In colored range counting (CRC), the input is a set of points where each point is assigned a “color” (or a “category”) and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data. However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exists or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through a reduction from the orthogonal vectors problem.

In colored range counting (CRC), the input is a set of points where each point is assigned a "color" (or a "category") and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data. However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exists or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through reductions from the orthogonal vectors problem.

In this paper we study lower bounds for a variant of jumbled-indexing problem: given an input string S on an alphabet Σ = {σ1,…, σλ}, store it in a data structure such that given frequencies f1,…, fλ, one can find all the substrings S’ of S where the frequency of the character σi is fi, for 1 ≤ i ≤ λ. This is a very interesting and challenging text indexing problem and it has received significant attention lately, both in the string-indexing community as well as in the theory community. It is known that when λ = 2, one can use a linear-size data structure to decide whether there exists a substring that matches the query in constant time [10]. It is also known [1] that for constant λ ≥ 3, then there exists a constant αλ such that it is not possible to achieve both O(n2−αλ) preprocessing time and O(n1−aλ) query time. We study a variant of the problem where the goal is to report all the substrings of S that match a given jumbled-indexing query. Assuming the data structure operates in the pointer machine model, we prove unconditional space lower bounds that also apply to binary alphabets: we show that if the data structure has the query time of O(n0.5−o(1) + k), where k is the output size of the query, then it must consume Ω(n2−o(1)) space. The o(1) term in our lower bound is at most where log(·) notation denotes the iterative log function (i.e., log(10) n = log log log log log log log log log log n). This result has a number of interesting consequences. First, it shows that reporting all the matches is significantly harder than deciding whether a match exists, at least for the binary alphabet. This was surprising for us since we believed that in order for jumbled-indexing to be difficult, the alphabet size must be at least 3. Second, from a technical point of view, we make connections between this problem and the area of additive combinatorics as well as random walks. Previously, Chan and Lewenstein [6] had shown the connections between this problem and the area of additive combinatorics from an upper bound point of view, however, we use fundamental theorems from additive combinatorics but these tools are quite different from the theorems used in [6]. In our opinion, the fact that additive combinatorics appears in our lower bound approach as well as in the upper bound approach of Chan and Lewenstein [6] is noteworthy and demands further investigation. We believe our results open up a few new venues for research. For example, we believe it is interesting to study whether it is possible to build a data structure that uses O(n) space s.t., it can report all the matches to a jumble-indexing query in O(n2/3 + k) time; this is even interesting for a binary alphabet.