Concept of an Algorithm. Correctness of Algorithms

Abstract

Algorithm is an exact prescription determining the computation process, leading from the varying source data to the sough result (data is the ordered set of characters). In other words, an algorithm describes the certain computation procedure, with the help of which the computation problem is solved. As a rule, an algorithm is used for solving some class of problems, rather than one certain problem.

Concept of an algorithm belongs to the basic, fundamental notions of mathematics. Many researchers use different wordings for the definition of an algorithm. However, all wordings explicitly or implicitly mean the following algorithm properties.

• 1. Discreteness. An algorithm shall represent a problem-solving process as a consecutive execution of separate steps. Execution of each step of an algorithm requires some time, and each operation is performed only in full and cannot be performed partially.

• 2. Elementary character of steps. The method of executing each command should be known and simple enough.

• 3. Determinateness. Each subsequent step of the algorithm operation is uniquely defined. The result should be the same for the same source data.

• 4. Directedness. It should be known, what is to be considered as the algorithm operation result.

• 5. Mass character. There should be a possibility to apply the algorithm to all collections of source data from a certain predetermined set.

As an example, the Warshall algorithm, which is used for calculating the reachability matrix of a specified directed graph D(V, E), is considered.