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Codechef Array Partition ARRPART Solution
You are given an array AA of NN integers, and an integer XX. For each KK in the range [1,N][1,N], determine the number of ways to partition the array into exactly KK non-empty subarrays such that the maxima of each of the subarrays are at least XX.
The number of ways can be large, so output it modulo 998244353998244353.
Note: The input and output are large, so use fast input-output methods.
Input Format
- The first line of input contains a single integer TT, denoting the number of test cases. The description of TT test cases follows.
- Each test case contains two lines of input.
- The first line of each test case consists of two space-separated integers NN and XX.
- The second line consists of NN space-separated integers denoting the elements of the array AA.
Output Format
For each test case, output a single line containing NN space-separated integers, the ii-th of which denotes the number of ways to partition the array into exactly ii non-empty subarrays such that the maxima of each of the ii subarrays are at least XX. Print the answer modulo 998244353998244353.
Constraints
- 1≤T≤7⋅1041≤T≤7⋅104
- 1≤N≤1061≤N≤106
- 1≤X≤1091≤X≤109
- 1≤Ai≤1091≤Ai≤109
- The sum of NN across all test cases does not exceed 106106
Subtasks
Subtask #1 (20 points):
- 1≤T≤2001≤T≤200
- 1≤N≤1031≤N≤103
- The sum of NN across all test cases does not exceed 2⋅1032⋅103
- Time Limit = 11 second
Subtask #2 (80 points):
- Original constraints
- Time Limit = 1.751.75 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][L,R] denotes the subarray consisting of elements AL,AL+1,…,ARAL,AL+1,…,AR.
Test case 11:
- The number of partitions having exactly 11 subarray is 11, which is the array itself.
- The number of partitions having exactly 22 subarrays is 44, which are [1,1][1,1]+[2,5][2,5], [1,2][1,2]+[3,5][3,5], [1,3][1,3]+[4,5][4,5], [1,4][1,4]+[5,5][5,5].
- The number of partitions having exactly 33 subarrays is 44, which are [1,1][1,1]+[2,3][2,3]+[4,5][4,5], [1,1][1,1]+[2,4][2,4]+[5,5][5,5], [1,2][1,2]+[3,3][3,3]+[4,5][4,5], [1,2][1,2]+[3,4][3,4]+[5,5][5,5].
- The number of partitions having 44 or 55 subarrays is 00.
Test case 22:
- The number of partitions having exactly 11 subarray is 11, which is the array itself.
- The number of partitions having exactly 22 subarrays is 22, which are [1,1][1,1]+[2,4][2,4], [1,2][1,2]+[3,4][3,4].
- The number of partitions having exactly 33 subarrays is 11, which is [1,1][1,1]+[2,2][2,2]+[3,4][3,4].
- The number of partitions having exactly 44 subarrays is 00.
Test Case 33:
- The number of partitions having exactly 11 subarray is 11, which is the array itself.
- The number of partitions having exactly 22 subarrays is 22, which are [1,1][1,1]+[2,3][2,3], [1,2][1,2]+[3,3][3,3].
- The number of partitions having exactly 33 subarrays is 11, which is [1,1][1,1]+[2,2][2,2]+[3,3][3,3].
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