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In the case where = (+ ()), there is a lower bound of () list labeling cost for deterministic algorithms. [6] Furthermore, the same lower bound holds for smooth algorithms, which are those whose only relabeling operation assigns labels evenly in a range of items [10] This lower bound is surprisingly strong in that it applies in the offline ...
The List Update or the List Access problem is a simple model used in the study of competitive analysis of online algorithms.Given a set of items in a list where the cost of accessing an item is proportional to its distance from the head of the list, e.g. a linked List, and a request sequence of accesses, the problem is to come up with a strategy of reordering the list so that the total cost of ...
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.
The optimal algorithm is by Andris Ambainis. [7] Yaoyun Shi first proved a tight lower bound when the size of the range is sufficiently large. [8] Ambainis [9] and Kutin [10] independently (and via different proofs) extended his work to obtain the lower bound for all functions.
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 ...
A sorting algorithm is stable if whenever there are two records R and S with the same key, and R appears before S in the original list, then R will always appear before S in the sorted list. When equal elements are indistinguishable, such as with integers, or more generally, any data where the entire element is the key, stability is not an issue.
It follows that every complexity of an algorithm, that is expressed with big O notation, is also an upper bound on the complexity of the corresponding problem. On the other hand, it is generally hard to obtain nontrivial lower bounds for problem complexity, and there are few methods for obtaining such lower bounds.
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.