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PAM Library
PAM (Parallel Augmented Maps) is an open-source parallel C++ library implementing the interface for sequence, ordered sets, ordered maps, and augmented maps. The library is available on GitHub. It uses the underlying balanced binary tree structure using join-based algorithms. PAM supports four balancing schemes, including AVL trees, red-black trees, treaps and weight-balanced trees. PAM is a parallel library and is also safe for concurrency. Its parallelism can be supported by cilk, OpenMP or the scheduler in PBBS. Theoretically, all algorithms in PAM are work-efficient and have polylogarithmic depth. PAM uses underlying persistent tree structure such that multi-versioning is allowed. PAM also supports efficient GC. Interface Sequences To define a sequence, users need to specify the key type of the sequence. PAM supports functions on sequences including construction, find an entry with a certain rank, first, last, next, previous, size, empty, filter, map-reduce, concatenatin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Ordering
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiversion Concurrency Control
Multiversion concurrency control (MCC or MVCC), is a concurrency control method commonly used by database management systems to provide concurrent access to the database and in programming languages to implement transactional memory. Description Without concurrency control, if someone is reading from a database at the same time as someone else is writing to it, it is possible that the reader will see a half-written or inconsistent piece of data. For instance, when making a wire transfer between two bank accounts if a reader reads the balance at the bank when the money has been withdrawn from the original account and before it was deposited in the destination account, it would seem that money has disappeared from the bank. Isolation is the property that provides guarantees in the concurrent accesses to data. Isolation is implemented by means of a concurrency control protocol. The simplest way is to make all readers wait until the writer is done, which is known as a read-write lock. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverted Index
In computer science, an inverted index (also referred to as a postings list, postings file, or inverted file) is a database index storing a mapping from content, such as words or numbers, to its locations in a table, or in a document or a set of documents (named in contrast to a forward index, which maps from documents to content). The purpose of an inverted index is to allow fast full-text searches, at a cost of increased processing when a document is added to the database. The inverted file may be the database file itself, rather than its index. It is the most popular data structure used in document retrieval systems, used on a large scale for example in search engines. Additionally, several significant general-purpose mainframe-based database management systems have used inverted list architectures, including ADABAS, DATACOM/DB, and Model 204. There are two main variants of inverted indexes: A record-level inverted index (or inverted file index or just inverted file) contains ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segment Tree
In computer science, a segment tree, also known as a statistic tree, is a tree data structure used for storing information about intervals, or segments. It allows querying which of the stored segments contain a given point. It is, in principle, a static structure and cannot be modified once built. A similar data structure is the interval tree. A segment tree for a set of ''n'' intervals uses ''O''(''n'' log ''n'') storage and can be built in ''O''(''n'' log ''n'') time. Segment trees support searching for all the intervals that contain a query point in time ''O''(log ''n'' + ''k''), ''k'' being the number of retrieved intervals or segments. Applications of the segment tree are in the areas of computational geometry, geographic information systems and machine learning. The segment tree can be generalized to higher dimension spaces. Definition Description Let ''S'' be a set of intervals, or segments. Let ''p''1, ''p''2, ..., ''pm'' be the list of distinct interval endpoin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sweepline Algorithm
In computational geometry, a sweep line algorithm or plane sweep algorithm is an algorithmic paradigm that uses a conceptual ''sweep line'' or ''sweep surface'' to solve various problems in Euclidean space. It is one of the key techniques in computational geometry. The idea behind algorithms of this type is to imagine that a line (often a vertical line) is swept or moved across the plane, stopping at some points. Geometric operations are restricted to geometric objects that either intersect or are in the immediate vicinity of the sweep line whenever it stops, and the complete solution is available once the line has passed over all objects. History This approach may be traced to scanline algorithms of rendering in computer graphics, followed by exploiting this approach in early algorithms of integrated circuit layout design, in which a geometric description of an IC was processed in parallel strips, because the entire description could not fit into memory. Applications Application ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Tree
In computer science, a range tree is an ordered tree data structure to hold a list of points. It allows all points within a given range to be reported efficiently, and is typically used in two or higher dimensions. Range trees were introduced by Jon Louis Bentley in 1979. Similar data structures were discovered independently by Lueker, Lee and Wong, and Willard. The range tree is an alternative to the ''k''-d tree. Compared to ''k''-d trees, range trees offer faster query times of (in Big O notation) O(\log^dn+k) but worse storage of O(n\log^ n), where ''n'' is the number of points stored in the tree, ''d'' is the dimension of each point and ''k'' is the number of points reported by a given query. Bernard Chazelle improved this to query time O(\log^ n + k) and space complexity O\left(n\left(\frac\right)^\right). Data structure A range tree on a set of 1-dimensional points is a balanced binary search tree on those points. The points stored in the tree are stored in the leav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Query
A range query is a common database operation that retrieves all records where some value is between an upper and lower boundary. For example, list all employees with 3 to 5 years' experience. Range queries are unusual because it is not generally known in advance how many entries a range query will return, or if it will return any at all. Many other queries, such as the top ten most senior employees, or the newest employee, can be done more efficiently because there is an upper bound to the number of results they will return. A query that returns exactly one result is sometimes called a singleton. Partial match query Match at least one of the requested keys. *B+ tree * k-d tree * R-tree See also * Range searching In computer science, the range searching problem consists of processing a set ''S'' of objects, in order to determine which objects from ''S'' intersect with a query object, called the ''range''. For example, if ''S'' is a set of points correspond ... References Dat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Tree
In computer science, an interval tree is a tree data structure to hold intervals. Specifically, it allows one to efficiently find all intervals that overlap with any given interval or point. It is often used for windowing queries, for instance, to find all roads on a computerized map inside a rectangular viewport, or to find all visible elements inside a three-dimensional scene. A similar data structure is the segment tree. The trivial solution is to visit each interval and test whether it intersects the given point or interval, which requires O(n) time, where n is the number of intervals in the collection. Since a query may return all intervals, for example if the query is a large interval intersecting all intervals in the collection, this is asymptotically optimal; however, we can do better by considering output-sensitive algorithms, where the runtime is expressed in terms of m, the number of intervals produced by the query. Interval trees have a query time of O(\log n + m) and an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Difference
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space. Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident to each of original objects. In this approach an intersection can be sometimes undefined, such as for parallel lines. In both cases the concept of intersection relies on logical conjunction. Alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero (0) sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. Union of two sets The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, :A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot have duplicate elements, so the union of the sets and is . Multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
