Algorithms Questions and Answers August 15, 2021

Calculate and type the final answer in the space. You may use either scientific notation or standard notation for the answer. Follow rules of sig figs and rounding correctly. 8.6510 45.217x1067.34 x1059.200102

An ancient recipe calls for 5.1 stones of butter, I would like to know how many pounds are needed. Give answer in standard notation with correct significant figures. 3 stone = 14.5 pounds 2) From your home to CNM is 1,790 steps, it takes you 15.9 minutes to do this walk, If you take 2,094 steps each mile, how many miles per hour are you walking?

I mentioned that large companies would like their employees to understand asymptotic notation and they ask such questions in job interviews. Why is it important Explain the idea of asymptotic notation and provide an example why in a real software development scenario it may be helpful.

1. Write asymptotic notation to tell total memory usage to store bipartite graph in adjacency matrix implementation. 2. Write asymptotic notation to tell the time need to enumerate all edges in bipartite graph in adjacency lists implementation.

The asymptotic notation in question is O(We^2) in the equation below. This is a fluid mechanics related question. I can't seem to attach the full research paper.

x 1532WeO(We)a Aspectratio, We Webernumber

Which grows faster asymptotically, log or (logn) Supply a rigorous proof, showing your work in detail. Use asymptotic notation to write your final conclusion in either the o() or O() form, as appropriate.

Identify using asymptotic notation the relationships between the functions and prove it. Let f(n)42n n and g(n) n.14 4 a) Provide a statement using asymptotic notation to identify the relationship based upon asymptotic speed between the following functions. Hint Which function is faster

The Binary Insertion Sort Algorithm is a variation of the Insertion Sort Algorithm that uses a binary search technique rather than a linear search technique to insert the ith element in the correct place among the previously sorted elements. (i) Express the Binary Insertion Sort Algorithm in pseudocode. (ii) Compare the number of comparisons of

write a program in python that takes the 1000 random numbers: (a):using insertion sort in ascending order. (b):using insertion sort in descending order. (c):in one program both ascending and descending order. (d):priority based ascending order.

(1) The Insertion Sort below can output an ascending array. Please modify the pseudocode to make it output a descending array. (10 points) INSERTIONSORT(A) for j 1 2 3 5678 2 to A.length Aj key Insert A into the sorted sequence A1..j1. j1 while 0 and Ai key Ai1 Ai 111 Ai1 key (2) Please give an array with 5 elements, which is the worst case of

Modify the algorithms to output the number of comparisons each algorithm makes 2. Discuss the runtime complexity of each algorithm,list , int length) for (int firstOutoforder 1 firstOutoforder length firstOutoforder) if (list firstOutoforder list firstOutoforder 1) void insertionsort ( temp list firstOutoforder int location firstOutoforder

A possible implementation of insertion sort is given by the following Python code def insertionsort (xs, keyidentity) for i in range(1, len (xs)) e xs 1 k key (e) ji1 while j 1 and key (xs j) k xs j1 xs j j j 1 xs j1 e Rigorously prove that the worstcase asymptotic time complexity of this insertion sort is T(n) (n).

2) Consider the following insertion sort algorithm. void insertionsort (element a, int n) Put a 0..an1 into ascending order by insertion sort. for (int k 1 k n k) At this point, a0..a k1 are already in order. Insert ak where it belongs among a 0..a k. You need to write code for this insertion as the body of the fork loop. endfor k a) Write

Also implement code to perform Dijkstras Algorithm on above graph.

Analyze the operations of the Dijkstras algorithm and the BellmanFord algorithm by simulating both algorithms on the below graph starting from vertex 0 and ending in vertex 5. Your simulation should illustrate all the steps for each algorithm from the start of the algorithm until the end of the algorithm. Clearly state the shortest path extracted

Assume you have two algorithms, A and B, which perform the same operation even though their implementations differ. Algorithm A has a running time of O (9n 12n 4) Algorithm B has a running time of O(11n 4) 1. What values of n will make algorithm A more efficient than algorithm B 2. What values of n will make algorithm A less efficient than

State and explain the three popular algorithms (given below) used in MD simulations Verlet Algorithm Velocity Verlet Algorithm Predictor-corrector algorithm

Lets assume that you are facing a programming problem which can be solved with two algorithms A and algorithm B. Both algorithms will work on same machine. For n inputs, the algorithm A time complexity is 8n and the time complexity for the algorithm B is 64nlogn. Answer the following questions 1. Which algorithm has higher Big O Complexity, please

1. Modify the algorithm to compute nh. 2. Modify the algorithm to compute n (n1) (n2)... 21 Exercise 2. Test (A,n) for i 1 to n2 do for j0 to ni do if AjAi1 swap(Aj.Aj1) Time complexity Total Read the algorithm given blow and answer these questions What does this algorithm do Calculate the number of steps performed by the algorithm and then

The Floyd-Warshall algorithm fails (does not work correctly) if the input graph has a negative weight cycle. (a) Draw a small graph with a negative-weight cycle. Demonstrate the result of running Floyd-Warshall on it (show D(k) for k = 1, . . . , n). (b) In general, how can you use the output of the Floyd-Warshall algorithm to detect the presence

Let Di, j, k be defined as in FloydWarshall algorithm (length of the shortest path from i to j using only vertices in 1, 2, ..., k on the path).,4 3 4 1 4 1 1 9 2 6 5,Consider the graph above. What is the value of D1, 4, 3,Consider the graph above. What is the value of D1, 4, 5

Show the distance matrix D(k) for 0 k n that results from applying the FloydWarshall algorithm for the graph shown in Figure 3. 12 7 3 3 4 17 2 1

Run the FloydWarshall algorithm on the weighted, directed graph below. Show the matrix D(K) that results for each iteration of the outer loop. 4 1 4 1 3 5 2 3 10 8 6

6. Simulating BellmanFord and FloydWarshall algorithms using the following graph. In the case of BellmanFord algorithm, use vertex 0 as the source. 0 1 3 1 2 2 5 1 3 4 4

Let dj, v) cost of shortest path from s to v that is j hops. dj, v) 7 00 X How to detect negative cycle 2 3 00 00 7 y EN u V X y j 1 0 8 8 8 80 2 3 4

Run the BellmanFord algorithm to find the shortest path (using each arc at most once) from node 1 to node 5. Show all of your work, and be sure to clearly show the progress of the algorithm at each intermediate iteration. Activate Wi Use the BellmanFord algorithm to answer the following question does the st digraph have any negativecost cycles

In Figure 6 Network Topology for BellmanFord algorithm, replace X and Y with your MITID, any two digits from last 4 digits. 15 Marks Fill the following Table2 for BellmanFord algorithms. First Cycle Second Cycle Figure 6 Network Topology for BellmanFord algorithm Third Cycle Table 2 BellmanFord algorithms A B PCP C PC P D E LL PCPC G P P H C

Apply BellmanFord algorithm on the following graph. Show all the steps. S 0 1 N 3 5 1 2 2 3 y

(a) Apply Dijkstras routing algorithm to the given network to generate a least cost path route to all nodes from the source node 5. Tabulate your results. (b) Apply BellmanFord algorithm to the given network to generate a leastcost path route to all nodes from the source node 5. Tabulate your results. (c) Will Dijkstras algorithm and the

Write pseudo code for MST, Prim and Kruskal With complexity. In addition, specify how minimum heap reduces complexity.

i) ii) Radix Sort Algorithm is created by performing stable Bucket Sort for each of the keys. Show, through an example, that without stability, sorting cannot be performed correctly. You must provide a reallife example. Does the first step of Radix Sort (that is the first bucket Sort operation) need to be stable Explain your answer.

Write a program using the bucket sort idea to create histograms of an election.

Derive expressions for the speedup, efficiency, and isoefficiency function of the bucket sort parallel formulation . Compare these expressions with the expressions for the other sorting algorithms like Quick sort, Bucket Sort, Bubble Sort. Which parallel formulations perform better than bucket sort, and which perform worse?

Illustrate and show how to sort the array A 6,8,3,1,7,4,4,5,3,2,7,1,8,4,3 using Counting sort and Bucket sort respectively.

Suppose that we are given a sequence A of n 32bit integers (a, a, ... a.) to sort such that 0 a N. The objective of this exercise is to produce the most efficient strategy to use Bucket Sort to sort Sequence A. 1) (5 points) Propose a strategy to adapt Bucket Sort to sort the above Sequence A. The objective is to design the most efficient

5. List the comparisons that are made when merging the two sorted arrays 1, 5, 7, 8, 9, 12, 14 and 2, 3, 4, 6, 10. 6. Sort the following array of 20 numbers by radix sort, using two runs of bucket sort to base 10. Write down the sequence after each run of bucket sort 42, 07, 44, 90, 89, 91, 09, 00, 51, 01, 92, 37, 21, 13, 73, 29, 10, 15, 98, 23

Given n uniformly distributed real numbers 0,1, a sorting algorithm called bucket sort can sort these numbers in expected linear time without assumptions on the precision. Illustrate the bucket sort using (0.08, 0.9234, 0.537, 0.33333, 0.8778, 0.1) and prove the expected time complexity.

a. Explain why Bucket Sort does not adhere to the (n log n) lower bound. b. Show the bucketed array for sorting 1, 5, 3, 5, 8, 2, 9, 10, 12, 13, 12, 1, 2, 7, 9, 13, 3, 1, with the upper bound of 15.

Given an unsorted sequence 11, 9, 7, 13, 8, 12, 10. Sort the sequence step by step, by using each algorithm shown in sorting.py, including 1. Bubble sort 2. Insertion sort 3. Selection sort 4. Quick sort 5. Merge sort 6. Heap sort 7. Radix sort 8. Bucket sort 9. Counting sort You need to show the sequence after each step to illustrate how each

Suppose 9 cards are punched as follows 349,153,361,423,638,128,321,542,366 Sort the following card numbers by using radix sort. Bucketsort (A) n length(A) for j 1 to n do Ai for i1 to n do sort Bi with insertion sort Concatenate the lists B0. B1... Bn1 Apply Bucket Sort on the following data 0.18, 0.17, 0.21, 0.23, 0.26 ,0.39, 0.98,0.72,

a) If the radix sort is O(kn), what would the constant be for 16bit binary numbers b How does the radix sort differ from all of the others for lots of large numbers

How does a radix sort algorithm work using any given base without converting the numbers to decimal? For example: A = [12,42,11,65,3,2] base 10 and B = [12,42,11,65,3,2] base 2 I have a function radix(aList,b) How does the radix sort work for these 2 list: radix(A,10) and radix(B,2)?

Writec#or C++ code for the Radix sort using queue to sort 2 digits numbers and sort these numbers using Radix sort and merge sort and compare both algorithms using 30000, 60000, and 90000 items

What is the time complexity of Build Max Heap Sort the following numbers using Radix sort algorithm 329, 457, 657, 839, 436, 720, 213, 582 What is the time complexity of Radix sort algorithm

elements You are given the following array consisting of 9 A (536, 822, 16, 012, 357, 210, 216, 316, 450) (a) (5 points) Use Radix Sort to sort the sequence of elements in A. Show each step of your procedure. (b) (3 points) Suppose that you modified Radix sort so that, instead of sorting from the least significant digit first (righttoleft), it

Sometimes we can assume the items to be sorted have keys which are integers in the range 1, . . . , m. In this case, the Counting Sort algorithm can sort n items in O(n + m) time. i. Describe the Counting Sort algorithm in detail. ii. Justify the O(n + m) running time bound. iii. Prove that the Counting Sort algorithm is stable, that is, it always

Sort the following list (in ascending order) using Counting Sort algorithm List3, 6, 2,0,3,6,1,0,4,5 5

(a) Explain a situation where it is better to use Counting Sort rather than Merge Sort to sort an array in ascending order. (b) Explain a situation where it is better to use Merge Sort rather than Counting Sort to sort an array in ascending order.

(a) Explain why counting sort may not be able to sort the integers in O(n) time. (b) Explain why radix sort may not be able to sort the integers in O(n) time. (c) Given (a) and (b), explain why we can typically regard counting sort and radix sort as (n) sorts when the keys are integers. (d) Compare the asymptotic running time of radix sort and

Write an algorithm for counting sort. Apply counting sort algorithm to sort elements in the array A 111 15, 19, 70, 8, 20, 19, 18, 15, 8, 3, 70. Explain the counting sort procedure step by step with results and also write down the limitations of counting sort.

(a) Write code to implement the Quicksort algorithm. Your code should print both the input and output arrays to the screen. (b) Using the list LArray 26, 14, 52, 7, 15, 37, 91, 2, 28, 73 illustrate how quicksort works. (c) Using the list LArray 37, 91, 2, 26, 14, 52, 7, 28, 15, 73 illustrate how quicksort works,ALGORITHM Quicksort(Al..r) Sorts a

write a java code for this pseudocode with an example of a quicksort.

QUICKSORT(A, p,r) 1 if p

Implement the following variations of quicksort algorithm: 1- Basic quicksort where the chosen pivot is the first element of an array. 2- Quicksort where the pivot is the median of three elements of an array (first, last, and middle). 3- Quicksort where the pivot is the median of three randomly chosen elements of an array. * Generate large arrays

Best case () Average case () Worst case () Basic Quicksort Theoretical Experimental efficiency efficiency (expected (avg of of operations operations) several for runs) Quicksort with pivot as median of first, last and middle elements Theoretical efficiency (expected of operations) Experimental efficiency (avg of operations for several runs)

The standard Partition method in QuickSort uses the first element in the (sub) array as a pivot element,which is used to split the array into left (smaller elements) and right(larger elements) The choice of pivot affects the runtime of the algorithm. a) Write the QuickSort method/function/procedure. public static void QuickSort (int|] A, int

Quicksort is a recursive, averagecase O(NlogN) sorting algorithm. The following is some pseudocode for Quicksort to sort an array of integers in nondecreasing order void quicksort (integer A, integer p, integer r) begin integer q if p r then q partition (A, p, r) quicksort (A, p, q) quicksort (A, q1, r) end if end quicksort integer partition

Define the function PartitionA () whose header is Partition (int data,int p,int r) Its definition should be identical to the definition of the PARTITION () algorithm from the Quicksort Define the function Partition () whose header is Partitions (int data), int p. int r) Its definition should be similar to the definition of the PARTITION ()

1. Describe QuickSort algorithm 2. Identify which algorithm technique QuickSort uses 3. Discuss QuickSort algorithm performance, including pivot selection, Cworst(n), Cavg(n), runtime and memory efficiency

(10 points) Given the following priority queue (both the logical and physical view are depicted), what will be the state of the priority queue after performing the operation deleteMin() 35 10 30 80 35 40 right Most 5 30 40 10 80

In Priority Queues explain the following a. Definition b. Operations: enqueue, dequeue. c. Differences between queues and priority queues d. Implementation of the ADT Priority Queue using a sorted lists, operations add and remove

1. Describe the similarity and difference(s) between a Priority Queue and a Heap. 2. Will any implementation of Priority Queue has better run time complexity than a Heap Explain your answer in details

Demonstrate your understanding of the C++ Priority Queue Abstract Data Type ( ADT ) by writing a basic program that will demonstrate the basic operations of a priority queue: Enqueue Dequeue Peek IsEmpty GetLength Demonstrate these operations with any program you would like to write.

Assume that there is a priority queue consisting of 6 items. Their arriving orders and priority values are as follows Arriving Sequence Symbol Priority Value R 6 2 X 3 3 H 1 4 U 5 s S 2 6 T 4 Suppose that the priority queue is implemented by using tree nodes. Draw the priority queue. Suppose that the priority queue is implemented by using an

Develop an ADT specification for a priority queue. A priority queue is like a FIFO queue except that items are ordered by some priority setting instead of time. In fact, you may think of a FIFO queue as a priority queue in which the time stamp is used to define priority.

Design a C class for PriorityQueue to implement a priority queue with the following methods. Your code should include, in addition to the implementation of the class, a main () function that creates an object of the class and invokes the methods implemented. a. PriorityQueue PriorityQueue () which constructs an empty priority queue b.

1. Give a name of an application whose implementation may require priority queue and describe what the application is about. No restriction on what the application should be. You can refer to a real application you know of, or even create a new application that makes sense to you. 2. Describe why priority queue is so critical in implementing the

For the following digraph, apply Dijkstras Algorithm to find the shortest path (i.c., with the minimum weight) from A to E. Show the initial value in the priority queue and the values in the priority queue after each pop() operation and subsequent updating of all neighbors. Use the format (VertexID, Dist, Parent) (Example, (B, 6, A)) for each

Write a Python class PQueue that implements a priority queue using an array list of PQElement s. In particular, you need to implement 5 functions init for creating an empty priority queue size for returning the size of the priority queue enq for enqueueing a new PQelement in the priority queue deg for dequeueing from the priority queue the

What are real life examples where we could use Priority queue and Binary heap give 2 examples for each? Give the differences between a Binary heap and Binary tree Is it possible to use Binary tree for Priority queue ?

Create a priority queue and add starting vertex (source) with distance of 0 Add remaining n1 vertices to priority queue with distance of infinity (i.e. sys.maxsize) Iterate over each item in the priority queue remove item from the priority queue get distances between this item and its adjacent vertices (plus current dist to source) if this

import numpy as np from scipy import stats import statistics def stats_values(arr): ''' Input : arr - Numpy array Output: Function must print the statistics of the array in the following order 1. Mean 2. Median 3. Standard Deviation 4. Variance 5. Mode 5. Inter-Quartile Range Note: All the answers must be of Float datatype.Round your answers to 2

Explain why the mean and the median are useful sample statistics (i.e. what does it tell us about data) When should use a mean versus a median Median versus mean (10 points) Suppose that Xrepresents the number of years a person goes to school, and Y represents the income they earn after finishing those years of school. Years of 1 schooling is

1. Find the mean, the median and the mode of the following the scores obtained by 10 students in Statistics class. 2. Given 60 75 80 75 85 75 90 98 95 90 X 1622 2329 3036 3743 44 50 5157 5864 f 225 11 15 32 2 Compute the mean, the median and the mode of the given distribution of scores.

The median income for a random sample of households in a school district was found to be $56,300. From this information, the school board decided that the median income for all households in the district was probably between about $52,000 and $60,000. Describe the difference between descriptive statistics and inferential statistics in the context

What do the descriptive statistics for year and month tell you? Are you surprised? What do the descriptive statistics for TSI tell you? What do the descriptive statistics for Aerosols tell you? Year Month MEI CO2 CH4 N2O CFC.11 CFC.12 TSI Min. :1983 Min. : 1.000 Min. :-1.6350 Min. :340.2 Min.

3. A continuous random variable Y 20 has cdf Fy(y) given by Fy(y) 1e Suppose that 5 sample values are taken from this distribution. Let M denote the median of these 5 numerical samples. What is the probability that M 1 Hint This is an order statistics problem. Which order statistic does M refer to in this context

Suppose that Y. Y...., Ys denotes a random sample from a uniform distribution defined on the interval (0, 1). That is, 0sysl. ro) 0 lo, elsewhere. Find the density function for the secondorder statistic. Also, give the joint density function for the second and fourthorder statistics. Let Y and Y be independent and uniformly distributed over the

(i) Find the quartiles and the median of the binomial distribution with parameters n 11 and p 0.3. (ii) Find the median of the binomial distribution with parameters n 8 and p 0.5.

The average salary in the workforce for a person with a BS/BSC in Statistics is $67,000 and the median salary for a Statistics graduate working as a Data Analyst is $58,328. Assume that the distribution for salaries for Statistics graduates is normally distributed with a mean = $67,000 and a standard deviation = $3, 000. What is the probability

11. 10 Suppose the score of statistics quiz is normally distributed with mean 70 and variance 100 Sample 5 statistics quiz scores and calculate of the following three statistics 3 X X X X X, X0.2x, 0.3x, 0.2X, 0.2X, 0.1X,,Mmedian 5 A Among these three estimator, which one is relatively efficient Why B. Show the X is the BLUE (best linear

Summarizing Data Sets with Statistics Assignment Active Calculating Statistics Use the set of data to calculate the measures that follow. Choose each correct measure. 0,2,3,3,3,3,3,4,4,4,4,5,5,6 Mean =hat (v) Median =hat (v) Range =hat (v) Interquartile range

3) 4) After entering test scores from her statistics class, the instructor then calculates the mean and the median of all the scores and finds that the mean is significantly smaller than the median. What might this imply (a description) about the shape of the distribution of students scores (2 points) Students in an archaeology class kept a record

The minimum and maximum value of the boxplot are from the median to the maximum value or from the median to the minimum value?

Data 50 45 40 35 30 Boxplot of LEG MAN, LEG WOMAN LEG MAN Summary Statistics Variable N Minimum Q1 Median Q3 LEG MAN 40 36.0000 40.5750 42.6500 44.4750 LEG WOMAN 40 27.0000 37.6250 39.0500 41.0000 LEG WOMAN Maximum 95 Median Cl 48.4000 (41.0411, 43.7535) 48.6000 (38.2000, 40.1178)

Let X(1) X(2) ... X(n) be the order statistics for a i.i.d. random sample X, X2, ..., Xn from an exponential distribution with the parameter 0. (a) Derive the probability distribution of X(1). (b) Derive the probability distribution of X(n). (c) For random samples of size n 2m 1 from this kind of population, derive the probability

Consider the two statistics about numerical variables mean and median. Asnwer the following (a) What does it mean for the variables distribution if mean and median values are close (b) What does it mean for the variables distribution if mean and median values are very different

Let X,..., XnCauchy(0) where the sample size is odd (to ensure uniqueness of the MLE) which has pdf f(x 0) 1 1 1 (x0) TER 0 ER (a) Show that the order statistics X(1) X(2) when are two polynomials over R identical) (b) Is the sample median median X, sufficient for Prove or disprove. X(n) are minimally sufficient for 0. (Hint

Find the mean, median, mode and range in each of the sets of data. 1) 61, 57, 49, 60, 45, 51, 57, 60, 53, 57, 55, 48, 65, 52 order 45, 2) order 3) order Mean Median Mode Range 129, 113, 110, 123, 112, 115, 110, 124, 121, 113, 115, 121, 115 Mean Mean Median Mode Range 14, 7, 4, 8, 12, 4, 2, 9, 7, 11, 15, 12, 11, 11, 8, 11, 7 Median Mode Range

The following list contains the number of hours that a sample of 9,13,20,13,5,11,16,14,14,20 Send data to a calculator Find the median number of hours for that sample hours.

Find the following statistics for the sample below d 0.1, 2.1, 1.1 (a) Mean, , and Median (b) Variance, S (c) Standard Deviation, S

Is the bucket sort algorithm in place? why or why not

public static void BSort(IntegerSequences) int n .size() IntegerSequence Rosation prec, sarc for (int i1 i succ.clement().intValue()) swap (prec. succ) 1th pass proc succ.

For each of the following prove correctness in your own words: Insertion sort Heap sort Quick sort Bucket sort

How to apply radix sort count sort bucket sort in matlab c and python?

Write a pseudocode, Python function, or Java method for the counting sort algorithm that sorts a list of positive integers taken from a small range. Write a pseudocode, Python function, or Java method for the radix sort algorithm that sorts a list of positive integers. Write pseudocode, Python function, or Java method for the bottomup dynamic

The priority queue is a proper generalization of the stack and the queue, because we can implement these data structures with priority queues. (a.) Explain how to use a priority queue ADT to implement a stack ADT. (b.) Explain how to use a priority queue ADT to implement a queue ADT.

Explain how heapsort algorithm is better than insertion sort and quicksort algorithms.

Convert 1.44 x 10 mgml to a. Standard notation (mgmL) b. Scientific notation (gL) Convert 7.38 x 107 mgmL to a. Standard notation (mgmL) b. Scientific notation (ngl)

Write the number in scientific notation 0.0314 x 10 -2 Write the number in standard notation 8.5 x 10 2 Write the numbers in scientific notation and perform the operations. Give the answer in scientific notation and standard notation 0.00000184 / 0.0004

What is motivation behind Asymptotic Analysis. Discuss briefly asymptotic notations with suitable diagrams?

Johnson Corporation has a current price of $22, is expected to pay a dividend of $2 in one year, and its expected price right after the dividend is $23. A. What is Johnsons expected dividend yield? B. What is Johnsons expected capital gain rate? C. What is Johnsons equity cost of capital (or expected return on its stock)?

(1) Describe how to use Breadth-First Search Algorithm to find a spanning tree. (2) Use Breadth-First Search Algorithm to find a spanning tree for the following graph.

A B S G D F E H

g. Recall the rod cutting problem. Is the following recurrence to find the maximum revenue correct r(n) max (Pi r(ni)), 1

In a rod cutting problem, given a table of prices p, the objective is to determine the maximum revenue r, obtainable by cutting up the given rod (of length n) and selling the pieces. The rod cutting problem exhibits optimal substructure property. The following table shows the price table for one particular situation Length 1 i Price 1 Pi 2 . 5 3 8