Encoding Nearest Larger Values

Abstract

In nearest larger value (NLV) problems, we are given an array $A[1..n]$ of numbers, and need to preprocess $A$ to answer queries of the following form: given any index $i \in [1,n]$, return a “nearest” index $j$ such that $A[j] > A[i]$. We consider the variant where the values in $A$ are distinct, and we wish to return an index $j$ such that $A[j] > A[i]$ and $|j-i|$ is minimized, the nondirectional NLV (NNLV) problem. We consider NNLV in the encoding model, where the array $A$ is deleted after preprocessing, and note that NNLV encoding problem has an unexpectedly rich structure: the effective entropy (optimal space usage) of the problem depends crucially on details in the definition of the problem. Using a new path-compressed representation of binary trees, that may have other applications, we encode NNLV in $1.9n + o(n)$ bits, and answer queries in $O(1)$ time.