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Finger Search Trees with Constant Insertion Time
 In Proc. 9th Annual ACMSIAM Symposium on Discrete Algorithms
, 1997
"... We consider the problem of implementing finger search trees on the pointer machine, i.e., how to maintain a sorted list such that searching for an element x, starting the search at any arbitrary element f in the list, only requires logarithmic time in the distance between x and f in the list. We pr ..."
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Cited by 15 (3 self)
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We consider the problem of implementing finger search trees on the pointer machine, i.e., how to maintain a sorted list such that searching for an element x, starting the search at any arbitrary element f in the list, only requires logarithmic time in the distance between x and f in the list. We present the first pointerbased implementation of finger search trees allowing new elements to be inserted at any arbitrary position in the list in worst case constant time. Previously, the best known insertion time on the pointer machine was O(log n), where n is the total length of the list. On a unitcost RAM, a constant insertion time has been achieved by Dietz and Raman by using standard techniques of packing small problem sizes into a constant number of machine words. Deletion of a list element is supported in O(log n) time, which matches the previous best bounds. Our data structure requires linear space. 1 Introduction A finger search tree is a data structure which stores a sorte...
Binary Search Trees of Almost Optimal Height
 ACTA INFORMATICA
, 1990
"... First we present a generalization of symmetric binary Btrees, SBB(k) trees. The obtained structure has a height of only \Sigma (1 + 1k) log(n + 1)\Upsilon, where k may be chosen to be any positive integer. The maintenance algorithms require only a constant number of rotations per updating operati ..."
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Cited by 11 (1 self)
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First we present a generalization of symmetric binary Btrees, SBB(k) trees. The obtained structure has a height of only \Sigma (1 + 1k) log(n + 1)\Upsilon, where k may be chosen to be any positive integer. The maintenance algorithms require only a constant number of rotations per updating operation in the worst case. These properties together with the fact that the structure is relatively simple to implement makes it a useful alternative to other search trees in practical applications. Then, by using an SBB(k)tree with a varying k we achieve a structure with a logarithmic amortized cost per update and a height of log n + o(log n). This result is an improvement of the upper bound on the height of a dynamic binary search tree. By maintaining two trees simultaneously the amortized cost is transformed into a worstcase cost. Thus, we have improved the worstcase complexity of the dictionary problem.
Ranksensitive data structures
 In Proc. 12th International Symposium on String Processing and Information Retrieval (SPIRE), LNCS v. 3772
, 2005
"... Abstract. Outputsensitive data structures result from preprocessing n items and are capable of reporting the items satisfying an online query in O(t(n) + ℓ) time, where t(n) is the cost of traversing the structure and ℓ ≤ n is the number of reported items satisfying the query. In this paper we foc ..."
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Cited by 9 (0 self)
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Abstract. Outputsensitive data structures result from preprocessing n items and are capable of reporting the items satisfying an online query in O(t(n) + ℓ) time, where t(n) is the cost of traversing the structure and ℓ ≤ n is the number of reported items satisfying the query. In this paper we focus on ranksensitive data structures, which are additionally given a ranking of the n items, so that just the top k bestranking items should be reported at query time, sorted in rank order, at a cost of O(t(n) + k) time. Note that k is part of the query as a parameter under the control of the user (as opposed to ℓ which is querydependent). We explore the problem of adding ranksensitivity to data structures such as suffix trees or range trees, where the ℓ items satisfying the query form O(polylog(n)) intervals of consecutive entries from which we choose the top k bestranking ones. Letting s(n) be the number of items (including their copies) stored in the original data structures, we increase the space by an additional term of O(s(n) lg ǫ n) memory words of space, each of O(lg n) bits, for any positive constant ǫ < 1. We allow for changing the ranking on the fly during the lifetime of the data structures, with ranking values in 0... O(n). In this case, query time becomes O(t(n)+k) plus O(lg n/lg lg n) per interval; each change in the ranking and each insertion/deletion of an item takes O(lg n) time; the additional term in space occupancy increases to O(s(n) lg n/lg lg n). 1
Fully Dynamic Transitive Closure in Plane Dags with One Source and One Sink
, 1994
"... We give an algorithm for the Dynamic Transitive Closure Problem for planar directed acyclic graphs with one source and one sink. The graph can be updated in logarithmic time under arbitrary edge insertions and deletions that preserve the embedding. Queries of the form `is there a directed path from ..."
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Cited by 2 (2 self)
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We give an algorithm for the Dynamic Transitive Closure Problem for planar directed acyclic graphs with one source and one sink. The graph can be updated in logarithmic time under arbitrary edge insertions and deletions that preserve the embedding. Queries of the form `is there a directed path from u to v?' for arbitrary vertices u and v can be answered in logarithmic time. The size of the data structure and the initialisation time are linear in the number of edges. We also give a lower bound of###26 n/ log log n) on the amortised complexity of the problem in the cell probe model with logarithmic word size.
Jaywalking your Dog – Computing the Fréchet Distance with Shortcuts ∗
, 2011
"... The similarity of two polygonal curves can be measured using the Fréchet distance. We introduce the notion of a more robust Fréchet distance, where one is allowed to shortcut between vertices of one of the curves. This is a natural approach for handling noise, in particular batched outliers. We comp ..."
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Cited by 2 (0 self)
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The similarity of two polygonal curves can be measured using the Fréchet distance. We introduce the notion of a more robust Fréchet distance, where one is allowed to shortcut between vertices of one of the curves. This is a natural approach for handling noise, in particular batched outliers. We compute a constant factor approximation to the minimum Fréchet distance over all possible such shortcuts. Our algorithm runs in O ( c 2 kn log 3 n) time if one is allowed to take at most k shortcuts and the input curves are cpacked. For the case where the number of shortcuts is unrestricted, we describe an algorithm which runs in O ( c 2 n log 3 n) time. To facilitate the new algorithm we develop several new datastructures, which we believe to be of independent interest: (i) for range reporting on a curve, and (ii) for preprocessing a curve to answer queries for the Fréchet distance between a subcurve and a line segment. 1
Cryptographic Accumulators for Authenticated Hash Tables ∗
, 2009
"... Hash tables are fundamental data structures that optimally answer membership queries. Suppose a client stores n elements in a hash table that is outsourced at a remote server. Authenticating the hash table functionality, i.e., verifying the correctness of queries answered by the server and ensuring ..."
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Cited by 1 (0 self)
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Hash tables are fundamental data structures that optimally answer membership queries. Suppose a client stores n elements in a hash table that is outsourced at a remote server. Authenticating the hash table functionality, i.e., verifying the correctness of queries answered by the server and ensuring the integrity of the stored data, is crucial because the server, lying outside the administrative control of the client, can be malicious. We design efficient and secure protocols for optimally authenticating (non)membership queries on hash tables, using cryptographic accumulators as our basic security primitive and applying them in a novel hierarchical way over the stored data. We provide the first construction for authenticating a hash table with constant query cost and sublinear update cost, strictly improving upon previous methods. Our first solution, based on the RSA accumulator, allows the server to provide a proof of integrity of the answer to a membership query in constant time and supports updates in O (n ǫ log n) time for any fixed constant 0 < ǫ < 1, yet keeping the communication and verification costs constant. It also lends itself to a scheme that achieves different tradeoffs—namely,
Exercise 23.16 in [6].
, 2000
"... 2: Array initialization Section III.8.1 of [15] contains a description of how a bitvector can be intitialized in worst case constant time. ..."
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2: Array initialization Section III.8.1 of [15] contains a description of how a bitvector can be intitialized in worst case constant time.
Maintaining alphabalanced Trees by Partial Rebuilding
 International Journal of Computer Mathematics
, 1991
"... The balance criterion defining the class of ffbalanced trees states that the ratio between the shortest and longest paths from a node to a leaf be at least ff. We show that a straightforward use of partial rebuilding for maintenance of ffbalanced trees requires an amortized cost of \Omega\Gamma ..."
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The balance criterion defining the class of ffbalanced trees states that the ratio between the shortest and longest paths from a node to a leaf be at least ff. We show that a straightforward use of partial rebuilding for maintenance of ffbalanced trees requires an amortized cost of \Omega\Gamma p n) per update. By slight modifications of the maintenance algorithms the cost can be reduced to O(log n) for any value of ff, 0 ! ff ! 1. KEY WORDS ffbalanced trees, partial rebuilding, search trees. CR CATEGORIES: E.1, F.2, I.1.2. 1 Introduction In his thesis Olivie [9] introduced a class of binary search trees, which he calls ffbalanced trees, or ffBBtrees. Let h(v) denote the length for the longest path from a node v to a leaf and let s(v) denote the length of the shortest path. We give a formal definition of ffbalanced trees below. Definition 1 A binary tree is ffbalanced if the following is true for each node v in the tree: s(v) h(v) ff; h(v) 1 1 \Gamma ff (1) h(v) \...
Literature Notes on Homeworks and the Takehome Exam
, 2000
"... lgorithm Exercise 2 of homework 4 in [13]. Prim's algorithm for computing a minimum spanning tree is, e.g., described in Section 24.2 of [6]. 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13]. Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n ..."
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lgorithm Exercise 2 of homework 4 in [13]. Prim's algorithm for computing a minimum spanning tree is, e.g., described in Section 24.2 of [6]. 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13]. Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution in [8]. 1 Algorithms January 17, 2000 Homework 3 1: Minimum Spanning Trees A linear time algorithm for nding a minimum spanning tree for planar graph was rst given in [5]. The O(m log n) time algorithm for nding a minimum spanning tree in a general graph was described in [7]the paper introducing Fibonacci heaps. The current best dertministic minimum spanning tree algorithms use time O(m(m;n)), where is an inverse of Ackerman's function [4, 17]. A randomized
Literature Notes on Homeworks
, 2001
"... cise 2 of homework 4 in [13]. Prim's algorithm for computing a minimum spanning tree is, e.g., described in Section 24.2 of [6]. 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13]. Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution ..."
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cise 2 of homework 4 in [13]. Prim's algorithm for computing a minimum spanning tree is, e.g., described in Section 24.2 of [6]. 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13]. Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution in [8]. 1 Algorithms February 13, 2001 Homework 3 1: Minimum Spanning Trees A linear time algorithm for nding a minimum spanning tree for planar graph was rst given in [5]. The O(m log n) time algorithm for nding a minimum spanning tree in a general graph was described in [7]the paper introducing Fibonacci heaps. The current best dertministic minimum spanning tree algorithms use time O(m(m;n)), where is an inverse of Ackerman's function [4, 17]. A randomized linear time algorit