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In data mining, k-means++ [1] [2] is an algorithm for choosing the initial values (or "seeds") for the k-means clustering algorithm. It was proposed in 2007 by David Arthur and Sergei Vassilvitskii, as an approximation algorithm for the NP-hard k-means problem—a way of avoiding the sometimes poor clusterings found by the standard k-means algorithm.
k-means clustering is a method of vector quantization, originally from signal processing, that aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean (cluster centers or cluster centroid), serving as a prototype of the cluster.
This relates directly to the k-median problem which is the problem of finding k centers such that the clusters formed by them are the most compact with respect to the 2-norm. Formally, given a set of data points x, the k centers c i are to be chosen so as to minimize the sum of the distances from each x to the nearest c i. The criterion ...
The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. [3]: ND22, ND23
NP-complete problems are in NP, the set of all decision problems whose solutions can be verified in polynomial time; NP may be equivalently defined as the set of decision problems that can be solved in polynomial time on a non-deterministic Turing machine.
A problem set, sometimes shortened as pset, [1] is a teaching tool used by many universities. Most courses in physics , math , engineering , chemistry , and computer science will give problem sets on a regular basis. [ 2 ]
The form comes with two worksheets, one to calculate exemptions, and another to calculate the effects of other income (second job, spouse's job). The bottom number in each worksheet is used to fill out two if the lines in the main W4 form. The main form is filed with the employer, and the worksheets are discarded or held by the employee.
Example of a worksheet for structured problem solving and continuous improvement. A3 problem solving is a structured problem-solving and continuous-improvement approach, first employed at Toyota and typically used by lean manufacturing practitioners. [1] It provides a simple and strict procedure that guides problem solving by workers.
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