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In the path minima problem on a tree, each edge is assigned a weight and a query asks for the edge with minimum weight on a path between two nodes. For the dynamic version of the problem, where the edge weights can be updated, we give data structures that achieve optimal query time in the comparison and the RAM models. These structures also support(More)
A k-ary cardinal tree is a rooted tree in which each node has at most k children, and each edge is labeled with a symbol from the alphabet {1,. .. , k}. We present a succinct representation for k-ary cardinal trees of n nodes where k = O(polylog(n)). Our data structure requires 2n + n log k + o(n log k) bits and performs the following operations in O(1)(More)
The two dimensional range minimum query problem is to preprocess a static m by n matrix (two dimensional array) A of size N=m⋅n, such that subsequent queries, asking for the position of the minimum element in a rectangular range within A, can be answered efficiently. We study the trade-off between the space and query time of the problem. We show that every(More)
We provide two succinct representations of binary trees that can be used to represent the Cartesian tree of an array A of size n. Both the representations take the optimal 2n + o(n) bits of space in the worst case and support range minimum queries (RMQs) in O(1) time. The first one is a modification of the representation of Farzan and Munro (SWAT 2008); a(More)
We consider the problem of encoding range minimum queries (RMQs): given an array A[1..n] of distinct totally ordered values, to pre-process A and create a data structure that can answer the query RMQ(i,j), which returns the index containing the smallest element in A[i..j], without access to the array A at query time. We give a data structure whose space(More)
Given a matrix of size N , two dimensional range minimum queries (2D-RMQs) ask for the position of the minimum element in a rectangular range within the matrix. We study trade-offs between the query time and the additional space used by indexing data structures that support 2D-RMQs. Using a novel technique—the discrepancy properties of Fibonacci lattices—we(More)
In the two-dimensional range minimum query problem an input matrix A of dimension m × n, m ≤ n, has to be preprocessed into a data structure such that given a query rectangle within the matrix, the position of a minimum element within the query range can be reported. We consider the space complexity of the encoding variant of the problem where queries have(More)
Cardinal trees (or tries of degree $$k$$ k ) are a fundamental data structure, in particular for many text-processing algorithms. Each node in a cardinal tree has at most $$k$$ k children, each labeled with a symbol from the alphabet $$\{1,\ldots , k\}$$ { 1 , … , k } . In this paper we introduce succinct representations for dynamic cardinal trees on $$n$$(More)
Given a set of n points in the plane, range diameter queries ask for the furthest pair of points in a given axis-parallel rectangular range. We provide evidence for the hardness of designing space-efficient data structures that support range diameter queries by giving a reduction from the set intersection problem. The difficulty of the latter problem is(More)