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13
Fullyfunctional succinct trees
 In Proc. 21st SODA
, 2010
"... We propose new succinct representations of ordinal trees, which have been studied extensively. It is known that any nnode static tree can be represented in 2n + o(n) bits and a large number of operations on the tree can be supported in constant time under the wordRAM model. However existing data s ..."
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Cited by 33 (12 self)
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We propose new succinct representations of ordinal trees, which have been studied extensively. It is known that any nnode static tree can be represented in 2n + o(n) bits and a large number of operations on the tree can be supported in constant time under the wordRAM model. However existing data structures are not satisfactory in both theory and practice because (1) the lowerorder term is Ω(nlog log n / log n), which cannot be neglected in practice, (2) the hidden constant is also large, (3) the data structures are complicated and difficult to implement, and (4) the techniques do not extend to dynamic trees supporting insertions and deletions of nodes. We propose a simple and flexible data structure, called the range minmax tree, that reduces the large number of relevant tree operations considered in the literature to a few primitives, which are carried out in constant time on sufficiently small trees. The result is then extended to trees of arbitrary size, achieving 2n + O(n/polylog(n)) bits of space. The redundancy is significantly lower than in any previous proposal, and the data structure is easily implemented. Furthermore, using the same framework, we derive the first fullyfunctional dynamic succinct trees. 1
Succinct Dynamic Data Structures
"... We develop succinct data structures to represent (i) a sequence of values to support partial sum and select queries and update (changing values) and (ii) a dynamic array consisting of a sequence of elements which supports insertion, deletion and access of an element at any given index. For the parti ..."
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Cited by 25 (2 self)
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We develop succinct data structures to represent (i) a sequence of values to support partial sum and select queries and update (changing values) and (ii) a dynamic array consisting of a sequence of elements which supports insertion, deletion and access of an element at any given index. For the partial sums problem...
Fullyfunctional static and dynamic succinct trees
, 2010
"... We propose new succinct representations of ordinal trees, which have been studied extensively. It is known that any nnode static tree can be represented in 2n + o(n) bits and various operations on the tree can be supported in constant time under the wordRAM model. However the data structures are c ..."
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Cited by 18 (11 self)
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We propose new succinct representations of ordinal trees, which have been studied extensively. It is known that any nnode static tree can be represented in 2n + o(n) bits and various operations on the tree can be supported in constant time under the wordRAM model. However the data structures are complicated and difficult to dynamize. We propose a simple and flexible data structure, called the range minmax tree, that reduces the large number of relevant tree operations considered in the literature, to a few primitives that are carried out in constant time on sufficiently small trees. The result is extended to trees of arbitrary size, achieving 2n + O(n/polylog(n)) bits of space. The redundancy is significantly lower than any previous proposal. For the dynamic case, where insertion/deletion of nodes is allowed, the existing data structures support very limited operations. Our data structure builds on the range minmax tree to achieve 2n + O(n / log n) bits of space and O(log n) time for all the operations. We also propose an improved data structure using 2n+O(n loglog n / logn) bits and improving the time to O(log n / loglog n) for most operations.
LinearTime Compression of BoundedGenus Graphs into InformationTheoretically Optimal Number of Bits
, 2002
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Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 12 (1 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
An Improved Succinct Representation for Dynamic kary Trees
"... Abstract. kary trees are a fundamental data structure in many textprocessing algorithms (e.g., text searching). The traditional pointerbased representation of trees is space consuming, and hence only relatively small trees can be kept in main memory. Nowadays, however, many applications need to st ..."
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Cited by 6 (2 self)
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Abstract. kary trees are a fundamental data structure in many textprocessing algorithms (e.g., text searching). The traditional pointerbased representation of trees is space consuming, and hence only relatively small trees can be kept in main memory. Nowadays, however, many applications need to store a huge amount of information. In this paper we present a succinct representation for dynamic kary trees of n nodes, requiring 2n + nlog k + o(nlog k) bits of space, which is close to the informationtheoretic lower bound. Unlike alternative representations where the operations on the tree can be usually computed in O(log n) time, our data structure is able to take advantage of asymptotically smaller values of k, supporting the basic operations parent and child in O(log k+log log n) time, which is o(log n) time whenever log k = o(log n). Insertions and deletions of leaves in the tree are supported log k in O((log k + log log n)(1 +)) amortized time. Our replog (log k+log log n) resentation also supports more specialized operations (like subtreesize, depth, etc.), and provides a new tradeoff when k = O(1) allowing faster updates (in O(log log n) amortized time, versus the amortized time of O((log log n) 1+ǫ), for ǫ> 0, from Raman and Rao [21]), at the cost of slower basic operations (in O(log log n) time, versus O(1) time of [21]). 1
A uniform approach towards succinct representation of trees
 In Proc. 11th Scandinavian Workshop on Algorithm Theory (SWAT), LNCS 5124
, 2008
"... Abstract. We propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decomposition of trees into subtrees. Recursive decomposition of a structure into substructures is a common technique in succinct data structures and has been shown ..."
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Cited by 4 (0 self)
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Abstract. We propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decomposition of trees into subtrees. Recursive decomposition of a structure into substructures is a common technique in succinct data structures and has been shown fruitful in succinct representation of ordinal trees [7, 10] and dynamic binary trees [16, 21]. We take an approach that simplifies the existing representation of ordinal trees while allowing the full set of navigational operations. The approach applied to cardinal (i.e. kary) trees yields a spaceoptimal succinct representation allowing cardinaltype operations (e.g. determining child labeled i) as well as the full set of ordinaltype operations (e.g. reporting the number of siblings to the left of a node). Existing spaceoptimal succinct representations had not been able to support both types of operations [2, 19]. We demonstrate how the approach can be applied to obtain a spaceoptimal succinct representation for the family of free trees where the order of children is insignificant. Furthermore, we show that our approach can be used to obtain entropybased succinct representations. We show that our approach matches the degreedistribution entropy suggested by Jansson et al. [13]. We discuss that our approach can be made adaptive to various other entropy measures. 1
Succinct Representations of Binary Trees for Range Minimum Queries
"... Abstract. We provide two succinct representations of binary trees that can be used to represent the Cartesian tree of an array A of size n. Both the representations take the optimal 2n + o(n) bits of space in the worst case and support range minimum queries (RMQs) in O(1) time. The first one is a mo ..."
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Cited by 4 (2 self)
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Abstract. We provide two succinct representations of binary trees that can be used to represent the Cartesian tree of an array A of size n. Both the representations take the optimal 2n + o(n) bits of space in the worst case and support range minimum queries (RMQs) in O(1) time. The first one is a modification of the representation of Farzan and Munro (SWAT 2008); a consequence of this result is that we can represent the Cartesian tree of a random permutation in 1.92n + o(n) bits in expectation. The second one uses a wellknown transformation between binary trees and ordinal trees, and ordinal tree operations to effect operations on the Cartesian tree. This provides an alternative, and more natural, way to view the 2DMinHeap of Fischer and Huen (SICOMP 2011). Furthermore, we show that the preprocessing needed to output the data structure can be performed in linear time using o(n) bits of extra working space, improving the result of Fischer and Heun who use n + o(n) bits working space. 1
Dynamic updates of succinct triangulations
, 2005
"... In a recent article, we presented a succinct representation of triangulations that supports efficient navigation operations. Here this representation is improved to allow for efficient local updates of the triangulations. Precisely, we show how a succinct representation of a triangulation with m tri ..."
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Cited by 2 (1 self)
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In a recent article, we presented a succinct representation of triangulations that supports efficient navigation operations. Here this representation is improved to allow for efficient local updates of the triangulations. Precisely, we show how a succinct representation of a triangulation with m triangles can be maintained under vertex insertions in O(1) amortized time and under vertex deletions/edge flips in O(lg 2 m) amortized time. Our structure achieves the information theory bound for the storage for the class of triangulations with a boundary, requiring asymptotically 2.17m + o(m) bits, and supports adjacency queries between triangles in O(1) time (an extra amount of O(g lg m) bits are needed for representing triangulations of genus g surfaces).
Succinct Geometric Indexes Supporting Point Location Queries
"... We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succi ..."
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Cited by 2 (1 self)
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We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time1. We also design three variants of this index. The first supports point location using lg n +2 √ lg n + O(lg 1/4 n) pointline comparisons. The second supports point location in o(lg n) time when the coordinates are integers bounded by U. The last variant can answer point location queries in O(H +1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy O(n) words or O(n lg n) bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in O(lg 2 n) time. 1