As we saw in lecture, hashing is very important for managing large data systems. For example, it is used…

As we saw in lecture, hashing is very important for managing large data systems. For example, it is used…

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As we saw in lecture, hashing is very important for managing large data systems.
For example, it is used to map from data/search keys to the location where that
data is stored (memory address, DB block, machine/disk). Another common
application is evenly dividing a set of inputs into buckets to process them in
parallel, for example when counting large numbers of tuples. In this problem, we’ll
get a little more intuition about hash functions and collision resolution by looking
at some examples.
Question 1.1 – Some Intuition about Hash Collisions
The details of how different hash functions work, including how they are
implemented and their statistical properties, are mostly beyond the scope of this
class. However, a feature common to all hash functions is collisions. When dealing
with hashing, it’s helpful to have some intuition about the frequency of collisions.
Recall that a collision occurs whenever for two distinct elements a and b and a hash
function h(x), h(a)=h(b).
Assume you have a hash function (the actual process by which it operates is
unknown) with the following properties:
• The outputs (hash values) are of size 8 bits (there are 256 possible hash
values/buckets).
The hash function distributes its outputs evenly across the entire output range
(this property is very desirable in hash functions and is called uniformity).
What is the probability of having at least one collision if we are hashing 5
inputs? Give your answer as a numerical value (not an equation) but show
some work/reasoning. A simple numerical answer with no reasoning will not NOTE: One of the interesting (and somewhat counter-intuitive) facts about
hash functions is that the probability of collision rises much faster than intuition
might suggest. Even when hashing relatively small numbers of inputs (w.r.t the
output range), the collision probability rapidly approaches 1. See
https://en.wikipedia.org/wiki/Birthday_problem for more interesting discussion on
this problem.
Question 1.2 -Thinking about Collision Resolution
In most applications of hashing, how collisions are handled is an important part of
the implementation. One example of this is in the implementation of hash partition
join, which we discussed in lecture. Consider the case where we wish to join two
relations R and S on a shared attribute A. The first step in the process, partitioning,
is done using a hash function. We hash tuples from both relations using their
attribute A in order to divide them up evenly into a finite number of buckets B.
In this process of partitioning, hash collisions may occur. Explain why
having many hash collisions in our buckets would harm the efficiency of the
hash partition join algorithm.
Considering this, a method is needed to address collisions. As described in
lecture, this can be done with double hashing. If the initial partition resulted in
collisions, these can be resolved by doing a second pass and re-hashing each
collided element with a new hash function (on attribute A). The result of this
second hash is used as an offset to determine how many bins to move the tuple
over. For example, if a tuple is hashed initially to bin 3 and there is a collision, if
h2(tuple) = 1 then the tuple will be moved to bin 3 + 1 = 4 (if this again results in a
collision, the tuple can be shifted again by h2(tuple) until the collision is resolved).
In practice, a common choice for this second hash function is Hash2(key) = K-(key
mod K), where K is a prime smaller than the number of buckets B.
You are given B = 5. You are also given the following relation R.
A
C
E
93
Red
28
Purple
70 Orange
11
545
Blue
6
88 Brown
The initial hash function is h₁(A) = A mod B. The second hash function is h₂(A)
= 3 – (A mod 3).
5
7
3
As part of the first step of a hash partition join, partition R using the given
hash function. Assume that the tuples are hashed into buckets in the order
they appear in the table (from top to bottom). Resolve any collisions with the
provided second hash function. You can indicate your answer in the form of
a table or by listing the tuple(s) in each bucket (B0, B1, …). Bonus Note: There are many other collision resolution strategies beyond those
mentioned here, each with different advantages and disadvantages which can be
analyzed probabilistically.
Question 1.3 – Computing Counts of Pairs
As seen above, hash functions are useful whenever we need to (relatively) evenly
distribute data. Another such application is dividing across multiple machines to
process in parallel.
Imagine you have created a music app. Users of your app can login, start a
session, and then play songs. The database which forms the backend to your app
contains a table with tuples (user_id, session id, song id) which represent every
time a user has played a song. You wish to see which pairs of songs are most
frequently listened to together in the same session.
You are given the following information:
• There are 10 million users
• There are 1 thousand songs
• There are 4 billion sessions
• The user IDs are 3 bytes
• The song IDs are 2 bytes
• The session IDs are 4 bytes
• The avg. number of unique songs played per session is 5
Assume all your data is stored on hard disks and use the values for disk
seek/scan times from the top of the assignment. Assume that data is stored
sequentially on disk.
Calculate the total time required to compute the counts for each distinct
pair. Do not count pairs with the same song (i.e. Song A – Song A) and assume
that when counting we do not count different orderings of the same songs as
distinct pairs (i.e. if Song A and Song B are played in the same session, we only
count the pair Song A – Song B and do not count the pair Song B – Song A).
Assume that it takes 1 hr. to sort all pairs using the external merge sort
algorithm. Please show some work/reasoning. A simple numerical answer
with no reasoning will not count for full points. Question 1.4 – Parallelizing Counting with Hashing
Now imagine you have a hash function which maps from any 64-bit input uniformly
to a 32-bit hash value.
Explain how you could use it to parallelize the counting across n
machines.
What would the total required time to compute the counts for each pair
be once you’ve used your hash function to divide the pair tuples across 6
separate disks (which can write in parallel)? Please show some
work/reasoning. A simple numerical answer with no reasoning will not
count.
Question 1.5 – Thinking a Little More About Hashing
Consider the application of hashing where we want to parallelize a task across n
machines (where n is reasonably small, say 100) versus the task of using hashing
as a way to map keys to a location (e.g. generating IDS).
For each of these two applications, is a low collision rate important? For
each of these two applications, is uniformity important? Why?

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