How does Duval's algorithm handle odd-length strings?

Question

Finding the Lexicographically minimal string rotation is a well known problem, for which a linear time algorithm was proposed by Jean Pierre Duval in 1983. This blog post is pro

Answer 1

One character can get a "bye", where it wins without participating in a "duel". The correctness of the algorithm does not rely on the specific duels that you perform; given any two distinct indices i and j, you can always conclusively rule out that one of them is the start-index of the lexicographically-minimal rotation (unless both are start-indices of identical lexicographically-minimal rotations, in which case it doesn't matter which one you reject). The reason to perform the duels in a specific order is performance: to get asymptotically linear time by ensuring that half the duels only need to compare one character, half of the rest only need to compare two characters, and so on, until the last duel only needs to compare half the length of the string. But a single odd character here and there doesn't change the asymptotic complexity, it just makes the math (and implementation) a little bit more complicated. A string of length 2ⁿ+1 still requires fewer "duels" than one of length 2ⁿ⁺¹.