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 11.4, Problem 4E
Program Plan Intro
To show that unsuccessful search for key k examines (1/d)th of the hash table before running to slot h1(k) under given condition.
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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.
Let T be a hash table in which collisions are resolved by chaining. Given a key k, briefly explain how a search operation for k in T works.
Given is the hash function h(k) = k mod 10. How many different insertion sequences of keys are there to generate the following hash table if closed hashing
with linear probing is used?
B0 = ∅ B1 = ∅ B2 = {32} B3 = {43} B4 = {54} B5 = {12} B6 = {76} B7 = {23} B8 = ∅ B9 = ∅
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- A hash map of size 12 has been constructed with Quadratic-Hashing by applying h(k) = (4ks +1 )mod 12. Perform Find(12) and mark in the hash-map below the cells which will be probed. Index 0 12 3 4 5 6 789 10 11 12 Value 3 44 36 11| 43arrow_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_forwardSuppose we are hashing integers with 7-bucket hash table using the hash function h(i) = I mod 7. Show the resulting closed hash table with linear resolution of collisions (i.e., handling collisions with separate chaining) if the perfect cubes 1, 8, 27, 64, 125, 216, 343 are inserted.arrow_forward
- Consider a hash table of size m = 8 with hash function h(k) = (3k+2) mod m. Draw the tablethat results after inserting, in the given order, the following values: 13, 49, 33, 26, 38, 87, 26,67 Handle collisions by linear probing.arrow_forwardConsider a hash table with 50 slots. Collisions are resolved using chaining. Assuming simple uniform hashing, what is the probability that the first 3 slots are unfilled after the first 3 insertions?(upto 2 decimal points)arrow_forwardFor the set of keys {17, 9, 34, 56, 11, 4, 71, 86, 55, 10, 39, 49, 52, 82, 31, 13, 22,35, 44, 20, 60, 28} obtain a hash table HT[0..8, 0..2] following quadratic probing.Make use of the hash function H(X) = X mod 9. What are your observations?arrow_forward
- Hey, 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_forwardWhat is the worst-case performance of a lookup operation in a hashmap and why? Group of answer choices A- O(1), hashmap always has a constant time lookup, and that is why we like using this associative data structure. B- O(lg(n)) hashmap has a log(n) lookup because we are able to perform a binary search on the keys because our hashmap always maintains a sorted order of entries added. C- O(n) because we can have a bad hash function that puts all of our items in the same bucket, thus we would have to iterate through all n items.arrow_forward1.Question: Consider a hash table of size m=7. Draw the table that results after inserting the following keys: 19, 26, 13, 48 and 17 for each case: a. collision resolved with chaining, with hash function h(k)=k%7 b. collision resolved by linear probing, with hash function h1(k,i)=[h(k)+i] % 7 (i=0,1,..,m-1) c. collision resolved by quadratic probing, with hash function h1(k,i)=[h(k)+i²] % 7 (i=0,1,..,m-1) d. collisions resolved by double hashing, with h2(k,i)=[h(k)+i*{5-(k%5)}]%7 (i=0,1,..,m-1)arrow_forward
- Given an array A {361,313, 163,179, 334, 659, 969, 934} and a hash functionh(x) = (x mod 11), show the results: (The answers must show where the collision occurs.) (1). Separate chaining hash table. (2). Hash table using linear probing. (3). Hash table using quadratic probing. (c1 = 0, c2 = 1) (4). Hash table with second hash function h2 (x) = 7 − (x mod 7).arrow_forwardA hash map of size 15 has been constructed with Double-Hashing by applying h; (k;) = [ h(k;)+jd(k;)] mod 15. The primary hashing function is h(k;) = k; mod 15. The secondary hashing function is d;(k;) = k; mod 7. In one sentence, describe why the secondary hashing function is poorly chosen.arrow_forwardConsider a linear hash table that uses 4-bit hash keys and stores two records per bucket. The capacity threshold is 75%; that is, we create a new bucket whenever the number of records is more than 3/2 the current number of buckets. Simulate the insertion, into an initially empty hash table, of records with (hash values of) keys 1111, 1110, 1101,..., 0001, 0000, in that order. Which of the following keys never appears alone in a bucket at any time during the insertion sequence? a) 1110 b) 1011 c) 0100 d) 1001arrow_forward
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