Algorithms Questions and Answers August 13, 2021

In this unsorted sequence 9,11,13,12,10,8,14. Sort the sequence using bucket sort. show all your steps and work. just draw pictures to show how the bucket sort is working.

write a c++ code to implement a radix sort the program should tell the user to Enter 10 numbers of 5 digits and the program will sort them using radix sort

Use radix sort on the following list of integers. Give intermediate answers for each digit (write the list again with them partially sorted). Remember that our version of radix sort is little-endian so starts from the lowest digit. 307 608 240 539 803 283 925 429 969 211

The Radix Sort introduced in the class works with array of positive integers. (a) Modify Radix Sort so that it can sort arrays of arbitrary numbers between given values m and M with upto two digits after decimal point. (b) Modify Radix Sort os that it can sort arrays of positive hexadecimal numbers.

Radix sort n integers of k digit (counting sort or bucket sort each digit and k is a constant) (a) Describe the most significant digit first radix sort algorithm. What is the time and space complexity 7 (b) Describe the least significant digit first radix sort algorithm. What is the time and space complexity 7 (c) Show that least significant digit

If Radix Sort is applied to the array of numbers [4567, 3210, 2345, 4321, 5678], show how these numbers will get rearranged after each of the four passes of the algorithm. Please use English, logic, mathematics, and/or pseudocode to formulate your answer.

a. Show the steps required to sort the following set of values using radix sort when using base 10 170, 45, 75, 90, 802, 24, 2, 66 b. In order for radix sort to work correctly, the sorting algorithm used to sort digits must be stable. Explain the previous statement c. What is the running time of radix sort Describe any variables you use other than

Sort the given numbers using Counting sort algorithm. Write the pseudocode. Give a Realtime example. 1,4, 1, 2, 7, 5, 2

Why if you change COUNTINGSORT line 10 (see picture) to for j 1 to A.length it makes counting sort not stable, and therefore cannot be used in radix sort in sorting the digits Counting Sort COUNTINGSORT (A, B, k) 1 let C0..k be a new array 0 to k Ci 0 for j 1 to A.length 2 for i 3 4 5 6 7 8 9 10 11 12 CAj CAj 1 Ci now contains the number of

Write a program that implements a Counting Sort that will sort an array of integers. Research how Counting Sort works before you implement it. We have the following list of numbers that are unordered: 90, 80, 30, 60, 50, 40, 70, 80, 40, 30, 60, 50, 70 Write a program that will arrange these numbers in ascending order.

and post count running sum. Upload the table as shown below. 5 3 Show a counting sort for the following. The 2 arrays in the center are for the count 1 8 2 3 1 6 4 0 9 8 1 2

Let A 3, 1,6,3,7,2. (a) 8 points Trace the execution of QUICKSORT(A, 1,6). Make sure to show all the intermediate steps of all calls to PARTITION.,(b) 2 points How many times in total is the PARTITION function called throughout the execu tion of QUICKSORT(A, 1,6)

Quicksort is one of the best sorting algorithms used in practice. You need to start from implementing a simple sequential version of Quicksort and making sure that it works correctly. Create a generic interface for inplace array sorting algorithms in which no new array is constructed. Implement the quicksort algorithm and test that its working

Consider the QUICKSORT and PARTITION algorithms below and suppose the initial call of QUICKSORT(X, 1,9), is X1..9 26,54,1,47,95,18,40,1,26 QUICKSORT(X,p,r) 1 2 3 1 PARTITION(X, p,r) 2 3 if p

Revise the HeapSort algorithm so that it sorts an array in descending order. Write down all the algorithmic steps. Next, use your algorithm and perform a HeapSort in descending order of the following array: 9, 12, 4, 8, 3. 11, 6, 15

Validate your heapsort program on arrays of small size (10s of elements) and compare the running times of your heapsort program with your implementation of the quicksort program (lab 7) on large integer arrays (at least 1000 entries). To aid you further in completing the assignment, here is a framework that you might want to follow to write a

Build the heap tree, then apply the HEAPSORT algorithm to sort the array A 9, 0, 8, 12, 11, 5, 4, 8, 1, 2, 7 in ascending order. (Note You should perform the HEAPSORT algorithm stepbystep and show your work

We are sorting the following array of integers via the HeapSort algorithm A 15, 8, 10, 17, 5, 9 Describe the evolution of the data in the array A after each iteration of the HeapSort algorithm.

Show how HeapSort process the input 142, 543, 123, 65, 453, 879, 572, 434, 111, 242, 811, 102. Show the intimal heap, and the heap after removing 1, 3, and 5 elements.

Asymptotic notation has been used for years without there being any need to change its purpose, which is to describe an algorithm's time complexity through mathematical expressions. Asymptotic notation is not always taken into account when programmers analyze a problem and start coding a solution. It is a key concept when the quality of the final

a) What do you mean by Asymptotic Analysis Why are Asymptotic Notations important Explain, different types of Asymptotic Notations with diagram. b) Using Master Method find the asymptotic bound for the following recurrences i) T(n)16T(n4)n ii) T(n)3T(n4)n iii) T(n) T(2n3)1

log x Rigorously prove that (log x) grows faster asymptotically than x 0() or o() asymptotic notation to represent the final conclusion. Use

Find a closed form for each of the following sums,I (2012 1 2013) (b) Assuming the following sum equals a polynomial in n, find the polynomial. Then verify by induction that the sum equals the polynomial you find.

Given that Insertion sort is worst case (n2) and Merge sort is worst case (nlg n): Under what conditions would Insertion sort outperform Merge sort? You can give a concrete example of this and show/plot the growth of the function as a graph. Describe scenario(s) in which Insertion sort may be preferable to Merge sort as it is implemented in the

Write a Java program to implement Insertion Sort algorithm that uses Binary Search to find the position where the next insertion should take place.

a) Outline how Algorithm Performance Analysis can be used to compare the quality of algorithms. Make sure to include a description of the order notation and a discussion of common orders. b) Explain in plain English the the Insertion Sort algorithm. c) Provide a pseudocode representation of the Insertion Sort algorithm. d) Provide an

Create String Search Algorithms using C++ With the BoyerMooreHorspool algorithm and the RabinKarp algorithm. Remember Your Code must use the following algorithms above and your coding must be in C++ Format only. IMPLEMENT BOTH algorithms - BoyerMooreHorspool algorithm and the RabinKarp algorithm IN THE CODING PROGRAM. Implement both algorithms

Given the following algorithms ( Algorithm A i1 while(i 0 and Aljl key Aj1 Aljl jj1 Aj1 key jj1 Find for both algorithms a) What is the general case T (n) b) What is the bestcase T (n) What is the worstcase T (n) Algorithm B int min 1000 i1 while(i arr.length) if(arri

Compute MST for the following algorithms (a) Prims algorithm (b) Dijkistras algorithm (Source a) (c) Kruskals algorithm C 4 g 5 2 6 3 a 4 d h k 5 7 3 1 6 8 b e i 3 4 5 T 2 3 6 9 f 5

8. Discuss briefly any two of the following a) Heap sort b) Dynamic Programming c) Height balanced tree d) Parallel algorithm

Discuss Strassen's matrix multiplication and derive it's time complexity.

b) Discuss Strassens matrix multiplication and derive time complexity.

Show the process that merge sort goes through to sort the following list 3 7 0 4 4. You do not have to show all of the steps of the merge, just what the merge sort does.

Given the array (2, 5, 16, 4, 10, 23, 39, 18, 26, 15) Show what it looks like after the 3rd recursive call of merge sort is complete. Use the merge sort and merge algorithms described in class.

Merge sort involves recursively splitting the array into 2 parts, sorting and finally merging them. A variant of merge sort is called 3way merge sort where instead of splitting the array into 2 parts we split it into 3 parts. As shown in the MergeSort code, it recursively breaks down the arrays to subarrays of size half. Similarly, 3way Merge sort

We consider the merge sort algorithm that uses a divideandconquer design strategy. The running time of a merge sort algorithm is given below T(A) 2T() n if n 2 if n2K, K 1 Prove that the expression giving the complex ity of the merge sort algorithm with which you are already familiar is also a solution to the above recurrence. Hint use

a) Both the merge sort and quick sort algorithms sort a list by partitioning it. Explain how the merge sort algorithm differs from the quick sort algorithm in partitioning the list. b) Show the steps how the following list of keys is sorted using either Merge sort or Quick sort. c) Show the number of comparisons required for any one the sorting

Draw the compare-and-exchange circuit configuration for the odd-even merge sort algorithm . Sort the following sequence by hand using the odd-even merge sort algorithm. 12 2 11 4 9 1 10 15 5 7 14 3 8 13 6 16

write code for merge sort algorithm for a 1*15 array without using recursive logic discuss about the computational cost of a non-recursive merge sort algorithm with a merge sort algorithm that is with a recursive logic.

Complete any implementation of Merge sort with an int array, where you are to sort in ascending order. Describe its worstcase time complexity afterwards.

)a) How many swaps are necessary to perform the Bubble Sort Algorithm on the input 3, 1, 8, 5, 6, 10, 4, 9 b) In the worst case, how many swaps would be necessary to Bubble Sort a list with 8 elements

Write a program that implements the following search algorithms: Selection Sort Bubble Sort Merge Sort Quick Sort Create an array of randomly selected whole numbers of 1,000 elements Each value is selected from the range 0-999 Sort that array using each method Make sure to copy the initial array into a new one so that the tests will be accurate

Write a C program that Take 10 integer inputs from user storing them in an array then Sorting the array in ascending order using Bubble sort Write functions Bubble sort Swap functions using pointer approach. The Bubble sort function should decrease number of comparisons every pass, make N1 comparisons on the second pass, N2 on the third pass,

1. Explain how Bubble Sort works with an example array Int a 23, 45, 59, 66, 68, 12 2. Write a Working code of Bubble Sort in C. Requirements I. Size of the array is decided by the user li. The elements in the array are input by the user. lii. Print the result of each pass of the algorithm, to highlight all the steps. 3. Analyse the best case and

Apply bubble sort to the following list to sort it in ascending order. Record your answer for the first two passes. 29 73 43 97 52 Original Pass 1 80 How many comparisons and exchanges are done in each pass of the bubble sort algorithm Passes Exchanges Comparisons Pass 1 Pass 2 Pass 2

Create a program that allows a user to input 10 grades. Leverage two functions BUBBLE SORT CODE IN C++ 2 FUNCTIONS 1. Function to bubble sort the array of grades 2. Function that prints to screen every step

The bubble sort is O(n2 ). Like selection sort, bubble sort wont perform any swaps if the list is already sorted. However, bubble sorts worst-case behavior for exchanges is greater than linear. The proof of this is left as an exercise for you. You can make a minor adjustment to the bubble sort to improve its best-case performance to linear. If no

1. For the following array x, show x after each of the first two passes of simple selec tion sort to arrange the elements in ascending order 2. 3. i 1 30 i 2 3 4 50 70 10 40 60 a. For the following array x, show x after each of the first two passes of bubble sort to arrange the elements in descending order i 1 2 3 60 50 70 10 5 1 20 6 4 5 6 40 20

Implement Bubble Sort Naive Implement Bubble Sort Optimized Handle Files and for each test case, call your bubble sort functions. Sample file given below. Sequential Bubble sort Pseudocode:

i N sorted false while ((i 1 ) (sorted)) sorted true for (int j1 j aj) temp aj1 aj1 aj aj temp sorted false i

Create a function that implements the bubble sort algorithm. Its input parameters are an array of numbers and arrays size. Create a program that store 25 random numbers in an array, and sort it using the function described above. The output of the program must display the content of the array after each pass of the bubble sort algorithm and, at

Find The Optimal Binary Search Tree (BST) By Dynamic Programming For The Following.. Key Pi Qi 2 10 1 al 53 3 8 1 4 12 4

Follow the Binary Search Tree (Years tableodes below) to answer the questions: 1/ Which year has the biggest value in the BINARY SEARCH TREE (Java)? 2/ Which year has the smallest value in the BINARY SEARCH TREE (Java)? 3/ Which year has the biggest value in the BINARY SEARCH TREE with root 2018 (Java)? 4/ Which year has the smallest value in the

2012 2017 2013 2018 2015 2019 2016 2014 2020 2022 2021 2024 2023 2025 2026 2027

Draw a binary search tree with 15 nodes that shows the worst case scenario O(n) Draw a binary search tree with 15 nodes that shows the average time scenario

4. (12 points) (a) Draw a picture of a binary search tree if nodes with the given keys are inserted in this order 25 47 73 36 29 14 42 20 8 23 81 75 (b) Draw the complete resulting binary search tree if node with key 36 is removed from the tree from part (a). (c) Draw the complete resulting binary search tree if node with key 25 is removed from

1. Given the following keys, 12 9 18 5 6 11 4 20 14 (a) Construct a binary search tree (b) Find the height of each tree (c) Use the height of binary search tree to calculate the approximated number of nodes. (d) Insert key of 15 into the binary search tree (BST). (e) What would be the time complexity of insert operation (f) List the leave nodes of

Add or remove an element from a binary search tree with maintaining the property of binary search tree add 21 to the binary search tree delete 30 from the binary search tree

7 10 25 18 27 28 30 31 32 34

(a) Determine the time complexity for efficiently finding an arbitrary item in an arbitrary binary search tree of n nodes (in enotation). (b) Determine the time complexity for efficiently deleting an arbitrary item in an arbitrary binary search tree of n nodes, once the item has been located (in 0notation). (c) Determine the maximum height of an

pre_order(L): creates a binary search tree from a LlST of preordered k,v tupIes, the input should be a list and the output should be the binary search tree post_order(L): creates a binary search tree from a LlST of post order k,v tupIes, the input should be a list and the output should be the binary search tree

3 3 1 1 00 2 2 Traversals of this tree yields the following lists of kv tuples preorder (3, 3), (1,1), (0, 0), (2, 2) postorder (0, 0), (2, 2), (1,1), (3, 3).

a) Draw the rooted tree with prefixed expression x + B-DF + AC. b) Draw the rooted tree with the posted expression ED-A + BC-Fx +.

A rooted tree is a tree T where one vertex has been distinguished and called the root. A root vertex is written as a square, not a circle. So the following tree has root v 4 5 2 3 (a) List all rooted trees with V 3. (b) Prove that the number of rooted trees with V n is n1.

Find the Prfer code for the following rooted tree on 10. 3 10 6 5 9 2

29. Draw the rooted tree T represented by the adjacency matrix shown below. Hint Node a is the root. a th PUP b d e f a 0 0 000 0 b 1 0 0 0 1 0 0 0 0 0 d 1 0 0 0 0 E 0 1 0 0 0 f 0 0 0 0 1 0

Define a function of getSmallestVar that accepts a vector of IntegerVariable object pointers and it will find and return the pointer to the object that has the smallest value.

Create two objects: a rectangle and a square. Also, create a rectangle pointer named rptr and initialize it to nullptr. Then: point the rptr pointer to the rectangle object and use it to call the getName() function. point the rptr pointer to the square object and use it to call the getName() function.

Write main function to answer the following questions. Only submit the main function for this problem. i. Create an Employee object using the constructor with parameters. [2pts] ii. Declare an Employee pointer and make it point to the Employee object created in part (i). [2pts] iii. Use the pointer to call the print() method of the object. [2pts]

Question 11 Consider the following text message between the double quotes aba bbaab bacbccbbb a cbbccb aaabaa aba bo 1. Build a Huffman coding tree that can be used to encode this message. 2. Show each symbol and its Huffman code representation. 3. Show the code for the transmitted Huffman coding tree, replacing the ASCII code of each character

A file contains the following characters with the frequencies as shown. Frequencies 10 15 12 4 Characteristics a i 3 u If Huffman Coding is used for data compression, determine a. Huffman Code for each character b. Average code length c. Length of Huffman encoded message (in bits) 13 S 1

1. Prove that the optimal substructure exists in Huffman coding 2. Prove that greedy-choice exists in Huffman coding

A source has an alphabet (a, az, a3, a4 with corresponding probabilities (0.1,0.2, 0.3, 0.4). (i) Find the entropy of the source. (ii) Identify what is the minimum required average code word length to represent this source for errorfree reconstruction (iii) Design a Huffman code for the source and compare the average length of the Huffman code

A DMS source is given with probabilities 0.27, 0.23, 0.2, 0.15, 0.10, 0.05). (a) Design a binary Huffman code and find the code efficiency. (b) Develop ternary Huffman code with maximum code efficiency.

A file contains the following characters with the frequencies as shown. If Huffman Coding is used for data compression, determine 1. Huffman Code for each character 2. Average code length 3. Length of Huffman encoded message (in bits) Characters Frequencies a e i 0 u S t 10 15 12 3 4 13 a. First let us construct the Huffman Tree and assign weight

A file contains the following characters with the frequencies as shown. If Huffman Coding is used for data compression, determine- Huffman Code for each character Average code length (for each character) Length of Huffman encoded message (in bits) Characters Frequencies a 10 e 15 i 12 o 3 u 4 s 13 t 1 Try Huffman encoding to encode the strings (i)

Starting from node "A", apply breadth-first search traversing all remaining nodes. If multiple alternative nodes are available at any step, your choice should be based on alphabetical order. In your solution, provide: a) The output list showing the order in which every node is visited. b) A tree search that indicates the resulting hierarchy from

1 131 C 41 L76 30 Q 107 45 H 196 D10141 N 160 90 G78 34 mawar mama tam T 127 110 30 41 S 164 4 es prese P40 3 M109 29R 122 K 75 F10838E 139 were 2 we 39. 0 160 B 152 A167

Draw a spanning tree formed by applying the breadth-first search algorithm to Kn. Draw a spanning tree formed by applying the breadth-first search algorithm to K m, n . Can a spanning tree formed by using the breadth-first search algorithm be a simple path?

Explain the significance of Breadth-First Search in graphs and its time complexity. List down any 4 (four) applications of Breadth-First Search in details.

1) Rod cutting problem A cutting factory buys long steel rods and then and cuts them into shorter rods, which it then sells. Each cut is free. The selling price is based on the length of the cut. The factory want to determine the best cuts in order to maximize its revenue. Below, the price that the factory can make for each length of rod is given.

A What is the Huffman code for C What is the Huffman code for R What is the Huffman code for A 2 Y 1 3 (space) 1 2 12 C 1 N Ex 011 3 D 1 G 1 4 O R 1

Consider a DMS where output is generated from an alphabet {a,b,c,d,e,f,g,h} with respective with probabilities {0.05, 0.1, 0.25, 0.02, 0.25, 0.03, 0.25, 0.05}.Design a Huffman code and sketch the corresponding coding tree. Explain the codewords for each character in the alphabet.Calculate the average length of the codeword if the Huffman code you

1. Show that Huffman coding is uniquely decipherable. 2. Show that Huffman coding is instantaneous. 3. Show that Huffman coding is not unique

Solve this problem and write the algorithm in Java language

6. Define Huffman coding technique. Construct a Huffman coding tree using the given Letters and frequencies below. Write the algorithm. Letter U Z CDE K L M Frequency 32 42 120 7 42 24 37 2

3. Prove the correctness of the rod cutting algorithm below using a loop invariant for the outer loop. Note that the algorithms inputs are a table of rod prices and the length of rod to be cut, and its output is the optimal price that can be obtained from cutting the given length of rod. cutrod (p, n) results array of size n1 results 0 0 for j1

6. (20 points) In the rod cutting problem, we are given a rod of length n, along with pi, which indicates the price of a rodpiece of length pi, for i 0, 1, 2, ...., n. Consider the following instance Po 0, P 2, P2 7,P3 15, P4 16 and n 4. Let r denote the maximum profit you can obtain by cutting a rod of length i into smaller pieces (or by

int se rma 4. (1) (7 points) Please find the Huffman code for each character in the table below. Character 0 m P u t e Frequency 2 3 30 4 22 11 28 Smy Huffman Code (2) (3 points) Please decode the Huffiman code string to a character string 00000100100010000111110

1.) Explain how one can identify connected components of a graph by using a. a depth-rst search. b. a breadth-rst search 2.) Explain how one can check a graphs acyclicity by using breadth-rst search

Consider a state space with branching factor 10. If a depth-first search to depth limited to 5 levels takes one minute to complete, approximately how long will depth-first search to depth limited to 6 levels take to complete?

1. 2. Trace the depthfirst search pseudocode below on the graph below and show only the sequence of vertices visited dfs (adj, s) n adj.last for (i 1 to n) visiti false dfsrecurs (adj, visit, s) dfsrecurs (adj, visit, v) visit v true ref adj v while (ref null) if (visit ref. data) dfsrecurs (adj, visit, ref.data) ref ref.next backtrack

(NPP) Write the modes in a depthfirst search (DFS) and a breadthfirst search (BFS), starting from the node s 0f the choose anode in an alphabetical order) a) Depthfirst search (DFS) b) Breadthfirst search (BFS) j

Apply the kruskals algorithm on the following graph ( Represented using the adjacency matrix) 1 2 3 5 7 0 8 1 6 6 3 2 3 6 6 3 1 4 8 6 5 3 10 6 9 Note Write and explain all the steps 6 3 1 8 8 5 6 5

Find the minimal spanning tree using Kruskals Algorithm for the following graph 743799 3 12 4 8 1 4 2 11 5 13 2 6 8 7 314637603813042023031120 10 5 9 9 037 Prove that (Vx) (P(x) Q(x)) logically implies (Vx)P(x) (Vx)Q(x). (793814442023 382012023031120,Find the minimal spanning tree using Kruskals Algorithm for the following graph 743799 3 12 4 8

4. Topological Sort What is the topological sort of the graph above Show the process with the removed, indegree, and zero degree tables as we did in class.,How many possibilities are there to select the first node in the topological sort above

Performing DFS starting at node A, what is the topological sorting of the following graph Draw the final topological sort. Remember that the topological sort is the decreasing ending times of the DFS traversal. When doing the traversal, follow the nodes in alphabetical order A D H E F

Consider the following directed graph. Draw a picture of how it might be represented in a computer using an adjacency list. Use the topological sorting algorithm label the nodes in a legal topological order. Show the calculations you made to execute the topological sorting algorithm.

Consider the graph shown at right S e) List any source or sink vertices if they exist. f) Perform a topological sort on this graph. W T (X,Consider the graph shown at right e) List any source or sink vertices if they exist. f) Perform a topological sort on this graph. E H K,Consider the graph shown at right 0.55 0.3 B 0.1 0.9 e) List any source or

A topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. Please give topological sort of following graph.

A C B Topological Sort D G E LL H

Find the topological sorting problem for the given graph. A B G D C E F,Find the topological sorting problem for the given graph (A) (15) Apply Merge sort and Quick to sort the list, L, I, B, R, A, R, Y in alphabetical order.

Using C++ need to implement topological sorting Requirenments: can only include or Represent the diagraph with Adjacency List Determine the data structures for the proccess of improved topological sorting Design a C++ programm implementing the improved topological sorting

Topological sort is an algorithm used to linearly order the vertices of a directed graph so that for every directed edge (u, v), vertex u comes before vertex v in the ordering. It applies only to directed acyclic graphs (DAGs) since cyclic graphs cannot have a valid topological ordering, since dependencies cannot be resolved. Further, multiple

1. Which of the following 3x3 matrices are in reduced rowechelon form (a) 100 010 001 2. (c) 0 1 0 001 000 (e) 100 000 001 (2) 110 010 000 001 000 000 (a) 100 010 001 (c) 100 010 020 15 01 (b) 100 010 0 0 (d) 100 001 000 (f) 000 (h) 0 1 07 100 000 O Which of the following 3x3 matrices are in rowechelon form (b) 120 010 000 013 000 00 230 000 000

1. Apply the DFSbased algorithm to solve the topological sorting problem for the following digraphs. 9 (a) (b) 8. Implement the two topological sorting algorithms in the language of your choice. Run an experiment to compare their running times.

Is the below graph, DAG? Why? Show a possible topological sort for the below graph. (Use stack data structure) Explain two applications of topological sort Is topological sort unique? Why? Explain it through an example.

1 2 3 4 5 6

Write a topological sort for the graph below. If the graph doesnt have a topological sort youshould indicate that in your answer 2 5 10