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computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, the range query problem consists of efficiently answering several queries regarding a given interval of elements within an array. For example, a common task, known as range minimum query, is finding the smallest value inside a given range within a list of numbers.


Definition

Given a function f that accepts an array, a range query f_q(l, r) on an array a= _1,..,a_n/math> takes two indices l and r and returns the result of f when applied to the subarray _l, \ldots, a_r/math>. For example, for a function \operatorname that returns the sum of all values in an array, the range query \operatorname_q(l, r) returns the sum of all values in the range , r/math>.


Solutions


Prefix sum array

Range sum queries may be answered in
constant time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
and linear space by pre-computing an array of same length as the input such that for every index , the element is the sum of the first elements of . Any query may then be computed as follows: \operatorname_q(l, r) = p_r - p_. This strategy may be extended to any other
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
f whose
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
f^ is well-defined and easily computable. It can also be extended to higher dimensions with a similar pre-processing. For example, if contains the sum of the first elements of , then \operatorname_q(l, r, t, b) = p_ - p_ - p_ + p_.


Dynamic range queries

A more difficult subset of the problem consists of executing range queries on dynamic data; that is, data that may mutate between each query. In order to efficiently update array values, more sophisticated data structures like the
segment tree In computer science, the segment tree is a data structure used for storing information about Interval (mathematics), intervals or segments. It allows querying which of the stored segments contain a given point. A similar data structure is the i ...
or Fenwick tree are necessary.


Examples


Semigroup operators

When the function of interest in a range query is a
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
operator, the notion of f^ is not always defined, so the strategy in the previous section does not work. Andrew Yao showed that there exists an efficient solution for range queries that involve semigroup operators. He proved that for any constant , a pre-processing of time and space \Theta(c\cdot n) allows to answer range queries on lists where is a semigroup operator in \theta(\alpha_c(n)) time, where \alpha_c is a certain functional inverse of the
Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total function, total computable function that is not Primitive recursive function, primitive recursive. ...
. There are some semigroup operators that admit slightly better solutions. For instance when f\in \. Assume f = \min then \min(A ..n returns the index of the
minimum In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
element of A ..n/math>. Then \min_(A) denotes the corresponding minimum range query. There are several data structures that allow to answer a range minimum query in O(1) time using a pre-processing of time and space O(n). One such solution is based on the equivalence between this problem and the
lowest common ancestor In graph theory and computer science, the lowest common ancestor (LCA) (also called least common ancestor) of two nodes and in a Tree (graph theory), tree or directed acyclic graph (DAG) is the lowest (i.e. deepest) node that has both and a ...
problem. The Cartesian tree T_A of an array A ,n/math> has as root a_i = \min\ and as left and right subtrees the Cartesian tree of A ,i-1/math> and the Cartesian tree of A +1,n/math> respectively. A range minimum query \min_(A) is the
lowest common ancestor In graph theory and computer science, the lowest common ancestor (LCA) (also called least common ancestor) of two nodes and in a Tree (graph theory), tree or directed acyclic graph (DAG) is the lowest (i.e. deepest) node that has both and a ...
in T_A of a_i and a_j. Because the lowest common ancestor can be solved in
constant time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
using a pre-processing of time and space O(n), range minimum query can as well. The solution when f = \max is analogous. Cartesian trees can be constructed in
linear time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations ...
.


Mode

The mode of an array is the element that appears the most in it. For instance the mode of a= ,5,6,7,4/math> is . In case of a tie, any of the most frequent elements might be picked as the mode. A range mode query consists in pre-processing A ,n/math> such that we can find the mode in any range of A ,n/math>. Several data structures have been devised to solve this problem, we summarize some of the results in the following table. Recently Jørgensen et al. proved a lower bound on the cell-probe model of \Omega\left(\tfrac\right) for any data structure that uses cells.


Median

This particular case is of special interest since finding the
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
has several applications. On the other hand, the median problem, a special case of the selection problem, is solvable in ''O''(''n''), using the median of medians algorithm. However its generalization through range median queries is recent. A range median query \operatorname(A,i,j) where ''A,i'' and ''j'' have the usual meanings returns the median element of A ,j/math>. Equivalently, \operatorname(A,i,j) should return the element of A ,j/math> of rank \frac. Range median queries cannot be solved by following any of the previous methods discussed above including Yao's approach for semigroup operators. There have been studied two variants of this problem, the
offline In computer technology and telecommunications, online indicates a state of connectivity, and offline indicates a disconnected state. In modern terminology, this usually refers to an Internet connection, but (especially when expressed as "on li ...
version, where all the ''k'' queries of interest are given in a batch, and a version where all the pre-processing is done up front. The offline version can be solved with O(n\log k + k \log n) time and O(n\log k) space. The following pseudocode of the quickselect algorithm shows how to find the element of rank in A ,j/math> an unsorted array of distinct elements, to find the range medians we set r=\frac. rangeMedian(A, i, j, r) Procedure partitions , using 's median, into two arrays and , where the former contains the elements of that are less than or equal to the median and the latter the rest of the elements of . If we know that the number of elements of A ,j/math> that end up in is and this number is bigger than then we should keep looking for the element of rank in ; otherwise we should look for the element of rank (r-t) in . To find , it is enough to find the maximum index m\leq i-1 such that a_m is in and the maximum index l\leq j such that a_l is in . Then t=l-m. The total cost for any query, without considering the partitioning part, is \log n since at most \log n recursion calls are done and only a constant number of operations are performed in each of them (to get the value of fractional cascading should be used). If a linear algorithm to find the medians is used, the total cost of pre-processing for range median queries is n\log k. The algorithm can also be modified to solve the
online In computer technology and telecommunications, online indicates a state of connectivity, and offline indicates a disconnected state. In modern terminology, this usually refers to an Internet connection, but (especially when expressed as "on lin ...
version of the problem.


Majority

Finding frequent elements in a given set of items is one of the most important tasks in data mining. Finding frequent elements might be a difficult task to achieve when most items have similar frequencies. Therefore, it might be more beneficial if some threshold of significance was used for detecting such items. One of the most famous algorithms for finding the majority of an array was proposed by Boyer and Moore which is also known as the Boyer–Moore majority vote algorithm. Boyer and Moore proposed an algorithm to find the majority element of a string (if it has one) in O(n) time and using O(1) space. In the context of Boyer and Moore’s work and generally speaking, a majority element in a set of items (for example string or an array) is one whose number of instances is more than half of the size of that set. Few years later, Misra and Gries proposed a more general version of Boyer and Moore's algorithm using O \left ( n \log \left ( \frac \right ) \right ) comparisons to find all items in an array whose relative frequencies are greater than some threshold 0<\tau<1. A range \tau-majority query is one that, given a subrange of a data structure (for example an array) of size , R, , returns the set of all distinct items that appear more than (or in some publications equal to) \tau , R, times in that given range. In different structures that support range \tau-majority queries, \tau can be either static (specified during pre-processing) or dynamic (specified at query time). Many of such approaches are based on the fact that, regardless of the size of the range, for a given \tau there could be at most O(1/\tau) distinct ''candidates'' with relative frequencies at least \tau. By verifying each of these candidates in constant time, O(1/\tau) query time is achieved. A range \tau-majority query is decomposable in the sense that a \tau-majority in a range R with partitions R_1 and R_2 must be a \tau-majority in either R_1or R_2. Due to this decomposability, some data structures answer \tau-majority queries on one-dimensional arrays by finding the
Lowest common ancestor In graph theory and computer science, the lowest common ancestor (LCA) (also called least common ancestor) of two nodes and in a Tree (graph theory), tree or directed acyclic graph (DAG) is the lowest (i.e. deepest) node that has both and a ...
(LCA) of the endpoints of the query range in a Range tree and validating two sets of candidates (of size O(1/\tau)) from each endpoint to the lowest common ancestor in constant time resulting in O(1/\tau) query time.


Two-dimensional arrays

Gagie et al. proposed a data structure that supports range \tau-majority queries on an m\times n array A. For each query \operatorname=(\operatorname, \tau) in this data structure a threshold 0<\tau<1 and a rectangular range \operatorname are specified, and the set of all elements that have relative frequencies (inside that rectangular range) greater than or equal to \tau are returned as the output. This data structure supports dynamic thresholds (specified at query time) and a pre-processing threshold \alpha based on which it is constructed. During the pre-processing, a set of ''vertical'' and ''horizontal'' intervals are built on the m \times n array. Together, a vertical and a horizontal interval form a ''block.'' Each block is part of a ''superblock'' nine times bigger than itself (three times the size of the block's horizontal interval and three times the size of its vertical one). For each block a set of candidates (with \frac elements at most) is stored which consists of elements that have relative frequencies at least \frac (the pre-processing threshold as mentioned above) in its respective superblock. These elements are stored in non-increasing order according to their frequencies and it is easy to see that, any element that has a relative frequency at least \alpha in a block must appear its set of candidates. Each \tau-majority query is first answered by finding the ''query block,'' or the biggest block that is contained in the provided query rectangle in O(1) time. For the obtained query block, the first \frac candidates are returned (without being verified) in O(1/\tau) time, so this process might return some false positives. Many other data structures (as discussed below) have proposed methods for verifying each candidate in constant time and thus maintaining the O(1/\tau) query time while returning no false positives. The cases in which the query block is smaller than 1/\alpha are handled by storing \log \left ( \frac \right ) different instances of this data structure of the following form: \beta=2^, \;\; i\in \left \ where \beta is the pre-processing threshold of the i-th instance. Thus, for query blocks smaller than 1/\alpha the \lceil\log (1 / \tau)\rceil-th instance is queried. As mentioned above, this data structure has query time O(1/\tau) and requires O \left ( m n(H+1) \log^2 \left ( \frac \right ) \right ) bits of space by storing a Huffman-encoded copy of it (note the \log(\frac) factor and also see
Huffman coding In computer science and information theory, a Huffman code is a particular type of optimal prefix code that is commonly used for lossless data compression. The process of finding or using such a code is Huffman coding, an algorithm developed by ...
).


One-dimensional arrays

Chan et al. proposed a data structure that given a one-dimensional arrayA, a subrange R of A (specified at query time) and a threshold \tau (specified at query time), is able to return the list of all \tau-majorities in O(1/\tau) time requiring O(n \log n) words of space. To answer such queries, Chan et al. begin by noting that there exists a data structure capable of returning the ''top-k'' most frequent items in a range in O(k) time requiring O(n) words of space. For a one-dimensional array A ,..,n-1/math>, let a one-sided top-k range query to be of form A ..i\text 0 \leq i \leq n-1. For a maximal range of ranges A ..i\text A ..j/math> in which the frequency of a distinct element e in A remains unchanged (and equal to f), a horizontal line segment is constructed. The x-interval of this line segment corresponds to ,j/math> and it has a y-value equal to f. Since adding each element to A changes the frequency of exactly one distinct element, the aforementioned process creates O(n) line segments. Moreover, for a vertical line x=i all horizonal line segments intersecting it are sorted according to their frequencies. Note that, each horizontal line segment with x-interval ell,r/math> corresponds to exactly one distinct element e in A, such that A elle. A top-k query can then be answered by shooting a vertical ray x=i and reporting the first k horizontal line segments that intersect it (remember from above that these line segments are already sorted according to their frequencies) in O(k) time. Chan et al. first construct a range tree in which each branching node stores one copy of the data structure described above for one-sided range top-k queries and each leaf represents an element from A. The top-k data structure at each node is constructed based on the values existing in the subtrees of that node and is meant to answer one-sided range top-k queries. Please note that for a one-dimensional array A, a range tree can be constructed by dividing A into two halves and recursing on both halves; therefore, each node of the resulting range tree represents a range. It can also be seen that this range tree requires O(n \log n) words of space, because there are O(\log n) levels and each level \ell has 2^ nodes. Moreover, since at each level \ell of a range tree all nodes have a total of n elements of A at their subtrees and since there are O(\log n) levels, the space complexity of this range tree is O(n \log n). Using this structure, a range \tau-majority query A ..j/math> on A ..n-1/math> with 0\leq i\leq j \leq n is answered as follows. First, the
lowest common ancestor In graph theory and computer science, the lowest common ancestor (LCA) (also called least common ancestor) of two nodes and in a Tree (graph theory), tree or directed acyclic graph (DAG) is the lowest (i.e. deepest) node that has both and a ...
(LCA) of leaf nodes i and j is found in constant time. Note that there exists a data structure requiring O(n) bits of space that is capable of answering the LCA queries in O(1) time. Let z denote the LCA of i and j, using z and according to the decomposability of range \tau-majority queries (as described above and in ), the two-sided range query A ..j/math> can be converted into two one-sided range top-k queries (from z to i and j). These two one-sided range top-k queries return the top-(1/\tau) most frequent elements in each of their respective ranges in O(1/\tau) time. These frequent elements make up the set of ''candidates'' for \tau-majorities in A ..j/math> in which there are O(1/\tau) candidates some of which might be false positives. Each candidate is then assessed in constant time using a linear-space data structure (as described in Lemma 3 in ) that is able to determine in O(1) time whether or not a given subrange of an array A contains at least q instances of a particular element e.


Tree paths

Gagie et al. proposed a data structure which supports queries such that, given two nodes u and v in a tree, are able to report the list of elements that have a greater relative frequency than \tau on the path from u to v. More formally, let T be a labelled tree in which each node has a label from an alphabet of size \sigma. Let label(u)\in ,\dots,\sigma/math> denote the label of node u in T. Let P_ denote the unique path from u to v in T in which middle nodes are listed in the order they are visited. Given T, and a fixed (specified during pre-processing) threshold 0<\tau<1, a query Q(u,v) must return the set of all labels that appear more than \tau, P_, times in P_. To construct this data structure, first (\tau n) nodes are ''marked''. This can be done by marking any node that has distance at least \lceil 1 / \tau\rceil from the bottom of the three (height) and whose depth is divisible by \lceil 1 / \tau\rceil. After doing this, it can be observed that the distance between each node and its nearest marked ancestor is less than 2\lceil 1 / \tau\rceil. For a marked node x, \log(depth(x)) different sequences (paths towards the root) P_i(x) are stored, P_(x)=\left\langle \operatorname(x), \operatorname(x), \operatorname^(x), \ldots, \operatorname^(x)\right\rangle for 0\leq i \leq \log(depth(x)) where \operatorname(x) returns the label of the direct parent of node x. Put another way, for each marked node, the set of all paths with a power of two length (plus one for the node itself) towards the root is stored. Moreover, for each P_i(x), the set of all majority ''candidates'' C_i(x) are stored. More specifically, C_i(x) contains the set of all (\tau/2)-majorities in P_i(x) or labels that appear more than (\tau/2).(2^i+1) times in P_i(x). It is easy to see that the set of candidates C_i(x) can have at most 2/\tau distinct labels for each i. Gagie et al. then note that the set of all \tau-majorities in the path from any marked node x to one of its ancestors z is included in some C_i(x) (Lemma 2 in ) since the length of P_i(x) is equal to (2^i+1) thus there exists a P_i(x) for 0\leq i \leq \log(depth(x)) whose length is between d_ \text 2 d_ where d_ is the distance between x and z. The existence of such P_i(x) implies that a \tau-majority in the path from x to z must be a (\tau/2)-majority in P_i(x), and thus must appear in C_i(x). It is easy to see that this data structure require O(n \log n) words of space, because as mentioned above in the construction phase O(\tau n) nodes are marked and for each marked node some candidate sets are stored. By definition, for each marked node O(\log n) of such sets are stores, each of which contains O(1/\tau) candidates. Therefore, this data structure requires O(\log n \times (1/\tau) \times \tau n)=O(n \log n) words of space. Please note that each node x also stores count(x) which is equal to the number of instances of label(x) on the path from x to the root of T, this does not increase the space complexity since it only adds a constant number of words per node. Each query between two nodes u and v can be answered by using the decomposability property (as explained above) of range \tau-majority queries and by breaking the query path between u and v into four subpaths. Let z be the lowest common ancestor of u and v, with x and y being the nearest marked ancestors of u and v respectively. The path from u to v is decomposed into the paths from u and v to x and y respectively (the size of these paths are smaller than 2\lceil 1 / \tau\rceil by definition, all of which are considered as candidates), and the paths from x and y to z (by finding the suitable C_i(x) as explained above and considering all of its labels as candidates). Please note that, boundary nodes have to be handled accordingly so that all of these subpaths are disjoint and from all of them a set of O(1/\tau) candidates is derived. Each of these candidates is then verified using a combination of the labelanc (x, \ell) query which returns the lowest ancestor of node x that has label \ell and the count(x) fields of each node. On a w-bit RAM and an alphabet of size \sigma, the labelanc (x, \ell) query can be answered in O\left(\log \log _ \sigma\right) time whilst having linear space requirements. Therefore, verifying each of the O(1/\tau) candidates in O\left(\log \log _ \sigma\right) time results in O\left((1/\tau)\log \log _ \sigma\right) total query time for returning the set of all \tau -majorities on the path from u to v .


Related problems

All the problems described above have been studied for higher dimensions as well as their dynamic versions. On the other hand, range queries might be extended to other data structures like
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only p ...
, such as the level ancestor problem. A similar family of problems are orthogonal range queries, also known as counting queries.


See also

* Level ancestor problem *
Lowest common ancestor In graph theory and computer science, the lowest common ancestor (LCA) (also called least common ancestor) of two nodes and in a Tree (graph theory), tree or directed acyclic graph (DAG) is the lowest (i.e. deepest) node that has both and a ...


References


External links


Open Data Structure - Chapter 13 - Data Structures for IntegersData Structures for Range Median Queries - Gerth Stolting Brodal and Allan Gronlund Jorgensen
{{CS-Trees Arrays