CF1705D Mark and Lightbulbs

Mark has just purchased a rack of $ n $ lightbulbs. The state of the lightbulbs can be described with binary string $ s = s_1s_2\dots s_n $ , where $ s_i=\texttt{1} $ means that the $ i $ -th lightbulb is turned on, while $ s_i=\texttt{0} $ means that the $ i $ -th lightbulb is turned off. Unfortunately, the lightbulbs are broken, and the only operation he can perform to change the state of the lightbulbs is the following: - Select an index $ i $ from $ 2,3,\dots,n-1 $ such that $ s_{i-1}\ne s_{i+1} $ . - Toggle $ s_i $ . Namely, if $ s_i $ is $ \texttt{0} $ , set $ s_i $ to $ \texttt{1} $ or vice versa. Mark wants the state of the lightbulbs to be another binary string $ t $ . Help Mark determine the minimum number of operations to do so.

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In the first test case, one sequence of operations that achieves the minimum number of operations is the following. - Select $ i=3 $ , changing $ \texttt{01}\color{red}{\texttt{0}}\texttt{0} $ to $ \texttt{01}\color{red}{\texttt{1}}\texttt{0} $ . - Select $ i=2 $ , changing $ \texttt{0}\color{red}{\texttt{1}}\texttt{10} $ to $ \texttt{0}\color{red}{\texttt{0}}\texttt{10} $ . In the second test case, there is no sequence of operations because one cannot change the first digit or the last digit of $ s $ .In the third test case, even though the first digits of $ s $ and $ t $ are the same and the last digits of $ s $ and $ t $ are the same, it can be shown that there is no sequence of operations that satisfies the condition. In the fourth test case, one sequence that achieves the minimum number of operations is the following: - Select $ i=3 $ , changing $ \texttt{00}\color{red}{\texttt{0}}\texttt{101} $ to $ \texttt{00}\color{red}{\texttt{1}}\texttt{101} $ . - Select $ i=2 $ , changing $ \texttt{0}\color{red}{\texttt{0}}\texttt{1101} $ to $ \texttt{0}\color{red}{\texttt{1}}\texttt{1101} $ . - Select $ i=4 $ , changing $ \texttt{011}\color{red}{\texttt{1}}\texttt{01} $ to $ \texttt{011}\color{red}{\texttt{0}}\texttt{01} $ . - Select $ i=5 $ , changing $ \texttt{0110}\color{red}{\texttt{0}}\texttt{1} $ to $ \texttt{0110}\color{red}{\texttt{1}}\texttt{1} $ . - Select $ i=3 $ , changing $ \texttt{01}\color{red}{\texttt{1}}\texttt{011} $ to $ \texttt{01}\color{red}{\texttt{0}}\texttt{011} $ .