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 17.4, Problem 1E
Program Plan Intro
To argue that the dynamic implementation of insertion in open-address hash table runs at
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Consider a hash table with `n` slots and a
hash function that distributes keys
uniformly across the slots. Suppose `m`
keys are inserted into the hash table,
where `m` is much larger than `n`. What is
the expected number of collisions?
Assume that each key is equally likely to
be hashed to any of the `n` slots, and that
each key is independent of all other keys.
Express your answer in terms of `n` and
`m`.
Consider 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 tables
a) Consider inserting the keys 10, 22, 31, 4, 15, 28, 17,
88, 59 into a hash table of length m = 11 using open
addressing with linear probing, we use the hash
function h(k, i) = (k+ i) mod m.
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- 10. Consider a hash table that uses the linear probing technique with the following hash function f(x) = (5x+4)%11. (The hash table is of size 11). If we insert the values 3, 9, 2, 1, 14, 6, and 25 into the table, in that order, show where these values would end up in the table? Show all the calculations and actions that needs to perform for the values to be stored. index 0 1 2 3 4 5 8 9 10 valuearrow_forwardGiven input {4371, 1323, 6173, 4199, 4344, 9679, 1989} and a hash function h(x) = x mod 10, show the resulting open addressing hash table: (ii) with second hash function h2(x) = 7 – (x mod 7). Please show me the stepsarrow_forwardConsider an extensible hash table that uses 4-bit hash keys and stores two records per bucket. 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 combination of i (the number of bits used to index the bucket array) and a set of of keys belong to one bucket at some time during the sequence of insertions? Group of answer choices i = 1; 1110 only. i = 3; 0101 only. i = 2; 1111 only. i = 3; 1000 only.arrow_forward
- Consider 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? Group of answer choices 1010 1001 1110 0011arrow_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_forwardConsider a hash table of capacity 5 that uses open addressing with linear probing. This hash table uses a hash function that takes the remainder when the hash code of a string is divided by the capacity of the hash table. (For example, "air" hashes 0.) The hashCode method has been overridden for strings, which simply returns 0 if a string begins with 'a', 1 if it begins with 'b', etc. /** Returns 0 if a String begins with 'a', 1 if it begins with 'b', etc. */ public int hashCode() { return this.charAt(0) - 'a'; } Assume that "apple", and "butter" have been inserted in this order into the hash table. "fig" is to be inserted into the hash table. (a) Which array slot/index should "fig" be placed? Answer: (b) What is the load factor after "fig" is inserted into the hash table? Answer:arrow_forward
- Consider a hash table of capacity 4 that uses open addressing with linear probing. This hash table uses a hash function that takes the remainder when the hash code of a string is divided by the capacity of the hash table. (For example, "air" hashes 0.) The hashCode method has been overridden for strings, which simply returns 0 if a string begins with 'a', 1 if it begins with 'b', etc. /** Returns 0 if a String begins with 'a', 1 if it begins with 'b', etc. */ public int hashCode() { return this.charAt(0) - 'a'; } Assume that "apple", and "cherry" have been inserted in this order into the hash table. "guava" is to be inserted into the hash table. (a) Which array slot/index should "guava" be placed? Answer: (b) What is the load factor after "guava" is inserted into the hash table? Answer:arrow_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_forwardSuppose we want to store a set S of n keys in a hash table T with size m, with m = n + 2, and a hash function that behaves uniformly. Should we resolve collisions with chaining or with open- addressing? Please explain why.arrow_forward
- Consider 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_forwardConsider Cuckoo hashing with a hash table H[0... 10] and the two hash functions: h₁(x) = (x +2) mod 11 h₂(x) = (4x + 1) mod 11. (When inserting, try h₁ first.) Consider the following sequence of operations: Insert(11) Insert(13) Insert(33) Which position is 33 placed in? 0 1 2 3 4 5 6 O O ос 7 8 9 10arrow_forwardHelp me pleasearrow_forward
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