How to Answer a Small Batch of RMQs or LCA Queries in Practice

Abstract

In the Range Minimum Query (RMQ) problem, we are given an array A of n numbers and we are asked to answer queries of the following type: for indices i and j between 0 and n − 1, query RMQ A (i, j) returns the index of a minimum element in the subarray A[i . . j]. Answering a small batch of RMQs is a core computational task in many realworld applications, in particular due to the connection with the Lowest Common Ancestor (LCA) problem. With small batch, we mean that the number q of queries is o(n) and we have them all at hand. It is therefore not relevant to build an Ω(n)-sized data structure or spend Ω(n) time to build a more succinct one. It is well-known, among practitioners and elsewhere, that these data structures for online querying carry high constants in their pre-processing and querying time. We would thus like to answer this batch efficiently in practice. With efficiently in practice, we mean that we (ultimately) want to spend n + O(q) time and O(q) space. We write n to stress that the number of operations per entry of A should be a very small constant. Here we show how existing algorithms can be easily modified to satisfy these conditions. The presented experimental results highlight the practicality of this new scheme. The most significant improvement obtained is for answering a small batch of LCA queries. A library implementation of the presented algorithms is made available.

Range Minimum Queries in Minimal Space

“…Their best result obtains competitive results against existing solutions, except possibly for small queries. Their results show query times essentially equal to ours and the algorithm of Alzamel, Charalampopoulos, Iliopoulos, and Pissis [2018] for large queries, but they also obtain 10 times slower performance for small queries.…”

“…We compared experimentally these approaches in Section 4. The results showed that out approach was competitive, both in terms of time and space, our prototype V was actually faster than the prototype B by Alzamel et al [2018]. We also showed that it was possible for our data structure to obtained O(1) amortized query time (Corolary 1), mostly for theoretical competitiveness.…”

“…We started by obtaining O(α(n)) amortized time and O(q) space, see Theorem 2. This result is in theory slower than the result by Alzamel, Charalampopoulos, Iliopoulos, and Pissis [2018], which obtained O(1) amortized query time. We compared experimentally these approaches in Section 4.…”

“…For a baseline comparison we used the ST-RMQ-CON algorithm by Alzamel, Charalampopoulos, Iliopoulos, a [2018]. We obtained the implementation from their github repository https://github.com/solonas13/rmqo.…”

“…A new approach was proposed by Alzamel, Charalampopoulos, Iliopoulos, and Pissis [2018] where no index data structure was created by a preprocessing step. Instead all the RMQs are batched together and solved in n + O(q) time and O(q) space.…”