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Kruskal's algorithm is a greedy algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph.
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Kruskal's and algorithm
* Kruskal's algorithm
Examples include Dijkstra's algorithm, Kruskal's algorithm, the nearest neighbour algorithm, and Prim's algorithm.
Where E is the number of edges in the graph and V is the number of vertices, Kruskal's algorithm can be shown to run in O ( E log E ) time, or equivalently, O ( E log V ) time, all with simple data structures.
Other algorithms for this problem include Kruskal's algorithm and Borůvka's algorithm.
Other algorithms for this problem include Prim's algorithm ( actually discovered by Vojtěch Jarník ) and Kruskal's algorithm.
This algorithm is a randomized version of Kruskal's algorithm.
It is also used for implementing Kruskal's algorithm to find the minimum spanning tree of a graph.
The simplest algorithm to find an EMST in two dimensions, given n points, is to actually construct the complete graph on n vertices, which has n ( n-1 )/ 2 edges, compute each edge weight by finding the distance between each pair of points, and then run a standard minimum spanning tree algorithm ( such as the version of Prim's algorithm or Kruskal's algorithm ) on it.
Since there are O ( n ) edges, this requires O ( n log n ) time using any of the standard minimum spanning tree algorithms such as Borůvka's algorithm, Prim's algorithm, or Kruskal's algorithm.
His two brothers, both eminent mathematicians, were Joseph Kruskal ( 1928-2010 ; discoverer of multidimensional scaling, the Kruskal tree theorem, and Kruskal's algorithm ) and William Kruskal ( 1919 – 2005 ; discoverer of the Kruskal – Wallis test ).
In computer science, his best known work is Kruskal's algorithm for computing the minimal spanning tree ( MST ) of a weighted graph.
* Kruskal's algorithm ( 1956 )
Kruskal's and is
A weaker result for trees is implied by Kruskal's tree theorem, which was conjectured in 1937 by Andrew Vázsonyi and proved in 1960 independently by Joseph Kruskal and S. Tarkowski.
* Embedding between finite trees with nodes labeled by elements of a wqo is a wqo ( Kruskal's tree theorem ).
Martin Kruskal's work in plasma physics is considered by some to be his most outstanding.
In statistics, Kruskal's most influential work is his seminal contribution to the formulation of multidimensional scaling.
In combinatorics, he is known for Kruskal's tree theorem ( 1960 ), which is also interesting from a mathematical logic perspective since it can only be proved nonconstructively.
Kruskal's and theory
Other measures of association are the distance correlation, tetrachoric correlation coefficient, Goodman and Kruskal's lambda, Tschuprow's T and Cramér's V. In information theory measures such as mutual information are used.
In the later part of his career, one of Kruskal's chief interests was the theory of surreal numbers.
Kruskal's and tree
* Kruskal's tree theorem ( 1960 )
# REDIRECT Kruskal's tree theorem
Kruskal's and .
* The Goodman and Kruskal's lambda in statistics indicates the proportional reduction in error when one variable's values are used to predict the values of another variable.
Specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs ( e. g., in connection with Friedman's various finite forms of Kruskal's theorem ).
Martin Kruskal's scientific interests covered a wide range of topics in pure mathematics and applications of mathematics to the sciences.
Kruskal's most widely known work was the discovery in the 1960s of the integrability of certain nonlinear partial differential equations involving functions of one spatial variable as well as time.
algorithm and is
The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields " yes " ( or true ) ( more accurately the number b in location B is greater than or equal to the number a in location A ) THEN the algorithm specifies B ← B − A ( meaning the number b − a replaces the old b ).
In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.
More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function.
While there is no generally accepted formal definition of " algorithm ," an informal definition could be " a set of rules that precisely defines a sequence of operations.
" For some people, a program is only an algorithm if it stops eventually ; for others, a program is only an algorithm if it stops before a given number of calculation steps.
A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers ; an example ( there are others ) is described by the flow chart above and as an example in a later section.
The concept of algorithm is also used to define the notion of decidability.
In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension.
Gurevich: "... Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage, an algorithm is a computational process defined by a Turing machine ".
Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and / or stored for further processing.
Stored data is regarded as part of the internal state of the entity performing the algorithm.
Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm.
In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended " target " computer ( s ), in order for the target machines to produce output from given input ( perhaps null ).
is the length of time taken to perform the algorithm.
Simulation of an algorithm: computer ( computor ) language: Knuth advises the reader that " the best way to learn an algorithm is to try it.
algorithm and greedy
The nearest neighbour ( NN ) algorithm ( or so-called greedy algorithm ) lets the salesman choose the nearest unvisited city as his next move.
The nearest neighbour algorithm is easy to implement and executes quickly, but it can sometimes miss shorter routes which are easily noticed with human insight, due to its " greedy " nature.
In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a connected weighted undirected graph.
However, if the set of numbers ( called the knapsack ) is superincreasing — that is, each element of the set is greater than the sum of all the numbers before it — the problem is ' easy ' and solvable in polynomial time with a simple greedy algorithm.
Then, using a simple greedy algorithm, the easy knapsack can be solved using O ( n ) arithmetic operations, which decrypts the message.
One practical routing algorithm is to pick the pin farthest from the center of the board, then use a greedy algorithm to select the next-nearest pin with the same signal name.
In particular, chapter II. 7 contains a list of methods for converting a vulgar fraction to an Egyptian fraction, including the greedy algorithm for Egyptian fractions, also known as the Fibonacci – Sylvester expansion.
Many approaches to the problem have been explored, including greedy algorithms, randomized search, genetic algorithms and A * search algorithm.
It is then easy to route a message to the owner of any key using the following greedy algorithm ( that is not necessarily globally optimal ): at each step, forward the message to the neighbor whose ID is closest to.
Typically, a greedy algorithm is used to solve a problem with optimal substructure if it can be proved by induction that this is optimal at each step ( Cormen et al.
So the algorithm can be compactified by a greedy strategy, as illustrated in the division below.
The simplest greedy algorithm places consecutive labels on the map in positions that result in minimal overlap of labels.
A 3-coloring may be found in linear time by a greedy coloring algorithm that removes any vertex of degree at most two, colors the remaining graph recursively, and then adds back the removed vertex with a color different from the colors of its two neighbors.
* Otakar Borůvka publishes Borůvka's algorithm, introducing the greedy algorithm.
UCLUST and CD-HIT use a greedy algorithm that identifies a representative sequence for each cluster and assigns a new sequence to that cluster if it is sufficiently similar to the representative ; if a sequence is not matched then it becomes the representative sequence for a new cluster.
Because the subdivision is formed by triangles, a greedy algorithm can find an independent set that contains a constant fraction of the vertices.
However, effective approximation algorithms are known with approximation ratios that are worse than this threshold ; for instance, a greedy algorithm that forms a maximal independent set by, at each step, choosing the minimum degree vertex in the graph and removing its neighbors achieves an approximation ratio of ( Δ + 2 )/ 3 on graphs with maximum degree Δ.
The problem of finding a maximal independent set can be solved in polynomial time by a trivial greedy algorithm.
This process of top-down induction of decision trees ( TDIDT ) is an example of a greedy algorithm, and it is by far the most common strategy for learning decision trees from data, but it is not the only strategy.
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