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Edge systems – such as mobile devices, IoT nodes, and embedded sensors – increasingly operate in dynamic environments while facing severe resource constraints. Among these, compute is often the primary bottleneck, limited not only by hardware capacity but also by thermal and energy constraints inherent to untethered operation. A common strategy in systems design is to trade memory for compute, e.g., via caching or memoization. But on the edge, memory is also scarce, making indiscriminate reuse infeasible. This thesis is built on a central insight: despite the dynamic inputs and behaviors observed at runtime, most deployed applications – from system software to ML pipelines – rely on underlying structural stability. This stability arises from how these systems are developed: through modular frameworks, predictable control paths, and abstractions that preserve behavior across updates. In this dissertation, we show how to dissect and extract these stable computational elements and selectively reuse them to reduce compute overhead, even when memory is tightly limited. This allows us to affirmatively answer a fundamental question: can edge systems, without the luxury of abundant resources, still benefit from the kind of reuse strategies common in richer environments? First, we present Floo, a system that memoizes function-level computations in mobile applications by leveraging persistent interaction flows and function signatures, enabling effective reuse with minimal overheads. Then, we introduce Remembrall, which reuses parts of neural network pipelines in video analytics workloads by identifying redundant structure in learned embedding spaces - enabling lightweight adaptation and compute reduction under memory constraints. Together, these systems demonstrate that judicious use of limited memory, grounded in structural stability, can enable efficient compute reuse on the edge.