### 779A - Pupils Redistribution

Problem setter: MikeMirzayanov

To solve this problem let's use array *cnt*[]. We need to iterate through first array with academic performances and for current performance *x* let's increase *cnt*[*x*] on one. In the same way we need to iterate through the second array and decrease *cnt*[*x*] on one.

If after that at least one element of array *cnt*[] is odd the answer is - 1 (it means that there are odd number of student with such performance and it is impossible to divide them in two. If all elements are even the answer is the sum of absolute values of array *cnt* divided by 2. In the end we need to divide the answer on 2 because each change will be counted twice with this way of finding the answer.

### 779B - Weird Rounding

Problem setter: MikeMirzayanov

To solve this problem we need to make *k* zeroes in the end of number *n*. Let's look on the given number as on the string and iterate through it beginning from the end (i.e. from the low order digit). Let *cnt* equals to the number of digits which we reviewed. If the current digit does not equal to zero we need to increase the answer on one. If *cnt* became equal to *k* and we reviewed not all digits we need to print the answer.

In the other case we need to remove from the string all digits except one, which equals to zero (if there are more than one such digit we left only one of them). Such digit always exists because the problem statement guaranteed that the answer exists.

### 779C - Dishonest Sellers

Problem setters: MikeMirzayanov and fcspartakm

To solve this problem we need at first to sort all items in increasing order of values *a*_{i} - *b*_{i}. Then let's iterate through sorted array. If for the current item *x* we did not buy *k* items now and if after discounts it will cost not more than now, we need to buy it now and pay *a*_{x}, in the other case we need to buy item *x* after discounts and pay *b*_{x}.

### 778A - String Game

Problem setter: firezi

In this problem we have to find the last moment of time, when *t* has *p* as a subsequence.

If at some moment of time *p* is a subsequence of *t* then at any moment before, *p* is also its subsequence. That's why the solution is binary search for the number of moves, Nastya makes. For binary search for a moment of time *m* we need to check, if *p* is a subsequence of *t*. We remove *a*_{1}, *a*_{2}, ... *a*_{m} and check if *p* is a subsequence greedily.

### 778B - Bitwise Formula

Problem setter: burakov28

Note that changing *i*-th bit of chosen number doesn't change any bits of any of the variables other than *i*-th one. Also note that the total number of values is greater, as more variables have 1 at *i*-th position.

Let's solve for every bit independently: learn, what is the value of *i*-th bit of chosen number. We can try both values and simulate the given program. Choose one of the values that makes more variables to have 1 at *i*-th position. If both 0 and 1 give equal number of variables to have 1 at *i*-th position, choose 0.

### 778C - Peterson Polyglot

Problem setters: niyaznigmatul and YakutovDmitriy

While erasing letters on position *p*, trie changes like the following: all the edges from one fixed vertex of depth *p* are merging into one. You can see it on the picture in the sample explanation. After merging of the subtrees we have the only tree — union of subtrees as the result.

Consider the following algorithm. For every vertex *v* iterate over all the subtrees of *v*'s children except for the children having largest subtree. There is an interesting fact: this algorithm works in in total.

Denote as *s*_{x} the size of the subtree rooted at vertex *x*. Let *h*_{v} be the *v*'s child with the largest subtree, i.e. *s*_{u} ≤ *s*_{hv} for every *u* — children of *v*. If *u* is a child of *v* and *u* ≠ *h*_{v} then . Let's prove that.

- Let . Then and .
- Otherwise, if , then we know that
*s*_{u}+*s*_{hv}<*s*_{v}. Therefore, .

Consider vertex *w* and look at the moments of time when we have iterated over it. Let's go up through the ancestors of *w*. Every time we iterate over *w* the size of the current subtree becomes twice greater. Therefore we could't iterate over *w* more than times in total. It proves that time complexity of this algorithm is .

Solution:

- Iterate over all integers
*p*up to the depth of the trie - For every vertex
*v*of depth*p* - Unite all the subtrees of
*v*with running over all of them except for the largest one.

How to unite subtrees? First method. Find the largest subtree: it has been already built. Try to add another subtree in the following way. Let's run over smaller subtree's vertices and add new vertices into respective places of larger subtree. As the result we will have the union of the subtrees of *v*'s children. All we need from this union is it's size. After that we need to roll it back. Let's remember all the memory cells, which were changed while merging trees, and their old values. After merging we can restore it's old values in reverse order.

Is it possible to implement merging without rolling back? Second method. Let's take all the subtrees except for the largest one and build their union using new memory. After that we should have two subtrees: the largest one and the union of the rest. We can find size of their union without any changes. Everything we need is to run over one of these trees examining another tree for the existence of respective vertices. After this we can reuse the memory we have used for building new tree.

### 778D - Parquet Re-laying

Problem setter: pashka

Let's assume that the width of the rectangle is even (if not, flip the rectangle). Convert both start and final configurations into the configuration where all tiles lie horizontally. After that, since all the moves are reversible, simply reverse the sequence of moves for the final configuration.

How to obtain a configuration in which all tiles lie horizontally. Let's go from top to bottom, left to right, and put all the tiles in the correct position. If the tile lie vertically, then try to turn it into the correct position. If it cannot be rotated, because the neighboring tile is oriented differently, proceed recursively to it. Thus, you get a "ladder", which can not go further than *n* tiles down. At the end of the ladder there will be two tiles, oriented the same way. Making operations from the bottom up, we'll put the top tile in a horizontal position.

### 778E - Selling Numbers

Problem setters: niyaznigmatul and VArtem

Because the target value for this problem is calculated independently for all digits, we'll use the dynamic programming approach. Define *dp*_{k, C} as the maximum possible cost of digits after we processed *k* least significant digits in *A* and *C* is the set of numbers having the carry in current digit. This information is sufficient to choose the digit in the current position in *A* and recalculate the *C* set and DP value for the next digit.

The key observation is that there are only *n* + 1 possible sets instead of 2^{n}. Consider last *k* digits of *A* and *B*_{i}. Sort all the length-*k* suffixes of *B*_{i} in descending lexicographical order. Because all these suffixes will be increased by the same value, the property of having the carry is monotone. That means that all possible sets *C* are the prefixes of length *m* (0 ≤ *m* ≤ *n*) of this sorted list of suffixes. This fact allows us to reduce the number of DP states to *O*(*n*·|*A*|). Sorting all suffixes of *B*_{i} can be accomplished using the radix sort, appending the digits to the left and recalculating the order.

The only thing that's left is to make all DP transitions in *O*(1) time. To do that, maintain the total cost of all digits and the amount of numbers that have the carry. After adding one more number with carry in current digit, these two values can be easily recalculated. After processing all digits in *A*, we have to handle the remaining digits in *B*_{i} (if there are any) and take the best answer. Total running time is .

somebody plzz tell how to do string game using binary search.i read tutorial but i didn't get it..

Let's consider the first test case, where you have this order of deleting characters:

Let's write down if we could obtain the target word after deleting characters order[0], order[1], ... order[i] of the original word (Note that we always delete characters in this order):

This sequence is monotonous and thus, we could do a binary search to find the maximum index, which brings "YES". :)

Hope that helps!

This problem has binary search property: Let i a arbitrary position of a[]. There are 2 cases: If you remove all the character of index a[1..i] and p is a subsequence of t. Then there will exist a position j in t such that 1<=i<=j<=|t| and if we remove all character from [1..j] then p is still a subsequence of t. Otherwise, then there will exist a position j in t such that 1<=j<=i<=|t| and if we remove all character from [1..j] then p is still a subsequence of t and there will be no position >=i that p is a subsequence of t. The first idea you maybe thinking of is to use complete search and search for the latest position that p is subsequence of t. unfortunately this approach take O(n^2) complexity (note that to check subsequence it take O(n)). So instead we can use binary search and search for the latest position and check if it's a subsequence or not.If it is then we move futher to the right, else we move back to the left. For more detail and better understanding see my submission 25052010.

Help me plzzz,What's wrong with my code ? (time limit on test 10) String game 25102212

It's a bruteforce, not a binary search.

Complete search is not fast enough for this problem because it's O(n^2) and n<=200000, try to use binary search to solve this problem.

Where's the editorial for Div. 1 C and Div. 1 D?

The way I implemented C was pretty much brute force, but if you calculate the complexity it's N * 26 Log(N).

For each node U, you count how many duplicate nodes you'll have in its subtree if you remove it and its outgoing edges. We add this to the number of removals for level(U) + 1.

So you'll have a dfs(vector v), where all vertices in v are equivalent with respect to U (same prefix in the sub-trie). Since all are equivalent you add v.size()-1 to the number of duplicates. Now for every letter c, you make the vector of nodes that are children of a node in v through an edge labelled with letter c. Now do (at most) 26 dfs calls, one for each new vector. If the current size of the node vector is 1 or 0, you stop since there won't be any extra duplicates. The initial call is dfs(children of U).

Submission

Your solution is also , because if the size of

vis 26, thennis distributed in this 26 subtrees, so every time you call`dfs3`

with`v.size() == k`

, you useknodes. You use only nodes, which are duplicate, that's why it's the same as if you only iterated over the small subtree.Can you elaborate on the the statement: "The complexity is N*26log(N)". That is the most vague part in your solution

778C: I dont know why you wrote about biggest child of vertex. I implemented your solution and then removed "lets merge all subtrees to the biggest subtree" rule — it became even faster (935ms 146Mb vs 732ms 124Mb)

So plain solution with rollbacks works.

No, your solution makes link to whole subtree, instead of copying it. That's why your merge function works in

min(s_{1},s_{2}) time, wheres_{1}ands_{2}are the sizes of the subtrees, which is equivalent to iterating over "all vertices in all subtrees except the largest one".пояснил по понятиям!!!

Could you please explain why they are equivalent? It really puzzled me.

Because

`dfs`

visits vertex for the first time if it's in both subtrees, that are being merged. Every next time the vertex is visited, it appears again in a new subtree. It means vertex is visitedk- 1 times, if it appeared inksubtrees. So, consider biggest subtree and for every vertex this - 1 fromk- 1 means "we don't visit biggest subtree".Got it. Thanks a lot.

what is wrong with my appraoch for "778C — Peterson Polyglot"? normal dfs when i enter vertex i try to kill its children and count the children of the children which occur more than once + number of its children and update kill[depth] after that iterate over kill get the largest one x to be the solution (n-x)... can any one help me http://codeforces.com/contest/778/submission/25119082

I got the same problem. Try this test case 7 1 2 a 1 3 b 2 4 b 4 6 c 3 5 b 5 7 c

This should've been in pretests before the big test case # 3.

Can someone explain the time complexity of the binary search solution to the String Game problem?

779C — Dishonest Sellers why i need to sort with respect to a[i]-b[i] ? what is the proof ?

why do you divide by 2 again at the end in pupils redistribution???