Algorithms Questions and Answers August 11, 2021

Consider the rooted tree of Fig. (a). Draw a figure showing its representation in the first-child/next-sibling form. Consider the rooted tree of Fig. 1(b) represented in the first-child/nextsibling form. Draw a figure showing the equivalent rooted tree

h T (a) root HUP ... h (b)

This question is about graph theory. Let T (V, E) be a tree, and let u, v E V be any two nodes. Prove that the height of the rooted tree (T, u) is at most twice the height of the rooted tree (T, v).

3. Let Tn be the set of all the rooted trees on the vertex set n with the root 1. A leaf of a tree T is a nonroot vertex that has degree 1. (a) Can a rooted tree in Tn have no leaf Justify your answer. (b) How many rooted trees in Tn have exactly one leaf Describe those trees. (Note that the root vertex 1, by definition, is not a leaf. In the

Let T (V, E) be a tree, and let r, r E V be any two nodes. Prove that the height of the rooted tree (T, r) is at most twice the height of the rooted tree (T, r).

a) Given the rooted tree shown in Figure 4. b 1. me go Figure 4 i) Find the following verticesvertex in the rooted tree by completing Table 1. Root Leaves Child of d Parent of q Descendent of c Siblings of n Ancestor of m Table 1 ii) How many vertices and edges does the rooted tree have

4. Answer the following. a. What can you say about a vertex in a rooted tree that has no ancestor b. What can you say about a vertex in a rooted tree that has no descendants 5. Find a spanning tree for the following. a SO g h 1 i k Ja 1 b d

Recall, that a rooted tree is a tree with a distinguished vertex called the root. (i) List all rooted trees on five vertices. (ii) Describe a recursive procedure for listing all rooted trees on n vertices.

Define size and height (see Exercise 8.10) for rooted ternery trees (RTT) and rooted trees (RT). (a) Prove For any rooted ternary tree (RTT) 7, size(7) (3height(7)11)3. Find such a bound for the size of a rooted tree (RT) in terms of its height or explain why there isnt one. (b) Prove For any rooted ternary tree (RTT), size(7) height(7) 1. Find

In this question two rooted trees are considered to be isomorphic if they are isomorphic as directed graphs. For example, the rooted trees TA and TB shown on the right are isomorphic because an isomorphism fof TA to TB is given by f(1) 5, f(2)7, f(3) 6,4) 8. Rooted trees that are not isomorphic are said to be nonisomorphic. (a) 6 T4 8 TB Draw

Suppose that preferences are quasilincar u(x, y) f(x) y, and monotonic f(r) 0. Denote the price of X by p and let Y be a numeraire p2 1. In parts (a) and (b) assume that income m is fixed and high enough for an interior solution. Also recall that with quasilinear preferences the indifference curves eventually intersect both axes. (a) (5pts)

4) Rooted Trees and Searches 1 2 5 6 HE 9 3 O a) Create a Rooted Tree with Root 5. Is the tree unbalanced or balanced b) Apply the search method you feel is most appropriate and write out the search order. c) Create a Rooted Tree with Root 7. Is the tree unbalanced or balanced d) Apply the search method you feel is most appropriate and write out

1 a. Shown below is an unrooted tree for 5 species. Draw all of the rooted trees that are compatible with this tree (3.5 points). B Tree 1 E b. If you have molecular data that allows you to construct an unrooted tree, what process or processes can you use to find the root What does providing a root do to an unrooted tree (1.5 points) A c. Shown

Determine the order in which the following traversals visit the vertices of the given ordered rooted tree. List the sequence of vertices in the order visited. (a) a preorder traversal (b) an inorder traversal (c) a postorder traversal b do a DOF Rh o of P

Declare a class "Fruits" with any two data Members and two Member functions. Derive 2 class Mango and Orange from this Fruits class with one data Member and Member function. Create two Pointer objects of Fruits class and convert one to the Mango Class Pointer Object and One to the Orange Class Pointer Object through explicit Cast

struct Student int id string name Student next 1. Using the struct definition above, create three Student objects by using three different pointers and new operator. 2. Using your pointers fill id and name information for each Student object. 3. By using your pointers and next link your objects and fill the next pointer of the last object with

Write a code in C++ using all the informations down below, precede every program with a comment explaining the purpose of the program (It should have .cpp and .hheader) Introduction to Structured Data and Classes. Outline: Structures Unions Enumerations Procedural and Object Oriented Programming Introduction to Classes. Pointers. Outline:

Smart pointers are designed to make dynamic memory more accessible and safe to use. They (i.e., smart pointers) are implemented as wrappers around raw pointers, providing behaviors reminiscent of the raw pointers youre so familiar with, while safeguarding against the pitfalls and opportunities for errors that youve probably experienced. We will

1 include 2 3 int main () 4 testing 5 stdcout Lets implement a UnqiuePtx stdendl 6 7 11 naiveuniquepointer driver.cc uniqueptr.hpp uniqueptr.cc Click run program to compile and run your code. Click grade code to submit and grade your code. CS 128 A Editor Run Grade 8 D

Give typed examples of the following and comment the code:

a. passing parameters by reference b. passing parameters by const reference c. pointer arithmetic d. dereferencing pointers e. pointers to baseclass objects f. pointers to derivedclass objects g. new ALWAYS returns a pointer to the memory allocated i. int tmp new int25 tmp MUST be a pointer

Write a program that defines an object of type double. Define a pointer that points to that object. Print the value of the pointed-to object by dereferencing a pointer.

Write code to instantiatecreate two Bicycle objects a default object (mybike) and a nondefault object (sisterbike). For the nondefault object you may hardcode (or make up) the required parameters (brand, price and wheel size). No console input is needed. 2. (5 points) Declare two pointers to my bike and sisterbike objects. Pointer names must

a. Can objects on the stack be shared by different functions? b. Is managing objects on the heap easier to claim and release than the objects on stack? c. Are objects on the heap managed via pointers and objects on stack be used through variable names and reference?

A6,5,8,7,3,4,2,1. Perform the Merge sort algorithm on this array, using a visual illustration of each step to generate the sorted array 1,2,3,4,5,6,7,8. Show that exactly 12 comparisons are needed to sort this input array A Directions Show step by step visualization of the merge sort algorithm on the given array.

What is Parallel Merge Sort and how it differs from regular merge sort what is the recurrence relation of parallel merge sort proof the running time of parallel merge sort.

you will implement divide and conquer algorithms: merge sort . For merge sort, you will just implement the algorithm on an array of ints. I have provided the pseudo code for algorithms. You will need to handle cases of all sizes, not just powers of 2. Implement Merge Sort Create a class called MergeSorter in the divideandconquer package. This

Merge Sort merge Sort (arr, start, end) if startend return mid midpoint of arr merge Sort (arr, start, mid) mergeSort (arr, mid1, end) merge the two parts. merge (arr, start, mid, end) merge (arr, start, mid, end) leftSize mid rightSize start 1 end mid left 0..leftSize right 0..rightSize for leftIndex0 to leftSize left leftIndex for

The merge sort is a recursive algorithm that can be used to sort an array by first sorting smaller versions of an initial list of object. In this discussion, provide a simple of code sample of Merge sort and describe the benefits of using the merge sort. What are the key steps that must be taken to ensure an efficient sorting of objects? In your

Iterative Deepening Search is the combination of BreadthFirst Search DepthFirstSearch, what benefits do we get in Iterative Deepening Search that lack in BreadthFirst Search DepthFirst Search

Let the nodes of a Breadthfirst search (BFS) model be the following bfs A B,C, B D, E, C F, D , E F, F0 a) Complete the code to perform the Breadthfirst search (BFS) algorithm. (10 marks) b) Calculate the distance of the node D from the main node. (5 marks) provide the output of the code too

Determine if the network of Figure 5 is balanced using breadthfirst search starting at node B. iii) Draw the layered graph resulting from the breadthfirst search among supernodes to ex plain your reasoning. Figure 5 A social network

Write a client program: a. Sequential client: Each client generates 20 requests; Each request consists of an integer, i.e., i th request is an integer "i" ranging from 1 to 20; display the received response on the terminal; close the connection after 20 requests. b. Concurrent client: For designs 2(b), 2(c), and 2(d), the client program is a

2. Consider the following graph Show the d and values that result from running breadthfirst search on the above graph. Start the traversal at vertex A, and resolve ties by the vertex in alphabetical order. What is the time complexity of breadthfirst search algorithm

4. Use the BreadthFirst Search Algorithm to construct the breadthfirst search tree with root u for the given graph W X 30 u y 0 Z S

Show the order of traversals: (a) A breadth-first search starting at 13. (b) A breadth-first search starting at 0. (c) A depth-first search starting at 13. (d) A depth-first search starting at 0.

10 4 13 1 8 5 3 9 2 7 12 6 Show the order of the following traversals 11

Consider the following graph 1. Perform a breadthfirst search (BFS) traversal on the graph. Use vertex O as the root node. 2. Perform a breadthfirst search (BFS) traversal on the graph. Use vertex 7 as the root node.

Apply Breadth-First Search to traverse the graph below. Create a table to present each node and its distance from the source in the Q for each pass of the Breadth-First Search traverse. Start with node s and go in c direction

a S d b f h i

10. The breadthfirst search algorithm can be applied to digraphs if slight changes are made. Modify the algorithm so that it can be used with digraphs. Apply your modified breadthfirst search algorithm to the following digraph. B AO H F G Activate Windo

14. Use the BreadthFirst Search Algorithm tree with root v for the graph G of Figure 13.25. G V3 Figure 13.25 15 V1 V2 U7 VA to construct the breadthfirst search V6

Perform a breadth-first search on the graph assuming that the vertices and adjacency lists are listed in alphabetical order. Show the breadth-first search tree that is generated.

Consider the following directed graph for each of the problems F LL C D H A B E K J

Breadthfirst search Assuming that each edge has the same length 1, show the distance that result from running breadthfirst search on this directed graph, using Vertex 3 as the source. 1 2 4 To V D from V3 3 5 6

A delta-star connected three phase transformer 50 Hz, has 24,000 Volt and 400 Volt as line voltages. The secondary side of the transformer is connected to a star balanced load at 80% power factor, which is in lagging nature. Assume the value of 6 A as the line current on the primary side of the transformer. Find the values of the current in

5. You are given the following graph 2 A F 4 1 3 D 3 2 B 2 6 E 5 H Do a breadthfirst search of the graph, starting from node A. There are several possible orders that a breadthfirst search could choose. Show two of these orders.

Given the directed graph below, show the traversal order (e.g., 1, 2, 3, 4, ....) when you apply (i) breadthfirst search starting from 2 and (ii) breadthfirst search starting from 10. 2 11 10 Activate Windows Go to Settings to activate W

During Shibuya Arc Gojo Satoru was sealed away in the Prison Realm by an ancient sorcerer named Kenjaku. Prison Realm is a special grade cursed object containing a barrier capable of sealing absolutely anything or anyone inside of an inescapable pocket dimension. While inside The Prison Realm, Gojos Six Eyes able to determine the structure of the

b C PROJ d a f e g b. Traverse the graph of Problem 4.a by breadthfirst search and construct the corresponding breadth first search tree. Start the traversal at vertex a and resolve ties by the vertex alphabetical order.

(a) Build the Huffman coding tree and determine the codes for the following set of letters and weights Letter A B CDE F G HI J K L Frequency 2 3 5 7 11 13 17 19 23 31 37 41 (b) What is the expected length in bits of a message containing n characters for this frequency distribution (c) Encode message Deed by using this Huffman code (d) Decode the

What are the differences between Huffman coding and Adaptive huffman coding ? You can explain from how they work, the advantages and the drawback.

Write the File compression and Decompression program using the Huffman Algorithm in C++. PROGRAM REQUIREMENTS: 1. read file and calculate number of appearance for each symbol in the file, 2. create tree with Huffman code 3. compress the file using Huffman code 4. decompress compressed file using Huffman code

Huffman coding is proved optimal, and is used many practical standards, such as JPEG and MPEG. Encode OTTAWA using Huffman coding algorithm.

Write the File compression and Decompression program using the Huffman Algorithm in C++. You need to: 1. read file and calculate number of appearance for each symbol in the file, 2. create tree with Huffman code 3. compress the file using Huffman code 4. decompress compressed file using Huffman code

Huffman coding . Consider the random variable X= x1 x2 x3 x4 x5 x6 x7 . 0.49 0.26 0.12 0.04 0.04 0.03 0.02 (a) Find a binary Huffman code for X. (b) Find the expected code length for this encoding. (c) Find a ternary Huffman code for X.

1. List the Huffman codes for several letters given a Huffman Tree 2. Develop a Huffman tree, given frequencies for several characters. List Huffman codes for several letters Using the following Huffman tree, list for code for each symbol A H S N

Using the alphabet and symbol probabilities shown in Table 1 create a Huffman code and find the average length of the Huffman code

Symbol (A) A B Symbol probability P(Ai) 0.5 0.0625 0.0625 Code Word K(A) K(B) 0.125 0.25 C D E Table 1. Alphabet and symbol probabilities for Huffman coding K(C) K(D) K(E)

A book containing 1000000 characters with the following probabilities. 0001 ol Character B 0.05 out A C 0.03 D 0.3 Weight 0.1 Huffman Code 0 I Draw its Huffman tree and assign Huffman codes to these characters 1 E 0.5 9000 F 0.02

Analyse the subproblems and comment on whether the subsubproblems of those subproblems overlap or not. length, 1 2 3 4 5 6 7 8 9 10 price, p. 1 5 8 10 14 8 9 7 4 13 Table 1 Length and associated price for an instance of a rod cutting problem

t you are asked to implement a dynamic programming algorithm for the Rod Cutting Problem . In the Rod Cutting problem, you are given an integer n21, along with a sequence of positive prices, P P2. Pn. where p, is the market price of a rod of length i. The goal is to figure out a best way of cutting the given rod of length n to generate the maximum

2 a) b) c) In a rod cutting problem, given a table of prices P, you are required to determine the maximum revenue I obtainable by cutting up a given rod of length n and selling the pieces. The following is a price matrix in one such situation. Lengthi 1 Price Pi 1 2 5 3 8 4 8 How can brute force approach be used to solve this problem. Is this

length 1 2 3 4 5 6 price 1 5 8 9 10 17 n 6 Goal Maximize Profit. Cut the Rod into pieces, can not reuse a cut length. Q1) Try solve using the Knapsack Dynamic Programing 01 Approach. Q2) Show the tree representation of this Rod lengths.

Consider the following variant of the Rod Cutting problem. As we cut the rod, it becomes weaker, so we can only make k

length 1 2 3 4 5 6 7 8 You are given the following prices of rod cutting. 9 10 price 2 5 6 9 9 17 17 18 20 22 longis price 2 2 3 5 6 56 9 2 To 1717182022 Compute the optimal rod cutting when the rod length is 10 using Dynamic programming table Activate W Go to Setting

Trace the bottomup tabulation Rod Cutting algorithm (below) on the following input the following input p1..5 1, 5, 8, 9, 10 and n 10. Draw the resulting table r (which stores the maximum revenue achievable for given rod length) and c which stores the first cut being made. A company buys long steel rods (of length n), cuts them into shorter

EXTENDEDBOTTOMUPCUTROD (p, n) 1 let r0..n and s0..n be new arrays 2 r0 0 3 for j1 to n 4 5 6 7 8 9 9 for i 1 to j if q pi rji q pi rji sj i rj q 10 return and s Store the optimal first piece size for rod length j Provide clear, concise answers to each of the following related questions. . (10 points). Consider an extension to the rod

What is the worst-case asymptotic time complexity (a.k.a. "running time") of the dynamic programming solution to the rod cutting problem?

Derive a recurrence relation to solve the rod cutting problem and show that the problem has the optimal substructure property.

State the rod cutting problem. Explain, with reasoning, if you can solve the problem using i) ii) Greedy algorithm Brute force approach Also explain any limitation of these approaches.

Rod cutting problem recurrence: T(n)=1+i=1nT(ni)=1+j=0n1T(j)T(n)=1+i=1nT(ni)=1+j=0n1T(j) T(0) = 0 Solution: T(n)=2^n

Anna Broderick is the dietitian for the State University football team, and she is attempting to determine a nutritious lunch menu for the team. She has set the following nutritional guidelines for each lunch serving Between 1,500 and 2,000 calories At least 5 mg of iron . At least 20 but no more than 60 g of fat . At least 30 g of protein At

explain rod cutting problem with example and write a codes in C++

State the rod cutting problem and write down the recurrence relation for finding the maximum revenue, r n , of a rod of length n. 5 marks b) Show that the rod cutting problem exhibits optimal substructure property 5 marks c) The following table shows the length and the corresponding prices for a particular kind of rod. Length / i 1 2 3 4 Price /

Solve the following two instances of the rod cutting problem. a) Length 1 2 3 4 5 6 7 8 Price 1 6 7 10 11 18 18 19 Length 1 2 3 4 5 6 7 8 3 5 8 9 9 10 11 15 b) Price

Given the following numbers (71, 23, 73, 97, 44, 77, 87,31) and a hash function h(x) (x 10). (a) Insert these numbers into an initially empty hash table using linear probing. Show the populated hash table. Assume the array that stores this hash table has 10 cells. (b) Insert these numbers into an initially empty hash table with collisions

create a Hash Table using C#. Do not use any C# intrinsic data structures, use an array you build yourself. Use Microsoft visual studio (windows app form) Table Size: user can enter the Hash Table size When the Create button is clicked, a new empty Hash Table of that size should be created. When the Resize button is clicked, a new Hash Table

Consider a hash table of size 8 and the following sequence of keys x (with hash values h(x)) to be inserted. X h(x) 37 6 85 17 73 27 9 5 6 5 1 1 3 1 a. Insert the keys into the hash table and resolve collisions by chaining. Draw the state of the hash table after every insertion. b. Insert the same keys into the hash table and resolve collisions by

1. Draw the hash table that results from adding the following integers (34 45 3 87 65 32 1 12 17) to a hash table of size 11 using the division method and linked chaining. 2. Draw the hash table from Q.1 using a hash table of size 11 and array chaining with a total array size of 20. 3. Draw the hash table from Q.1 using a table size of 17 and open

Perfect Hash Function (explain) Collision Resolution Techniques Advantages and Disadvantages Approaches for distributing data across a hash table Array of LLL for Hash tables Allocating memory and deallocating memory

Write a class for hash table using chaining. Class must contain following functions 1. Insert an element in hash table 2. Search an element in hash table 3. Delete an element in hash table 4. Return the number of chains currently present 5. Display hash table Write a main function to test class. Note Use STL list in your program.

Explain the concept of "Big O" notation in computer science. Provide an example of its application and describe what it represents in terms of algorithmic complexity.

Describe why the Bellman-Ford algorithm does not work when the given graph includes negative cycles. Describe how the Bellman-Ford algorithm detects the negative cycles. Provide an example graph with negative cycles and show how it can be detected.

3. The BellmanFord algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstras algorithm for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers. Perform the BellmanFord

Consider the following Graph. Run the Bellman-Ford algorithm on the above directed graph, using vertex a as the start source. Write down your steps and describe it briefly.

1 a d 3 2 5. 1 2 f

6. Given a graph G (V, E) with positive edge weights, the Bellman Ford algorithm and Dijkstras algorithm can produce different shortestpath trees despite always producing the same shortest path weights. Explanation 8. BellmanFord Algorithm a) For iteration 1, fill arrays d and p. 5 2 Edge Weight 01 5 02 2 1 0 3 13 23 30 1 02 2 Edge Weight 01 5

Draw a graph with at least 4 vertices and at least 4 edges in such a way that, if the BellmanFord algorithm was run on the graph, it would return FALSE. Describe what it means when the BellmanFord algorithm returns TRUE.

Run Prim (starting from vertex "b") and Kruskal algorithms on the graph below - draw a table using Prim's algorithm showing the vertices in the queue at each iteration and list the order in which edges are added to the tree.

Run Prim (starting from vertex b) and Kruskal algorithms on the graph below a 2 3 b d 4 3 2, 1 d 6 4 5 JO 5 h 4 Ac Go

for the following graph, write a program to find the minimum spanning tree, using either Kruskal or Prim algorithms. Calculate the time for your method. A F 9 1 D 11 3 10 B 5 G 12 8 E 13 2 C H

Perform the complexity analysis by writing the pseudo-code of the Prim algorithm. Create MST by applying Prim and Kruskal algorithms on the graph below. The operations should be shown step by step in the table.

7 a C 2 4 5 e 4 3 5 b 6 d

1.Show the Prim MST algorithm starting at 3, and the Kruskal algorithm both applied to the following graph

What is Kruskals algorithm and what is the main purpose of using Kruskal algorithm and which algorithm approach is used in Kruskals algorithm. For the graph given below i) Apply Kruskal algorithm in detail explain each step of algorithm on given graph. 0 8 8

. Generate random instances of DAGS with various sizes and densities. Write the algorithms to find the topological sort using Kahns algorithm and depthfirst search (DFS) algorithm Measure the performance of each algorithm when applied to solve the topological sort problem of various size and degree (you may need to generate a very large problem to

A B G D (E) F LL,Trace through a topological sort of this graph starting from vertex A using depthfirst search and a stack. Why would we not be able to topologically sort the graph if the direction of the edge between vertices B and E were switched

a.) What is the topological sort of the graph above b.) How many possibilities are there to select the first node in the topological sort above

To analysis topological sort algorithm, write an algorithm of decrease and conquer of topological sort by using any language, and any compiler tools to run the code and study the output.

Show the ordering of vertices produced by the topological sort algorithm when it is run on the following DAG A R M H Figure 2 Figure of the DAG. Show all steps of of the topological sort algorithm. Is the ordering unique If yes explain, if provide another ordering without running the algorithm again. Activate otsdows Go to Settings to activat

Give an ordering of the vertices resulting from a Topological Sort of the graph shown in Figure 2. A B E F C Activate Go to Sett

Write down a topological sort of the following graph using edgeremoval algorithm. Also draw the resulting topologically sorted graph. At any point in the traversal, if more than one vertex can be selected, visit them alphabetically. B F E D A

You will perform two topological sorts on the directed graph to the right When the processing of a vertex creates more than one new pending vertex, add the new pending vertices to your set of pending vertices in alphabetical order (e.g., push (X). push (Y), push (2)). Indegree For each topological sort, use the provided tables to compute the

a) Describe the algorithm for a topological sort. b) what are the conditions on a graph to do a topological sort,c) Write pseudo code or code for a topological sort d) Give three examples of data you would use a topological sort to order, and describe why you do that order. e) What is the efficiency of your topological sort Act

Using the topological sort algorithm, find the shortest paths from node A to all other nodes in the following directed acyclic graph. Please show the procedures and result for each of the steps below. Step 1 Find a topological sort order Step 2 Follow the topological sort order in Step 1 and perform edge relaxation. N 2 5 9 1 2 4 3 6 2 7 3 5 1

Write a java or python programme that supports the following requirements Create your own programme and submit an implementation of a simple Depth First Search (DFS) for the above graph. Write a java or python programme that supports the following requirements Create your own programme and submit an implementations of Topological Sorting for the

Apply the DFSbased algorithm to solve the topological sorting problem for the following digraphs f 9 O Q (b) d a. Prove that the topological sorting problem has a solution if and only if it is a dag. b. For a digraph with n vertices, what is the largest number of distinct solutions the topological sorting problem can have Go to Settings to

Consider the directed graph of Figure 3. 5 2 3 Figure 3 4 Now answer the questions. (i) Can we perform topological sort in the directed graph Why (ii) Find topological sort using indegree approach. (iii) Find the topological sort using DFS. Activ Go to

In the search tree , the twoline codes indicate the target, and the numbers on the edge indicate the cost of the search. Using each of the following blind methods, write the order of the nodes to be searched to reach the first target a) Breathfirst Search (BFS) b) Depthfirst Search (DFS) c) Iterative Depthfirst Search Act Go t, 4 X C, 4 X C,In

A) What is the topological sort algorithm? B) List one application of topological sort algorithm in real life with an example. How does the topological sort algorithm help in your real-life example?