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In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
In number theory, the Kempner function [1] is defined for a given positive integer to be the smallest number such that divides the factorial!. For example, the number 8 {\displaystyle 8} does not divide 1 ! {\displaystyle 1!} , 2 ! {\displaystyle 2!} , or 3 ! {\displaystyle 3!} , but does divide 4 ! {\displaystyle 4!} , so S ( 8 ) = 4 ...
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points ), together with a set of unordered pairs of these ...
Shortest path (A, C, E, D, F), blue, between vertices A and F in the weighted directed graph. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph.If the graph is connected, it finds a minimum spanning tree.It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. [2]
Python's standard library includes heapq.nsmallest and heapq.nlargest functions for returning the smallest or largest elements from a collection, in sorted order. The implementation maintains a binary heap , limited to holding k {\displaystyle k} elements, and initialized to the first k {\displaystyle k} elements in the collection.
Division into a larger number of communities can be achieved by repeated bisection or by using multiple eigenvectors corresponding to the smallest eigenvalues. [12] The examples in Figures 1,2 illustrate the spectral bisection approach. Figure 1: The graph G = (5,4) is analysed for spectral bisection. The linear combination of the smallest two ...
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. [1] [2] For example, a 7-smooth number is a number in which every prime factor is at most 7.