Description
1.1 Description
In a queue, people are supposed to stand in ascending order of their heights. However, due to some confusion, the order of the queue gets disrupted. Your task is to determine the number of disorder pairs in the queue. A disorder pair is defined as a pair of people (pi , pj ) such that i < j and pi is taller than pj in the queue, which means their order is reversed.
For example, consider a queue with 5 people: [180, 160, 175, 170, 170]. In this case, there are 6 disorder pairs: (180, 160), (180, 175), (180, 170), (180, 170), (175, 170) and (175, 170). Please note that (170, 170) is not considered as a disorder pair. Write a program to calculate the number of disorder pairs in a given queue.
1.2 Input
The first line of input contains an integer N (1 ≤ N ≤ 106 ), representing the number of people in the queue. The second line contains N space-separated integers p1, p2, . . . , pN (1 ≤ pi ≤ 109 ), representing the heights of people in the queue. 1.3 Output Output a single integer, the number of disorder pairs in the queue.
Sample Input 1 6 1 2 3 4 5 6 Sample Output 1 0 Sample Input 2 5 180 160 175 170 170 Sample Output 2 6 4 Problem Scale & Subtasks Test Case No. Constraints 1-4 N ≤ 1000 5-8 N ≤ 10000 9-10 N ≤ 106 Hint For C/C++ and Java users, an int type stores integers range from -2,147,483,648 to 2,147,483,647. It may be too small for this problem.
You need other data types, such as long long for C/C++ and long for Java. They store integers ranging from -9,223,372,036,854,775,808 to 9,223,372,036,854,775,807. Use scanf(“%lld”,&n) for C, cin>>n for C++ and n = scanner.nextLong() for Java to get the input n. And the other operations for long and long long are quite same as int. 2 Star Sequence (50% of this assignment)
2.1 Description
Renko Usami observes space through a telescope when she notices a fantastic phenomenon – the number of stars in the fields follows a mathematical pattern. Specifically, let’s denote the number of stars in the Nth field by fN , then fN satisfies the following expression, and a, b are given positive integers. fN = afN−1 + bfN−2 (N ≥ 2)
Now Renko Usami is curious about how many stars in the nth field, but the nth field is too far away to be observed through her cheap astronomical telescope. Since there are so many stars, she only cares about the value of the number of stars modulo m. In other words, she want to know fn mod m.
Fortunately, Renko Usami is able to observe the two closest star fields to her, and the numbers of stars in fields are f0 and f1. Unfortunately, she is going to be late again for her appointment with Merry. Can you write a program for her to calculate fn mod m?
2.2 Input
The only line of the input contains 6 integers n(1 ≤ n ≤ 1018), a, b, f0, f1(1 ≤ a, b, f0, f1 ≤ 100), m(1 ≤ m ≤ 2 × 109 ). – the meanings of these numbers are shown in the problem description.
2.3 Output
Output a single integer – fn mod m. Sample Input 1 4 1 1 1 1 1000 Sample Output 1 5 Sample Input 2 468908 34 29 33 30 998244353 Sample Output 2 829261643 5 Problem Scale & Subtasks For 100% of the test cases, 1 ≤ n ≤ 1018, 1 ≤ a, b, f0, f1 ≤ 100, 1 ≤ m ≤ 2 × 109 . Test Case No. Constraints 1-2 n ≤ 10 3-5 n ≤ 106 6-10 No additional constraints Hint Hint1 : For C/C++ and Java users, an int type stores integers range from -2,147,483,648 to 2,147,483,647. It may be too small for this problem.
You need other data types, such as long long for C/C++ and long for Java. They store integers ranging from -9,223,372,036,854,775,808 to 9,223,372,036,854,775,807. Use scanf(“%lld”,&n) for C, cin>>n for C++ and n = scanner.nextLong() for Java to get the input n. And the other operations for long and long long are quite same as int.
Hint2 : Here’s a simple definition of the modulo operation. Your output should be the remainder of the Euclidean division of fn by m, where fn is the dividend and m is the divisor. And for the modulo operation the following equation holds: (x + y) mod m = ((x mod m) + (y mod m)) mod m Hint3 : When a and b are 1, fn is Fibonacci Sequence.
Here is a way to compute the Fibonacci Sequence by using matrices: 1 1 1 0 f1 f0 = f2 f1 The Matrix 1 1 1 0 is called transition matrix. Also, note that matrix multiplication is associative. By multiplying the transition matrix n − 1 times, we can get the fn. 6
