Memristor is one of the most promising emerging technology to solve the von Neumann bottleneck problem due to its non-volatile and binary characteristics. This paper studies the design method of high efficiency logic circuit based on memristor. First, a Multiple Input Multiple Output (MIMO) logic circuit design scheme based on IMPLY and AND logic is proposed, which can derive multiple new efficient logic operation methods, and complete complex logic with fewer steps and memristors. Second, in order to perform rapid interactive operations between different rows, an alternating crossbar array structure is designed, which can quickly complete cross-row logic operations. Finally, a high-efficient Full Adder (FA) based on MIMO logic and alternating crossbar array is proposed. To accomplish 32-bit add operation, the proposed FA needs 160 memristors and only 41 steps. Compared with the state of art FA, our work has faster execution speed and fewer memristors.