With the help of an example, explain the following:
i) Algorithm,
ii) Input and output for an algorithm
iii) Running time of an algorithm (5)
b) Illustrate the operations of partition in quick sort on the array
.
An algorithm is a step-by-step procedure to solve a problem or perform a task. It provides a clear sequence of instructions that can be followed to achieve the desired output. For example, an algorithm to find the largest number in an array could be: 1. Start with the first element as the largest. 2. Compare each element of the array with the current largest. 3. If an element is greater, update the largest. 4. Continue until all elements are checked. 5. Return the largest number. Input and output for an algorithm: * Input is the data provided to the algorithm to process. For the example above, the input is an array of numbers, e.g., `[4, 11, 9, 6, 13]`. * Output is _________ ___ ____ _______ _____ _______ _____ ________ ______ ________ ____ ___.
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Construct a binary search tree with the following keys, inserting them in the given order:
110, 8, 19, 6, 9, 12, 32.
Use Dijkstra's algorithm to find the shortest paths in the graph given below with a as the source vertex.
For the following set of points, describe how the CLOSET-PAIR algorithm finds a closest pair of points:
(1,2),(2,−1),(3,−3),(3,2),(−3,4),(−1,2),(2,3),(1,1),(−5,−2).
Find an optimal parenthesisation of a matrix chain product whose sequence of dimensions is (4,8,7,2,3).
Using bipartite checking algorithm find out whether the graph given below is bipartite or not.
Show step by step how you will construct a binary search tree by inserting keys in the order 12, 6, 4, 7, 8, 10, 7, 5, 13, 16, 11, starting from an empty binary tree. Explain step by step how you will delete the node with the key 13 from the binary tree. (5)
b) Illustrate the operations of performing counter sort on the array .
Write an algorithm to find the first two largest numbers in an in array of integers. For example, given the input ,2 ,3 ,7 − ,4 ,1,5 the algorithm should output ,7 .5 State precisely a loop invariant for you algorithm. Prove that your loop invariant holds and hence conclude that your algorithm works.