Compact dictionaries for variable-length keys and data with applications

Abstract

We consider the problem of maintaining a dynamic dictionary T of keys and associated data for which both the keys and data are bit strings that can vary in length from zero up to the length w of a machine word. We present a data structure for this variable-bit-length dictionary problem that supports constant time lookup and expected amortized constant time insertion and deletion. It uses O(m + 3n − n log 2 n) bits, where n is the number of elements in T , and m is the total number of bits across all strings in T (keys and data). Our dictionary uses an array A[1 . . . n] in which locations store variable-bit-length strings. We present a data structure for this variable-bit-length array problem that supports worst-case constant-time lookups and updates and uses O(m + n) bits, where m is the total number of bits across all strings stored in A.The motivation for these structures is to support applications for which it is helpful to efficiently store short varying length bit strings. We present several applications, including representations for semi-dynamic graphs, order queries on integers sets, cardinal trees with varying cardinality, and simplicial meshes of d dimensions. These results either generalize or simplify previous results.

An optimal algorithm for the distinct elements problem

“…For the former, we use a "variable-bitlength array" data structure [7], and for the latter we use an approach inspired by the technique of deamortization of global rebuilding (see [29,Ch. 5]).…”

“…Definition 1 (Blandford, Blelloch [7]). A variablebit-length array (VLA) is a data structure implementing an array C1, .…”

“…Theorem 8 (Blandford and Blelloch [7]). There is a VLA data structure using O(n+ P i len(Ci)) space to store n elements, supporting worst-case O(1) updates and reads, under the assumptions that (1) len(Ci) ≤ w for all i, and (2) w ≥ log(M).…”

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