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時間:2021-01-25
Exercise Sheet 1: Hashing and Bloom filters
COMS31900 Advanced Algorithms 2019/2020
Please feel free to discuss these problems on the unit discussion board. If you would like to
have your answers marked, please either hand them in in person at the lecture or email them
to me with the email subject ”Problem sheet 1” by the deadline stated.
1 Weakly-universal Hashing
A hash function family H = {h1, h2, . . . } is weakly-universal iff for randomly and uniformly
chosen h ∈ H, we have Pr(h[x] = h[y]) ≤ 1/m for any distinct x, y ∈ U . Consider the following
hash function families. For each one, prove that it is weakly universal or give a counter-example.
1. Let p be a prime number and m be an integer, p ≥ m. Consider the hash function
family where you pick at random a ∈ {1, . . . , p? 1} and then define ha : {0, . . . , p? 1} →
{0, . . . ,m? 1} as ha(x) = (ax mod p) mod m.
2. Let p be a prime and m be an integer such that p ≥ m. Consider the hash function
family where you pick at random b ∈ {0, . . . , p? 1} and then define hb : {0, . . . , p? 1} →
{0, . . . ,m? 1} as hb(x) = ((x + b) mod p) mod m.
3. Let p be a multiple of m. Consider the hash function family where you pick at random
a ∈ {1, . . . ,m ? 1} and b ∈ {0, . . . ,m ? 1}. Define ha,b : {0, . . . , p ? 1} → {0, . . . ,m ? 1}
as ha,b(x) = ((ax + b) mod p) mod m).
2 Cuckoo Hashing
1. This question is about cuckoo hashing. Consider a small variant of cuckoo hashing where
we use two tables T1 and T2 of the same size and hash function h1 and h2. When inserting
a new key x, we first try to put x at position h1(x) in T1. If this leads to a collision,
then the previously stored key y is moved to position h2(y) in T2. If this leads to another
collision, then the next key is again inserted at the appropriate position in T1, and so on.
In some cases, this procedure continues forever, i.e. the same configuration appears after
some steps of moving the keys around to dissolve collisions.
(a) Consider two tables of size 5 each and two hash functions h1(k) = k mod 5 and
h2(k) = bk5c mod 5. Insert the keys 27, 2, 32 in this order into initially empty hash
tables, and show the result.
(b) Find another key such that its insertion leads to an infinite sequence of key displace-
ments.
2. In order to use cuckoo hashing under an unbounded number of key insertions, we cannot
have a hash table of fixed size. The size of the hash table has to scale with the number
1
of keys inserted. Suppose that we never delete a key that has been inserted. Consider
the following approach with Cuckoo hashing. When the current hash table fills up to its
capacity, a new hash table of doubled size is created. All keys are then rehashed to the
new table. Argue that the average time it takes to resize and rebuild the hash table, if
spread out over all insertions, is constant in expectation. That is, the expected amortised
cost of rebuilding is constant.
3 Bloom Filters
1. Answer the following three questions about Bloom filters:
(a) What operations do we perform on Bloom filters?
(b) What is the difference between hash tables and Bloom filters in terms of which data
we can access?
(c) Why is there is a problem when deleting elements from a Bloom filter?
2. Suppose you have two Bloom filters A and B (each having the same number of cells and
the same hash functions) representing the two sets A and B. Let C = A&B be the Bloom
filter formed by computing the bitwise Boolean and of A and B.
(a) C may not always be the same as the Bloom filter that would be constructed by
adding the elements of the set (A intersect B) one at a time. Explain why not.
(b) Does C correctly represent the set (A intersect B), in the sense that it gives a positive
answer for membership queries of all elements in this set? Explain why or why not.
(c) Suppose that we want to store a set S of n = 20 elements, drawn from a universe
of U = 10000 possible keys, in a Bloom filter of exactly N = 100 cells, and that we
care only about the accuracy of the Bloom filter and not its speed. For this problem
size, what is the best choice of the number of hash functions (the parameter r in the
lecture)? (That is what value of r gives the smallest possible probability that a key
not in S is a false positive?) What is the probability of a false positive for this choice
of r?
4 Perfect Hashing
This question is about perfect hashing:
1. Our perfect hashing scheme assumed the set of keys stored in the table is static. Suppose
instead that we want to add a few new items to our table after the initial construction.
Suggest a way to modify our intitial construction so that we can insert these new items
using no new space and without making significant changes to our existing table (in
particular, we don’t want to change our initial hash function). Your scheme should still
do lookups of all items in O(1) time, but you may use a bit more initial space.
2. Suppose now that we want to delete some of our initial items. Describe a simple way to
support deletions in our perfect hashing scheme.
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