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Search spaces hallmark the advancement of Neural Architecture Search (NAS). Large and complex search spaces with versatile building operators and structures provide more opportunities to brew promising architectures, yet pose severe challenges on efficient exploration and exploitation. Subsequently, several search space shrinkage methods optimize by selecting a single sub-region that contains some well-performing networks. Small performance and efficiency gains are observed with these methods but such techniques leave room for significantly improved search performance and are ineffective at retaining architectural diversity. We propose LISSNAS, an automated algorithm that shrinks a large space into a diverse, small search space with SOTA search performance. Our approach leverages locality, the relationship between structural and performance similarity, to efficiently extract many pockets of well-performing networks. We showcase our method on an array of search spaces spanning various sizes and datasets. We accentuate the effectiveness of our shrunk spaces when used in one-shot search by achieving the best Top-1 accuracy in two different search spaces. Our method achieves a SOTA Top-1 accuracy of 77.6% in ImageNet under mobile constraints, best-in-class Kendal-Tau, architectural diversity, and search space size.