# Rday6

## Connecting Vertices

There are n points marked on the plane. The points are situated in such a way that they form a regular polygon (marked points are its vertices, and they are numbered in counter-clockwise order). You can draw n - 1 segments, each connecting any two marked points, in such a way that all points have to be connected with each other (directly or indirectly).

But there are some restrictions. Firstly, some pairs of points cannot be connected directly and have to be connected undirectly. Secondly, the segments you draw must not intersect in any point apart from the marked points (that is, if any two segments intersect and their intersection is not a marked point, then the picture you have drawn is invalid).

How many ways are there to connect all vertices with n - 1 segments? Two ways are considered different iff there exist some pair of points such that a segment is drawn between them in the first way of connection, but it is not drawn between these points in the second one. Since the answer might be large, output it modulo 109 + 7.

`Input`
The first line contains one number n (3 ≤ n ≤ 500) — the number of marked points.

Then n lines follow, each containing n elements. ai, j (j-th element of line i) is equal to 1 iff you can connect points i and j directly (otherwise ai, j = 0). It is guaranteed that for any pair of points ai, j = aj, i, and for any point ai, i = 0.

`Output`
Print the number of ways to connect points modulo 109 + 7.

`题解`:

f[i][j]f[i][j]表示ii到jj有边且构成树的方案数
g[i][j]g[i][j]表示ii到jj无边且构成树的方案数

f[i][j]+=∑j−1k=i(f[i][k]+g[i][k])∗(f[k+1][j]+g[k+1][j])f[i][j]+=∑k=ij−1(f[i][k]+g[i][k])∗(f[k+1][j]+g[k+1][j])
g[i][j]+=∑j−1k=i+1f[k][j]∗(f[i][k]+g[i][k])g[i][j]+=∑k=i+1j−1f[k][j]∗(f[i][k]+g[i][k])

code:

## Local Extrema

You are given an array a. Some element of this array ai is a local minimum iff it is strictly less than both of its neighbours (that is, ai < ai - 1 and ai < ai + 1). Also the element can be called local maximum iff it is strictly greater than its neighbours (that is, ai > ai - 1 and ai > ai + 1). Since a1 and an have only one neighbour each, they are neither local minima nor local maxima.

An element is called a local extremum iff it is either local maximum or local minimum. Your task is to calculate the number of local extrema in the given array.

`Input`
The first line contains one integer n (1 ≤ n ≤ 1000) — the number of elements in array a.

The second line contains n integers a1, a2, …, an (1 ≤ ai ≤ 1000) — the elements of array a.

`Output`
Print the number of local extrema in the given array.

`题解`:

code:

## Xor-MST

You are given a complete undirected graph with n vertices. A number ai is assigned to each vertex, and the weight of an edge between vertices i and j is equal to ai xor aj.

Calculate the weight of the minimum spanning tree in this graph.

`Input`
The first line contains n (1 ≤ n ≤ 200000) — the number of vertices in the graph.

The second line contains n integers a1, a2, …, an (0 ≤ ai < 230) — the numbers assigned to the vertices.

`Output`
Print one number — the weight of the minimum spanning tree in the graph.

`题意`:

dalao’s code:

## Buggy Robot

Ivan has a robot which is situated on an infinite grid. Initially the robot is standing in the starting cell (0, 0). The robot can process commands. There are four types of commands it can perform:

U — move from the cell (x, y) to (x, y + 1);
D — move from (x, y) to (x, y - 1);
L — move from (x, y) to (x - 1, y);
R — move from (x, y) to (x + 1, y).
Ivan entered a sequence of n commands, and the robot processed it. After this sequence the robot ended up in the starting cell (0, 0), but Ivan doubts that the sequence is such that after performing it correctly the robot ends up in the same cell. He thinks that some commands were ignored by robot. To acknowledge whether the robot is severely bugged, he needs to calculate the maximum possible number of commands that were performed correctly. Help Ivan to do the calculations!

`Input`
The first line contains one number n — the length of sequence of commands entered by Ivan (1 ≤ n ≤ 100).

The second line contains the sequence itself — a string consisting of n characters. Each character can be U, D, L or R.

`Output`
Print the maximum possible number of commands from the sequence the robot could perform to end up in the starting cell.

`题意`:

code:

## K-Dominant Character

You are given a string s consisting of lowercase Latin letters. Character c is called k-dominant iff each substring of s with length at least k contains this character c.

You have to find minimum k such that there exists at least one k-dominant character.

`Input`
The first line contains string s consisting of lowercase Latin letters (1 ≤ |s| ≤ 100000).

`Output`
Print one number — the minimum value of k such that there exists at least one k-dominant character.

`题意`:

## Maximum Subsequence

You are given an array a consisting of n integers, and additionally an integer m. You have to choose some sequence of indices b1, b2, …, bk (1 ≤ b1 < b2 < … < bk ≤ n) in such a way that the value of is maximized. Chosen sequence can be empty.

Print the maximum possible value of .

`Input`
The first line contains two integers n and m (1 ≤ n ≤ 35, 1 ≤ m ≤ 109).

The second line contains n integers a1, a2, …, an (1 ≤ ai ≤ 109).

`Output`
Print the maximum possible value of .

`题意`:

## Alomost Identity Permutations

A permutation p of size n is an array such that every integer from 1 to n occurs exactly once in this array.

Let’s call a permutation an almost identity permutation iff there exist at least n - k indices i (1 ≤ i ≤ n) such that pi = i.

Your task is to count the number of almost identity permutations for given numbers n and k.

`Input`
The first line contains two integers n and k (4 ≤ n ≤ 1000, 1 ≤ k ≤ 4).

`Output`
Print the number of almost identity permutations for given n and k.

`题意`:

k≤4，枚举即可。

code:

## Alyona and Spreadsheet

During the lesson small girl Alyona works with one famous spreadsheet computer program and learns how to edit tables.

Now she has a table filled with integers. The table consists of n rows and m columns. By ai, j we will denote the integer located at the i-th row and the j-th column. We say that the table is sorted in non-decreasing order in the column j if ai, j ≤ ai + 1, j for all i from 1 to n - 1.

Teacher gave Alyona k tasks. For each of the tasks two integers l and r are given and Alyona has to answer the following question: if one keeps the rows from l to r inclusive and deletes all others, will the table be sorted in non-decreasing order in at least one column? Formally, does there exist such j that ai, j ≤ ai + 1, j for all i from l to r - 1 inclusive.

Alyona is too small to deal with this task and asks you to help!

`Input`
The first line of the input contains two positive integers n and m (1 ≤ n·m ≤ 100 000) — the number of rows and the number of columns in the table respectively. Note that your are given a constraint that bound the product of these two integers, i.e. the number of elements in the table.

Each of the following n lines contains m integers. The j-th integers in the i of these lines stands for ai, j (1 ≤ ai, j ≤ 109).

The next line of the input contains an integer k (1 ≤ k ≤ 100 000) — the number of task that teacher gave to Alyona.

The i-th of the next k lines contains two integers li and ri (1 ≤ li ≤ ri ≤ n).

`Output`
Print “Yes” to the i-th line of the output if the table consisting of rows from li to ri inclusive is sorted in non-decreasing order in at least one column. Otherwise, print “No”.

`题意`:

## Shell Game

Bomboslav likes to look out of the window in his room and watch lads outside playing famous shell game. The game is played by two persons: operator and player. Operator takes three similar opaque shells and places a ball beneath one of them. Then he shuffles the shells by swapping some pairs and the player has to guess the current position of the ball.

Bomboslav noticed that guys are not very inventive, so the operator always swaps the left shell with the middle one during odd moves (first, third, fifth, etc.) and always swaps the middle shell with the right one during even moves (second, fourth, etc.).

Let’s number shells from 0 to 2 from left to right. Thus the left shell is assigned number 0, the middle shell is 1 and the right shell is 2. Bomboslav has missed the moment when the ball was placed beneath the shell, but he knows that exactly n movements were made by the operator and the ball was under shell x at the end. Now he wonders, what was the initial position of the ball?

`Input`
The first line of the input contains an integer n (1 ≤ n ≤ 2·109) — the number of movements made by the operator.

The second line contains a single integer x (0 ≤ x ≤ 2) — the index of the shell where the ball was found after n movements.

`Output`
Print one integer from 0 to 2 — the index of the shell where the ball was initially placed.

`题意`:

code:

## Hanoi Factory

Of course you have heard the famous task about Hanoi Towers, but did you know that there is a special factory producing the rings for this wonderful game? Once upon a time, the ruler of the ancient Egypt ordered the workers of Hanoi Factory to create as high tower as possible. They were not ready to serve such a strange order so they had to create this new tower using already produced rings.

There are n rings in factory’s stock. The i-th ring has inner radius ai, outer radius bi and height hi. The goal is to select some subset of rings and arrange them such that the following conditions are satisfied:

Outer radiuses form a non-increasing sequence, i.e. one can put the j-th ring on the i-th ring only if bj ≤ bi.
Rings should not fall one into the the other. That means one can place ring j on the ring i only if bj > ai.
The total height of all rings used should be maximum possible.
`Input`
The first line of the input contains a single integer n (1 ≤ n ≤ 100 000) — the number of rings in factory’s stock.

The i-th of the next n lines contains three integers ai, bi and hi (1 ≤ ai, bi, hi ≤ 109, bi > ai) — inner radius, outer radius and the height of the i-th ring respectively.

`Output`
Print one integer — the maximum height of the tower that can be obtained.

`题意`:

code:

## Cloud of Hashtags

Vasya is an administrator of a public page of organization “Mouse and keyboard” and his everyday duty is to publish news from the world of competitive programming. For each news he also creates a list of hashtags to make searching for a particular topic more comfortable. For the purpose of this problem we define hashtag as a string consisting of lowercase English letters and exactly one symbol ‘#’ located at the beginning of the string. The length of the hashtag is defined as the number of symbols in it without the symbol ‘#’.

The head administrator of the page told Vasya that hashtags should go in lexicographical order (take a look at the notes section for the definition).

Vasya is lazy so he doesn’t want to actually change the order of hashtags in already published news. Instead, he decided to delete some suffixes (consecutive characters at the end of the string) of some of the hashtags. He is allowed to delete any number of characters, even the whole string except for the symbol ‘#’. Vasya wants to pick such a way to delete suffixes that the total number of deleted symbols is minimum possible. If there are several optimal solutions, he is fine with any of them.

`Input`
The first line of the input contains a single integer n (1 ≤ n ≤ 500 000) — the number of hashtags being edited now.

Each of the next n lines contains exactly one hashtag of positive length.

It is guaranteed that the total length of all hashtags (i.e. the total length of the string except for characters ‘#’) won’t exceed 500 000.

`Output`
Print the resulting hashtags in any of the optimal solutions.

`题意`:

code:

## Game of Credit Cards

After the fourth season Sherlock and Moriary have realized the whole foolishness of the battle between them and decided to continue their competitions in peaceful game of Credit Cards.

Rules of this game are simple: each player bring his favourite n-digit credit card. Then both players name the digits written on their cards one by one. If two digits are not equal, then the player, whose digit is smaller gets a flick (knock in the forehead usually made with a forefinger) from the other player. For example, if n = 3, Sherlock’s card is 123 and Moriarty’s card has number 321, first Sherlock names 1 and Moriarty names 3 so Sherlock gets a flick. Then they both digit 2 so no one gets a flick. Finally, Sherlock names 3, while Moriarty names 1 and gets a flick.

Of course, Sherlock will play honestly naming digits one by one in the order they are given, while Moriary, as a true villain, plans to cheat. He is going to name his digits in some other order (however, he is not going to change the overall number of occurences of each digit). For example, in case above Moriarty could name 1, 2, 3 and get no flicks at all, or he can name 2, 3 and 1 to give Sherlock two flicks.

Your goal is to find out the minimum possible number of flicks Moriarty will get (no one likes flicks) and the maximum possible number of flicks Sherlock can get from Moriarty. Note, that these two goals are different and the optimal result may be obtained by using different strategies.

`Input`
The first line of the input contains a single integer n (1 ≤ n ≤ 1000) — the number of digits in the cards Sherlock and Moriarty are going to use.

The second line contains n digits — Sherlock’s credit card number.

The third line contains n digits — Moriarty’s credit card number.

`Output`
First print the minimum possible number of flicks Moriarty will get. Then print the maximum possible number of flicks that Sherlock can get from Moriarty.

`题意`:
S 某,M某分别有一张序号长度为n的信用卡，他们定了一个规则：比赛分为n局，每局S某，M某从n个数字中分别不重复的取出一个数字，谁的数字小谁得一分，平局不算分。

code:

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