Algorithms and complexity

The study of simple stochastic games (SSGs) was initiated by Condon for analyzing the computational power of randomized space-bounded alternating Turing machines. The game is played by two players, MAX and MIN, on a directed multigraph, and when the play terminates at a sink vertex s, MAX wins from MIN a payoff p(s)∈[0,1]. Condon proved that the problem SSG-VALUE—given a

This paper describes several traditional algorithms for making Chinese and Italian noodles and classifies each according to its computational complexity. It examines machines for doing each algorithm. It cites a world speed record for making a large number of noodles using the algorithm with the maximal complexity. It dissects mysteries about the legend that Marco Polo brought the technology of making noodles to Italy from China. It determines that both Chinese and Italian ways of eating food can be applied to both Chinese and Italian noodle dishes. It compares the power of the algorithms. It considers the nature of variations of the traditional algorithms. It concludes by mentioning avenues for further studies.

There are a number of news sources available on the Internet. Searching through all these sources for facts related to a certain subject would be exhaustive for a user. Queries that require exact substring matching in order to obtain very relevant results are not currently possible. We developed a sentence level search engine, News Fact Finder, using suffix arrays to filter out opinions and to provide the users with factual information relevant to the topic of the query. We classified sentences in a document as facts and opinions by training a Naïve Bayes classifier on a set of fact and opinion sentences, which were manually classified by judges. The News Fact Finder produces 81% success rate of providing relevant fact based results, which is 8% higher than that of the News Fact Finder Word List (NFF-WL) method [1].

This paper describes several traditional algorithms for making Chinese and Italian noodles and classifies each according to its computational complexity. It examines machines for doing each algorithm. It cites a world speed record for making a large number of noodles using the algorithm with the maximal complexity. It dissects mysteries about the legend that Marco Polo brought the technology of making noodles to Italy from China. It determines that both Chinese and Italian ways of eating food can be applied to both Chinese and Italian noodle dishes. It compares the power of the algorithms. It considers the nature of variations of the traditional algorithms. It concludes by mentioning avenues for further studies.

In mathematics and computer science, the concept of Divide and Conquer is an algorithm-design algorithm. Divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type (divide), until these become simple enough to be solved directly (conquer). The solutions to the sub-problems are then combined to give a solution to the original problem.
This divide and conquer technique is the basis of efficient algorithms for all kinds of problems, such as sorting (e.g., quick sort, merge sort), multiplying large numbers (e.g. Karatsuba), syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform (FFTs).
Understanding and designing D&C algorithms is a complex skill that requires a good understanding of the nature of the underlying problem to be solved. As when proving a theorem by induction, it is often necessary to replace the original problem with a more general or complicated problem in order to initialize the recursion, and there is no systematic method for finding the proper generalization. These D&C complications are seen when optimizing the calculation of a Fibonacci number with efficient double recursion.
The correctness of a divide and conquer algorithm is usually proved by mathematical induction, and its computational cost is often determined by solving recurrence relations.
By Definition,  it is the  process of dividing  a complex problem  into  two  or more  sub-problems of the  same  type  until  the  sub-problems are easy enough to be solved directly  and recursively,  and then  these solutions  are combined  to give a solution  to the initial  problem.  So, to clarify the main steps of the divide and conquer algorithm is as follows:

(1) Divide the problem into two (or more) sub-problems


(2) Conquer these sub problems recursively


(3) Combine  the  solutions  to  the  sub  problems  in order  to  get  a solution  to  the original problem.
for many  different problems,  including  sorting  problems  and  the  discrete  Fourier transform. But when exactly was the divide and conquer algorithm first used?