
Covering Codes for Insertions and Deletions
A covering code is a set of codewords with the property that the union o...
read it

Listdecodable Codes and Covering Codes
The listdecodable code has been an active topic in theoretical computer...
read it

Size bounds and query plans for relational joins
Relational joins are at the core of relational algebra, which in turn is...
read it

The Stretch Factor of HexagonDelaunay Triangulations
The problem of computing the exact stretch factor (i.e., the tight bound...
read it

Box Covers and Domain Orderings for Beyond WorstCase Join Processing
Recent beyond worstcase optimal join algorithms Minesweeper and its gen...
read it

The equational theory of the natural join and inner union is decidable
The natural join and the inner union operations combine relations of a d...
read it

Approximations and Bounds for (n, k) ForkJoin Queues: A Linear Transformation Approach
Compared to basic forkjoin queues, a job in (n, k) forkjoin queues onl...
read it
Covering the Relational Join
In this paper, we initiate a theoretical study of what we call the join covering problem. We are given a natural join query instance Q on n attributes and m relations (R_i)_i ∈ [m]. Let J_Q = _i=1^m R_i denote the join output of Q. In addition to Q, we are given a parameter Δ: 1≤Δ≤ n and our goal is to compute the smallest subset 𝒯_Q, Δ⊆ J_Q such that every tuple in J_Q is within Hamming distance Δ  1 from some tuple in 𝒯_Q, Δ. The join covering problem captures both computing the natural join from database theory and constructing a covering code with covering radius Δ  1 from coding theory, as special cases. We consider the combinatorial version of the join covering problem, where our goal is to determine the worstcase 𝒯_Q, Δ in terms of the structure of Q and value of Δ. One obvious approach to upper bound 𝒯_Q, Δ is to exploit a distance property (of Hamming distance) from coding theory and combine it with the worstcase bounds on output size of natural joins (AGM bound hereon) due to Atserias, Grohe and Marx [SIAM J. of Computing'13]. Somewhat surprisingly, this approach is not tight even for the case when the input relations have arity at most two. Instead, we show that using the polymatroid degreebased bound of Abo Khamis, Ngo and Suciu [PODS'17] in place of the AGM bound gives us a tight bound (up to constant factors) on the 𝒯_Q, Δ for the arity two case. We prove lower bounds for 𝒯_Q, Δ using wellknown classes of errorcorrecting codes e.g, ReedSolomon codes. We can extend our results for the arity two case to general arity with a polynomial gap between our upper and lower bounds.
READ FULL TEXT
Comments
There are no comments yet.