By Lerma M.
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Venn diagrams and arrows can be used for representing relations between given sets. 14 represents the relation from A = {a, b, c, d} to B = {1, 2, 3, 4} given by R = {(a, 1), (b, 1), (c, 2), (c, 3)}. In the diagram an arrow from x to y means that x is related to y. This kind of graph is called directed graph or digraph. 3. 14. Relation. 15, which represents the divisibility relation on the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. 15. Binary relation of divisibility. Matrix of a Relation. Another way of representing a relation R from A to B is with a matrix.
A function f (n) is said to be of order g(n), written f (n) = Θ(g(n)), if f (n) = O(g(n)) and f (n) = Ω(g(n)). Remark : All logarithmic functions are of the same order: loga n = Θ(logb n) for any a, b > 1, because loga n = logb n/ logb a, so they always differ in a multiplicative constant. As a consequence, if the execution time of an algorithm is of order a logarithmic function, we can just say that its time is “logarithmic”, we do not need to specify the base of the logarithm. The following are several common growth functions: Order Θ(1) Θ(log log n) Θ(log n) Θ(n log n) Θ(n) Θ(n2 ) Θ(n3 ) Θ(nk ) Θ(an ) Name Constant Log log Logarithmic n log n Linear Quadratic Cubic Polynomial Exponential Let’s see now how we find the complexity of algorithms like bubble sort and merge sort.
5. Output. The algorithm produces output. 6. Generality. The algorithm applies to a set of inputs. Basically an algorithm is the idea behind a program. Conversely, programs are implementations of algorithms. 1. Pseudocode. Pseudocode is a language similar to a programming language used to represent algorithms. The main difference respect to actual programming languages is that pseudocode is not required to follow strict syntactic rules, since it is intended to be just read by humans, not actually executed by a machine.