # ^ |

Thanks :)

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goldflake

8 years ago, # |

← Rev. 2 →

-10

problem:G... first I am finding in string s, all the possible positions where the string t can end and storing in boolean array, eg for (win???dwin,win) f=[0,0,1,0,0,1,0,0,0,1] . And then I am finding the LPS for string s. And then using this DP state :(lt=len of string t,ls =len of string s)
for(i=lt;i<ls;i++) { if(flag[i]) dp[i]=dp[ i-lt+lps[lt-1] ] + 1 ; else dp[i]=dp[i-1]; } I am getting right answer for most of the test cases...except test cases 56,60,64. Can anyone pl explain why am I getting WA for these test cases.

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SegmentationRight

8 years ago, # |

there is another solution for problem E. we can sort the array and select front item greedily, then do some fix to generate right answer. 27342219

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anmol23

8 years ago, # |

Why cant E be solved by simple 0-1 knapsack problem like this?

http://ideone.com/EiWMGd

What is wrong in this approach?

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8 years ago, # ^ |

It's O(n*W) which is too slow for this task.

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