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 11.5, Problem 1E
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To show that the probability of no collision is
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Suppose that 50 keys are to be inserted into an initially empty hash table using quadratic probing. What should be the size of the hash table to guarantee that all the collisions are resolved?
For a n-bit hash function and m-bit messages, there are 2m-n (I,e; Two to the power m-n) messages per hash value. For example, m=1024 and n=128 there are 2 896 (two to the power 896) messages per hash value.
Still. Why is it difficult to find hash collision?
Suppose that keys are t-bit integers. For a modular hash function with prime M, prove that each key bit has the property that there exist two keys differing only in that bit that have different hash values.
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- Consider a hash table of size M using separate chaining with ordered lists and the hash function hash(k) = k mod M. a) Assume N items have already been inserted in the table. What is, in average, the cost (number of operations) needed for searching an item in the table? Justify your answer. b) Assume N items have already been inserted in the table. What is, in average, the cost (number of operations) needed for inserting the next (N+1)th item? Justify your answer.arrow_forwardConsider a CBHT (closed bucket hash table) in which the keys are student identifiers. Assume that the number of buckets m = 100 and hash function hash (id) = first two digits of id. #Starting with an empty hash table, show the effect of successively adding the following student identifiers: 000014, 990021, 990019, 970036, 000015, 970012, and 970023. Show the effect of deleting 000014 from the hash table.arrow_forwardUsing quadratic probing, create a hash table HT[0..8, 0..2] for the collection of keys 17, 9, 34, 56, 11, 4, 71, 86, 55, 10, 39, 49, 52, 82, 31, 13, 22, 35, 44, 20, 60, and 28.Use the hashing formula H(X) = X mod 9. What conclusions have you drawn?arrow_forward
- Demonstrate what happens when we insert the keys 5; 28; 19; 15; 20; 33; 12; 17; 10into a hash table with collisions resolved by chaining. Let the table have 9 slots,and let the hash function be h.k/ D k mod 9.arrow_forwardConsider inserting the keys 10, 22, 31, 4, 15, 28, 17, 88, 59 into a hash table of length m = 11 using open addressing with the auxiliary hash function h'(k) = k. Illustrate the result of inserting these keys using linear probing, using quadratic probing with c1 = 1 and c2 = 3, and using double hashing with h1(k) = k and h2(k) = 1 + (k mod(m − 1)).arrow_forwardGiven the following keys {336,121,613, 93, 63, 17, 10, 53} for insertion into a hash table, and using a hash function h(k)= k mod m, m = 11, show the resulting:(i) Quadratic probing hash table(ii) Linear probing hash table(iii) Separate chaining hash tablearrow_forward
- Insert the keys <13, 19, 35, 71, 31, 6, 23, 4> into hash table of size m=11 using linear probing hashing. Here, h(k, i) =((k mod m) + i) mod m, i=0,1,2,…. How many times you have to increment i to resolve collisions?arrow_forwardI need help in this HASHTABLE question Let n be the number of keys, and m be the size of the hash table. Show that if n > m, the collision is inevitable.arrow_forward#. Consider a hash table of size 11 with hash function h(x) = x mod 11.Draw both hash tables (Closed-Bucket and Open-Bucket) that results after inserting, in the given order, the following values: 45, 12, 35, 67, 25, 52, 88, 70, 68, 53, 55arrow_forward
- Given:• a hash function: h(x) = | 3x + 1 | mod M• bucket array of capacity 'N'• set of objects with the folloeing keys: 12, 44, 13, 88, 23, 94, 11, 39, 20, 16, 5 (to input from left to right) 1. What would be the hash table where M=N=11 and collisions are taken care of using linear probing? 2. What would be the hash table where M=N=11 and collisions are taken care of using separate chaining? 3. Would a size N for the bucket array be able to exist, so that no collisions happen with thehash function h(x) = | 2x + 5 | mod 11 and the keys above?arrow_forwardProve the theorem Suppose that a hash function h is chosen randomly from a universal collection of hash functions and has been used to hash n keys into a table T of size m, using chaining to resolve collisions. If key k is not in the table, then the expected length E Œnh.k/ of the list that key k hashes to is at most the load factor ˛ D n=m. If key k is in the table, then the expected length E Œnh.k/ of the list containing key k is at most 1 C ˛.arrow_forwardDoes a hash table of size m contain the same number of linked lists at all times? No matter how hard I attempt, I cannot identify the purpose of a hash function. Provide an example to illustrate your point.arrow_forward
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