Problem p01611 - Generation 1

Orig Description

K-th String
Problem Statement
The suffix array SA for a string S of length N is defined as follows by the following procedure that produces a permutation of positive integers not exceeding N. Here, a substring of S from the ith to jth characters is referred to as S[i..j]. The indices are 1-indexed.
Sort each suffix S[i..N] (i=1,2,...,N) in lexicographic order (ascending order).
Arrange each sorted suffix's starting position i in order.
For example, the suffix array SA for S="mississippi" is as follows:
Given a suffix array SA for a string of length N, find the K-th (1-indexed) lexicographically smallest string that can be obtained from SA. The original string consists of at most A characters, which are numbered in alphabetical order from 'a'.
Input
The input is given in the following format. All given integers are integers.
N A K
SA_1
SA_2
...
SA_N
Constraints
1≦N≦10^5
1≦A≦26
1≦K≦10^{18}
1≦SA_i≦N
i \neq j implies SA_i \neq SA_j.
Output
Output the K-th lexicographically smallest string that can be obtained from SA, as a single line. If the K-th string does not exist, output "Impossible".
Sample Input 1
3 4 2
2
1
3
Output for the Sample Input 1
bad
If A=4, then the four strings that can be obtained from SA={2,1,3} are "bac", "bad", "cad", and "cbd". Therefore, the second smallest string in lexicographic order, "bad", is the answer.
Sample Input 2
18 26 10275802967
10
14
9
13
7
8
2
6
11
18
12
1
4
3
16
5
17
15
Output for the Sample Input 2
ritsumeicampdaytwo

Extracted Specification

An integer N (1 ≤ N ≤ 10^5), representing some quantity or size.
An integer A (1 ≤ A ≤ 26), representing some quantity or size.
An integer K (1 ≤ K ≤ 10^18), representing some quantity.
A list of integers SA of size N, where each integer is between 1 and N and all integers are unique.

### Example Input:

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

### Function Signature:
Write a function f(N, A, K, SA) that takes in the input.
def f(N: int, A: int, K: int, SA: List[int]):
    '''
    N: an integer
    A: an integer
    K: an integer
    SA: a list of integers
    '''

Test Case Generator

Sure! To create a test case generator for the "K-th String" problem, we need to generate values for \(N\), \(A\), and \(K\), as well as the suffix array \(SA\). Here is a Python function to generate such test cases:

```python
import random

def generate_suffix_array(N):
    suffix_array = list(range(1, N + 1))
    random.shuffle(suffix_array)
    return suffix_array

def tcgen():
    N = random.randint(1, 10**5)
    A = random.randint(1, 26)
    # For large N, to avoid extremely large K, we limit K to a reasonable factor of N.
    K_max = 10**18 // (26**N)
    K = random.randint(1, K_max if K_max > 1 else 1)
    SA = generate_suffix_array(N)
    
    return N, A, K, SA

# Example usage:
N, A, K, SA = tcgen()
print(N)
print(A)
print(K)
for sa in SA:
    print(sa)

```

### Explanation:
1. **N Generation**: We generate \(N\) as a random integer between 1 and \(10^5\).
2. **A Generation**: We generate \(A\) as a random integer between 1 and 26.
3. **K Generation**: The value of \(K\) can be extremely large (up to \(10^{18}\)). To avoid generating an infeasible test case, \(K\) is limited based on \(N\) and \(A\). This will ensure that \(K\) remains within computable bounds.
4. **Suffix Array Generation**: The suffix array \(SA\) is generated as a shuffled list of integers from 1 to \(N\).

This function generates random but valid test cases for the problem statement. You can call `tcgen()` to get a new test case each time.

Please note that the constraints are preserved, and the suffices are unique and range from 1 to \(N\). The value of \(K\) is kept within a feasible range to ensure the problem remains solvable.

Extract Arguments

def extract_arguments(fh):
    N, A, K = map(int, fh.readline().strip().split())
    SA = [int(fh.readline().strip()) for _ in range(N)]
    return N, A, K, SA