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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 algorithm exploits that there are also C n−1 possible triangulations of a polygon with n+1 sides. This image illustrates possible triangulations of a regular hexagon. These ...
For typical serial sorting algorithms, good behavior is O(n log n), with parallel sort in O(log 2 n), and bad behavior is O(n 2). Ideal behavior for a serial sort is O(n), but this is not possible in the average case. Optimal parallel sorting is O(log n). Swaps for "in-place" algorithms. Memory usage (and use of other computer resources).
It has a O(n 2) time complexity, which makes it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity and has performance advantages over more complicated algorithms in certain situations, particularly where auxiliary memory is limited.
Can the edit distance between two strings of length n be computed in strongly sub-quadratic time? (This is only possible if the strong exponential time hypothesis is false.) Can X + Y sorting be done in o(n 2 log n) time? What is the fastest algorithm for matrix multiplication? Can all-pairs shortest paths be computed in strongly sub-cubic time ...
This restriction then means that an algorithm can find a solution in polynomial time that is correct within a factor of (1-ε) of the optimal solution. [26] algorithm FPTAS is input: ε ∈ (0,1] a list A of n items, specified by their values, , and weights output: S' the FPTAS solution P := max {} // the highest item value K := ε for i from 1 ...
Karatsuba multiplication is an O(n log 2 3) ≈ O(n 1.585) divide and conquer algorithm, that uses recursion to merge together sub calculations. By rewriting the formula, one makes it possible to do sub calculations / recursion. By doing recursion, one can solve this in a fast manner.
To improve this search time from complexity O(n) to O(1), Knuth implemented a sparse matrix where only 1's are stored. At all times, each node in the matrix will point to the adjacent nodes to the left and right (1's in the same row), above and below (1's in the same column), and the header for its column (described below).
A further relaxation requiring only a list of the k smallest elements, but without requiring that these be ordered, makes the problem equivalent to partition-based selection; the original partial sorting problem can be solved by such a selection algorithm to obtain an array where the first k elements are the k smallest, and sorting these, at a total cost of O(n + k log k) operations.