#### Approach #1: Expand Around Center [Accepted]

Intuition

Let N be the length of the string. The middle of the palindrome could be in one of 2N - 1 positions: either at letter or between two letters.

For each center, let's count all the palindromes that have this center. Notice that if [a, b] is a palindromic interval (meaning S[a], S[a+1], ..., S[b] is a palindrome), then [a+1, b-1] is one too.

Algorithm

For each possible palindrome center, let's expand our candidate palindrome on the interval [left, right] as long as we can. The condition for expanding is left >= 0 and right < N and S[left] == S[right]. That means we want to count a new palindrome S[left], S[left+1], ..., S[right].

Complexity Analysis

• Time Complexity: where is the length of S. Each expansion might do work.

• Space Complexity: .

#### Approach #2: Manacher's Algorithm [Accepted]

Intuition

Manacher's algorithm is a textbook algorithm that finds in linear time, the maximum size palindrome for any possible palindrome center. If we had such an algorithm, finding the answer is straightforward.

What follows is a discussion of why this algorithm works.

Algorithm

Our loop invariants will be that center, right is our knowledge of the palindrome with the largest right-most boundary with center < i, centered at center with right-boundary right. Also, i > center, and we've already computed all Z[j]'s for j < i.

When i < right, we reflect i about center to be at some coordinate j = 2 * center - i. Then, limited to the interval with radius right - i and center i, the situation for Z[i] is the same as for Z[j].

For example, if at some time center = 7, right = 13, i = 10, then for a string like A = '@#A#B#A#A#B#A#＄', the center is at the '#' between the two middle 'A''s, the right boundary is at the last '#', i is at the last 'B', and j is at the first 'B'.

Notice that limited to the interval [center - (right - center), right] (the interval with center center and right-boundary right), the situation for i and j is a reflection of something we have already computed. Since we already know Z[j] = 3, we can quickly find Z[i] = min(right - i, Z[j]) = 3.

Now, why is this algorithm linear? The while loop only checks the condition more than once when Z[i] = right - i. In that case, for each time Z[i] += 1, it increments right, and right can only be incremented up to 2*N+2 times.

Finally, we sum up (v+1) / 2 for each v in Z. Say the longest palindrome with some given center C has radius R. Then, the substring with center C and radius R-1, R-2, R-3, ..., 0 are also palindromes. Example: abcdedcba is a palindrome with center e, radius 4: but e, ded, cdedc, bcdedcb, and abcdedcba are all palindromes.

We are dividing by 2 because we were using half-lengths instead of lengths. For example we actually had the palindrome a#b#c#d#e#d#c#b#a, so our length is twice as big.

Complexity Analysis

• Time Complexity: where is the length of S. As discussed above, the complexity is linear.

• Space Complexity: , the size of A and Z.

Analysis written by: @awice.