I/O-Efficient Range Minima Queries

Abstract

In this paper we study the offline (batched) range minima query (RMQ) problem in the external memory (EM) and cache-oblivious (CO) models. In the static RMQ problem, given an array A, a query rmq _A (i,j) returns the smallest element in the range A[i,j].

If B is the size of the block and m is the number of blocks that fit in the internal memory in the EM and CO models, we show that Q range minima queries on an array of size N can be answered in O$({{{N}\over{B}} + {{Q}\over{B}}\log_{m}{{Q}\over{B}}}) = {\rm O}{({\rm scan}({N}) + {\rm sort}({Q}))}$ I/Os in the CO model and slightly better O$({{\rm scan}({N}) + {{Q}\over{B}} \log_m \min\{{{Q}\over{B}}, {{N}\over{B}}\}})$ I/Os in the EM model and linear space in both models. Our cache-oblivious result is new and our external memory result is an improvement of the previously known bound. We also show that the EM bound is tight by proving a matching lower bound. Our lower bound holds even if the queries are presorted in any predefined order.

In the batched dynamic RMQ problem, the queries must be answered in the presence of the updates (insertions/deletions) to the array. We show that in the EM model we can solve this problem in O$({{\rm sort}({N}) + {\rm sort}{Q}\log_m {{N}\over{B}}})$ I/Os, again improving the best previously known bound.