Maintaining maximal independent set in dynamic graph is a fundamental open problem in graph theory and the first sublinear time deterministic algorithm was came up by Assadi, Onak, Schieber and Solomon(STOC18), which achieves $O(m^{3/4})$ amortized update time. We have two main contributions in this paper. We present a new simple deterministic algorithm with $O(m^{2/3}sqrt{log m})$ amortized update time, which improves the previous best result. And we also present the first randomized algorithm with expected $O(sqrt{m}log^{1.5}m)$ amortized time against an oblivious adversary.
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