Optimization: Implementation and Analysis

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En provenance du cours de University of California San Diego

Data Structures

1470 notes

University of California San Diego

Data Structures

1470 notes

Cours 2 sur 6 dans Specialization Data Structures and Algorithms

A good algorithm usually comes together with a set of good data structures that allow the algorithm to manipulate the data efficiently. In this course, we consider the common data structures that are used in various computational problems. You will learn how these data structures are implemented in different programming languages and will practice implementing them in our programming assignments. This will help you to understand what is going on inside a particular built-in implementation of a data structure and what to expect from it. You will also learn typical use cases for these data structures. A few examples of questions that we are going to cover in this class are the following: 1. What is a good strategy of resizing a dynamic array? 2. How priority queues are implemented in C++, Java, and Python? 3. How to implement a hash table so that the amortized running time of all operations is O(1) on average? 4. What are good strategies to keep a binary tree balanced? You will also learn how services like Dropbox manage to upload some large files instantly and to save a lot of storage space!

À partir de la leçon

Hash Tables

In this module you will learn about very powerful and widely used technique called hashing. Its applications include implementation of programming languages, file systems, pattern search, distributed key-value storage and many more. You will learn how to implement data structures to store and modify sets of objects and mappings from one type of objects to another one. You will see that naive implementations either consume huge amount of memory or are slow, and then you will learn to implement hash tables that use linear memory and work in O(1) on average! In the end, you will learn how hash functions are used in modern disrtibuted systems and how they are used to optimize storage of services like Dropbox, Google Drive and Yandex Disk!

Search Pattern in Text7:09

Rabin-Karp's Algorithm9:45

Optimization: Precomputation9:25

Optimization: Implementation and Analysis5:57

Rencontrer les enseignants

Alexander S. Kulikov
Visiting Professor
Department of Computer Science and Engineering
Michael Levin
Lecturer
Computer Science
Daniel M Kane
Assistant Professor
Department of Computer Science and Engineering / Department of Mathematics
Neil Rhodes
Adjunct Faculty
Computer Science and Engineering

Hi, in this video we'll use the precomputed hashes from the previous video

to improve the running time of the RabinKarp cell algorithm.

And here is the pseudo code.

Actually it is very similar to the pseudo code of the initial

RabinKarp algorithm and only a few lines changed.

So again, choose a very big prime number p and we choose a random number

x from 1 to p- 1 to choose a random Hash function from the polynomial family.

We initialize the result with an positions.

And we compute the hash of the pattern in the variable pHash

directly using our implementation of polynomial hash.

And then we call the PrecomputeHashes function from

the previous video to precompute big H, an array with hash values

of all sub strings of the text with length equal to the pattern p.

We need them to check whether it makes sense to compare pattern to

a sub string if their hashes are the same.

Or maybe if their hashes are different then,

there is no point comparing them character by characte,r because it means that

pattern is definitely different from the substream.

So, then our main for loop goes for

all i, starting positions for the pattern, from 0 to length of text

minus length of pattern as in the previous version of the RabinKarp's algorithm.

And the main thing that changed is that.

We compare the hash of the pattern, not with a vary

of hash functions computed on the fly, but with the pre-computed value of the hash

function for the substream starting in position i, H[i].

If they are different, it means that the pattern is definitely different from

the substream starting in position i and we don't need to compare them character

by character, so we just continue to the next iteration of the for loop.

Otherwise, if the hash value of the pattern is the same

as the hash value of the substring,

we need to actually compare them on the quality, character by character.

And to do that, we call function AreEqual for the substring and the pattern.

If they are actually equal, we append position i to the result

to the list of all the occurrences of pattern indexed.

Otherwise, we proceed to the next iteration.

And in the end, we return result,

the list of all positions in which pattern occurs in the text.

Let's analyze the running time of this version of RabinKarp's algorithm.

First we compute the hash value of the pattern in time

proportional to its length.

Then we call the PrecomputeHashes function,

which we estimated in the previous video around

in time proportional to the sum of length of text and the pattern.

And then the only other thing that we do is we compare the hashes,

and for some of the substrings we call function AreEqual.

And we already know from the previous videos that the total time spent in

AreEqual is on average, proportional to q multiplied by length of the pattern,

where q is the number of occurrences of pattern and text.

Why is that?

Because we only compared pattern to a substring if they're equal or

if there's a collision.

You can compare such a big prime P, that the collisions have very low probability,

and on average, they won't influence the running time.

So, on average, total time spent in AreEqual is proportional to q

multiplied by length of the pattern.

And then the total average running time, is proportional to length of the text,

plus q plus 1, multiplied by length of the pattern.

And this is actually much better, than the time for the algorithm, because usually

q is very small, q is the number of times you actually found pattern in text.

If you are, for example, searching for your name on a website or

for infected code pattern in the binary code of the program,

there will be no or only a few places where you actually find it.

And that their number is q and it is usually much, much less than the total

number of positions in the test which is length of the test.

So the second sum of [q plus 1 multiplied by 1 looks like by length of the pattern

is much smaller than length of the text multiply it by length of the pattern.

And if pattern is sufficiently long, then the first summoned is also

much smaller than length of the text multiplied by length of the pattern.

So we improved our running time for most practical purposes very significantly.

Of course it's only an average, but in practice, this will work really well.

And to conclude. In this module, we cited hash tables and

hash functions, and we learned that hash tables are useful for storing sets of

objects and mappings from one type of object to another one.

And we managed to do it in such a way that you can search and

modify keys and values of the hash tables in constant time on average.

And to do so, you must use good hash families, and

you must select random hash functions from good hash families.

And you also learned that hashes are not only useful for

storing something, but they're also useful while working with strings and

texts, for finding patterns in long texts.

And actually, there are a lot more applications of hashing

in distributed systems, for example, and in data science.

And I'll tell you about some applications and

distributive systems in the next few optional videos.

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