Instance
In mathematics and computer science, an instance refers to a specific occurrence or example of a problem or concept. This concept is commonly used in the field of computational complexity theory, where it is used to describe the input of a decision problem.
Decision problems are mathematical problems that require a yes or no answer, and the input for such problems is typically a set or sequence of numbers, letters, or symbols. An instance of a decision problem is a specific input for the problem that is being solved.
For example, the decision problem "Is this number prime?" takes a single integer as input. An instance of this problem could be the number 17. Another example of a decision problem is the "Traveling Salesman Problem," which involves finding the shortest possible route that visits a given set of cities exactly once. An instance of this problem could be a specific set of cities and the distances between them.
Instances are also used in the field of computational complexity theory, where they are used to analyze the time and space complexity of algorithms. The time complexity of an algorithm refers to the amount of time it takes to solve a problem, while the space complexity refers to the amount of memory required to run the algorithm.
For example, let's consider the sorting problem, which involves arranging a set of numbers in ascending or descending order. The time complexity of an algorithm refers to the number of comparisons and swaps required to sort the numbers, while the space complexity refers to the amount of memory required to store the numbers and the intermediate results of the algorithm.
In conclusion, the concept of an instance is a fundamental concept in mathematics and computer science, particularly in the field of computational complexity theory. Instances are used to define and analyze decision problems, as well as to analyze the time and space complexity of algorithms. Understanding the concept of an instance is essential to solving complex mathematical and computational problems.
