Python - Optimize code for finding the largest palindrome number that can be formed from the given digits

Question

I wrote code for task, but get error: time-limit-exceeded from testing system. I need to get advice how I can write this code faster and more precise

Code:

# input
n = int(input())
seq = input()
pairs = []
seq = list(seq)

# find pairs
counted = []
for i, item in enumerate(seq):
    for j, num in enumerate(seq):
        if (i != j) and (item == num):
            if (i not in counted) and (j not in counted):
                pairs.append((item, num))
                counted.append(i)
                counted.append(j)

# remove pairs from seq
for pair in pairs:
    seq.remove(pair[0])
    seq.remove(pair[1])

# create a palindrome

start = []
end = []

pairs = sorted(pairs)
pairs = list(reversed(pairs))

for item in pairs:
    start.append(item[0])
    end.append(item[1])

end = list(reversed(end))

if len(seq) != 0:
    seq = [int(item) for item in seq]
    max_el = list(sorted(seq))[-1]
    start.append(max_el)

final_s = start + end
# output
output = ''.join([str(item) for item in final_s])
print(output)

Answer 1

It's an interesting problem and not completely trivial. First, I think the input can only have odd count of a single digit, otherwise it cannot be formed into a palindrome. For example, 11333 is a valid input, but 113334 is not (both 3 and 4 have odd counts). It should also be noted that we cannot just dump the odd-count digits in the middle of the output. For example, we might be tempted to do 1335551 -> 3155513, but the correct answer (largest palindrome) is 5315135.

Given these constraints, here's my attempt at a solution. It uses collections.Counter to count the digit pairs, which are then sorted in descending order and mirrored to create the output. The possible odd-count digit is handled by treating it as a single digit (which goes into the middle of the output), plus a bunch of paired digits.

I tested it for input sizes of 10^5 digits and it didn't seem to take much time at all.

from collections import Counter

def biggest_pal(n):
    c = Counter(str(n))
    s = ''
    evens = {k: v for k, v in c.items() if not v % 2}
    odds = {k: v for k, v in c.items() if v % 2}
    vodd = ''
    if len(odds) > 1:
        raise ValueError('Invalid input')
    elif odds:
        vodd, nodd = odds.popitem()
        if nodd > 1:
            evens[vodd] = nodd - 1
    for k, v in sorted(evens.items(), key=lambda p: -int(p[0])):
        s += k * int(v/2)
    return s + vodd + s[::-1]

Some test inputs:

biggest_pal(112)  # 121
biggest_pal(1122)  # 2112
biggest_pal(1234123)  # 3214123
biggest_pal(1331555)  # 5315135
biggest_pal(112212)  # ValueError: invalid input