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2. matching closed parenthesis, is the nth Catalan number. The number of ways to cut an n+2-sided convex polygon in a plane into triangles by connecting vertices with3. straight, non-intersecting lines is the nth Catalan number. This is the application in which Euler was interested. Using zero-based numbering, the n th Catalan number is given directly in terms of binomial coefficients by the following equation. Example of Catalan Number: Here value of n = 4.(Best Example - From Wikipedia) Auxiliary Space:# Page 107Algorithms Notes for Professionals 102# Page 108Algorithms Notes for Professionals 103Chapter 17: polynomial-time bounded algorithm for Minimum Vertex CoverMeaning G Input connected un-directed graph X Set of vertices C Final set of vertices This is a polynomial algorithm for getting the minimum vertex cover of connected undirected graph. The time complexity of this algorithm is O(n2) Section 17.1: Algorithm Pseudo CodeAlgorithm PMinVertexCover (graph G) Input connected graph G Output Minimum Vertex Cover Set Cnew Set < Vertex > ( ) X <- G. getAllVerticiesArrangedDescendinglyByDegree ( ) for v in X do List < Vertex > adjacentVertices1 <- G. getAdjacent ( v )! C contains any of adjacentVertices1 thenC. add ( v ) for vertex in C do List < vertex > adjacentVertices2 <- G. adjacentVertecies ( vertex )if C contains any of adjacentVertices2 thenreturn C C is the minimum vertex cover of graph G we can use bucket sort for sorting the vertices according to its degree because the maximum value of degrees is (n-1) where n is the number of vertices then the time complexity of the sorting will be O(n)# Page 109Algorithms Notes for Professionals 104Chapter 18: Multithreaded Algorithms Examples for some multithreaded algorithms. Section 18.1: Square matrix multiplication multithreadmultiply square matrix parallel ( A, B ) n = A. lines//create a new matrix n*n parallel for i =Section 18.2: Multiplication matrix vector multithread