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In machine translation (MT) that involves translating between two languages with significant differences in word order, determining the correct word order of translated words is a major challenge. The dependency parse tree of a source sentence can help to determine the correct word order of the translated words. In this paper, we present a novel reordering approach utilizing a neural network and dependency-based embeddings to predict whether the translations of two source words linked by a dependency relation should remain in the same order or should be swapped in the translated sentence. Experiments on Chinese-to-English translation show that our approach yields a statistically significant improvement of 0.57 BLEU point on benchmark NIST test sets, compared to our prior state-of-the-art statistical MT system that uses sparse dependency-based reordering features.