Abstract

We consider range queries that search for low-frequency elements (least frequent elements and $$\alpha $$?-minorities) in arrays. An $$\alpha $$?-minority of a query range has multiplicity no greater than an $$\alpha $$? fraction of the elements in the range. Our data structure for the least frequent element range query problem requires $$O(n)$$O(n) space, $$O(n^{3/2})$$O(n3/2) preprocessing time, and $$O(\sqrt{n})$$O(n) query time. A reduction from boolean matrix multiplication to this problem shows the hardness of simultaneous improvements in both preprocessing time and query time. Our data structure for the $$\alpha $$?-minority range query problem requires $$O(n)$$O(n) space, supports queries in $$O(1/\alpha )$$O(1/?) time, and allows $$\alpha $$? to be specified at query time.

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