Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 11.2, Problem 6E
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To describe a procedure that selects a key at random from among the keys in hash table and returns it in expected time
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Let S be the set consisting of all 10 digits and all upper- and lowercase letters. Let U be the set of all (ordered) strings consisting of exactly four elements from S. Construct an injective hash function h: U -> N with the smallest possible value of maxUh(u). Modify your hash function to give a simply uniform map h' to the set {0, ..., 30}. In other words, if U is a probability space with P(u) = P(u') for all u, u' in U, then the random variable h': U -> {0, ..., 30} should have a uniform distribution..
Regarding the Hash algorithm, which one(s) of the following is/are correct, check all that apply.
Given a hash of a piece of information, it is hard to recover the original information itself.
Given a piece of information, it is hard to find another piece of information that can generate the same hash.
O It is theoretically possible, but practically infeasible, to find two pieces of information that generates the same hash.
O Hash is unique to a piece of information, there are no any two piece of information that can generate the same hash unless these two pieces of information are identical.
Given a sorted array of n comparable items A, and a search value key, return the position (array index) of key in A if it is present, or -1 if it is not present. If key is present in A, your algorithm must run in order O(log k) time, where k is the location of key in A. Otherwise, if key is not present, your algorithm must run in O(log n) time.
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- code in python attached please Define a hash table with an associated hash function ℎ(?)h(k) mapping keys ?k to their associated hash value. b) In simple uniform hashing, each key is assumed to have equal probability to map to any of the hashes in a given table of size m. Given an open-address table of size 500500 and 22 random keys, what is the probability that they hash to the same value? What is the probability that they hash to different values?arrow_forwardHey, Given is a hash table with an initial size of 1000 and a hash function that ensures ahashing, where the keys are chosen randomly under the uniformity assumption.me are chosen. After how many insertions do you have to expect a collision probability of more than 80%?of more than 80%?To keep the number of collisions low during hashing, one can reduce the size of the has-hash table after a certain number of insertions. After which number n of inserted elements must the table be increased for the first time, if no collision occurred in the previous n - 1 elements and the probability of a collision in the n-th insertion should be less than 20%?arrow_forwardYou are given k > 2 linked lists, each containing n > k natural numbers sorted in increasing order. All n · k numbers in these lists are distinct. Describe an algorithm that finds the k-th smallest element among all n k numbers. Your algorithm should run in O(k log k) time. Analyze the running time of your algorithm and prove its correctness.arrow_forward
- security pr As we mentioned in class, a universal hash function is a function UH(K, M) that takes a key K, a message M and produces a fixed length digest. The universal hash is defined to be "secure" if for any two messages M₁ and M2, if K is selected uniformly at random, then the probability that UH(K, M₁) UH(K, M₂) is approximately zero. == Suppose that H is a secure hash function. Is UH(K, M) = H(K||M) a secure universal hash function? Either prove the answer is "yes" using the security properties of H (which can be assumed), or show how the security of UH could be violated. VOarrow_forwardSuppose you want to find documents that contain at least k of a given set of n keywords. Suppose also you have a keyword index that gives you a (sorted) list of identifiers of documents that contain a specified keyword. Give an efficient algorithm to find the desired set of documents.arrow_forwardConsider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.arrow_forward
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