TY - GEN

T1 - Improved submatrix maximum queries in Monge matrices

AU - Gawrychowski, Paweł

AU - Mozes, Shay

AU - Weimann, Oren

PY - 2014

Y1 - 2014

N2 - We present efficient data structures for submatrix maximum queries in Monge matrices and Monge partial matrices. For n x n Monge matrices, we give a data structure that requires O(n) space and answers submatrix maximum queries in O(logn) time. The best previous data structure [Kaplan et al., SODA'12] required O(n logn) space and O(log2 n) query time. We also give an alternative data structure with constant query-time and O(n1+ε) construction time and space for any fixed ε < 1. For n x n partial Monge matrices we obtain a data structure with O(n) space and O(logn·α (n)) query time. The data structure of Kaplan et al. required O(n logn·α(n)) space and O(log2 n) query time. Our improvements are enabled by a technique for exploiting the structure of the upper envelope of Monge matrices to efficiently report column maxima in skewed rectangular Monge matrices. We hope this technique will be useful in obtaining faster search algorithms in Monge partial matrices. In addition, we give a linear upper bound on the number of breakpoints in the upper envelope of a Monge partial matrix. This shows that the inverse Ackermann α(n) factor in the analysis of the data structure of Kaplan et. al is superfluous.

AB - We present efficient data structures for submatrix maximum queries in Monge matrices and Monge partial matrices. For n x n Monge matrices, we give a data structure that requires O(n) space and answers submatrix maximum queries in O(logn) time. The best previous data structure [Kaplan et al., SODA'12] required O(n logn) space and O(log2 n) query time. We also give an alternative data structure with constant query-time and O(n1+ε) construction time and space for any fixed ε < 1. For n x n partial Monge matrices we obtain a data structure with O(n) space and O(logn·α (n)) query time. The data structure of Kaplan et al. required O(n logn·α(n)) space and O(log2 n) query time. Our improvements are enabled by a technique for exploiting the structure of the upper envelope of Monge matrices to efficiently report column maxima in skewed rectangular Monge matrices. We hope this technique will be useful in obtaining faster search algorithms in Monge partial matrices. In addition, we give a linear upper bound on the number of breakpoints in the upper envelope of a Monge partial matrix. This shows that the inverse Ackermann α(n) factor in the analysis of the data structure of Kaplan et. al is superfluous.

UR - http://www.scopus.com/inward/record.url?scp=84904176635&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-43948-7_44

DO - 10.1007/978-3-662-43948-7_44

M3 - Conference contribution

AN - SCOPUS:84904176635

SN - 9783662439470

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 525

EP - 537

BT - Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings

PB - Springer Verlag

T2 - 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014

Y2 - 8 July 2014 through 11 July 2014

ER -