## LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations ⋆

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Citations: | 3 - 2 self |

### BibTeX

@MISC{Barbay_lrm-trees:compressed,

author = {Jérémy Barbay and Johannes Fischer and Gonzalo Navarro},

title = {LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations ⋆},

year = {}

}

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### Abstract

Abstract. LRM-Trees are an elegant way to partition a sequence of values into sorted consecutive blocks, and to express the relative position of the first element of each block within a previous block. They were 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 succinct indices for range minimum queries on partially sorted arrays; a new adaptive sorting algorithm; and a compressed succinct data structure for permutations supporting direct and inverse application in time inversely proportional to the permutation’s compressibility. 1

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Citation Context ...ll be used in Thms. 4 and 5.i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A[i] 15 8 13 7 11 16 1 10 9 14 2 12 3 6 5 4 −∞ 15(1) 8(2) 7(4) 1(7) 13(3) 11(5) 10(8) 9(9) 16(6) 2(11) 14(10) 12(12) 3(13) 6(14) 5=-=(15)-=- 4(16) Fig. 1. An example of an array and its LRM-Tree. Lemma 1. Given an array A[1, n] of totally ordered objects, there is an algorithm computing its LRM-Tree in at most 2(n − 1) comparisons within ... |

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Citation Context ... can be10 achieved for some parameterized classes of permutations. Petersson and Moffat [29] summarized many of them; we describe here a few that are relevant to our work. For a permutation π, Knuth =-=[23]-=- considered Runs (contiguous ascending subsequences), counted by ρ = 1 + |{i, 1 ≤ i < n, πi+1 < πi}|; Levcopoulos and Petersson [24] introduced Shuffled Up-Sequences and its generalization Shuffled Mo... |

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Citation Context ...utations from specific classes to less than the information-theoretic lower bound of n lg n bits. Furthermore they used the similarity of the execution of the merge-sort algorithm with a Wavelet Tree =-=[14]-=-, to support the application of π() and its inverse π−1 () in time logarithmic in the disorder of the permutation π (as measured by |Runs|, |SRuns|, |SUS|, |SSUS| or |SMS|) in the worst case. We summa... |

193 | Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets
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Citation Context ...ses to A and O(1) accesses to the index. Proof. We mark the beginnings of the runs in A with a 1 in a bit-vector B[1, n], and represent B with the compressed succinct data structure from Raman et al. =-=[24]-=-, using ⌈lg ( ) n ′ ⌉ + o(n) bits. Further, we define A as the (conceptual) |SRuns| array consisting of the heads of A’s runs (A′[i] = A[select1(B, i)]). We build the LRM-Tree from Lemma 2 on A ′ usin... |

180 | Opportunistic Data Structures with Application
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Citation Context ....g., the first-order entropy of a string is no larger than its zero-order entropy), but in many cases they are incomparable. Formally,aCompressed Data Structure(alsocalled“opportunisticdatastructure” =-=[11]-=- or “ultra-succinct data structure” [21]) for a compressibility measure µ is a data structure that requires lgf(n,µ)+o(lgf(n)) bits to encode any instance of size n and compressibility µ. Note that, w... |

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Citation Context ...er way to construct the LRM-Tree in at most 2(n−1) comparisons within the array and overall linear time, which will be used in Theorems 4 and 5. The construction is similar to that of Cartesian Trees =-=[15]-=-. Lemma 1. Given an array A[1..n] of n totally ordered objects, there is an algorithm computing its LRM-Tree in at most 2(n−1) comparisons within A and O(n) total time. Proof. The computation of the L... |

173 | Compressed full-text indexes - Navarro, Mäkinen |

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Citation Context ...used in Thms. 4 and 5.i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A[i] 15 8 13 7 11 16 1 10 9 14 2 12 3 6 5 4 −∞ 15(1) 8(2) 7(4) 1(7) 13(3) 11(5) 10(8) 9(9) 16(6) 2(11) 14(10) 12(12) 3(13) 6(14) 5(15) 4=-=(16)-=- Fig. 1. An example of an array and its LRM-Tree. Lemma 1. Given an array A[1, n] of totally ordered objects, there is an algorithm computing its LRM-Tree in at most 2(n − 1) comparisons within A and ... |

93 |
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Citation Context ...torsrank0(B,i)andselect0(B,i)aredefinedanalogously4 for 0-bits. A succinct data structure for this abstract data type requires n+o(n) bits of space and supports all those operations in constant time =-=[9,26]-=-. On the other hand, a data structure that does not support the access operator can use less space than the information theory lower bound. Its space usage can still be expressed as a function of the ... |

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Citation Context ...of leaf siblings to the left of a node, and25 finding the p-th leaf child of a node. We show next how to extend the structure to support these. Janssonetal.’s structure[21]encodesaDFUDSrepresentation=-=[5]-=-ofthetree, where the nodes are represented in preorder and each node with d children is representedasd1s followed bya0:“1···10”. Inourexample, this representation yields the bit sequence (where we add... |

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Citation Context ...ious data structures and algorithms, including text indexing [14], pattern matching [10], and more elaborate kinds of range queries [8]. We define them as follows: Definition 2 (Range Minimum Queries =-=[6]-=-). Consider an array A[1..n] of n ordered objects. A Range Minimum Query consists of a pair of integers i and j such that 1 ≤ i ≤ j ≤ n, and its answer rmq A(i,j) is the leftmost position of a minimum... |

50 | Practical entropy-compressed rank/select dictionary
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Citation Context ...bitmap will be nH0 ∈ O(nlglgn/lg 2 n) ∈ o(n). In this case, the compressed data structure mentioned [31] has a redundancy of Θ(nlglgn/lgn) ⊂ ω(nH0). A fully compressed data structure for this problem =-=[28]-=- requires nH0+O(n1) bits of space (and supports the access, rank and select operators in super-constant time). This is fully compressed as long as n1 ∈ o(n); otherwise nH0 ∈ Θ(n) and the previous stru... |

42 |
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Citation Context ...eparating the encoding from the index was introduced to prove lower bounds on the trade-off between space and operation times of succinct data structures [17] (it had only been implicitly used before =-=[33]-=-), it has other advantages. In particular, the concept of succinct index is additive in the sense that, if D is a succinct data structure, then the data structure formed by the union of D and I is a s... |

41 | Succinct indexes for strings, binary relations and multilabeled trees
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Citation Context ...e root of the tree has value A[0] = −∞ (smaller than all elements), the algorithm always terminates. The construction algorithm performs at most 2(n − 1) comparisons: the first two elements A[0] and A=-=[1]-=- can be inserted without any comparison as a simple path of two nodes (so A[1] will be charged only once). For the remaining elements, we charge the last comparison performed during the insertion of a... |

36 |
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Citation Context ...〈3, 6〉, 〈5〉, and 〈4〉 in A. The first four runs have a PSV of A[0] = −∞ for their corresponding head elements, the next two head-PSVs point to A[7] = 1, thenext one to A[11] = 2, and the last two to A=-=[13]-=- = 3. Hence, the heads of the runs “destroy” exactly four of the n − n0 + 1 potential degree-1 nodes in the tree, so n1 = n − n0 − 4 + 1 = 16 − 9 − 3 = 4. Now TA, with degree-distribution n0, . . . , ... |

36 | Ultra-succinct representation of ordered trees
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Citation Context ...TA (i, j). Then if ℓ = i, rmq A(i, j) is i, otherwise, rmq A(i, j) is given by the child of ℓ that is on the path from ℓ to j [9]. Since there are succinct data structures supporting the LCA operator =-=[9,17]-=-. in succinctly encoded trees in constant time, this yields a succinct index (which we improve with Thms. 1 and 3). Lemma 2 (Fischer [9]). Given an array A[1, n] of totally ordered objects, there is a... |

36 | Succinct representations of permutations - MUNRO, RAMAN, et al. - 2003 |

35 | Lowest common ancestors in trees and directed acyclic graphs
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Citation Context ...mparisons within the array and overall linear time, which will be used in Thms. 4 and 5.i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A[i] 15 8 13 7 11 16 1 10 9 14 2 12 3 6 5 4 −∞ 15(1) 8(2) 7(4) 1(7) 13=-=(3)-=- 11(5) 10(8) 9(9) 16(6) 2(11) 14(10) 12(12) 3(13) 6(14) 5(15) 4(16) Fig. 1. An example of an array and its LRM-Tree. Lemma 1. Given an array A[1, n] of totally ordered objects, there is an algorithm c... |

34 | Fully-functional succinct trees
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Citation Context ...ility. 1 Introduction Introduced by Fischer [9] as an indexing data structure which supports Range Minimum Queries (RMQs) in constant time with no access to the main data, and by Sadakane and Navarro =-=[26]-=- to support navigation operators on ordinal trees, Left-to-Right-Minima Trees (LRM-Trees) are an elegant way to partition a sequence of values into sorted consecutive blocks, and to express the relati... |

30 | The cell probe complexity of succinct data structures
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Citation Context ...ear time, which will be used in Thms. 4 and 5.i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A[i] 15 8 13 7 11 16 1 10 9 14 2 12 3 6 5 4 −∞ 15(1) 8(2) 7(4) 1(7) 13(3) 11(5) 10(8) 9(9) 16(6) 2(11) 14(10) 12=-=(12)-=- 3(13) 6(14) 5(15) 4(16) Fig. 1. An example of an array and its LRM-Tree. Lemma 1. Given an array A[1, n] of totally ordered objects, there is an algorithm computing its LRM-Tree in at most 2(n − 1) c... |

29 | Implicit compression boosting with applications to self-indexing - MÄKINEN, NAVARRO |

22 | Optimal succinctness for range minimum queries
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Citation Context ...compressed succinct data structure for permutations supporting direct and inverse application in time inversely proportional to the permutation’s compressibility. 1 Introduction Introduced by Fischer =-=[9]-=- as an indexing data structure which supports Range Minimum Queries (RMQs) in constant time with no access to the main data, and by Sadakane and Navarro [26] to support navigation operators on ordinal... |

20 |
A framework for adaptive sorting
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Citation Context ...rison model requires Θ(nlgn) comparisons in the worst case over permutations of n elements. Yet, better results can be10 achieved for some parameterized classes of permutations. Petersson and Moffat =-=[29]-=- summarized many of them; we describe here a few that are relevant to our work. For a permutation π, Knuth [23] considered Runs (contiguous ascending subsequences), counted by ρ = 1 + |{i, 1 ≤ i < n, ... |

19 | Compressed representations of permutations, and applications
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Citation Context ...ported by a DFG grant (German Research Foundation).3. Based on this measure, we propose a new sorting algorithm and its adaptive analysis, asymptotically superior to sorting algorithms based on runs =-=[2]-=-, and on many instances faster than sorting algorithms based on subsequences [19]. 4. We design a compressed succinct data structure for permutations based on this measure, which supports the access o... |

18 | Alphabet partitioning for compressed rank/select and applications
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Citation Context ...itymeasureµ,whichisadatastructurerequiringlgf(n,µ)+o(lgf(n,µ)) bitsonanyinstanceofsizenandcompressibilityµ.Whiletheo(·)termisasymptotic in n, it is useful to allow µ to depend on n too. Barbay et al. =-=[1]-=- gave an example of such a structure for strings s[1..n] over an alphabet of size σ, supporting the access, rank and select operators in time O(lglgσ) while using nH0(s)+o(nH0(s)) bits of space, where... |

17 | Faster Entropy-Bounded Compressed Suffix Trees
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Citation Context ... ≤ n): Definition 2 (Range Minimum Queries). rmq A(i, j) = position of a minimum in A[i, j]. RMQs have a wide range of applications for various data structures and algorithms, including text indexing =-=[11]-=-, pattern matching [7], and more elaborate kinds of range queries [6]. For two given nodes i and j in a tree T , let lcaT (i, j) denote their Lowest Common Ancestor (LCA), that is, the deepest node th... |

16 | On the range maximumsum segment query problem”, Discrete Applied Mathematics
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Citation Context ... a minimum in A[i, j]. RMQs have a wide range of applications for various data structures and algorithms, including text indexing [11], pattern matching [7], and more elaborate kinds of range queries =-=[6]-=-. For two given nodes i and j in a tree T , let lcaT (i, j) denote their Lowest Common Ancestor (LCA), that is, the deepest node that is an ancestor of both i and j. Now let TA be the LRM-Tree of A. F... |

11 | T.: Improved algorithms for the range next value problem and applications
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Citation Context ...nge Minimum Queries). rmq A(i, j) = position of a minimum in A[i, j]. RMQs have a wide range of applications for various data structures and algorithms, including text indexing [11], pattern matching =-=[7]-=-, and more elaborate kinds of range queries [6]. For two given nodes i and j in a tree T , let lcaT (i, j) denote their Lowest Common Ancestor (LCA), that is, the deepest node that is an ancestor of b... |

11 |
Partitioning permutations into increasing and decreasing subsequences
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Citation Context ... techniques to adapt our compressed data structure to this new setting. Since finding a minimum-size partitioning into up- and down-sequences when considering general subsequences [24] is NP-complete =-=[22]-=-, RC-Sorting seems a much desirable improvement on merging ascending and descending runs, as well as a more practical alternative to a hypothetical exponential-time SMS-Sorting algorithm, in an even s... |

8 | On space efficient two dimensional range minimum data structures
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Citation Context ...ons within the array and overall linear time, which will be used in Thms. 4 and 5.i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A[i] 15 8 13 7 11 16 1 10 9 14 2 12 3 6 5 4 −∞ 15(1) 8(2) 7(4) 1(7) 13(3) 11=-=(5)-=- 10(8) 9(9) 16(6) 2(11) 14(10) 12(12) 3(13) 6(14) 5(15) 4(16) Fig. 1. An example of an array and its LRM-Tree. Lemma 1. Given an array A[1, n] of totally ordered objects, there is an algorithm computi... |

7 | Sorting and selection in posets
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Citation Context ...thin the array and overall linear time, which will be used in Thms. 4 and 5.i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A[i] 15 8 13 7 11 16 1 10 9 14 2 12 3 6 5 4 −∞ 15(1) 8(2) 7(4) 1(7) 13(3) 11(5) 10=-=(8)-=- 9(9) 16(6) 2(11) 14(10) 12(12) 3(13) 6(14) 5(15) 4(16) Fig. 1. An example of an array and its LRM-Tree. Lemma 1. Given an array A[1, n] of totally ordered objects, there is an algorithm computing its... |

7 |
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Citation Context ... propose a new sorting algorithm and its adaptive analysis, asymptotically superior to sorting algorithms based on runs [2], and on many instances faster than sorting algorithms based on subsequences =-=[19]-=-. 4. We design a compressed succinct data structure for permutations based on this measure, which supports the access operator and its inverse in time inversely proportional to the permutation’s preso... |

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5 |
H.M.: Practical entropy bounded schemes for O(1)range minimum queries
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Citation Context ...all linear time, which will be used in Thms. 4 and 5.i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A[i] 15 8 13 7 11 16 1 10 9 14 2 12 3 6 5 4 −∞ 15(1) 8(2) 7(4) 1(7) 13(3) 11(5) 10(8) 9(9) 16(6) 2(11) 14=-=(10)-=- 12(12) 3(13) 6(14) 5(15) 4(16) Fig. 1. An example of an array and its LRM-Tree. Lemma 1. Given an array A[1, n] of totally ordered objects, there is an algorithm computing its LRM-Tree in at most 2(n... |

3 | On compressing permutations and adaptive sorting
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Citation Context ...e parent node of i. The children of each node are ordered in increasing order from left to right.8 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A[i] 15 8 13 7 11 16 1 10 9 14 2 12 3 6 5 4 −∞ 15(1) 8(2) 7=-=(4)-=- 1(7) 13(3) 11(5) 10(8) 9(9) 16(6) 2(11) 14(10) 12(12) 3(13) 6(14) 5(15) 4(16) Fig.1. An example of an array and its LRM-Tree. The numbers at the nodes are the array values, and the smaller numbers in... |