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 1E
Program Plan Intro
To determine expected number of collisions hash n distinct keys into an array of length m.
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Suppose we use a hash function h to hash n distinct keys into an array T oflength m. Assuming simple uniform hashing, what is the expected number ofcollisions? More precisely, what is the expected cardinality of ffk; lg W k ¤ l andh.k/ D h.l/g?
Given a hash function h(x), insert the following keys into the hash table below according to the
order of the keys. Use Linear Probing when a collision occurs.
h (x) = |40 – x|mod 11,
Keys = { 15, 40, 57, 46, 24, 32, 80, 70, 22, 28 }
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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- Prove 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_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_forward3. Suppose you are given a hash function h : {0, 1}* → {0, 1}100, and a machine M that can compute 230 hashes a minute. Using M, how long does it take to find a collision, using the birthday attack, on h?arrow_forward
- Answer the following questions for a hash table of size T storing n items: What are the main operations that a hash table supports and what is the average and worst time complexity, respectively, of each operation?arrow_forwardFor a 16-bits hash, if someone gives you an R such that the hash of R is H(R) < 512, explain what expected number of hashes they have tried as a measure of work ("expected" means in an average case).arrow_forwardFor 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?arrow_forward
- Given a known set of keys, a perfect hash function is one that translates every key to adifferent bucket index. For example, given the set of keys {'alpha', 'beta', 'gamma', 'delta’},the following is a perfect hash function:m = 4; hash(k) = (initial letter of k - 'a') modulo 4a. Under what circumstances does it make sense to seek a perfect hash function? Explain.b. How can a perfect hash function be exploited in implementing a hash table? Explain.c. Given the following set of keys:{CA, MX, US}find a perfect hash function with m = 3,d. Given the following set of keys:{AT, BE, DE, DK, ES, FI, FR, GR, IE, IT, LU, NL, PT, SE, UK}find a perfect hash function with m < 20.arrow_forwardSuppose you have a hash table of size N = 64, and you are using quadratic probing. The keys in your hash are 4-digit integers (0000 through 9999) and your hash function is h(k) = (the sum of the digits in k). Assuming keys are uniformly random over the range 0000 to 9999, is this a good hash function? (Answer true for yes, false for no). True Falsearrow_forward1. 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.)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_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_forwardConsider a hash table of size 11 with hash function h(x) = x mod 11. Draw the tablethat results after inserting, in the given order, the following values: 26, 57, 40, 50,13, 33, and 72 for each of the three scenarios below: i. When collisions are handled by separate chaining. ii. When collisions are handled by double hashing using a second hash function ℎ’(?) = (? ??? 7) + 1. Hint, the overall (combined) hash function is ?(?) = ( ℎ(?) + ? × ℎ′(?) ) ??? 11, where i = 0, 1, 2, 3, … iii. When collisions are handled by quadratic probing with a quadratic probe function ℎ′(?,?) = (ℎ(?) + 0.5 ? + 0.5 ?2) ??? 11 where ? = 1, 2, 3,….arrow_forward
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