We demonstrate shape analyses that can achieve a state space reduction exponential in the number of threads compared to the state-of-the-art analyses, while retaining sufficient precision to verify sophisticated properties such as linearizability. The key idea is to abstract the global heap by decomposing it into (not necessarily disjoint) subheaps, abstracting away some correlations between them. These new shape analyses are instances of an analysis framework based on heap decomposition. This framework allows rapid prototyping of complex static analyses by providing efficient abstract transformers given user-specified decomposition schemes. Initial experiments confirm the value of heap decomposition in scaling concurrent shape analyses.