Succinct representations of weighted trees supporting path queries

Patil, Manish; Shah, Rahul; Thankachan, Sharma V.

doi:10.1016/j.jda.2012.08.003

Cited by 13 publications

(10 citation statements)

References 16 publications

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“…This is the case for range/path minimum [13], range/path selection [28,29,37], and range/path top-k queries (this paper, Sect. 7).…”

Section: Discussion and Directions For Future Researchmentioning

confidence: 91%

“…A variety of other range query problems have been examined on arrays and trees, including range minimum/extrema, for which optimal data structures exist requiring O(n) space and O(1) query time (e.g., on arrays [5,[12][13][14]21] and on trees [13]) and range selection/median, for which optimal data structures exist requiring O(n) space and O(log n/ log log n) query time (e.g., on arrays [7,25,26,31,34] and on trees [28,29,37]). Jørgensen and Larsen [31] proved a matching lower bound of Ω(log n/ log log n) on the worst-case time required for range median query on arrays by any O(n)-space data structure.…”

Section: Related Workmentioning

confidence: 99%

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Linear-Space Data Structures for Range Frequency Queries on Arrays and Trees

et al. 2014

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We present O(n)-space data structures to support various range frequency queries on a given array A[0 : n − 1] or tree T with n nodes. Given a query consisting of an arbitrary pair of pre-order rank indices (i, j), our data structures return a least frequent element, mode, α-minority, or top-k colors (values) of the multiset of elements in the unique path with endpoints at indices i and j in A or T . We describe a data structure that supports range least frequent element queries on arrays in O( √ n/w) time, improving the Θ( √ n) worst-case time required by the data structure of Chan et al. (SWAT 2012), where w ∈ Ω(log n) is the word size in bits. We describe a data structure that supports path mode queries on trees in O(log log n √ n/w) time, improving the Θ( √ n log n) worst-case time required by the data structure of Krizanc et al. (ISAAC 2003). We describe the first data structures to support path least frequent 123 Algorithmica element queries, path α-minority queries, and path top-k color queries on trees in O(log log n √ n/w), O(α −1 log log n), and O(k) time, respectively, where α ∈ [0, 1] and k ∈ {1, . . . , n} are specified at query time.

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“…This is the case for range/path minimum [13], range/path selection [28,29,37], and range/path top-k queries (this paper, Sect. 7).…”

Section: Discussion and Directions For Future Researchmentioning

confidence: 91%

Section: Related Workmentioning

confidence: 99%

Linear-Space Data Structures for Range Frequency Queries on Arrays and Trees

et al. 2014

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“…The types of queries we consider are PM, PC, and PR. The theoretical foundation of our work are the data structures and algorithms developed in [40,27,28]. The succinct data structure by He et al [28] is optimal both in space and time in the RAM model.…”

Section: Our Workmentioning

confidence: 99%

Path Query Data Structures in Practice

He,

Kazi

2020

Preprint

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We perform an experimental study on data structures that answer path median, path counting, and path reporting queries in weighted trees. These query problems generalize the well-known problems of the range median query problem in arrays, as well as the 2d orthogonal range counting and reporting in planar point sets, to tree structured data. We propose practical realizations of the latest theoretical results on path queries. Our data structures, which use tree extraction, heavy-path decomposition and wavelet trees, are implemented in both succinct and plain pointer-based form. Our succinct data structures are further specialized into entropycompressed and plain forms. Through a set of experiments on large datasets, we show that succinct data structures for path queries present a viable alternative to standard pointer-based realizations in practical scenarios. Compared to naïve approaches which do not preprocess the tree but rather compute the answer by explicitly traversing the query path, our succinct data structures are several times faster in path median queries and perform comparably in path counting and path reporting queries, while being several times more space-efficient. Plain pointer-based realizations of our data structures, requiring a few times more space than the naïve structures, yield a 30-100-times speed-up over them. In addition, our succinct data structures provide more functionality within the little space they use than their plain pointer-based counterparts. ACM Subject Classification Information systems → Data structuresKeywords and phrases path query path median path counting path reporting weighted tree Digital Object Identifier 10.4230/LIPIcs...

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“…He et al [17] introduced and solved path reporting problem using linear space and O((1 + k) lg σ) query time, and O(n lg lg σ) words of space but O(lg σ + k lg lg σ) query time, in the word-RAM model; henceforth we reserve k for the size of the output. Patil et al [23] presented a succinct data structure for path reporting with n lg σ + 6n + O(n lg σ) bits of space and O((lg n + k) lg σ) query time. An optimal-space solution with nH(W T ) + O(n lg σ) bits of space and O((1 + k)( lg σ lg log n + 1)) reporting time is due to He et al [17].…”

Section: Previous Workmentioning

confidence: 99%

Path and Ancestor Queries on Trees with Multidimensional Weight Vectors

He,

Kazi

2019

Preprint

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We consider an ordinal tree T on n nodes, with each node assigned a d-dimensional weight vector w ∈ {1, 2, . . . , n} d , where d ∈ N is a constant. We study path queries as generalizations of wellknown orthogonal range queries, with one of the dimensions being tree topology rather than a linear order. Since in our definitions d only represents the number of dimensions of the weight vector without taking the tree topology into account, a path query in a tree with d-dimensional weight vectors generalize the corresponding (d + 1)-dimensional orthogonal range query. We solve ancestor dominance reporting problem as a direct generalization of dominance reporting problem, in timewhere k is the size of the output, for d ≥ 2. We also achieve a tradeoff of O(n lg d−2+ n) words of space, with query time of O((lg d−1 n)/(lg lg n) d−2 + k), for the same problem, when d ≥ 3. We solve path successor problem in O(n lg d−1 n) words of space and time O(lg d−1+ n) for d ≥ 1 and an arbitrary constant > 0. We propose a solution to path counting problem, with O(n(lg n/ lg lg n) d−1 ) words of space and O((lg n/ lg lg n) d ) query time, for d ≥ 1. Finally, we solve path reporting problem in O(n lg d−1+ n) words of space and O((lg d−1 n)/(lg lg n) d−2 + k) query time, for d ≥ 2. These results match or nearly match the best tradeoffs of the respective range queries. We are also the first to solve path successor even for d = 1. 2012 ACM Subject Classification Information systems → Data structures; Theory of computation → Data structures design and analysis; Information systems → Multidimensional range search Keywords and phrases path queries range queries algorithms data structures theory Digital Object Identifier 10.4230/LIPIcs...

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Succinct representations of weighted trees supporting path queries

Cited by 13 publications

References 16 publications

Linear-Space Data Structures for Range Frequency Queries on Arrays and Trees

Linear-Space Data Structures for Range Frequency Queries on Arrays and Trees

Path Query Data Structures in Practice

Path and Ancestor Queries on Trees with Multidimensional Weight Vectors

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