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Splay Trees
A splay tree is a binary search tree with the additional property that recently accessed elements are quick to access again. Like self-balancing binary search trees, a splay tree performs basic operations such as insertion, look-up and removal in big O notation, O(log ''n'') amortized analysis, amortized time. For random access patterns drawn from a non-uniform random distribution, their amortized time can be faster than logarithmic, proportional to the Entropy (information theory), entropy of the access pattern. For many patterns of non-random operations, also, splay trees can take better than logarithmic time, without requiring advance knowledge of the pattern. According to the unproven dynamic optimality conjecture, their performance on all access patterns is within a constant factor of the best possible performance that could be achieved by any other self-adjusting binary search tree, even one selected to fit that pattern. The splay tree was invented by Daniel Sleator and Rob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Daniel Dominic Sleator
Daniel Dominic Kaplan Sleator (born 10 December 1953) is a professor of computer science at Carnegie Mellon University, Pittsburgh, United States. In 1999, he won the Association for Computing Machinery, ACM Paris Kanellakis Award (jointly with Robert Tarjan) for the splay tree data structure. He was one of the pioneers in amortized analysis of algorithms, early examples of which were the analyses of the Move-to-front transform, move-to-front heuristic, and splay trees. He invented many data structures with Robert Tarjan, such as splay trees, link/cut trees, and skew heaps. The Sleator and Tarjan paper on the move-to-front heuristic first suggested the idea of comparing an online algorithm to an optimal offline algorithm, for which the term Competitive analysis (online algorithm), competitive analysis was later coined in a paper of Anna Karlin, Karlin, Manasse, Rudolph, and Sleator. Sleator also developed the theory of link grammars, and the Serioso music analyzer for analyzing m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Zigzag
A zigzag is a pattern made up of small corners at variable angles, though constant within the zigzag, tracing a path between two parallel lines; it can be described as both jagged and fairly regular. In geometry, this pattern is described as a Infinite skew polygon, skew apeirogon. From the point of view of symmetry, a regular zigzag can be generated from a simple motif like a line segment by repeated application of a glide reflection. Although the origin of the word is unclear, its first printed appearances were in French-language books and ephemera of the late 17th century. Examples of zigzags * The trace of a triangle wave or a sawtooth wave is a zigzag. * Pinking shears are designed to cut cloth or paper with a zigzag edge, to lessen fraying. * In sewing, a ''zigzag stitch'' is a sewing machine, machine stitch in a zigzag pattern. * The zigzag arch is an architectural embellishment used in Islamic architecture, Islamic, Byzantine architecture, Byzantine, Norman archite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Link/cut Tree
A link/cut tree is a data structure for representing a forest, a set of rooted trees, and offers the following operations: * Add a tree consisting of a single node to the forest. * Given a node in one of the trees, disconnect it (and its subtree) from the tree of which it is part. * Attach a node to another node as its child. * Given a node, find the root of the tree to which it belongs. By doing this operation on two distinct nodes, one can check whether they belong to the same tree. The represented forest may consist of very deep trees, so if we represent the forest as a plain collection of parent pointer trees, it might take us a long time to find the root of a given node. However, if we represent each tree in the forest as a link/cut tree, we can find which tree an element belongs to in ''O''(log(''n'')) amortized time. Moreover, we can quickly adjust the collection of link/cut trees to changes in the represented forest. In particular, we can adjust it to merge (link) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Iacono's Working Set Structure
In computer science, Iacono's working set structure is a comparison based dictionary. It supports insertion, deletion and access operation to maintain a dynamic set of n elements. The working set of an item x is the set of elements that have been accessed in the structure since the last time that x was accessed (or inserted if it was never accessed). Inserting and deleting in the working set structure takes O(\log n) time while accessing an element x takes O(\log w(x)). Here, w(x) represents the size of the working set of x. Structure To store a dynamic set of n elements, this structure consists of a series of ''Red–black trees'', or other ''Self-balancing binary search trees'' T_1, T_2, \ldots, T_k, and a series of '' deques'' (Double-ended queues) Q_1, Q_2, \ldots Q_k, where k = \lceil \log\log n\rceil. For every 1\leq i\leq k, tree T_i and deque Q_i share the same contents and pointers are maintained between their corresponding elements. For every i < k , the size ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometry Of Binary Search Trees
In computer science, one approach to the dynamic optimality problem on online algorithms for binary search trees involves reformulating the problem geometrically, in terms of augmenting a set of points in the plane with as few additional points as possible to avoid rectangles with only two points on their boundary. Access sequences and competitive ratio As typically formulated, the online binary search tree problem involves search trees defined over a fixed key set \. An ''access sequence'' is a sequence x_1, x_2, ... where each access x_i belongs to the key set. Any particular algorithm for maintaining binary search trees (such as the splay tree algorithm or Iacono's working set structure) has a ''cost'' for each access sequence that models the amount of time it would take to use the structure to search for each of the keys in the access sequence in turn. The cost of a search is modeled by assuming that the search tree algorithm has a single pointer into a binary search tree, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finger Tree
In computer science, a finger tree is a purely functional data structure that can be used to efficiently implement other functional data structures. A finger tree gives amortized constant time access to the "fingers" (leaves) of the tree, which is where data is stored, and concatenation and splitting logarithmic time in the size of the smaller piece. It also stores in each internal node the result of applying some associative operation to its descendants. This "summary" data stored in the internal nodes can be used to provide the functionality of data structures other than trees. Overview Ralf Hinze and Ross Paterson state a finger tree is a functional representation of persistent sequences that can access the ends in amortized constant time. Concatenation and splitting can be done in logarithmic time in the size of the smaller piece. The structure can also be made into a general purpose data structure by defining the split operation in a general form, allowing it to act a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
B-tree
In computer science, a B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree generalizes the binary search tree, allowing for nodes with more than two children. Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as databases and file systems. History While working at Boeing Research Labs, Rudolf Bayer and Edward M. McCreight invented B-trees to efficiently manage index pages for large random-access files. The basic assumption was that indices would be so voluminous that only small chunks of the tree could fit in the main memory. Bayer and McCreight's paper ''Organization and maintenance of large ordered indices'' was first circulated in July 1970 and later published in '' Acta Informatica''. Bayer and McCreight never explained what, if anything, the ''B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
AVL Tree
In computer science, an AVL tree (named after inventors Adelson-Velsky and Landis) is a self-balancing binary search tree. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Lookup, insertion, and deletion all take time in both the average and worst cases, where n is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations. The AVL tree is named after its two Soviet inventors, Georgy Adelson-Velsky and Evgenii Landis, who published it in their 1962 paper "An algorithm for the organization of information". It is the first self-balancing binary search tree data structure to be invented. AVL trees are often compared with red–black trees because both support the same set of operations and take \text(\log n) time for the basic operations. For lookup-intensiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Succinct Data Structure
In computer science, a succinct data structure is a data structure which uses an amount of space that is "close" to the information-theoretic lower bound, but (unlike other compressed representations) still allows for efficient query operations. The concept was originally introduced by Jacobson to encode bit vectors, (unlabeled) trees, and planar graphs. Unlike general lossless data compression algorithms, succinct data structures retain the ability to use them in-place, without decompressing them first. A related notion is that of a compressed data structure, insofar as the size of the stored or encoded data similarly depends upon the specific content of the data itself. Suppose that Z is the information-theoretical optimal number of bits needed to store some data. A representation of this data is called: * '' implicit'' if it takes Z + O(1) bits of space, * ''succinct'' if it takes Z + o(Z) bits of space, and * ''compact'' if it takes O(Z) bits of space. For example, a data ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Double-ended Queue
In computer science, a double-ended queue (abbreviated to deque, ) is an abstract data type that generalizes a queue, for which elements can be added to or removed from either the front (head) or back (tail). It is also often called a head-tail linked list, though properly this refers to a specific data structure ''implementation'' of a deque (see below). Naming conventions ''Deque'' is sometimes written ''dequeue'', but this use is generally deprecated in technical literature or technical writing because ''dequeue'' is also a verb meaning "to remove from a queue". Nevertheless, several libraries and some writers, such as Aho, Hopcroft, and Ullman in their textbook ''Data Structures and Algorithms'', spell it ''dequeue''. John Mitchell, author of ''Concepts in Programming Languages,'' also uses this terminology. Distinctions and sub-types This differs from the queue abstract data type or ''first in first out'' list ( FIFO), where elements can only be added to one end and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Key-independent Optimality
Key-independent optimality is a property of some binary search tree data structures in computer science proposed by John Iacono John Iacono is an American computer scientist specializing in data structures, algorithms and computational geometry. He is one of the inventors of the tango tree, the first known competitive binary search tree data structure. Iacono obtained h .... Suppose that key-value pairs are stored in a data structure, and that the keys have no relation to their paired values. A data structure has key-independent optimality if, when randomly assigning the keys, the expected performance of the data structure is within a constant factor of the optimal data structure. Key-independent optimality is related to dynamic optimality. Definitions There are many binary search tree algorithms that can look up a sequence of m keys X = x_1, x_2, \cdots, x_m, where each x_i is a number between 1 and n. For each sequence X, let \textit(X) be the fastest binary search t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Average-case Analysis
In analysis of algorithms, probabilistic analysis of algorithms is an approach to estimate the computational complexity of an algorithm or a computational problem. It starts from an assumption about a probabilistic distribution of the set of all possible inputs. This assumption is then used to design an efficient algorithm or to derive the complexity of a known algorithm. This approach is not the same as that of probabilistic algorithms, but the two may be combined. For non-probabilistic, more specifically deterministic, algorithms, the most common types of complexity estimates are the average-case complexity and the almost-always complexity. To obtain the average-case complexity, given an input distribution, the expected time of an algorithm is evaluated, whereas for the almost-always complexity estimate, it is evaluated that the algorithm admits a given complexity estimate that almost surely holds. In probabilistic analysis of probabilistic (randomized) algorithms, the distributi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
