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In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
The closed form follows from the master theorem for divide-and-conquer recurrences. The number of comparisons made by merge sort in the worst case is given by the sorting numbers. These numbers are equal to or slightly smaller than (n ⌈lg n⌉ − 2 ⌈lg n⌉ + 1), which is between (n lg n − n + 1) and (n lg n + n + O(lg n)). [6]
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Divide et impera is the third of three political maxims in Immanuel Kant's Perpetual Peace (1795), Appendix I, the others being Fac et excusa ("Act now, and make excuses later") and Si fecisti, nega ("If you commit a crime, deny it"): [4] Kant refers this tactic when describing the traits of a "political moralist."
It is a specific type of divide and conquer algorithm. A well-known example is binary search. [3] Abstractly, a dichotomic search can be viewed as following edges of an implicit binary tree structure until it reaches a leaf (a goal or final state).
This technique can be used to improve the efficiency of many eigenvalue algorithms, but it has special significance to divide-and-conquer. For the rest of this article, we will assume the input to the divide-and-conquer algorithm is an real symmetric tridiagonal matrix . The algorithm can be modified for Hermitian matrices.
Figure 1. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (among other paths, not shown, sharing the same two vertices); the bold line is the overall shortest path from start to goal.
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