ONLINE ALGORITHMS

Amortized Complexity and Online Algorithms

Amortized Complexity and Online Algorithms

Introduction

The notion of "algorithm" has long been familiar, not only for mathematicians. It is a conceptual framework for a variety of information processing. The ability to automate these processes is ensured by appropriate algorithms. On first acquaintance algorithms are at the beginning flax-school in the study of arithmetic with natural numbers. In a simplified understanding of the "algorithm" - is something that can be programmed on a computer.

Word of the algorithm includes in its composition transformed geographical name Khorezm. The term "algorithm" owes its origin to the great scholar of the medieval East - Muhammad ibn Musa al-Khwarizmi (Mohammed, the son of Moses and Khorezm). He lived from about 783 to 850 years., And in 1983 was noted in 1200 anniversary of his birth in the city of Urgench - the regional center of modern Khorezm region of Uzbekistan. In Latin translations from Arabic arithmetic treatise by al-Khwarizmi's name transcribed as algorismi. Hence the word "algorithm" - first to describe the algorithms of digital computing decimal positional arithmetic, and then to refer to arbitrary processes in which the unknown quantities are problems to be solved sequentially from the input data according to certain rules and regulations.

Up until the 30s of this century, the concept of the algorithm had an intuitive, which had rather a methodological and not a mathematical value. Thus, the beginning of the twentieth century. many bright examples given algebra and number theory. Among them we mention the Euclidean algorithm for finding the greatest common divisor of two integers or two integer polynomials, the Gauss algorithm solving systems of linear equations over the field, an algorithm for finding the rational roots of polynomials of one variable with rational coefficients, the Sturm algorithm for determining the number of real roots of a polynomial with real coefficients on an interval real numbers, the algorithm is the decomposition of the polynomial of one variable over finite fields into irreducible factors. These algorithmic problems are solved by specifying a specific permitting procedures. To obtain the results of this type rather intuitive concept of algorithm. However, in the early twentieth century. algorithmic problems have been formulated, a positive solution which seemed unlikely. The solution of these problems required the use of new logical devices. It's one thing to prove the existence of a permissive algorithm - this can be done using the intuitive concept of algorithm. Another thing - to prove the nonexistence of an algorithm - for this you need to know for sure - that this algorithm.

In the science of computing the algorithms are more important than the LP or the computer , solving a problem by making use of computers requires one hand an algorithm or method of resolution and the other a program or coding the algorithm in a LP. Both components are important, but the algorithm is absolutely essential, we know that an algorithm is a sequence of steps to solve a problem, its features are:

Independent: The LP and machine.

Defined: In clear and concrete steps.

Finite: The number of steps ...