We present a novel algorithm to conservatively check whether a (recursive) heap-manipulating program terminates. Our algorithm can be used as a post-processing phase of any shape analysis satisfying some natural properties. The running time of the post-processing phase is linear in the size of the output of the chosen shape analysis. The main idea is to partition the (unbounded but finite) set of allocated objects in every state into a bounded set of regions, and track the flow of objects between heap regions in every step of the program. The algorithm proves the existence of the well-founded relation over states by showing that in every loop iteration at least one object (which was allocated before entering the loop) moves to a strictly lower-ranked heap region. The partitioning of objects into regions, the flow of objects between regions, and the ranks of regions are computed automatically from the output of the underlying shape analysis. Our algorithm extends the state of the art in terms of complexity, the class of supported data structures, and its generality. We successfully applied a prototype of our analysis to prove termination of a suite of benchmarks from existing literature, including (looping, recursive, and concurrent) list manipulating programs, looping list-sorting programs, and looping programs that manipulate trees and graphs. The overhead of the termination phase in our experiments is at most 14% of the overall analysis time.