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 3E
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To give the upper bounds on the expected number of probes in an unsuccessful search and on the expected number of probes in a successful search when the load factor is ¾ and when it is 7/8.
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Consider an open-address hash table with
uniform hashing. Give upper bounds on the
expected number of probes in an
unsuccessful search when the load factor is
5
0/0
6°
Insert into a hash table with open addressing - linear probing with size 12 and hashing function h(x) = 5x mod 12:
31, 24, 51, 73, 89, 4, 103, 21, 33, 55, 81, 62
1. Consider an open hash (that is, one that deals with collisions by using a linked list) with a million entries. The hash algorithm takes 200ns to run, each comparison of objects takes 10ns, and everything else involved in the lookup takes trivial time. How large does the hash table need to be to use less time on chained comparisons than on the initial hash lookup? (Remember that you always have to compare once.)
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- 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 open addressing with 11 slots. Using the hash function h(x) = x mod 11, insert the keys (52,44,56,61,64) into the table in the same order. Assume that keys 0,1,8,9 already in the table .Use Linear probing and Quadratic probing for collision resolution Show the results in the two separate tablesarrow_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
- 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.arrow_forwardA hash-map has been constructed with quadratic-hashing. The hashing function is h(k;) = (3 * k; + 7) mod 17 and the table length is N = 17. How many cells will be probed by the call of insert(2) 1 4 5 6 7 8 10 11 12 13 14 15 16 26 89 11 35 70 59 Note: First row contains the indices and the second row contains the items.arrow_forwardGiven input {71, 22, 21, 99, 53, 69, 39} and a hash function h(x) = x mod 10, show the resulting: Hash table with double hash function (Use: h(x)=5-(x mod 5) ) and give the time complexity of search and insert operation with big-oh notation.arrow_forward
- Consider 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_forwardGiven 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 linear probing hash table of length 10 uses the hash function h(x) = x mod 10 + 1. mod is the reminder operator. For example the h(42) = 42 mod 10 + 1 = 2 + 1 = 3 so we insert 42 in the position 3 in the array. After inserting 6 integer keys into an initially empty hash table, the array of keys is…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_forwardA 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_forwardpython In simple uniform hashing, each key is assumed to have equal probability to map to any ofthe hashes in a given table of size m. Given an open-address table of size 100 and 2random keys, what is the probability that they hash to the same value? What is theprobability that they hash to different values?arrow_forward
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