分类： 赛事资讯

Farmer John is expanding his farm! He has identified the perfect location in the Red-Black Forest, which consists of N trees (1≤N≤10⁵  ) on a number line, with the i-th tree at position x_i (−109≤x_i ≤10⁹).

Environmental protection laws restrict which trees Farmer John can cut down to make space for his farm. There are K restrictions (1≤K≤10⁵ ) specifying that there must always be at least t_i trees (1≤t_i ≤N) in the line segment [l_i,r_i]
, including the endpoints (−109≤l_i≤r_i≤10⁹). It is guaranteed that the Red-Black Forest initially satisfies these restrictions.

Farmer John wants to make his farm as big as possible. Please help him compute the maximum number of trees he can cut down while still satisfying all the restrictions!

INPUT FORMAT (input arrives from the terminal / stdin):

Each input consists of T (1≤T≤10) independent test cases. It is guaranteed that the sums of all N and of all K within an input each do not exceed 3⋅10⁵.

The first line of input contains T. Each test case is then formatted as follows:

The first line contains integers N and K.
The next line contains the N integers x₁,…,x_N.
Each of the next K lines contains three space-separated integers: l_i, r_i and t_i.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, output a single line with an integer denoting the maximum number of trees Farmer John can cut down.

SAMPLE INPUT:

3
7 1
8 4 10 1 2 6 7
2 9 3
7 2
8 4 10 1 2 6 7
2 9 3
1 10 1
7 2
8 4 10 1 2 6 7
2 9 3
1 10 4

SAMPLE OUTPUT:

4
4
3

For the first test case, Farmer John can cut down the first 4 trees, leaving trees at

x_i = 2,6,7 to satisfy the restriction.

For the second test case, the additional restriction does not affect which trees Farmer John can cut down, so he can cut down the same trees and satisfy both restrictions.

For the third test case, Farmer John can only cut down at most 3 trees because there are initially 7 trees but the second restriction requires him to leave at least 4 trees uncut.

SCORING:

Input 2: N , K≤16
Inputs 3-5: N, K≤1000
Inputs 6-7:t_i=1 for all i=1,…,K.
Inputs 8-11: No additional constraints.

Problem credits: Tina Wang, Jiahe Lu, Benjamin Qi

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Bessie and Elsie have discovered a row of N cakes (2≤N≤5⋅10⁵ ,N even)
cakes, with sizes a₁,a₂,…,a_N in that order (1≤a_i≤10⁹).

Each cow wants to eat as much as possible. However, being very civilized cows, they decided to play a game to split it! The game proceeds with both cows alternating turns. Each turn consists of one of the following:

1.Bessie chooses two adjacent cakes and stacks them, creating a new cake with the sum of the sizes.
2.Elsie chooses either the leftmost or rightmost cake and puts it in her stash.

When only one cake remains, Bessie eats it, and Elsie eats all cakes in her stash. If both cows play optimally to maximize the amount of cake they eat and Bessie plays first, how much cake will each cow eat?

INPUT FORMAT (input arrives from the terminal / stdin):

Each input consists of T (1≤T≤10) independent test cases. It is guaranteed that the sum of all N within an input does not exceed 10⁶.

Each test case is formatted as follows. The first line contains N . The next line contains N space-separated integers, a₁,a₂,…,a_N.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, output a line containing b and e, representing the amounts of cake Bessie and Elsie respectively will consume if both cows play optimally.

SAMPLE INPUT:

2
4
40 30 20 10
4
10 20 30 40

SAMPLE OUTPUT:

60 40
60 40

For the first test case, under optimal play,

1.Bessie will stack the middle two cakes. The cakes now have sizes [40,50,10].
2.Elsie will eat the leftmost cake. The remaining cakes now have sizes [50,10].
3.Bessie stacks the remaining two cakes.

Bessie will eat 30+20+10=60 cake, while Elsie will eat 40 cake.

The second test case is the reverse of the first, so the answer is the same.

SCORING:

Input 2: All a_i  are equal.
Input 3: N≤10
Inputs 4-7: N≤5000
Inputs 8-11: No additional constraints.

Problem credits: Linda Zhao, Agastya Goel, Gavin Ye

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Bessie the cow has N (1≤N≤2⋅10⁵) jobs for you to potentially complete. The i-th one, if you choose to complete it, must be started at or before time s_i and takes t_i time to complete (0≤s_i ≤10¹⁸,1≤t_i ≤10¹⁸).

What is the maximum number of jobs you can complete? Time starts at 0
, and once you start a job you must work on it until it is complete, without starting any other jobs in the meantime.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains T, the number of independent test cases (1≤T≤10). Each test case is formatted as follows.

The first line contains N.

Each of the next N lines contains two integers s_i and t_i . Row i+1 has the details for the ith job.

It is guaranteed that the sum of N over all test cases does not exceed 3⋅10⁵.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, the maximum number of jobs you can complete, on a new line.

SAMPLE INPUT:

3
2
1 4
1 2
2
2 3
1 2
3
1 4
2 3
1 2

SAMPLE OUTPUT:

1
2
2

For the first test case, you can only complete one of the jobs. After completing one job, it will then be time 2 or later, so it is too late to start the other job, which must be started at time 1 or earlier.

For the second test case, you can start the second job at time 0 and finish at time 2, then start the first job at time 2 and finish at time 5.

SCORING:

Inputs 2: Within the same test case, all t_i  are equal.
Inputs 3-4: N≤2000, s_i ,t_i ≤2000
Inputs 5-8: N≤2000
Inputs 9-16: No additional constraints.

Problem credits: Benjamin Qi

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It's the year 3000 , and Bessie became the first cow in space! During her journey between the stars, she found a number line with N (2≤N≤5⋅10⁵) points, numbered from 1 to N. All points are initially colored white. She can perform the following operation any number of times.

Choose a position i within the number line and a positive integer x. Then, color all the points in the interval [i,i+x−1] red and all points in [i+x , i+2x−1]
blue. All chosen intervals must be disjoint (i.e. no points in [ i, i+2x−1]
can be already colored red or blue). The entire interval must also fall within the number line (i.e. 1≤i≤i+2x−1≤N).

Farmer John gives Bessie a string s of length N consisting of characters R, B
, and X. The string represents Farmer John's color preferences for each point: s_i =R means the i'th point must be colored red, s_i =B means the i'th point must be colored blue, and s_i =X means there is no constraint on the color for the i'th point.

Help Bessie count the number of distinct ways for the number line to be colored while satisfying Farmer John's preferences. Two colorings are different if there is at least one corresponding point with a different color. Because the answer may be large, output it modulo 10⁹+7.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains an integer N.

The following line contains string s.

OUTPUT FORMAT (print output to the terminal / stdout):

Output the number of distinct ways for the number line to be colored while satisfying Farmer John's preferences modulo 10⁹+7.

SAMPLE INPUT:

6
RXXXXB

SAMPLE OUTPUT:

5

Bessie can choose i=1,x=1(i.e. color point 1 red and point 2 blue) and i=3,x=2
(i.e. color points 3,4 red and points 5,6 blue) to produce the coloring RBRRBB.

The other colorings are RRBBRB, RBWWRB, RRRBBB, and RBRBRB.

SAMPLE INPUT:

6
XXRBXX

SAMPLE OUTPUT:

6

The six colorings are WWRBWW, WWRBRB, WRRBBW, RBRBWW, RBRBRB, and RRRBBB.

SAMPLE INPUT:

12

XBXXXXRXRBXX

SAMPLE OUTPUT:

18

SCORING:

Input 4: N≤500
Inputs 5-6: N≤10⁴
Inputs 7-13: All but at most 100 characters in s are X.
Inputs 14-23: No additional constraints

Problem credits: Chongtian Ma, Alex Liang

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Farmer John's N (1≤N≤10⁵ ) cows have been arranged into a line. The i
th cow has label a_i (1≤a_i ≤N). A group of cows can form a friendship group if they all have the same label and each cow is within x cows of all the others in the group, where x is an integer in the range [1,N]. Every cow must be in exactly one friendship group.

For each x from 1 to N , calculate the minimum number of friendship groups that could have formed.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line consists of an integer N.

The next line contains a₁...a_N , the labels of each cow.

OUTPUT FORMAT (print output to the terminal / stdout):

For each x from 1 to N , output the minimum number of friendship groups for that x on a new line.

SAMPLE INPUT:

9
1 1 1 9 2 1 2 1 1

SAMPLE OUTPUT:

7
5
4
4
4
4
4
3
3

Here are examples of how to assign cows to friendship groups for x=1
and x=2 in a way that minimizes the number of groups. Each letter corresponds to a different group.

SCORING:

Inputs 2-3: N≤5000
Inputs 4-7: a_i≤10 for all i
Inputs 8-11: No label appears more than 10 times.
Inputs 12-20: No additional constraints.

Problem credits: Chongtian Ma

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Moo! You are given an integer N (2≤N≤2000). Consider all permutations p=[p₀,p₁,…,p_N−1]of [0,1,2…,N−1]. Let f(p)=|p_i−p_i+1|denote the minimum absolute difference between any two consecutive elements of p. and let S denote the set of all p that achieve the maximum possible value of f(p).

You are additionally given K (0≤K≤N) constraints of the form p_i=j (0≤i, j<N). Count the number of permutations in S satisfying all constraints, modulo 10⁹+7.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains T (1≤TN≤2⋅10⁴) and N, meaning that you will need to solve T independent test cases, each specified by a different set of constraints.

Each test case starts with K, followed by K lines each containing i and j. It is guaranteed that

The same i does not appear more than once within the same test case.
The same j does not appear more than once within the same test case.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, the answer modulo 10⁹+7 on a separate line.

SAMPLE INPUT:

3 4
0
1
1 1
2
0 2
2 3

SAMPLE OUTPUT:

2
0
1

The maximum possible value of f (p) is 2, and S={[2,0,3,1],[1,3,0,2]}.

SAMPLE INPUT:

9 11
2
0 5
6 9
3
0 5
6 9
1 0
4
0 5
6 9
1 0
4 7
5
0 5
6 9
1 0
4 7
2 6
6
0 5
6 9
1 0
4 7
2 6
9 3
7
0 5
6 9
1 0
4 7
2 6
9 3
5 2
8
0 5
6 9
1 0
4 7
2 6
9 3
5 2
7 4
9
0 5
6 9
1 0
4 7
2 6
9 3
5 2
7 4
3 1
10
0 5
6 9
1 0
4 7
2 6
9 3
5 2
7 4
3 1
8 10

SAMPLE OUTPUT:

6
6
1
1
1
1
1
1
1

p=[5,0,6,1,7,2,9,4,10,3,8] should be counted for all test cases.

SAMPLE INPUT:

10 11
0
1
3 8
2
3 8
5 7
3
3 8
5 7
4 2
4
3 8
5 7
4 2
10 6
5
3 8
5 7
4 2
10 6
8 10
6
3 8
5 7
4 2
10 6
8 10
1 9
7
3 8
5 7
4 2
10 6
8 10
1 9
7 5
8
3 8
5 7
4 2
10 6
8 10
1 9
7 5
2 3
9
3 8
5 7
4 2
10 6
8 10
1 9
7 5
2 3
6 0

SAMPLE OUTPUT:

160
20
8
7
2
1
1
1
1
1

p=[4,9,3,8,2,7,0,5,10,1,6]should be counted for all test cases.

SAMPLE INPUT:

5 987
3
654 321
543 210
432 106
2
654 321
543 210
1
654 321
1
0 493
0

SAMPLE OUTPUT:

0
538184948
693625420
932738155
251798971
Make sure to output the answer modulo 10⁹+7.

SCORING:

Input 5: N=15
Input 6: N=2000
Inputs 7-9: For all test cases, the constraint p₀=⌊N/2⌋ appears.
Inputs 10-13: For all test cases, there exists some constraint p_i =j with j equal to ⌊N/2⌋.
Inputs 14-20: No additional constraints.

Problem credits: Benjamin Qi

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Bessie has a string of length N(1≤N≤3⋅10⁵) consisting solely of the characters M and O. For each position i of the string, there is a cost c_i to change the character at that position to the other character (1≤c_i≤10⁸).

Bessie thinks the string will look better if it contains more moos of length L
(1≤L≤min(N,3)). A moo of length L is an M followed by L−1 Os.

For each positive integer k from 1 to ⌊N/L⌋ inclusive, compute the minimum cost to change the string to contain at least k substrings equal to a moo of length L.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains L and N.

The next line contains Bessie's length-N string, consisting solely of Ms and Os.

The next line contains space-separated integers c₁…c_N.

OUTPUT FORMAT (print output to the terminal / stdout):

Output ⌊N/L⌋ lines, the answer for each k in increasing order.

SAMPLE INPUT:

1 4
MOOO
10 20 30 40

SAMPLE OUTPUT:

0
20
50
90

SAMPLE INPUT:

3 4
OOOO
50 40 30 20

SAMPLE OUTPUT:

40

SAMPLE INPUT:

2 20
OOOMOMOOOMOOOMMMOMOO
44743602 39649528 94028117 50811780 97338107 30426846 94909807 22669835 78498782 18004278 16633124 24711234 90542888 88490759 12851185 74589289 54413775 21184626 97688928 10497142

SAMPLE OUTPUT:

0
0
0
0
0
12851185
35521020
60232254
99881782
952304708

SAMPLE INPUT:

3 20
OOOMOMOOOMOOOMMMOMOO
44743602 39649528 94028117 50811780 97338107 30426846 94909807 22669835 78498782 18004278 16633124 24711234 90542888 88490759 12851185 74589289 54413775 21184626 97688928 10497142

SAMPLE OUTPUT:

0
0
0
44743602
119332891
207066974

SCORING:

Input 5: L=3,N≤5000
Input 6: L=1
Inputs 7-10: L=2
Inputs 11-18: L=3

Problem credits: Benjamin Qi

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Farmer John's N (1≤N≤5⋅10⁵) cows are each assigned a bitstring of length K that is not all zero (1≤K≤20). Different cows may be assigned the same bitstring.

The Jaccard similarity of two bitstrings is defined as the number of set bits in their bitwise intersection divided by the number of set bits in their bitwise union. For example, the Jaccard similarity of the bitstrings 11001 and 11010 would be 2/4.

For each cow, output the sum of her bitstring's Jaccard similarity with each of the N cows' bitstrings including her own, modulo 10⁹+7. Specifically, if the sum is equal to a rational number a/b where a and b are integers sharing no common factors, output the unique integer x in the range [0,10⁹+7) such that bx−a is divisible by 10⁹+7.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains N and K.

The next N lines each contain an integer i∈(0,2^K), representing a cow associated with the length-K binary representation of i.

INPUT FORMAT (input arrives from the terminal / stdin):

Output the sum modulo 10⁹+7 for each cow on a separate line.

SAMPLE INPUT:

4 2
1
1
2
3

SAMPLE OUTPUT:

500000006
500000006
500000005
500000006

The cows are associated with the following bitstrings: [01,01,10,11].

For the first cow, the sum is  sim(1,1)+sim(1,1)+sim(1,2)+sim(1,3)=1+1+0+1/2≡500000006(mod 10⁹+7).

The second cow's bitstring is the same as the first cow's, so her sum is the same as above.

For the third cow, the sum is

sim(2,1)+sim(2,1)+sim(2,2)+sim(2,3)=0+0+1+1/2≡500000005(mod 10⁹+7).

SCORING:

Inputs 2-15: There will be two test cases for each of K∈{10,15,16,17,18,19,20}.

Problem credits: Benjamin Qi

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