Construct a MAXHEAP tree on the following list of numbers

15, 22, 7, 10, 43, 52 13, 16,

 Use Dijkstra's algorithm to find the shortest paths in the graph given below with a as the source vertex.

 Use Kruskal's algorithm to find a minimal spanning tree in the graph given below

Apply DFS algorithm to find a spanning tree of the following graph. Also find all the fundamental cycles of the spanning tree.

 Determine an LCS of (1, 1, 1, 1, 0, 1, 1, 0, 1, 0) and (1, 1, 1, 0, 0, 1, 1, 0, 1, 0).

 Taking  v₁ as the root, find a spanning tree of the following graph using the breadth first search (BFS) algorithm.

  Find an optimal parenthesisation of a matrix chain product whose sequence of dimensions is (4,8,7,2,3).

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).

Show step by step how the merge procedure of merge sort will merge the arrays ,1 ,3 ,4 ,7 11,9 13, 14, and ,2 ,5 ,6 10,8 12, . 

Using bipartite checking algorithm find out whether the graph given below is bipartite or not. 

 Show the results of inserting the keys
   C, R, A, U, J, S, F, T, O, P, M, L, N, W, Q

in order into an empty B-tree with minimum degree 2. Only draw the configurations of the tree just before some node must split, and also draw the final configuration. 

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 . 

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
    . 

Analyse the algorithm to find the upper bound for run time of the above algorith.

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.

) Given an image f(x,y) with Fourier transform F(u,v) obtain the fourier transform of .
c) Given that the 2-D Fourier Tranform is real and even, obtain the constraints on the image characteristics.

Define the following with suitable examples
i) Unsharp marking

ii) High boost filtering

iii) High frequency filterin

 Filter the given  gray level image with

i)  mean filter using zero padding

ii) 3×3 weighted mean filter using zero padding with mask

iii) 3×3 median filter processing only such pixels that have all the needed neighbours.
iv) Laplacian filter with the given mask and reflecting the border pixels.

 Describe the homographic filtering. Explain why the filtering scheme is effective for the applications it is used.

Given that

where f and g are real images and h is a spatial filter.

Obtain G(u,v), in terms of F(u,v), and H(u,v), the 2–D Fourier transform of g(x,y).