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Sorting a set of unlabelled weights by weight using only a balance scale requires a comparison sort algorithm. A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation (often a "less than or equal to" operator or a three-way comparison) that determines which of two elements should occur first in the final sorted list.
Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide-and-conquer algorithms, data structures such as heaps and binary trees, randomized algorithms, best, worst and average ...
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 ...
In this case, the existence of numerous comparison-sorting algorithms having this time complexity, such as mergesort and heapsort, demonstrates that the bound is tight. [2]: 91 This argument does not use anything about the type of query, so it in fact proves a lower bound for any sorting algorithm that can be modeled as a binary decision tree.
A nonlinear lower bound of () is known for the number of comparisons needed for a sorting algorithm. Thus the best sorting algorithms are optimal, as their complexity is (). This lower bound results from the fact that there are n! ways of ordering n objects.
A well-known lower bound for unstructured sorting, in the decision tree model, is based on the factorial number of sorted orders that an unstructured list may have. Because each comparison can at best reduce the number of possible orderings by a factor of two, sorting requires a number of comparisons at least equal to the binary logarithm of ...
For these models, an algorithm based on comparison sort solves the problem within a constant factor of the best possible number of comparisons. The same lower bound applies as well to the expected number of comparisons in the randomized algebraic decision tree model. [3] [4]