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Let S be a finite, ordered alphabet, and let x = x_1 x_2 ... x_n be a stringover S. A "secondary index" for x answers alphabet range queries of the form:Given a range [a_l,a_r] over S, return the set I_{[a_l;a_r]} = {i |x_i \in[a_l; a_r]}. Secondary indexes are heavily used in relational databases andscientific data analysis. It is well-known that the obvious solution, storing adictionary for the position set associated with each character, does not alwaysgive optimal query time. In this paper we give the first theoretically optimaldata structure for the secondary indexing problem. In the I/O model, the amountof data read when answering a query is within a constant factor of the minimumspace needed to represent I_{[a_l;a_r]}, assuming that the size of internalmemory is (|S| log n)^{delta} blocks, for some constant delta > 0. The spaceusage of the data structure is O(n log |S|) bits in the worst case, and wefurther show how to bound the size of the data structure in terms of the 0-thorder entropy of x. We show how to support updates achieving various time-spacetrade-offs.  We also consider an approximate version of the basic secondary indexingproblem where a query reports a superset of I_{[a_l;a_r]} containing eachelement not in I_{[a_l;a_r]} with probability at most epsilon, where epsilon >0 is the false positive probability. For this problem the amount of data thatneeds to be read by the query algorithm is reduced to O(|I_{[a_l;a_r]}|log(1/epsilon)) bits.

Secondary Indexing in One Dimension: Beyond B-trees and Bitmap Indexes