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Movements, leaders, victories and the continuing fight for equality.
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Meet the people whose lives, choices and achievements shaped the journey.
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Katherine Johnson’s mathematical calculations helped guide some of America’s most important early space missions while she confronted the racial and gender barriers faced by Black women in twentieth-century America.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
 | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (March 2021) |
| Exponential tree | ||||||||||||||||||||||||
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| Type | tree | |||||||||||||||||||||||
| Invented | 1995 | |||||||||||||||||||||||
| Invented by | Arne Andersson | |||||||||||||||||||||||
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An exponential tree is a type of search tree where the number of children of its nodes decreases doubly-exponentially with increasing depth. Values are stored only in the leaf nodes. Each node contains a splitter, a value less than or equal to all values in the subtree which is used during search. Exponential trees use another data structure in inner nodes containing the splitters from children, allowing fast lookup.
Exponential trees achieve optimal asymptotic complexity on some operations. They have mainly theoretical importance.
An exponential tree is a rooted tree where every node contains a splitter and every leaf node contains a value. The value may be different from the splitter. An exponential tree with values is defined recursively:
An additional condition is that searching for a value using the splitters must yield the correct node (i.e. the one containing the value). Therefore, if a root of a subtree contains the splitter and its right sibling contains the splitter , then this subtree can only contain keys in the range .
The tree uses a static data structure in every inner node to allow fast lookup of values. It must be possible to build this structure with values in time . The lookup time in this structure is denoted .
A Fusion tree can be used as this data structure.
The exponential tree can be searched in the same way as a normal search tree. In each node, the local data structure can be used to find the next child quickly.
Let denote the time complexity of the search. Then it satisfies the following recurrence:
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An exponential tree is a type of search tree where the number of children of its nodes decreases doubly-exponentially with increasing depth. Values are stored only in the leaf nodes. Each node contains a splitter, a value less than or equal to all values in the subtree which is used during search. Exponential trees use another data structure in inner nodes containing the splitters from children, allowing fast lookup. Exponential trees achieve optimal asymptotic complexity on some operations. They have mainly theoretical importance.
This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics. Accelerating change Approximating natural exponents (log base e) Artin–Hasse exponential Bacterial growth Baker–Campbell–Hausdorff formula Barometric formula Beer–Lambert law Carlitz exponential Catenary Cell growth Characterizations of the exponential function Compound interest C0-semigroup De Moivre's formula Derivative of the exponential map Doléans-Dade exponential Double exponential function Doubling time e-folding Elimination half-life Error exponent Euler's formula Euler's identity e (mathematical constant) Exponent Exponent bias Exponential (disambiguation) Exponential backoff Exponential decay Exponential dichotomy Exponential discounting Exponential diophantine equation Exponential dispersion model Exponential distribution Exponential error Exponential factorial Exponential family Exponential field Ordered exponential field Exponential formula Exponential function Exponential generating function Exponential-Golomb coding Exponential growth Exponential hierarchy Exponential integral Exponential integrator Exponential map (Lie theory) Exponential map (Riemannian geometry) Exponential map (discrete dynamical systems) Exponential notation Exponential object (category theory) Exponential polynomials—see also Touchard polynomials (combinatorics) Exponential response formula Exponential sheaf sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential type Exponentially equivalent measures Exponentiating by squaring Exponentiation Forgetting curve Four exponentials conjecture Fourier analysis List of Fourier analysis topics Gaussian function Gudermannian function Half-exponential function Half-life Hyperbolic function Inflation, inflation rate Interest Lambert W function Lifetime (physics) Limiting factor Lindemann–Weierstrass theorem List of integrals of exponential functions List of integrals of hyperbolic functions Lyapunov exponent Malthusian catastrophe Malthusian growth model Marshall–Olkin exponential distribution Matrix exponential Modular exponentiation Moore's law Nachbin's theorem Ordered exponential Piano key frequencies Plethystic exponential p-adic exponential function Power law Proof that e is irrational Proof that e is transcendental q-exponential Radioactive decay Rule of 70, Rule of 72 Schanuel's conjecture Scientific notation Six exponentials theorem Spontaneous emission Super-exponentiation Tetration Transseries Versor Weber–Fechner law Wilkie's theorem Zenzizenzizenzic
In computer science, a fusion tree is a type of tree data structure that implements an associative array on w-bit integers on a finite universe, where each of the input integers has size less than 2w and is non-negative. When operating on a collection of n key–value pairs, it uses O(n) space and performs searches in O(logw n) time, which is asymptotically faster than a traditional self-balancing binary search tree, and also better than the van Emde Boas tree for large values of w. It achieves this speed by using certain constant-time operations that can be done on a machine word. Fusion trees were invented in 1990 by Michael Fredman and Dan Willard. Several advances have been made since Fredman and Willard's original 1990 paper. In 1999 it was shown how to implement fusion trees under a model of computation in which all of the underlying operations of the algorithm belong to AC0, a model of circuit complexity that allows addition and bitwise Boolean operations but does not allow the multiplication operations used in the original fusion tree algorithm. A dynamic version of fusion trees using hash tables was proposed in 1996 which matched the original structure's O(logw n) runtime in expectation. Another dynamic version using exponential tree was proposed in 2007 which yields worst-case runtimes of O(logw n + log log n) per operation. Finally, it was shown that dynamic fusion trees can perform each operation in O(logw n) time deterministically. This data structure implements add key, remove key, search key, and predecessor (next smaller value) and successor (next larger value) search operations for a given key. The partial result of most significant bit locator in constant time has also helped further research. Fusion trees utilize word-level parallelism to be efficient, performing computation on several small integers, stored in a single machine word, simultaneously to reduce the number of total operations.
A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry. Displaying hierarchical data as a tree suffers from visual clutter as the number of nodes per level can grow exponentially. For a simple binary tree, the maximum number of nodes at a level n is 2n, while the number of nodes for trees with more branching grows much more quickly. Drawing the tree as a node-link diagram thus requires exponential amounts of space to be displayed. One approach is to use a hyperbolic tree, first introduced by Lamping et al. Hyperbolic trees employ hyperbolic space, which intrinsically has "more room" than Euclidean space. For instance, linearly increasing the radius of a circle in Euclidean space increases its circumference linearly, while the same circle in hyperbolic space would have its circumference increase exponentially. Exploiting this property allows laying out the tree in hyperbolic space in an uncluttered manner: placing a node far enough from its parent gives the node almost the same amount of space as its parent for laying out its own children. Displaying a hyperbolic tree commonly utilizes the Poincaré disk model of hyperbolic geometry, though the Klein-Beltrami model can also be used. Both display the entire hyperbolic plane within a unit disk, making the entire tree visible at once. The unit disk gives a fish-eye lens view of the plane, giving more emphasis to nodes which are in focus and displaying nodes further out of focus closer to the boundary of the disk. Traversing the hyperbolic tree requires Möbius transformations of the space, bringing new nodes into focus and moving higher levels of the hierarchy out of view. Hyperbolic trees were patented in the U.S. by Xerox in 1996, but the patent has since expired.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Shirley Chisholm, elected in 1968.
