The problem of merging sorted lists in the least number of pairwise comparisons has been solved completely only for a few special cases. Graham and Karp taocp independently discovered that the tape merge algorithm is optimal in the worst case when the two lists have the same size. In the seminal papers, Stockmeyer and Yao yao , Murphy and Paull 3k3 , and Christen christen1978optimality independently showed when the lists to be merged are of size @math and @math satisfying @math , the tape merge algorithm is optimal in the worst case. This paper extends this result by showing that the tape merge algorithm is optimal in the worst case whenever the size of one list is no larger than 1.52 times the size of the other. The main tool we used to prove lower bounds is Knuth's adversary methods taocp . In addition, we show that the lower bound cannot be improved to 1.8 via Knuth's adversary methods. We also develop a new inequality about Knuth's adversary methods, which might be interesting in its own right. Moreover, we design a simple procedure to achieve constant improvement of the upper bounds for @math .
Besides the worst-case complexity, the average-case complexity has also been investigated for merge problems. Tanner @cite_24 designed an algorithm which uses at most @math comparisons on average. The average case complexity of insertive merging algorithms as well as binary merge has also been investigated @cite_29 @cite_19 .
@cite_24 @cite_19 @cite_29
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A simple partitioning algorithm for merging two disjoint linearly ordered sets is given, and an upper bound on the average number of comparisons required is established. The upper bound is @math , where n is the number of elements in the larger of the two sets, m the number of the smaller, and @math . An immediate corollary is that any sorting problem can be done with an average number of comparisons within @math of the information theoretic bound using repeated merges; it does not matter what the merging order used is. Although the provable bound is @math over the lower bound, computations indicate that the algorithm will asymptotically make only @math more comparisons than the lower bound. The algorithm is compared with the Hwang-Lin algorithm, and modifications to improve average efficiency of this well known algorithm are given. This paper introduces a simple intersection algorithm for two sorted sequences that is fast on average. It is related to the multiple searching problem and to merging. We present the worst and average case analysis, showing that in the former, the complexity nicely adapts to the smallest list size. In the later case, it performs less comparisons than the total number of elements on both inputs when n = αm (α > 1). Finally, we show its application to fast query processing in Web search engines, where large intersections, or differences, must be performed fast. Existing statistical approaches to natural language problems are very coarse approximations to the true complexity of language processing. As such, no single technique will be best for all problem instances. Many researchers are examining ensemble methods that combine the output of successful, separately developed modules to create more accurate solutions. This paper examines three merging rules for combining probability distributions: the well known mixture rule, the logarithmic rule, and a novel product rule. These rules were applied with state-of-the-art results to two problems commonly used to assess human mastery of lexical semantics|synonym questions and analogy questions. All three merging rules result in ensembles that are more accurate than any of their component modules. The dierences among the three rules are not statistically signicant, but it is suggestive that the popular mixture rule is not the best rule for either of the two problems. In this paper a parallel algorithm to solve the stable marriage problem is given. The worst case performance of this algorithm is stated. A theoretical analysis shows that the probability of the occurrence of this worst case is extremely small. For instance, if there are sixteen men and sixteen women involved, then the probability that the worst case occurs is only 10−45. Possible future research is also discussed in this paper.