Problem Description

You are given a string num that consists only of digits. The task is to construct the largest palindromic integer (representing it as a string) by reordering digits taken from num. A palindrome is a sequence that reads the same backward as forward (like "121" or "1331"). The resulting palindrome should not have leading zeroes, meaning it should not start with the digit '0' unless the palindrome is simply "0". It is also important to note that you can choose to use some or all the digits from num, but you must use at least one digit to construct the palindrome.

Intuition

The intuition behind the solution is to strategically utilize the digits in num to create the largest possible palindrome. To achieve this, we consider several factors:

1. The largest digit should be placed in the middle of the palindrome for odd-length palindromes.
2. Even-length palindromes are formed by placing mirrored digits around the center.
3. Leading zeroes can be avoided by ensuring that we don't start the construction of the palindrome with zeroes, except if the largest palindrome is "0" itself.

With this in mind, the following steps are taken in the solution:

1. Count the frequency of each digit in the string.
2. Starting from the largest digit (9) and moving to the smallest (0), look for a digit that has an odd count.
   • This digit, if any, can be used as the central character in an odd-length palindrome.
   • Decrease its count by one and store it.
3. Then, for each digit from 0 to 9, append half of the remaining count of that digit to both sides of the palindrome.
   • This creates the mirrored effect around the center.
4. Finally, remove any leading zeroes (if they are not the only character) from the resulting string to maintain the 'no leading zero' constraint.
5. If the resulting string is empty (which can happen if we start with lots of zeroes), return "0".

By following these steps, we utilize the counts of digits in such a way to maximize the integer value of the resulting palindrome.

Learn more about Greedy patterns.

Solution Approach

The solution implemented above uses a greedy approach and some basic understanding of how palindromes are structured. Here's the step-by-step explanation of the solution:

1. Import the Counter class from Python's collections module to count the frequency of each digit in the input string num. Counter(num) provides a dictionary-like object where keys are unique digits from num, and the values are counts of those digits.

2. Initialize an empty string ans to accumulate the palindrome constructed so far.

3. Iterate through the digits in descending order, from '9' to '0', to find a digit that occurs an odd number of times.

   • When such a digit is found (cnt[v] % 2), use it as the central digit of the palindrome (ans = v) only if the palindrome is meant to be of odd length. This digit will not have a mirrored counterpart.
   • Decrease the count of that digit by 1 and break the loop as we are interested in only one such digit for the center of the palindrome.

4. Iterate through the digits again, this time starting from '0'. For each digit v:

   • Check if the digit has a non-zero count in cnt.
   • Divide the count by 2 (cnt[v] //= 2) because we place half on each side of the palindrome.
   • Create a string s by repeating the digit v, cnt[v] times.
   • Update the palindrome string ans by placing string s before and after the current ans.

5. Before returning the result, use ans.strip('0') to ensure that there are no leading zeroes, unless the palindrome is '0'. If ans is an empty string at this point (meaning it was made up of only zeroes), simply return '0'.

In summary, the algorithm employs a counter to keep track of frequency, a greedy approach for constructing the largest palindrome from the center outwards, and simple string manipulation to assemble the final palindromic string, making sure to adhere to the constraints laid out in the problem description.

Example Walkthrough

Let's illustrate the solution approach with a small example where the input string num is "310133".

Following the solution steps:

1. We count the frequency of each digit:

   '1': 1, '3': 2, '0': 1

2. We find that digit '1' occurs an odd number of times. It can be the central digit of the palindrome.

   After this step, our palindrome under construction looks like this:

   "_ _ 1 _ _"

3. Next, we iterate from digit ‘9’ to ‘0’ and add the digits in pairs around the central digit:

   • We skip '9', '8', '7', '6', '5', '4', and '2' because their count in num is zero.
   • For digit '3' (which has a count of 2), we will take one to place on each side of '1'.

   After updating, the palindrome is:

   "_ 3 1 3 _"

   • Since '1' is already used as the central digit, we move to '0'. There's one '0', so we cannot form a pair (it's left out).

     Our palindrome is currently:

   "3 1 3"

   • We've added all digits where possible. No further digits can be placed.

4. There’s no need to trim leading zeroes since the palindrome does not start or end with '0'.

5. If our constructed palindrome were empty (e.g., if num was just "0"s), we would return "0".

As a result, for the given num "310133", the largest palindromic integer we can construct is "313".

Solution Implementation

Time and Space Complexity

The given code snippet aims to find the largest palindromic number that can be formed by rearranging the digits of the given number num. The code uses a Counter to store the frequency of each digit and then constructs the palindrome.

• Time Complexity:

  The time complexity of the code is determined by a few operations:

  1. Creating a Counter from the string num takes O(n) where n is the length of num, as each character in the string needs to be read once.
  2. The first loop runs a constant 10 times (digits 9 to 0) which is O(1) since it doesn't depend on the length of num.
  3. The second loop also runs a constant 10 times, and inside this loop, it performs string concatenation. Assuming that the Python string concatenation in this case takes O(k) time (where k is the length of the string being concatenated), the maximum length of s will be n/2. Therefore, the overall work done here is proportional to n.

  Since these operations occur sequentially, the time complexity is the sum of their individual complexities, resulting in O(n).

• Space Complexity:

  The space complexity is determined by:

  1. The Counter cnt, which can potentially store a count for each different digit, thus having a space complexity of O(10) or O(1) since the number of possible different digits (0-9) is constant and does not grow with n.
  2. The string ans, which can grow up to a length of n in the worst case when each character is identical. Thus it has a space complexity of O(n).

  Therefore, the total space complexity is O(n), where n is the length of the input number num.

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