Problem p01611 - Generation 2

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 limit or range.
An integer K (1 ≤ K ≤ 10^18), representing some position or index.
A list of integers SA of size N, where each integer SA[i] (1 ≤ SA[i] ≤ N) represents some index or position. All integers in SA 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

Certainly! Below is a test case generator for the given "K-th String" problem statement. The generator will create random inputs for \(N\), \(A\), \(K\), and the suffix array \(SA\) while adhering to the constraints provided.

```python
import random

def tcgen():
    # Generate N
    N = random.randint(1, 10**5)
    
    # Generate A
    A = random.randint(1, 26)
    
    # Generate K
    K = random.randint(1, 10**18)
    
    # Generate SA as a permutation of numbers from 1 to N
    SA = list(range(1, N + 1))
    random.shuffle(SA)
    
    return N, A, K, SA

def format_test_case(N, A, K, SA):
    test_case = f"{N} {A} {K}\n"
    test_case += "\n".join(map(str, SA))
    return test_case

# Example usage
N, A, K, SA = tcgen()
test_case = format_test_case(N, A, K, SA)
print(test_case)
```

### Explanation:
1. **N**: The length of the string is randomly chosen between 1 and \(10^5\).
2. **A**: The number of distinct characters in the string is randomly chosen between 1 and 26.
3. **K**: The position of the lexicographically smallest string is randomly chosen between 1 and \(10^{18}\).
4. **SA**: The suffix array is generated as a permutation of the numbers from 1 to \(N\).

The function `tcgen` generates the raw values, and `format_test_case` formats them into the required input format for the problem statement. This ensures that the test cases are both valid and diverse, covering a wide range of possible inputs.

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