The Geometry of Binary Search Trees

Demaine, Erik D.; Harmon, Dion; Iacono, John; Kane, Daniel M.; Pâtraşcu, Mihai

doi:10.1137/1.9781611973068.55

Cited by 44 publications

(161 citation statements)

References 14 publications

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“…It seems likely that the ways in which this permutation evolves could be captured by a forbidden substructure argument. Haeupler et al proved their algorithm runs in Opn [7] demonstrated that a greedy offline binary search tree could be simulated online using a technique that resembles path compression. Is it possible to prove that this algorithm has logarithmic access time using some forbidden substructure argument?…”

Section: Discussionmentioning

confidence: 99%

“…The number of type (iii) rotations is at most n since each such rotation reduces the number of nodes on the left path of an inactive block. 7 The number of type (iv) rotations may be bounded inductively. In the analysis below we build an n{B ¢ n{B period-block incidence matrix A, which is initially zero.…”

Section: Sequential Access In Splay Treesmentioning

confidence: 99%

“…The above analysis shows that T pnq ¤ 6 The left depth of a node x is the number of its strict ancestors greater than x. 7 The cost of (iii) is already counted in (i) but we keep it distinct nonetheless. nplog B 2q 2°n {B i1 T p|Z i |q.…”

Section: Sequential Access In Splay Treesmentioning

confidence: 99%

“…Demaine et al [7] showed an interesting equivalence between the intrinsic cost of an access sequence in a binary search tree and a problem on 0-1 matrices that does not involve forbidden substructure.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Applications of Forbidden 0–1 Matrices to Search Tree and Path Compression-Based Data Structures

Pettie

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we improve, reprove, and simplify several theorems on the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (non-geometric) algorithms and data structures, namely, to bound the complexity of an object via its forbidden substructures.To analyze an algorithm or data structure in the forbidden substructure framework one proceeds in three discrete steps. First, one transcribes the behavior of the algorithm as some combinatorial object M ; for example, M may be a graph, sequence, permutation, matrix, set system, or tree. (The size of M should ideally be linear in the running time.) Second, one shows that M excludes some forbidden substructure P , and third, one bounds the size of any object avoiding this substructure. The power of this framework derives from the fact that M lies in a more pristine environment and that upper bounds on the size of a P -free object M may be reused in different contexts.Among our results, we present the first asymptotically sharp bound on the length of arbitrary path compressions on arbitrary trees, improving analyses of Tarjan [35] and Seidel and Sharir [31]. We reprove the linear bound on postordered path compressions, due to Lucas [23] and Loebel and Nesetȓil [22], the linear bound on deque-ordered path compressions, due to Buchsbaum, Sundar, and Tarjan [5], and the sequential access theorem for splay trees, originally due to Tarjan [38]. We disprove a conjecture of Aronov et al. [3] related to the efficiency of their data structure for half-plane proximity queries and provide a significantly cleaner analysis of their structure. With the exception of the sequential access theorem, all our proofs are exceptionally simple. Notably absent are calculations of any kind.

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Section: Discussionmentioning

confidence: 99%

Section: Sequential Access In Splay Treesmentioning

confidence: 99%

Section: Sequential Access In Splay Treesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Applications of Forbidden 0–1 Matrices to Search Tree and Path Compression-Based Data Structures

Pettie

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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“…An earlier, similar, concept is competitive analysis [12], whose differences from instance optimality are nicely explained in [17]. Instance optimal algorithms have been designed for many other problems, such as manipulating binary search trees [15], approximating the distance from a point to a curve [8], computing the union/intersection of sorted lists [16], finding the convex hull of polygons [9], to mention just a few. The most recent work to our knowledge is [1], which proposes instance optimal algorithms for several computational geometry problems.…”

Section: Problem and Related Workmentioning

confidence: 99%

Finding maximum degrees in hidden bipartite graphs

Tao

Sheng

2010

Proceedings of the 2010 ACM SIGMOD International Conference on Management of Data

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An (edge) hidden graph is a graph whose edges are not explicitly given. Detecting the presence of an edge requires expensive edgeprobing queries. We consider the k most connected vertex problem on hidden bipartite graphs. Specifically, given a bipartite graph G with independent vertex sets B and W , the goal is to find the k vertices in B with the largest degrees using the minimum number of queries. This problem can be regarded as a top-k extension of a semi-join, and is encountered in many applications in practice (e.g., top-k spatial join with arbitrarily complex join predicates).If B and W have n and m vertices respectively, the number of queries needed to solve the problem is nm in the worst case. This, however, is a pessimistic estimate on how many queries are necessary on practical data. In fact, on some easy inputs, the problem can be efficiently settled with only km+n edges, which is significantly lower than nm for k n. The huge difference between km + n and nm makes it interesting to design an adaptive algorithm that is guaranteed to achieve the best possible performance on every input G. We give such an algorithm, and prove that it is instance optimal among a broad class of solutions. This means that, for any G, our algorithm can perform more queries than the optimal solution (which is currently unknown) by only a constant factor, which can be shown to be at most 2. Extensive experiments demonstrate that, in practice, the number of queries required by our technique is far less than nm, and agrees with our theoretical findings very well.

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O(log log n)-Competitive Binary Search Tree

Wang¹,

Sleator²

2016

Encyclopedia of Algorithms

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The Geometry of Binary Search Trees

Cited by 44 publications

References 14 publications

Applications of Forbidden 0–1 Matrices to Search Tree and Path Compression-Based Data Structures

Applications of Forbidden 0–1 Matrices to Search Tree and Path Compression-Based Data Structures

Finding maximum degrees in hidden bipartite graphs

O(log log n)-Competitive Binary Search Tree

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