Skyline computation aims at looking for the set of tuples that are not worse than any other tuples in all dimensions from a multidimensional database. In this paper, we present SDI (Skyline on Dimension Index), a dimension indexing conducted general framework to skyline computation. We prove that to determine whether a tuple belongs to the skyline, it is enough to compare this tuple with a bounded subset of skyline tuples in an arbitrary dimensional index, but not with all existing skyline tuples. Base on SDI, we also show that any skyline tuple can be used to stop the whole skyline computation process with outputting the complete set of all skyline tuples. We develop an efficient algorithm SDI-RS that significantly reduces the skyline computation time, of which the space and time complexity can be guaranteed. Our experimental evaluation shows that SDI-RS outperforms the baseline algorithms in general and is especially very efficient on high-dimensional data.