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The lower bound on worst-case running time of output-sensitive convex hull algorithms was established to be Ω(n log h) in the planar case. [1] There are several algorithms which attain this optimal time complexity. The earliest one was introduced by Kirkpatrick and Seidel in 1986 (who called it "the ultimate convex hull algorithm").
In fact all bounds (lower and upper) currently known for the average case are precisely matched by this lower bound. For example, this gives the new result that the Janson-Knuth upper bound is matched by the resulting lower bound for the used increment sequence, showing that three pass Shellsort for this increment sequence uses Θ ( N 23 / 15 ...
lower_bound: lower_bound: lower_bound: lower_bound: Returns an iterator to the first element with a key not less than the given value. upper_bound: upper_bound: upper_bound: upper_bound: Returns an iterator to the first element with a key greater than a certain value. Observers key_comp: key_comp: key_comp: key_comp: Returns the key comparison ...
Binary search Visualization of the binary search algorithm where 7 is the target value Class Search algorithm Data structure Array Worst-case performance O (log n) Best-case performance O (1) Average performance O (log n) Worst-case space complexity O (1) Optimal Yes In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search ...
As a general guide, if the last few stages of the tour are comparable in length to the first stages, then the tour is reasonable; if they are much greater, then it is likely that much better tours exist. Another check is to use an algorithm such as the lower bound algorithm to estimate if this tour is good enough.
The cost of the solution produced by the algorithm is within 3/2 of the optimum. To prove this, let C be the optimal traveling salesman tour. Removing an edge from C produces a spanning tree, which must have weight at least that of the minimum spanning tree, implying that w(T) ≤ w(C) - lower bound to the cost of the optimal solution.
The following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function f. [3] To obtain an actual algorithm from this, one requires a bounding function bound, that computes lower bounds of f on nodes of the search tree, as well as a problem-specific branching rule.
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: