Problem p01611 - Generation 3

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

Canonicalized Input Description:

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 or size.
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.
```python
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

Creating a test case generator for the "K-th String" problem involves generating random values for \( N \), \( A \), \( K \), and a permutation of suffix array indices. Below is a Python function to generate test cases for this problem:

```python
import random

def tcgen():
    # Generate N within the given constraints
    N = random.randint(1, 10**5)
    
    # Generate A within the given constraints
    A = random.randint(1, 26)
    
    # Generate K within the given constraints
    K = random.randint(1, 10**18)
    
    # Generate a permutation of numbers from 1 to N for the suffix array
    SA = random.sample(range(1, N + 1), N)
    
    # Print the generated test case
    print(N, A, K)
    for sa in SA:
        print(sa)

# Example usage
tcgen()
```

### Explanation:
1. **Generating \( N \)**: 
   - \( N \) is an integer between 1 and \( 10^5 \). The function `random.randint(1, 10**5)` is used to generate this random value.

2. **Generating \( A \)**: 
   - \( A \) is an integer between 1 and 26. The function `random.randint(1, 26)` is used to generate this random value.

3. **Generating \( K \)**: 
   - \( K \) is a very large integer between 1 and \( 10^{18} \). The function `random.randint(1, 10**18)` is used to generate this random value.

4. **Generating Suffix Array (SA)**:
   - The suffix array is a permutation of numbers from 1 to \( N \). The function `random.sample(range(1, N + 1), N)` generates a random permutation of these numbers.

5. **Printing the Test Case**:
   - The generated values are printed in the required format.

You can use this generator function to create multiple test cases for testing purposes. Adjust the function if you need to generate test cases with specific properties or constraints.

Extract Arguments

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