Fast algorithms for finding nearest common ancestors
SIAM Journal on Computing
On finding lowest common ancestors: simplification and parallelization
SIAM Journal on Computing
Highly parallelizable problems
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Algorithms on strings, trees, and sequences: computer science and computational biology
Algorithms on strings, trees, and sequences: computer science and computational biology
Scaling and related techniques for geometry problems
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Lowest common ancestors in trees and directed acyclic graphs
Journal of Algorithms
Theoretical and practical improvements on the RMQ-Problem, with applications to LCA and LCE
CPM'06 Proceedings of the 17th Annual conference on Combinatorial Pattern Matching
A new succinct representation of RMQ-information and improvements in the enhanced suffix array
ESCAPE'07 Proceedings of the First international conference on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies
On Cartesian Trees and Range Minimum Queries
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Data structures for range minimum queries in multidimensional arrays
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On space efficient two dimensional range minimum data structures
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
A quick tour on suffix arrays and compressed suffix arrays
Theoretical Computer Science
Encoding 2d range maximum queries
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Two dimensional range minimum queries and fibonacci lattices
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Space-efficient data-analysis queries on grids
Theoretical Computer Science
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We consider the two-dimensional Range Minimum Query problem: for a static (m × n)-matrix of size N = mn which may be preprocessed, answer on-line queries of the form "where is the position of a minimum element in an axis-parallel rectangle?". Unlike the onedimensional version of this problem which can be solved in provably optimal time and space, the higher-dimensional case has received much less attention. The only result we are aware of is due to Gabow, Bentley and Tarjan [1], who solve the problem in O(N logN) preprocessing time and space and O(log N) query time. We present a class of algorithms which can solve the 2-dimensional RMQ-problem with O(kN) additional space, O(N log[k+1] N) preprocessing time and O(1) query time for any k 1, where log[k+1] denotes the iterated application of k + 1 logarithms. The solution converges towards an algorithm with O(N log* N) preprocessing time and space and O(1) query time. All these algorithms are significant improvements over the previous results: query time is optimal, preprocessing time is quasi-linear in the input size, and space is linear. While this paper is of theoretical nature, we believe that our algorithms will turn out to have applications in different fields of computer science, e.g., in computational biology.