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Test case prioritization (TCP) aims to reorder the regression test suite with a goal of increasing the fault detection rate. Various TCP techniques have been proposed based on different prioritization strategies. Among them, the greedy-based techniques are the most widely-used TCP techniques. However, existing greedy-based techniques usually reorder all candidate test cases in prioritization iterations, resulting in both efficiency and effectiveness problems. In this paper, we propose a generic partial attention mechanism, which adopts the previous priority values (i.e., the number of additionally-covered code units) to avoid considering all candidate test cases. Incorporating the mechanism with the additional-greedy strategy, we implement a novel coverage-based TCP technique based on partition ordering (OCP). OCP first groups the candidate test cases into different partitions and updates the partitions on the descending order. We conduct a comprehensive experiment on 19 versions of Java programs and 30 versions of C programs to compare the effectiveness and efficiency of OCP with six state-of-the-art TCP techniques: total-greedy, additional-greedy, lexicographical-greedy, unify-greedy, art-based, and search-based. The experimental results show that OCP achieves a better fault detection rate than the state-of-the-arts. Moreover, the time costs of OCP are found to achieve 85%-99% improvement than most state-of-the-arts.