LRM-Trees: Compressed indices, adaptive sorting, and compressed permutations

LRM-Trees are an elegant way to partition a sequence of values into sorted consecutive blocks. They have been used to encode ordinal trees and to index integer arrays in order to support range minimum queries on them. We describe how they yield many other convenient results in a variety of areas: compressed indices for range minimum queries on partially sorted arrays, a new adaptive sorting algorithm, and a compressed data structure for permutations supporting direct and inverse application in time inversely proportional to the compressibility of the permutation.

[1]  Rajeev Raman,et al.  Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets , 2007, ACM Trans. Algorithms.

[2]  Anna Gál,et al.  The cell probe complexity of succinct data structures , 2007, Theor. Comput. Sci..

[3]  Gonzalo Navarro,et al.  Compressed full-text indexes , 2007, CSUR.

[4]  Maxime Crochemore,et al.  Improved Algorithms for the Range Next Value Problem and Applications , 2008, STACS.

[5]  Uzi Vishkin,et al.  Recursive Star-Tree Parallel Data Structure , 1993, SIAM J. Comput..

[6]  David Richard Clark,et al.  Compact pat trees , 1998 .

[7]  Gonzalo Navarro,et al.  Compressed Representations of Permutations, and Applications , 2009, STACS.

[8]  Gonzalo Navarro,et al.  On compressing permutations and adaptive sorting , 2011, Theor. Comput. Sci..

[9]  Rajeev Raman,et al.  Representing Trees of Higher Degree , 2005, Algorithmica.

[10]  Gonzalo Navarro,et al.  Fully-functional succinct trees , 2010, SODA '10.

[11]  S. Srinivasa Rao,et al.  Succinct indexes for strings, binary relations and multi-labeled trees , 2007, SODA '07.

[12]  Rajeev Raman,et al.  Succinct indexable dictionaries with applications to encoding k-ary trees and multisets , 2002, SODA '02.

[13]  Gonzalo Navarro,et al.  Faster entropy-bounded compressed suffix trees , 2009, Theor. Comput. Sci..

[14]  Kunihiko Sadakane,et al.  Practical Entropy-Compressed Rank/Select Dictionary , 2006, ALENEX.

[15]  Volker Heun,et al.  Practical Entropy-Bounded Schemes for O(1)-Range Minimum Queries , 2008, Data Compression Conference (dcc 2008).

[16]  S. Srinivasa Rao,et al.  On Space Efficient Two Dimensional Range Minimum Data Structures , 2010, ESA.

[17]  Roberto Grossi,et al.  High-order entropy-compressed text indexes , 2003, SODA '03.

[18]  Chi Wang,et al.  Partitioning Permutations into Increasing and Decreasing Subsequences , 1996, J. Comb. Theory, Ser. A.

[19]  Alistair Moffat,et al.  A Framework for Adaptive Sorting , 1995, Discret. Appl. Math..

[20]  Robert E. Tarjan,et al.  Scaling and related techniques for geometry problems , 1984, STOC '84.

[21]  Guy Jacobson,et al.  Space-efficient static trees and graphs , 1989, 30th Annual Symposium on Foundations of Computer Science.

[22]  Gonzalo Navarro,et al.  Alphabet Partitioning for Compressed Rank/Select and Applications , 2010, ISAAC.

[23]  Roberto Grossi,et al.  Squeezing succinct data structures into entropy bounds , 2006, SODA '06.

[24]  Gonzalo Navarro,et al.  Implicit Compression Boosting with Applications to Self-indexing , 2007, SPIRE.

[25]  D. Huffman.  A Method for the Construction of Minimum-Redundancy Codes , 1952 .

[26]  Kunihiko Sadakane,et al.  Ultra-succinct representation of ordered trees , 2007, SODA '07.

[27]  Johannes Fischer,et al.  Optimal Succinctness for Range Minimum Queries , 2008, LATIN.

[28]  Giovanni Manzini,et al.  Opportunistic data structures with applications , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[29]  Kuan-Yu Chen,et al.  On the Range Maximum-Sum Segment Query Problem , 2004, ISAAC.

[30]  Christos Levcopoulos,et al.  Sorting Shuffled Monotone Sequences , 1994, Inf. Comput..

[31]  Alexander Golynski.  Optimal lower bounds for rank and select indexes , 2007, Theor. Comput. Sci..