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This article introduces a technique to accurately perform static prediction of resource usage for ML-like functional programs with references and arrays. Previous research successfully integrated the potential method of amortized analysis with a standard type system to automatically derive parametric resource bounds. The analysis is naturally compositional and the resource consumption of functions can be abstracted using potential-annotated types. The soundness theorem of the analysis guarantees that the derived bounds are correct with respect to the resource usage defined by a cost semantics. Type inference can be efficiently automated using off-the-shelf LP solvers, even if the derived bounds are polynomials. However, side effects and aliasing of heap references make it notoriously difficult to derive bounds that depend on mutable structures, such as arrays and references. As a result, existing automatic amortized analysis systems for ML-like programs cannot derive bounds for programs whose resource consumption depends on data in such structures. This article extends the potential method to handle mutable structures with minimal changes to the type rules while preserving the stated advantages of amortized analysis. To do so, we introduce a swap operation for references and arrays that users can use to make programs suitable for automatic analysis. We prove the soundness of the analysis introducing a potential-annotated memory typing, which gathers all unique locations reachable from a reference. Apart from the design of the system, the main contribution is the proof of soundness for the extended analysis system.