By offloading intensive computation tasks to the edge cloud located at the cellular base stations, mobile-edge computation offloading (MECO) has been regarded as a promising means to accomplish the ambitious millisecond-scale end-to-end latency requirement of the fifth-generation networks. In this paper, we investigate the latency-minimization problem in a multi-user time-division multiple access MECO system with joint communication and computation resource allocation. Three different computation models are studied, i.e., local compression, edge cloud compression, and partial compression offloading. First, closed-form expressions of optimal resource allocation and minimum system delay for both local and edge cloud compression models are derived. Then, for the partial compression offloading model, we formulate a piecewise optimization problem and prove that the optimal data segmentation strategy has a piecewise structure. Based on this result, an optimal joint communication and computation resource allocation algorithm is developed. To gain more insights, we also analyze a specific scenario where communication resource is adequate while computation resource is limited. In this special case, the closed-form solution of the piecewise optimization problem can be derived. Our proposed algorithms are finally verified by numerical results, which show that the novel partial compression offloading model can significantly reduce the end-to-end latency.