Compact dictionaries for variable-length keys and data with applications

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 + 3 n − 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 semidynamic 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.