Invertible Bloom Lookup Tables with Listing Guarantees

The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, Steiner systems, and covering arrays as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory consumption. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations

A construction for Steiner 3-designs

Let q be a prime power. For every ν satisfying necessary arithmetic conditions we construct a Steiner 3-design S(3, q + 1; ν · q^n + 1) for every n sufficiently large. Starting with a Steiner 2-design S(2, q; ν), this is extended to a 3-design S_λ(3, q + 1; ν + 1), with index λ = q^d for some d, such that the derived design is λ copies of the Steiner 2-design. The 3-design is used, by a generalization of a construction of Wilson, to form a group-divisible 3-design GD(3, {q, q + 1}, νp^d) with index one. The structure of the derived design allows a circle geometry S(3, q + 1; q^d + 1) to be combined with the group-divisible design to form, via a method of Hanani, the desired Steiner 3-design S(3, q + 1; νq^n + 1), for all n ⩾ n_0