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The k-way merge problem consists of merging k sorted arrays to produce a single sorted array with the same elements. Denote by n the total number of elements. n is equal to the size of the output array and the sum of the sizes of the k input arrays. For simplicity, we assume that none of the input arrays is empty.
In the merge sort algorithm, this subroutine is typically used to merge two sub-arrays A[lo..mid], A[mid+1..hi] of a single array A. This can be done by copying the sub-arrays into a temporary array, then applying the merge algorithm above. [1] The allocation of a temporary array can be avoided, but at the expense of speed and programming ease.
A spreadsheet's concatenate ("&") function is used to assemble a complex text string—in this example, XML code for an SVG "circle" element. In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball".
The other n sides of the polygon, in the clockwise direction, represent the matrices. The vertices on each end of a side are the dimensions of the matrix represented by that side. With n matrices in the multiplication chain there are n−1 binary operations and C n−1 ways of placing parentheses, where C n−1 is the (n−1)-th Catalan number.
The n × n matrices that have an inverse form a group under matrix multiplication, the subgroups of which are called matrix groups. Many classical groups (including all finite groups ) are isomorphic to matrix groups; this is the starting point of the theory of group representations .
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
The dip is great served with pita chips or an array of fresh veggies. Serve it at a summer party, the Super Bowl , or on-the-go as the ultimate picnic snack. Get the Creamy White Bean Dip recipe .
COBOL uses the STRING statement to concatenate string variables. MATLAB and Octave use the syntax "[x y]" to concatenate x and y. Visual Basic and Visual Basic .NET can also use the "+" sign but at the risk of ambiguity if a string representing a number and a number are together. Microsoft Excel allows both "&" and the function "=CONCATENATE(X,Y)".