Let Σ be a finite, ordered alphabet, and consider a string x = x1x2... xn ∈ Σn. A secondary index for x answers alphabet range queries of the form: Given a range [al, ar] ⊆ Σ, return the set I [al;ar] = {i | xi ∈ [al; ar]}. Secondary indexes are heavily used in relational databases and scientific data analysis. It is well-known that the obvious solution, storing a dictionary for the set ⋃ i {xi} with a position set associated with each character, does not always give optimal query time. In this paper we give the first theoretically optimal data structure for the secondary indexing problem. In the I/O model, the amount of data read when answering a query is within a constant factor of the minimum space needed to represent the set I [al;ar], assuming that the size of internal memory is (|Σ | lg n) δ blocks, for some constant δ> 0. The space usage of the data structure is O(n lg |Σ|) bits in the worst case, and we further show how to bound the size of the data structure in terms of the 0th order entropy of x. We show how to support updates achieving various time-space trade-offs. We also consider an approximate version of the basic secondary indexing problem where a query reports a superset of I [al;ar] containing each element not in I [al;ar] with probability at most ε, where ε> 0 is the false positive probability. For this problem the amount of data that needs to be read by the query algorithm is reduced to O(|I [al;ar] | lg(1/ε)) bits. The main ideas for this work were conceived during

Abstract—Set intersection is the core in a variety of problems, e.g. frequent itemset mining and sparse boolean matrix multiplication. It is well-known that large speed gains can, for some computational problems, be obtained by using a graphics processing unit (GPU) as a massively parallel computing device. However, GPUs require highly regular control flow and memory access patterns, and for this reason previous GPU methods for intersecting sets have used a simple bitmap representation. This representation requires excessive space on sparse data sets. In this paper we present a novel data layout, BATMAP, that is particularly well suited for parallel processing, and is compact even for sparse data. Frequent itemset mining is one of the most important applications of set intersection. As a case-study on the potential of BATMAPs we focus on frequent pair mining, which is a core special case of frequent itemset mining. The main finding is that our method is able to achieve speedups over both Apriori and FP-growth when the number of distinct items is large, and the density of the problem instance is above 1%. Previous implementations of frequent itemset mining on GPU have not been able to show speedups over the best single-threaded implementations. Keywords-Set intersection; Frequent itemset mining; Sparse boolean matrix multiplication; Data layout; GPU

Set intersection is a fundamental operation in information retrieval and database systems. This paper introduces linear space data structures to represent sets such that their intersection can be computed in a worst-case efficient way. In general, given k (preprocessed) sets, with totally n elements, we will show how to compute their intersection in expected time O(n / √ w + kr), where r is the intersection size and w is the number of bits in a machine-word. In addition,we introduce a very simple version of this algorithm that has weaker asymptotic guarantees but performs even better in practice; both algorithms outperform the state of the art techniques for both synthetic and real data sets and workloads. 1.

Let Σ be a finite, ordered alphabet, and let x = x1x2... xn ∈ Σn. A secondary index for x answers alphabet range queries of the form: Given a range [al, ar] ⊆ Σ, return the set I [al;ar] = {i | xi ∈ [al; ar]}. Secondary indexes are heavily used in relational databases and scientific data analysis. It is wellknown that the obvious solution, storing a dictionary for the set ⋃ i {xi} with a position set associated with each character, does not always give optimal query time. In this paper we give the first theoretically optimal data structure for the secondary indexing problem. In the I/O model, the amount of data read when answering a query is within a constant factor of the minimum space needed to represent I [al;ar], assuming that the size of internal memory is (|Σ | lg n) δ blocks, for some constant δ> 0. The space usage of the data structure is O(n lg |Σ|) bits in the worst case, and we further show how to bound the size of the data structure in terms of the 0th order entropy of x. We show how to support updates achieving various time-space trade-offs. We also consider an approximate version of the basic secondary indexing problem where a query reports a superset of I [al;ar] containing each element not in I [al;ar] with probability at most ε, where ε> 0 is the false positive probability. For this problem the amount of data that needs to be read by the query algorithm is reduced to O(|I [al;ar] | lg(1/ε)) bits. The main ideas for this work were conceived during Dagstuhl seminar No. 08081 on Data