# Optimal On The Fly Index Selection in Polynomial Time

The index selection problem (ISP) is an important problem for accelerating the execution of relational queries, and it has received a lot of attention as a combinatorial knapsack problem in the past. Various solutions to this very hard problem have been provided. In contrast to existing literature, we change the underlying assumptions of the problem definition: we adapt the problem for systems that store relations in memory, and use complex specification languages, e.g., Datalog. In our framework, we decompose complex queries into primitive searches that select tuples in a relation for which an equality predicate holds. A primitive search can be accelerated by an index exhibiting a worst-case run-time complexity of log-linear time in the size of the output result of the primitive search. However, the overheads associated with maintaining indexes are very costly in terms of memory and computing time. In this work, we present an optimal polynomial-time algorithm that finds the minimal set of indexes of a relation for a given set of primitive searches. An index may cover more than one primitive search due to the algebraic properties of the search predicate, which is a conjunction of equalities over the attributes of a relation. The index search space exhibits a exponential complexity in the number of attributes in a relation, and, hence brute-force algorithms searching for solutions in the index domain are infeasible. As a scaffolding for designing a polynomial-time algorithm, we build a partial order on search operations and use a constructive version of Dilworth's theorem. We show a strong relationship between chains of primitive searches (forming a partial order) and indexes. We demonstrate the effectiveness and efficiency of our algorithm for an in-memory Datalog compiler that is able to process relations with billions of entries in memory.

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