Abstract Data Type

Abstract Data Type - Trees

A tree is an abstract data type that stores elements hierarchically. Each element in a tree has a parent element and zero or more children elements. The top element of the tree is called root.

A tree can also be defined as a set of nodes storing elements in a parent child relationship.

The children nodes of the same parent are called siblings. The node which has children is known as internal. The node which has no children is known as external or leaves.

A free is ordered if there is a linear ordering defined for the chi1dren of each node. e.g, structured documents such as Books.

A binary tree is an ordered tree in which every node has at most two children.

The depth of a node is the number of ancestors of the node excluding the node itself. The depth of the root of a tree is 0.

The height of a tree is equal to the maximum depth of an external node of the tree. The height of the external node is 0.

A traversal of a tree is a systematic way of accessing all the

nodes of the tree. The traversal schemes for trees are preorder and postorder traversals. The root of the tree is visited first in the preorder whereas the children of a tree is visited first in postorder traverse.

The binary tree support a traversal method called inorder traversal.

Gaussian Elimination - Row reduction Algorithm

Gaussian elimination is a method for solving matrix equations of the form, Ax=b. This method is also known as the row reduction algorithm. Back Substitution Solving the last equation for the variable and then work backward into the first equation to solve it. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Pivot row The row that is used to perform elimination of a variable from other rows is called the pivot row. Example: Solving a linear equation The augmented matrix for the above equation shall be The equation shall be solved using back substitution. The eliminating the first variable (x1) in the first row (Pivot row) by carrying out the row operation. As the second row become zero, the row will be shifted to bottom by carrying out partial pivoting. Now, the second variable (x2) shall be eliminated by carrying out the row operation again. ...

Exercise 2 - Amdahl's Law

A programmer has parallelized 99% of a program, but there is no value in increasing the problem size, i.e., the program will always be run with the same problem size regardless of the number of processors or cores used. What is the expected speedup on 20 processors? Solution As per Amdahl's law, the speedup, N - No of processors = 20 f - % of parallel operation = 99% = 1 / (1 - 0.99) + (0.99 / 20) = 1 / 0.01 + (0.99 / 20) = 16.807 The expected speedup on 20 processors is 16.807

Minor, Cofactor, Determinant, Adjoint & Inverse of a Matrix

Consider a matrix Minor of a Matrix I n the above matrix A, the minor of first element a 11 shall be Cofactor The Cofactor C ij of an element a ij shall be When the sum of row number and column number is even, then Cofactor shall be positive, and for odd, Cofactor shall be negative. The determinant of an n x n matrix can be defined as the sum of multiplication of the first row element and their respective cofactors. Example, For a 2 x 2 matrix Cofactor C 11 = m 11 = | a 22 | = a 22 = 2 Determinant The determinant of A is |A| = (3 x 2) - (1 x 1) = 5 Adjoint or Adjucate The Adjoint matrix of A , adjA is the transpose of its cofactor matrix. Inverse Matrix A matrix should be square matrix to have an inverse matrix and also its determinant should not be zero. The multiplication of matrix and its inverse shall be Identity matrix. The square matrix has no inverse is called Singular. Inv A = adjA / |A| [ adjoint A / determ...

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