Problem Description

The task here is to find certain special numbers called "positive palindromes." A positive palindrome is a number that reads the same both forward and backward, and it does not have any leading zeros. Given two inputs: an array of integers named queries and a positive integer intLength, our goal is to determine the intLength-digit palindrome corresponding to each query.

For each integer in queries, we interpret it as the queries[i]-th smallest positive palindrome of length intLength. If this palindrome exists, we add it to our answer array; otherwise, we append -1 to indicate there's no such palindrome for that query.

If we think about the nature of palindromes, we can realize that for a palindrome of a given length, the first half of the number dictates the second half due to the mirror-like property of palindromes. Therefore, finding a palindrome can be done by constructing its first half and then mirroring it to create the second half.

The problem requires us to handle the fact that palindromes of even and odd lengths behave slightly differently: an odd-length palindrome will have a single digit in the middle that isn't mirrored.

Intuition

The solution utilizes the pattern that palindromes of a certain length can be constructed by taking a base number and mirroring its digits. The base number is essentially the first half of the palindrome.

• For intLength that is odd, the middle digit is part of the base.
• For intLength that is even, the base directly mirrors itself to form the whole palindrome.

To construct the smallest intLength-digit palindrome, we need to start with the smallest base possible that, once mirrored, forms a palindrome of that length. This smallest base number is 10^(l-1) where l is the length of the base. The base itself is half of intLength: l = (intLength + 1) // 2.

The scope of possible bases ranges from this smallest base to 10^l - 1. This is because 10^l would result in a palindrome exceeding intLength digits, violating the problem constraints.

With this understanding, the solution consists of the following steps:

1. Calculate the length l of the base number for the palindrome. If intLength is even, l is half of intLength. If intLength is odd, l is half of intLength, rounded up.

2. Determine the starting point (start) and the endpoint (end) for possible bases—the starting point being 10^(l-1) and the endpoint being 10^l - 1.

3. Iterate over each query in queries:

   • Calculate a tentative palindrome base v by adding the 0-based index of the query (q - 1) to the starting point.
   • If this base is greater than the range's endpoint (end), append -1 to the answers array (ans), signifying no such palindrome exists.
   • Otherwise, create a string (s) for the first half of the palindrome (which is v), mirror it, and append it to itself. For odd intLength, ignore the last digit of the mirrored portion to prevent duplication of the middle digit.
   • Convert the resulting string back to an integer and append it to ans.

4. Return the completed ans list, which now contains the queries[i]-th smallest palindrome or -1 for each query tested.

Learn more about Math patterns.

Solution Approach

The solution approach leverages simple integer and string operations to generate palindromes of a specific length. Here's a more detailed breakdown of the implementation:

1. Determine Base Length (l):

   • The first step in the algorithm involves calculating the length of the base number l, from which the entire palindrome can be constructed. This length is found by dividing intLength by 2 and rounding up if necessary. It uses the right shift operator >> 1 which is equivalent to dividing by 2.

   l = (intLength + 1) >> 1

2. Define Start and End Range for Bases:

   • To form palindromes of intLength, the starting point is the smallest possible positive integer of half that length, which is 10**(l - 1). The endpoint is the largest integer of that half-length, 10**l - 1. These define the range of numbers that can be the first half of a palindrome.

   start, end = 10 ** (l - 1), 10**l - 1

3. Iterate Over Queries and Construct Palindromes:

   • The main logic happens in a loop iterating over each query. For each query, a base value v for the palindrome is created by adding the query's 0-based index to the start of the range. Then the solution checks whether v exceeds the end, in which case it adds -1 to the answer array, indicating that no such palindrome exists within the given length constraint.

   v = start + q - 1
   if v > end:
       ans.append(-1)

4. Constructing the Full Palindrome String:

   • When a valid base v is found, it is converted to a string s. To construct the whole palindrome, s is concatenated with its reverse (s[::-1]). For palindromes of an odd length, the middle character should not be duplicated, so the slice [intLength % 2:] trims the first character from the reversed string before concatenation if intLength is odd.

   s += s[::-1][intLength % 2 :]

5. Finalize and Return the Answer:

   • After constructing the full palindrome string for each query, it is converted back to an integer and appended to the answer array ans. Once all queries are processed, ans is returned as the result containing the requested palindromes or -1 when no such palindrome exists.

   ans.append(int(s))

The algorithm effectively uses string manipulation to leverage the inherent symmetry of palindromes, constructing them in an optimal manner by only dealing with half the number and mirroring it to get the full length. This means the time complexity is primarily determined by the number of queries and the computational complexity of string manipulation, both of which are managed efficiently within the provided problem constraints.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach with queries = [1, 2, 4] and intLength = 3.

1. Determine the Base Length (l):

   • Since intLength is 3, which is odd, we calculate l as (3 + 1) >> 1, yielding l = 2.

2. Define Start and End Range for Bases:

   • The starting point start is 10**(2 - 1), which equals 10.
   • The endpoint end is 10**2 - 1, which equals 99.

3. Iterate Over Queries and Construct Palindromes:

   • The algorithm will loop over the queries [1, 2, 4] to find the corresponding palindromes.

4. Constructing Palindromes:

   • For the first query (1):

     • We calculate v as start + 1 - 1, which is 10.
     • v is within the range, so we proceed to construct the palindrome.
     • The string representation of v is '10', and because intLength is 3 (odd), we don't repeat the middle digit when reversing.
     • The palindrome becomes '101', and we append the integer 101 to our answer ans.

   • For the second query (2):

     • We calculate v as start + 2 - 1, which is 11.
     • This is also within range, so the palindrome would be '111', and we add 111 to ans.

   • For the fourth query (4):

     • We calculate v as start + 4 - 1, which is 13.
     • This base is not greater than end, so we construct the palindrome '131' and add 131 to ans.

5. Finalize and Return the Answer:

   • Since there is no third query, we do not perform any action for it as it was not given in queries.
   • We also note that no queries in this example exceed our range, so there is no need to add -1 at any point.
   • Our final answer ans is [101, 111, 131].

The algorithm has successfully determined the queries[i]-th smallest positive palindrome of length intLength for each provided query. The palindromes are [101, 111, 131] for the 1st, 2nd, and 4th smallest palindromes of length 3, respectively.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be broken down into a couple of key operations that occur within the loop running once for each element in queries.

1. Calculation of the half-length: This is done only once before the loop, taking a constant time, so it does not affect the total time complexity.

2. The loop: The main loop runs once for each query in queries, hence, if there are n queries, the loop runs n times.

3. String operations within the loop:

   • Creating the s string: This involves creating a string representation of an integer which is O(L) where L is the number of digits in the integer.
   • String slicing and concatenation: The slicing s[::-1] takes O(L) time and the concatenation s[::-1][intLength % 2 :] also takes O(L) time, as the length of the slice is proportional to L.
   • Creating the ans list append operation: Appending to the list is O(1).

Since L (length of the palindrome) is at most half of intLength and intLength itself is a constant with respect to the length of queries, the operations inside the loop that depend on intLength can be considered to occur in constant time as well. Thus, the total time complexity for the loop can be approximated as O(n) where n is the number of queries.

Space Complexity

For space complexity, we are mostly concerned with additional space that the program uses aside from the input and output.

1. The ans list: This list increases proportionally with the number of queries n, so the space complexity contribution here is O(n).

2. Temporary variables v and s: These are used for each query and do not depend on the number of queries. They do not increase the space complexity beyond O(n).

Therefore, the overall space complexity is O(n) where n is the number of queries.

Learn more about how to find time and space complexity quickly using problem constraints.