Range searching and lowest common ancestors

Two-dimensional range-searching using a Cartesian tree: the bottom point (red in the figure) within a three-sided region with two vertical sides and one horizontal side (if the region is nonempty) can be found as the nearest common ancestor of the leftmost and rightmost points (the blue points in the figure) within the slab defined by the vertical region boundaries. The remaining points in the three-sided region can be found by splitting it by a vertical line through the bottom point and recursing.

Cartesian trees form part of an efficient data structure for range minimum queries. An input to this kind of query specifies a contiguous subsequence of the original sequence; the query output should be the minimum value in this subsequence.^[7] In a Cartesian tree, this minimum value can be found at the lowest common ancestor of the leftmost and rightmost values in the subsequence. For instance, in the subsequence (12,10,20,15,18) of the example sequence, the minimum value of the subsequence (10) forms the lowest common ancestor of the leftmost and rightmost values (12 and 18). Because lowest common ancestors can be found in constant time per query, using a data structure that takes linear space to store and can be constructed in linear time, the same bounds hold for the range minimization problem.^[8]

Bender & Farach-Colton (2000) reversed this relationship between the two data structure problems by showing that data structures for range minimization could also be used for finding lowest common ancestors. Their data structure associates with each node of the tree its distance from the root, and constructs a sequence of these distances in the order of an Euler tour of the (edge-doubled) tree. It then constructs a range minimization data structure for the resulting sequence. The lowest common ancestor of any two vertices in the given tree can be found as the minimum distance appearing in the interval between the initial positions of these two vertices in the sequence. Bender and Farach-Colton also provide a method for range minimization that can be used for the sequences resulting from this transformation, which have the special property that adjacent sequence values differ by one. As they describe, for range minimization in sequences that do not have this form, it is possible to use Cartesian trees to reduce the range minimization problem to lowest common ancestors, and then to use Euler tours to reduce lowest common ancestors to a range minimization problem with this special form.^[9]

The same range minimization problem may also be given an alternative interpretation in terms of two dimensional range searching. A collection of finitely many points in the Cartesian plane can be used to form a Cartesian tree, by sorting the points by their ${\displaystyle x}$-coordinates and using the ${\displaystyle y}$-coordinates in this order as the sequence of values from which this tree is formed. If ${\displaystyle S}$ is the subset of the input points within some vertical slab defined by the inequalities ${\displaystyle L\leq x\leq R}$, ${\displaystyle p}$ is the leftmost point in ${\displaystyle S}$ (the one with minimum ${\displaystyle x}$-coordinate), and ${\displaystyle q}$ is the rightmost point in ${\displaystyle S}$ (the one with maximum ${\displaystyle x}$-coordinate) then the lowest common ancestor of ${\displaystyle p}$ and ${\displaystyle q}$ in the Cartesian tree is the bottommost point in the slab. A three-sided range query, in which the task is to list all points within a region bounded by the three inequalities ${\displaystyle L\leq x\leq R}$ and ${\displaystyle y\leq T}$, can be answered by finding this bottommost point ${\displaystyle b}$, comparing its ${\displaystyle y}$-coordinate to ${\displaystyle T}$, and (if the point lies within the three-sided region) continuing recursively in the two slabs bounded between ${\displaystyle p}$ and ${\displaystyle b}$ and between ${\displaystyle b}$ and ${\displaystyle q}$. In this way, after the leftmost and rightmost points in the slab are identified, all points within the three-sided region can be listed in constant time per point.^[3]

The same construction, of lowest common ancestors in a Cartesian tree, makes it possible to construct a data structure with linear space that allows the distances between pairs of points in any ultrametric space to be queried in constant time per query. The distance within an ultrametric is the same as the minimax path weight in the minimum spanning tree of the metric.^[10] From the minimum spanning tree, one can construct a Cartesian tree, the root node of which represents the heaviest edge of the minimum spanning tree. Removing this edge partitions the minimum spanning tree into two subtrees, and Cartesian trees recursively constructed for these two subtrees form the children of the root node of the Cartesian tree. The leaves of the Cartesian tree represent points of the metric space, and the lowest common ancestor of two leaves in the Cartesian tree is the heaviest edge between those two points in the minimum spanning tree, which has weight equal to the distance between the two points. Once the minimum spanning tree has been found and its edge weights sorted, the Cartesian tree can be constructed in linear time.^[11]

As a binary search tree

The Cartesian tree of a sorted sequence is just a path graph, rooted at its leftmost endpoint. Binary searching in this tree degenerates to sequential search in the path. However, a different construction uses Cartesian trees to generate binary search trees of logarithmic depth from sorted sequences of values. This can be done by generating priority numbers for each value, and using the sequence of priorities to generate a Cartesian tree. This construction may equivalently be viewed in the geometric framework described above, in which the ${\displaystyle x}$-coordinates of a set of points are the values in a sorted sequence and the ${\displaystyle y}$-coordinates are their priorities.^[12]

This idea was applied by Seidel & Aragon (1996), who suggested the use of random numbers as priorities. The self-balancing binary search tree resulting from this random choice is called a treap, due to its combination of binary search tree and min-heap features. An insertion into a treap can be performed by inserting the new key as a leaf of an existing tree, choosing a priority for it, and then performing tree rotation operations along a path from the node to the root of the tree to repair any violations of the heap property caused by this insertion; a deletion can similarly be performed by a constant amount of change to the tree followed by a sequence of rotations along a single path in the tree.^[12] A variation on this data structure called a zip tree uses the same idea of random priorities, but simplifies the random generation of the priorities, and performs insertions and deletions in a different way, by splitting the sequence and its associated Cartesian tree into two subsequences and two trees and then recombining them.^[13]

If the priorities of each key are chosen randomly and independently once whenever the key is inserted into the tree, the resulting Cartesian tree will have the same properties as a random binary search tree, a tree computed by inserting the keys in a randomly chosen permutation starting from an empty tree, with each insertion leaving the previous tree structure unchanged and inserting the new node as a leaf of the tree. Random binary search trees have been studied for much longer than treaps, and are known to behave well as search trees. The expected length of the search path to any given value is at most ${\displaystyle 2\ln n}$, and the whole tree has logarithmic depth (its maximum root-to-leaf distance) with high probability. More formally, there exists a constant ${\displaystyle C}$ such that the depth is ${\displaystyle \leq C\ln n}$ with probability tending to one as the number of nodes tends to infinity. The same good behavior carries over to treaps. It is also possible, as suggested by Aragon and Seidel, to reprioritize frequently-accessed nodes, causing them to move towards the root of the treap and speeding up future accesses for the same keys.^[12]

In sorting

Pairs of consecutive sequence values (shown as the thick red edges) that bracket a sequence value (the darker blue point). The cost of including this value in the sorted order produced by the Levcopoulos–Petersson algorithm is proportional to the logarithm of this number of bracketing pairs.

Levcopoulos & Petersson (1989) describe a sorting algorithm based on Cartesian trees. They describe the algorithm as based on a tree with the maximum at the root,^[14] but it can be modified straightforwardly to support a Cartesian tree with the convention that the minimum value is at the root. For consistency, it is this modified version of the algorithm that is described below.

The Levcopoulos–Petersson algorithm can be viewed as a version of selection sort or heap sort that maintains a priority queue of candidate minima, and that at each step finds and removes the minimum value in this queue, moving this value to the end of an output sequence. In their algorithm, the priority queue consists only of elements whose parent in the Cartesian tree has already been found and removed. Thus, the algorithm consists of the following steps:^[14]

1. Construct a Cartesian tree for the input sequence
2. Initialize a priority queue, initially containing only the tree root
3. While the priority queue is non-empty:
   • Find and remove the minimum value in the priority queue
   • Add this value to the output sequence
   • Add the Cartesian tree children of the removed value to the priority queue

As Levcopoulos and Petersson show, for input sequences that are already nearly sorted, the size of the priority queue will remain small, allowing this method to take advantage of the nearly-sorted input and run more quickly. Specifically, the worst-case running time of this algorithm is ${\displaystyle O(n\log k)}$, where ${\displaystyle n}$ is the sequence length and ${\displaystyle k}$ is the average, over all values in the sequence, of the number of consecutive pairs of sequence values that bracket the given value (meaning that the given value is between the two sequence values). They also prove a lower bound stating that, for any ${\displaystyle n}$ and (non-constant) ${\displaystyle k}$, any comparison-based sorting algorithm must use ${\displaystyle \Omega (n\log k)}$ comparisons for some inputs.^[14]

In pattern matching

The problem of Cartesian tree matching has been defined as a generalized form of string matching in which one seeks a substring (or in some cases, a subsequence) of a given string that has a Cartesian tree of the same form as a given pattern. Fast algorithms for variations of the problem with a single pattern or multiple patterns have been developed, as well as data structures analogous to the suffix tree and other text indexing structures.^[15]