(a)
To show that the expected value represented by the counter after n INCREMENT operations that have been performed is exactly n.
(a)
Explanation of Solution
Given Information: The INCREMENT operation works on a counter containing the value i in a probabilistic manner. If
Explanation:
The counter after n INCREMENT operations is performed exactly n times for the expected value.
Consider that the initial value of the counter is i and increasing the number representation from
The expected increase is calculated as,
Hence, the expected increment represented by the counter is 1.
(b)
To calculate the variance in the value represented by the register after n INCREMENT operations have been performed.
(b)
Explanation of Solution
Given Information: Consider a simple case:
Explanation:
Consider that
Since
The value represented increases by 100. Therefore, by the equation (C.27)
Now, adding the variances of the
Want to see more full solutions like this?
Chapter 5 Solutions
Introduction to Algorithms
- Find the values of N for which interpolation search in a symbol table of size N becomes 1, 2, and 10 times faster than binary search, assuming the keys to be random 32-bit integers . Predict the values with analysis, and verify them experimentallyarrow_forwardCorrect answer will be upvoted else downvoted. Computer science. Allow us to signify by d(n) the amount of all divisors of the number n, for example d(n)=∑k|nk. For instance, d(1)=1, d(4)=1+2+4=7, d(6)=1+2+3+6=12. For a given number c, track down the base n to such an extent that d(n)=c. Input The principal line contains one integer t (1≤t≤104). Then, at that point, t experiments follow. Each experiment is characterized by one integer c (1≤c≤107). Output For each experiment, output: "- 1" in case there is no such n that d(n)=c; n, in any case.arrow_forwardWrite a Python program to create a Markov model of order k, and then use that model to generate text. Our Markov model will be stored in a dictionary. The keys of the dictionary will be k-grams, and the value for each key will also be a dictionary, storing the number of occurrences of each character that follows the k-gram. Note how, for instance, the key 'ga' in the dictionary has the value {'g': 4, 'a': 1}. That is because, following 'ga' in the input text, the letter 'g' appears four times and the letter 'a' appears one time. write the functions: get_grams(text, k): Returns a dictionary of k-grams as described above, using the input string text and the given positive integer k. Do not form k-grams for the last k characters of the text. combine_grams(grams1, grams2): Takes two k-gram dictionaries and combines them, returning the new combined dictionary. All key-value pairs from both dictionaries should be added into the new dictionary. If a key exists in both dictionaries, then…arrow_forward
- Diffie-Hellman Key exchange is based on the difficulty of solving the Discrete Logarithm Problem. For instance, Alice and Bob agree on a prime number p=7 and a number (base) g=4. This is the public aspect of the system. Using this algorithm, find the kprivate.arrow_forwardProve using the concept of interpretations and the value of a formula vI under an interpretation I that [∀x p(x) ∨ ∃x q(x)] → ∃x[p(x) ∨ q(x)] is valid.arrow_forwardThis paper deals with Extended Huffman Codes. For instance, for a source emitting two symbols A and B, the second order extension involves coding messages AA, AB, BA and BB (2^2 in number). The third order extension involves messages such as AAA, AAB, etc. (2^3 in number). The probabilities of such strings are computed by multiplying the individual probabilities. Calculate third, fourth and fifth order extensions of a source message. 1. Choose an alphabet a set of at least six (6) symbols with assigned probabilities. It can be assigned as [ A=0.3; B= 0.7; C=0.1;D=0.2;E=0.5;F=0.15]. Compute the third, fourth and fifth order extension probabilities. Using the built-in algorithm, derive the Huffman Code for each extension. Compute the following quantities: (i) Average length of the codeword; (ii) The code efficiency; (iii) The Compression Ratio. Note: I am assuming but needs to be verified that if we need are using 6 symbols A to F. 6^3=216, the 6 to the third power comes…arrow_forward
- Using semantic tableaux prove the following first-order statement is valid. ⊢ (∀x.p(x) ∨ ∀x.q(x)) → ∀x.(p(x) ∨ q(x)) Clearly indicate which rule was applied at each step.arrow_forwardMake it clear that the Chandy-Lamport algorithm is secure enough to pass the test.arrow_forwardf(n) = n3 - (n2 log2 n) + 2n f(n) ∈ O(n3) by giving the constants (c, n0) and arguing that your constants hold as n goes to infinity by either "chaining up", "chaining down", carefully treating the inequality as an equality and doing some algebra, or even weak induction. Also, what is the minimum value for c (in the definition of O() ) forj this problem and is it an exclusive or inclusive minimum/bound.arrow_forward
- In our shuffling algorithm, let's say that you select a random integer between 0 and N-1 as opposed to one between i and N-1. Demonstrate that none of the N! possibilities are equally probable to be the resulting order. Run the test for this version of the prior exercise.arrow_forwardMinimize the number of states in the following DFA D using the table-filling algorithm studied in the lecture. Show all your work and explain. Σ = {a, b}.arrow_forwardModeling the spread of a virus like COVID-19 using recursion. Let N = total population (assumed constant, disregarding deaths, births, immigration, and emigration). S n = number who are susceptible to the disease at time n (n is in weeks). I n = number who are infected (and contagious) at time n. R n = number who are recovered (and not contagiuous) at time n. The total population is divided between these three groups: N = S n + I n + R n There are several hidden assumptions here that may or may not apply to COVID-19, such as a recovered person is assumed to not be able to get the disease a second time, at least within the time window being examined. On week 0 (the start), you assume a certain small number of people have the infection (just to get things going). Everyone else is initially susceptible, and no one is recovered. There are two constants of interest: Let period = time period that it takes for an infected person to recover (recover meaning they become not infectious to…arrow_forward
- Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
- C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education