Pattern masking for dictionary matching

Abstract

Data masking is a common technique for sanitizing sensitive data maintained in database systems, and it is also becoming increasingly important in various application areas, such as in record linkage of personal data. This work formalizes the Pattern Masking for Dictionary Matching (PMDM) problem. In PMDM, we are given a dictionary of d strings, each of length , a query string q of length , and a positive integer z, and we are asked to compute a smallest set K \xe2\x8a\x86 {1,\xe2\x80\xa6,}, so that if q[i] is replaced by a wildcard for all i \xe2\x88\x88 K, then q matches at least z strings from . Solving PMDM allows providing data utility guarantees as opposed to existing approaches. \nWe first show, through a reduction from the well-known k-Clique problem, that a decision version of the PMDM problem is NP-complete, even for strings over a binary alphabet. We thus approach the problem from a more practical perspective. We show a combinatorial ((d)^{|K|/3}+d)-time and (d)-space algorithm for PMDM for |K| = (1). In fact, we show that we cannot hope for a faster combinatorial algorithm, unless the combinatorial k-Clique hypothesis fails [Abboud et al., SIAM J. Comput. 2018; Lincoln et al., SODA 2018]. We also generalize this algorithm for the problem of masking multiple query strings simultaneously so that every string has at least z matches in . \nNote that PMDM can be viewed as a generalization of the decision version of the dictionary matching with mismatches problem: by querying a PMDM data structure with string q and z = 1, one obtains the minimal number of mismatches of q with any string from . The query time or space of all known data structures for the more restricted problem of dictionary matching with at most k mismatches incurs some exponential factor with respect to k. A simple exact algorithm for PMDM runs in time (2^ d). We present a data structure for PMDM that answers queries over in time (2^{/2}(2^{/2}+\xcf\x84)) and requires space (2^ d\xc2\xb2/\xcf\x84\xc2\xb2+2^{/2}d), for any parameter \xcf\x84 \xe2\x88\x88 [1,d]. \nWe complement our results by showing a two-way polynomial-time reduction between PMDM and the Minimum Union problem [Chlamt\xc3\xa1\xc4\x8d et al., SODA 2017]. This gives a polynomial-time (d^{1/4+\xce\xb5})-approximation algorithm for PMDM, which is tight under a plausible complexity conjecture. </p