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Faster JoinProjects and Sparse Matrix Multiplications
, 2009
"... Computing an equijoin followed by a duplicate eliminating projection is conventionally done by performing the two operations in serial. If some join attribute is projected away the intermediate result may be much larger than both the input and the output, and the computation could therefore potenti ..."
Abstract

Cited by 8 (5 self)
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Computing an equijoin followed by a duplicate eliminating projection is conventionally done by performing the two operations in serial. If some join attribute is projected away the intermediate result may be much larger than both the input and the output, and the computation could therefore potentially be performed faster by a direct procedure that does not produce such a large intermediate result. We present a new algorithm that has smaller intermediate results on worstcase inputs, and in particular is more efficient in both the RAM and I/O model. It is easy to see that joinproject where the join attributes are projected away is equivalent to boolean matrix multiplication. Our results can therefore also be interpreted as improved sparse, outputsensitive matrix multiplication.
Worstcase Optimal Join Algorithms
"... Efficient join processing is one of the most fundamental and wellstudied tasks in database research. In this work, we examine algorithms for natural join queries over many relations and describe a novel algorithm to process these queries optimally in terms of worstcase data complexity. Our result b ..."
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Efficient join processing is one of the most fundamental and wellstudied tasks in database research. In this work, we examine algorithms for natural join queries over many relations and describe a novel algorithm to process these queries optimally in terms of worstcase data complexity. Our result builds on recent work by Atserias, Grohe, and Marx, who gave bounds on the size of a full conjunctive query in terms of the sizes of the individual relations in the body of the query. These bounds, however, are not constructive: they rely on Shearer’s entropy inequality which is informationtheoretic. Thus, the previous results leave open the question of whether there exist algorithms whose running time achieve these optimal bounds. An answer to this question may be interesting to database practice, as we show in this paper that any projectjoin plan is polynomially slower than the optimal bound for some queries. We construct an algorithm whose running time is worstcase optimal for all natural join queries. Our result may be of independent interest, as our algorithm also yields a constructive proof of the general fractional cover bound by Atserias, Grohe, and Marx without using Shearer’s inequality. In addition, we show that this bound is equivalent to a geometric inequality by Bollobás and Thomason, one of whose special cases is the famous LoomisWhitney inequality. Hence, our results algorithmically prove these inequalities as well. Finally, we discuss how our algorithm can be used to compute a relaxed notion of joins.
3SUM, 3XOR, Triangles
, 2013
"... We show that if one can solve 3SUM on a set of size n in time n1+ɛ then one can list t triangles in a graph with m edges in time Õ(m1+ɛt1/3+ɛ ′ ) for any ɛ ′> 0. This is a reversal of Pǎtra¸scu’s reduction from 3SUM to listing triangles (STOC ’10). We then reexecute both Pǎtra¸scu’s reduction and o ..."
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We show that if one can solve 3SUM on a set of size n in time n1+ɛ then one can list t triangles in a graph with m edges in time Õ(m1+ɛt1/3+ɛ ′ ) for any ɛ ′> 0. This is a reversal of Pǎtra¸scu’s reduction from 3SUM to listing triangles (STOC ’10). We then reexecute both Pǎtra¸scu’s reduction and our reversal for the variant 3XOR of 3SUM where integer summation is replaced by bitwise xor. As a corollary we obtain that if 3XOR is solvable in linear time but 3SUM requires quadratic randomized time, or vice versa, then the randomized time complexity of listing m triangles in a graph with m edges is m 4/3 up to a factor m α for any α> 0. Our results are obtained building on and extending works by the Paghs (PODS ’06) and by Vassilevska and Williams (FOCS ’10).