Array Partition ARRPART Solution Codechef

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Array Partition ARRPART Solution

You are given an array A of N integers, and an integer X. For each K in the range [1,N], determine the number of ways to partition the array into exactly K non-empty subarrays such that the maxima of each of the subarrays are at least X.

The number of ways can be large, so output it modulo 998244353.

Note: The input and output are large, so use fast input-output methods.

Input Format

  • The first line of input contains a single integer T, denoting the number of test cases. The description of T test cases follows.
  • Each test case contains two lines of input.
  • The first line of each test case consists of two space-separated integers N and X.
  • The second line consists of N space-separated integers denoting the elements of the array A.

Output Format

For each test case, output a single line containing N space-separated integers, the i-th of which denotes the number of ways to partition the array into exactly i non-empty subarrays such that the maxima of each of the i subarrays are at least X. Print the answer modulo 998244353.

Constraints

  • 1≤T≤7⋅104
  • 1≤N≤106
  • 1≤X≤109
  • 1≤Ai≤109
  • The sum of N across all test cases does not exceed 106

Subtasks

Subtask #1 (20 points):

  • 1≤T≤200
  • 1≤N≤103
  • The sum of N across all test cases does not exceed 2⋅103
  • Time Limit = 1 second

Subtask #2 (80 points):

  • Original constraints
  • Time Limit = 1.75 seconds

Sample Input 1

3
5 3
4 1 7 1 6
4 2
2 2 2 1
3 4
5 6 7

Sample Output 1

1 4 4 0 0 
1 2 1 0 
1 2 1

Explanation

In the below explanation, [L,R] denotes the subarray consisting of elements AL,AL+1,…,AR.

Test case 1:

  • The number of partitions having exactly 1 subarray is 1, which is the array itself.
  • The number of partitions having exactly 2 subarrays is 4, which are [1,1]+[2,5], [1,2]+[3,5], [1,3]+[4,5], [1,4]+[5,5].
  • The number of partitions having exactly 3 subarrays is 4, which are [1,1]+[2,3]+[4,5], [1,1]+[2,4]+[5,5], [1,2]+[3,3]+[4,5], [1,2]+[3,4]+[5,5].
  • The number of partitions having 4 or 5 subarrays is 0.

Test case 2:

  • The number of partitions having exactly 1 subarray is 1, which is the array itself.
  • The number of partitions having exactly 2 subarrays is 2, which are [1,1]+[2,4], [1,2]+[3,4].
  • The number of partitions having exactly 3 subarrays is 1, which is [1,1]+[2,2]+[3,4].
  • The number of partitions having exactly 4 subarrays is 0.

Test Case 3:

  • The number of partitions having exactly 1 subarray is 1, which is the array itself.
  • The number of partitions having exactly 2 subarrays is 2, which are [1,1]+[2,3], [1,2]+[3,3].
  • The number of partitions having exactly 3 subarrays is 1, which is [1,1]+[2,2]+[3,3].

January Long Challenge 2022 Solution

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