**3**Comments September 03, 2008

# MIT's Introduction to Algorithms, Lectures 7 and 8: Hashing

This is the fifth post in an article series about MIT's lecture course "**Introduction to Algorithms**." In this post I will review lectures seven and eight, which are on the topic of **Hashing**.

Many applications require a dynamic set that supports dictionary operations **insert**, **search**, and **delete**. For example, a compiler for a computer language maintains a symbol table, in which the keys of elements are arbitrary strings that correspond to identifiers in the language. A hash table is an effective data structure for implementing dictionaries.

Lectures seven and eight cover various implementation techniques of hash tables and hash functions.

## Lecture 7: Hashing I

Lecture seven starts with the **symbol-table problem** -- given a table S, we'd like to insert an element into S, delete an element from S and search for an element in S. We'd also like these operations to take constant time.

The simplest solution to this problem is to use a **direct-access (or direct-address) table**. To represent the dynamic set, we use an array, or direct-address table, denoted by **T**, in which each position, or slot, corresponds to a key.

Using direct-address table, the dictionary operations are trivial to implement.

Direct-Address-Search(T, k) return T[k] Direct-Address-Insert(T, x) T[key[x]] = x Direct-Address-Delete(T, x) T[key[x]] = NIL

Direct addressing is applicable when we can afford to allocate an array that has one position for every possible key. It is not applicable when the range of keys can be large (as it requires a lot of space for the array T). This is where **hashing** comes in.

The lecture continues with explanation of what hashing is. Hashing uses a **hash function** h(k) that maps keys k randomly into slots of hash-table T. There is one hitch: two keys may hash to the same slot. We call this situation a **collision**. Fortunately, there are effective techniques for resolving the conflict created by collisions.

One of the simplest collision resolution techniques is called **chaining**. In chaining, we put all the elements that hash to the same slot in a linked list:

Collision resolution by chaining. Each hash-table slot T[j] contains a linked list of all the keys whose hash value is j. For example, h(k_{1}) = h(k_{4}).

Professor Leiserson then analyzes the running time of insert, delete and search operations. It is concluded that the expected running time operations is still O(1), under assumption that the number of hash-table slots is at least proportional to the number of elements in the table.

The other half of the lecture is devoted to hash functions and another way of resolving collisions -- resolving collisions by **open addressing**, and probing strategies (search) for open addressing -- **linear probing** and **double hashing**.

A good hash function should distribute the keys uniformly into the slots of the table and the regularity in the key distributions should not affect uniformity.

Two hash function creating methods are introduced - the **division method**, which defines h(k) = k mod m, and the **multiplication method**, where h(k) = (A·k mod 2^{w})>>(w-r), where w is bits in a word, A is an odd integer between 2^{w-1} and 2^{w}, and r is lg(m).

You're welcome to watch lecture seven:

Topics covered in lecture seven:

- [00:30] Symbol-table problem.
- [02:05] Symbol-table operations: insert, delete, search.
- [04:35] Direct-address table (direct addressing).
- [09:45] Hashing.
- [14:30] Resolving hash function collisions by chaining.
- [17:05] Worst-case analysis of chaining.
- [19:15] Average-case analysis of chaning.
- [29:30] Choosing a hash function.
- [30:55] Division method hash function.
- [39:05] Multiplication method hash function.
- [46:30] Multiplication method explained with a modular wheel.
- [50:12] Resolving hash function collisions by open addressing.
- [59:00] Linear probing strategy.
- [01:01:30] Double hashing probing strategy.
- [01:04:20] Average-case analysis of open addressing.

Lecture seven notes:

## Lecture 8: Hashing II

Lecture eight starts with addressing a **weakness of hashing** -- for any choice of hash function, there exists a set of keys that all hash to the same value. This weakness can lead to denial of service attacks on the application using hashing.

The idea of addressing this problem is to choose a hash function at random! This is called **universal hashing**.

The lecture then moves to a mathematically rigorous the definition of universal hashing and explains one of many ways to construct a universal hash function.

The other half of the lecture is devoted to **perfect hashing**. Perfect hashing solves a problem of constructing a static hash table (such as a hash table stored on a CD), so that searches take O(1) time guaranteed (worst-case). The key idea in creating such hash table is to use 2-level universal hashing, so that no collisions occur in level 2.

Video of lecture eight:

Topics covered in lecture eight:

- [00:30] Fundamental weakness of hashing.
- [05:12] Universal hashing.
- [20:10] Constructing a universal hash function.
- [49:45] Perfect hashing.
- [54:30] Example of perfect hashing.
- [01:06:27] (Markov inequality)
- [01:14:30] Analysis of storage requirements for perfect hashing.

Lecture eight notes:

Have fun hashing! The next post will be about random binary search trees and red-black trees!

PS. This course is taught from the CLRS book (also called "Introduction to Algorithms"):

A friend of mine recently showed me his version of extracting ip address(es) from **ifconfig**. The ifconfig tool on Linux is used to list IP addresses and configure network interfaces. It is sometimes necessary to capture your IP in a variable, like when writing some firewall rules.

I decided to show off and golf the extraction of IP address with four commonly used tools -- Awk, sed, a version of Perl everyone has, and the latest version of Perl 5.10.

His original version was pretty ugly:

$ ifconfig | perl -ple 'print $_ if /inet addr/ and $_ =~ s/.*inet addr:((?:\d+\.){3}\d+).*/$1/g ;$_=""' | grep -v ^\s*$

My own version on my firewall was:

$ ifconfig | grep inet | awk -F: '{ print $2 }' | awk '{ print $1 }'

This looks nicer but I was uselessly using grep, and calling awk twice.

## Golfing in Perl

My first attempt was to do it with grep and Perl.

$ ifconfig | grep inet | perl -ple '($_) = /addr:([^ ]+)/' 192.168.1.1 127.0.0.1

In a split second I realized that Perl does grep itself:

$ ifconfig | perl -nle '/addr:([^ ]+)/ and print $1' 192.168.1.1 127.0.0.1

Then I noticed that 'dr:' matches the same lines as 'addr:'; also 'and' can be replaced with '&&':

$ ifconfig | perl -nle '/dr:([^ ]+)/ && print $1' 192.168.1.1 127.0.0.1

The regular expression '([^ ]+)' can be replaced with '(\S+)', which matches non-whitespace characters:

$ ifconfig | perl -nle '/dr:(\S+)/ && print $1' 192.168.1.1 127.0.0.1

Cutting the whitespace, the final version for Perl is 37 characters long:

$ ifconfig|perl -nle'/dr:(\S+)/&&print$1' 192.168.1.1 127.0.0.1

## Golfing in Perl 5.10

It can be made shorter with Perl 5.10, which introduces the new say function:

$ ifconfig|perl -nE'/dr:(\S+)/&&say$1' 192.168.1.1 127.0.0.1

The result for Perl 5.10 is 34 characters.

## Golfing in sed

Then I also tried doing the same with sed:

$ ifconfig | sed -n '/dr:/{;s/.*dr://;s/ .*//;p;}' 192.168.1.1 127.0.0.1

I modified it to strip off everything that was not numbers and dots:

$ ifconfig | sed -n '/dr:/{;s/.*dr:\([0-9.]\+\) .*/\1/;p;}' 192.168.1.1 127.0.0.1

This turned out to be much longer, so I tried getting rid of backslashes, by enabling extended regexes in sed with -r argument:

$ ifconfig | sed -rn '/dr:/{;s/.*dr:([0-9.]+) .*/\1/;p;}' 192.168.1.1 127.0.0.1

I forgot that I had used the ([^ ]+) regex before. Another my friend reminded me this, shortening the sed script to:

$ ifconfig | sed -rn 's/.*r:([^ ]+) .*/\1/p' 192.168.1.1 127.0.0.1

Dropping the whitespace, sed version turned out to be 40 characters long.

## Golfing in Awk

My final attempt was to optimize my original Awk version:

$ ifconfig | awk '/dr:/{gsub(/.*:/,"",$2);print$2}' 192.168.1.1 127.0.0.1

Cutting the whitespace, the Awk version is 43 characters.

## The Winner

The winner in this golf tournament is Perl 5.10 with 34 chars!

## Can You Do Better?

Can you golf this even shorter?

Around a year ago I wrote a **YouTube video downloader** in GNU Awk. As I explained, I did it to explore the corners of Awk language.

The key idea in writing this program was to figure out how to do networking in Awk. The original version of Awk does not have any networking capabilities, but it turned out that GNU version of Awk does! There is even a manual on "TCP/IP Networking With Gnu Awk"!

One of my blog readers, **Werner Illchmann**, suggested that I add a progress bar and sent me a patch. I improved it a little and here is the result:

$ chmod +x get_youtube_vids.awk $ ./get_youtube_vids.awk http://www.youtube.com/watch?v=4bQOSRm9YiQ Parsing YouTube video urls/IDs... Getting video information for video: 4bQOSRm9YiQ... Downloading: Premature Optimization is the Root of All Evil!... Saving video to file 'Premature_Optimization_is_the_Root_of_All_Evil.flv' (size: 227.81kb)... <strong>Done: 85121/233280 bytes (36%, 83.13kb/227.81kb)</strong>

Here is the source code of the program:

#!/usr/bin/gawk -f # # 2007.07.10 v1.0 - initial release # 2007.10.21 v1.1 - youtube changed the way it displays vids # 2008.03.01 v1.2 - youtube changed the way it displays vids # 2008.08.28 v1.3 - added a progress bar and removed need for --re-interval # 2009.08.25 v1.4 - youtube changed the way it displays vids # # Peteris Krumins (peter@catonmat.net) # http://www.catonmat.net -- good coders code, great reuse # # Usage: gawk -f get_youtube_vids.awk <http://youtube.com/watch?v=ID1 | ID1> ... # or just ./get_youtube_vids.awk <http://youtube.com/watch?v=ID1 | ID1> # BEGIN { if (ARGC == 1) usage(); BINMODE = 3 delete ARGV[0] print "Parsing YouTube video urls/IDs..." for (i in ARGV) { vid_id = parse_url(ARGV[i]) if (length(vid_id) < 6) { # havent seen youtube vids with IDs < 6 chars print "Invalid YouTube video specified: " ARGV[i] ", not downloading!" continue } VIDS[i] = vid_id } for (i in VIDS) { print "Getting video information for video: " VIDS[i] "..." get_vid_info(VIDS[i], INFO) if (INFO["_redirected"]) { print "Could not get video info for video: " VIDS[i] continue } if (!INFO["video_url"]) { print "Could not get video_url for video: " VIDS[i] print "Please goto my website, and submit a comment with an URL to this video, so that I can fix it!" print "Url: http://www.catonmat.net/blog/downloading-youtube-videos-with-gawk/" continue } if ("title" in INFO) { print "Downloading: " INFO["title"] "..." title = INFO["title"] } else { print "Could not get title for video: " VIDS[i] print "Trying to download " VIDS[i] " anyway" title = VIDS[i] } download_video(INFO["video_url"], title) } } function usage() { print "Downloading YouTube Videos with GNU Awk" print print "Peteris Krumins (peter@catonmat.net)" print "http://www.catonmat.net -- good coders code, great reuse" print print "Usage: gawk -f get_youtube_vids.awk <http://youtube.com/watch?v=ID1 | ID1> ..." print "or just ./get_youtube_vids.awk <http://youtube.com/watch?v=ID1 | ID1> ..." exit 1 } # # function parse_url # # takes a url or an ID of a youtube video and returns just the ID # for example the url could be the full url: http://www.youtube.com/watch?v=ID # or it could be www.youtube.com/watch?v=ID # or just youtube.com/watch?v=ID or http://youtube.com/watch?v=ID # or just the ID # function parse_url(url) { gsub(/http:\/\//, "", url) # get rid of http:// part gsub(/www\./, "", url) # get rid of www. part gsub(/youtube\.com\/watch\?v=/, "", url) # get rid of youtube.com... part if ((p = index(url, "&")) > 0) # get rid of &foo=bar&... after the ID url = substr(url, 1, p-1) return url } # # function get_vid_info # # function takes the youtube video ID and gets the title of the video # and the url to .flv file # function get_vid_info(vid_id, INFO, InetFile, Request, HEADERS, matches, escaped_urls, fmt_urls, fmt) { delete INFO InetFile = "/inet/tcp/0/www.youtube.com/80" Request = "GET /watch?v=" vid_id " HTTP/1.1\r\n" Request = Request "Host: www.youtube.com\r\n\r\n" get_headers(InetFile, Request, HEADERS) if ("Location" in HEADERS) { INFO["_redirected"] = 1 close(InetFile) return } # fix this bug: # http://www.youtube.com/watch?v=nb1u7wMKywM while ((InetFile |& getline) > 0) { if (match($0, /"fmt_url_map": "([^"]+)"/, matches)) { escaped_urls = url_unescape(matches[1]) split(escaped_urls, fmt_urls, /,?[0-9]+\|/) for (fmt in fmt_urls) { if (fmt_urls[fmt] ~ /itag=5/) { # fmt number 5 is the best video INFO["video_url"] = fmt_urls[fmt] close(InetFile) return } } close(InetFile) return } else if (match($0, /<title>YouTube - ([^<]+)</, matches)) { # lets try to get the title of the video from html tag which is # less likely a subject to future html design changes INFO["title"] = matches[1] } } close(InetFile) } # # function url_unescape # # given a string, it url-unescapes it. # charactes such as %20 get converted to their ascii counterparts. # function url_unescape(str, nmatches, entity, entities, seen, i) { nmatches = find_all_matches(str, "%[0-9A-Fa-f][0-9A-Fa-f]", entities) for (i = 1; i <= nmatches; i++) { entity = entities[i] if (!seen[entity]) { if (entity == "%26") { # special case for gsub(s, r, t), when r = '&' gsub(entity, "\\&", str) } else { gsub(entity, url_entity_unescape(entity), str) } seen[entity] = 1 } } return str } # # function find_all_matches # # http://awk.freeshell.org/FindAllMatches # function find_all_matches(str, re, arr, j, a, b) { j=0 a = RSTART; b = RLENGTH # to avoid unexpected side effects while (match(str, re) > 0) { arr[++j] = substr(str, RSTART, RLENGTH) str = substr(str, RSTART+RLENGTH) } RSTART = a; RLENGTH = b return j } # # function url_entity_unescape # # given an url-escaped entity, such as %20, return its ascii counterpart. # function url_entity_unescape(entity) { sub("%", "", entity) return sprintf("%c", strtonum("0x" entity)) } # # function download_video # # takes the url to video and saves the movie to current directory using # santized video title as filename # function download_video(url, title, filename, InetFile, Request, Loop, HEADERS, FOO) { title = sanitize_title(title) filename = create_filename(title) parse_location(url, FOO) InetFile = FOO["InetFile"] Request = "GET " FOO["Request"] " HTTP/1.1\r\n" Request = Request "Host: " FOO["Host"] "\r\n\r\n" Loop = 0 # make sure we do not get caught in Location: loop do { # we can get more than one redirect, follow them all get_headers(InetFile, Request, HEADERS) if ("Location" in HEADERS) { # we got redirected, let's follow the link close(InetFile) parse_location(HEADERS["Location"], FOO) InetFile = FOO["InetFile"] Request = "GET " FOO["Request"] " HTTP/1.1\r\n" Request = Request "Host: " FOO["Host"] "\r\n\r\n" if (InetFile == "") { print "Downloading '" title "' failed, couldn't parse Location header!" return } } Loop++ } while (("Location" in HEADERS) && Loop < 5) if (Loop == 5) { print "Downloading '" title "' failed, got caught in Location loop!" return } print "Saving video to file '" filename "' (size: " bytes_to_human(HEADERS["Content-Length"]) ")..." save_file(InetFile, filename, HEADERS) close(InetFile) print "Successfully downloaded '" title "'!" } # # function sanitize_title # # sanitizes the video title, by removing ()'s, replacing spaces with _, etc. # function sanitize_title(title) { gsub(/\(|\)/, "", title) gsub(/[^[:alnum:]-]/, "_", title) gsub(/_-/, "-", title) gsub(/-_/, "-", title) gsub(/_$/, "", title) gsub(/-$/, "", title) gsub(/_{2,}/, "_", title) gsub(/-{2,}/, "-", title) return title } # # function create_filename # # given a sanitized video title, creates a nonexisting filename # function create_filename(title, filename, i) { filename = title ".flv" i = 1 while (file_exists(filename)) { filename = title "-" i ".flv" i++ } return filename } # # function save_file # # given a special network file and filename reads from network until eof # and saves the read contents into a file named filename # function save_file(Inet, filename, HEADERS, done, cl, perc, hd, hcl) { OLD_RS = RS OLD_ORS = ORS ORS = "" # clear the file print "" > filename # here we will do a little hackery to write the downloaded data # to file chunk by chunk instead of downloading it all to memory # and then writing # # the idea is to use a regex for the record field seperator # everything that gets matched is stored in RT variable # which gets written to disk after each match # # RS = ".{1,512}" # let's read 512 byte records RS = "@" # I replaced the 512 block reading with something better. # To read blocks I had to force users to specify --re-interval, # which made them uncomfortable. # I did statistical analysis on YouTube video files and # I found that hex value 0x40 appears pretty often (200 bytes or so)! # cl = HEADERS["Content-Length"] hcl = bytes_to_human(cl) done = 0 while ((Inet |& getline) > 0) { done += length($0 RT) perc = done*100/cl hd = bytes_to_human(done) printf "Done: %d/%d bytes (%d%%, %s/%s) \r", done, cl, perc, bytes_to_human(done), bytes_to_human(cl) print $0 RT >> filename } printf "Done: %d/%d bytes (%d%%, %s/%s) \n", done, cl, perc, bytes_to_human(done), bytes_to_human(cl) RS = OLD_RS ORS = OLD_ORS } # # function get_headers # # given a special inet file and the request saves headers in HEADERS array # special key "_status" can be used to find HTTP response code # issuing another getline() on inet file would start returning the contents # function get_headers(Inet, Request, HEADERS, matches, OLD_RS) { delete HEADERS # save global vars OLD_RS=RS print Request |& Inet # get the http status response if (Inet |& getline > 0) { HEADERS["_status"] = $2 } else { print "Failed reading from the net. Quitting!" exit 1 } RS="\r\n" while ((Inet |& getline) > 0) { # we could have used FS=": " to split, but i could not think of a good # way to handle header values which contain multiple ": " # so i better go with a match if (match($0, /([^:]+): (.+)/, matches)) { HEADERS[matches[1]] = matches[2] } else { break } } RS=OLD_RS } # # function parse_location # # given a Location HTTP header value the function constructs a special # inet file and the request storing them in FOO # function parse_location(location, FOO) { # location might look like http://cache.googlevideo.com/get_video?video_id=ID if (match(location, /http:\/\/([^\/]+)(\/.+)/, matches)) { FOO["InetFile"] = "/inet/tcp/0/" matches[1] "/80" FOO["Host"] = matches[1] FOO["Request"] = matches[2] } else { FOO["InetFile"] = "" FOO["Host"] = "" FOO["Request"] = "" } } # function bytes_to_human # # given bytes, converts them to human readable format like 13.2mb # function bytes_to_human(bytes, MAP, map_idx, bytes_copy) { MAP[0] = "b" MAP[1] = "kb" MAP[2] = "mb" MAP[3] = "gb" MAP[4] = "tb" map_idx = 0 bytes_copy = int(bytes) while (bytes_copy > 1024) { bytes_copy /= 1024 map_idx++ } if (map_idx > 4) return sprintf("%d bytes", bytes, MAP[map_idx]) else return sprintf("%.02f%s", bytes_copy, MAP[map_idx]) } # # function file_exists # # given a path to file, returns 1 if the file exists, or 0 if it doesn't # function file_exists(file, foo) { if ((getline foo <file) >= 0) { close(file) return 1 } return 0 }

If you decide to learn Awk programming language, I suggest that you take a look at the Awk Cheat Sheet that I have made.

Download GNU Awk YouTube Video Downloader

Download link: gawk youtube video downloader

Total downloads: 37089 times

**8**Comments August 27, 2008

# MIT's Introduction to Algorithms, Lecture 6: Order Statistics

This is the fourth post in an article series about MIT's lecture course "**Introduction to Algorithms**." In this post I will review lecture six, which is on the topic of **Order Statistics**.

The problem of order statistics can be described as following. Given a set of N elements, find k-th smallest element in it. For example, the minimum of a set is the first order statistic (k = 1) and the maximum is the N-th order statistic (k = N).

Lecture six addresses the problem of selecting the k-th order statistic from a set of N distinct numbers.

The lecture begins with discussing the naive algorithm for finding k-th order statistic -- sort the given set (or array) A of numbers and return k-th element A[k]. Running time of such algorithm at best is O(n·log(n)).

Erik Demaine (professor) presents a randomized divide and conquer algorithm for this problem (my second post covers divide and conquer algorithms), which runs in expected linear time.

The key idea of this algorithm is to call Randomized-Partition (from previous lecture) subroutine on the given array and then recursively conquer on the partitioned elements on the left and right of the chosen pivot element.

Here is the pseudo code of this algorithm:

Randomized-Select(A, p, q, k) // finds k-th smallest element in array A[p..q] if p == q return A[p] r = Randomized-Partition(A, p, q) n = r - p + 1 if k == n return A[r] if k < n return Randomized-Select(A, p, r-1, k) else return Randomized-Select(A, r+1, q, k-n)

The lecture continues with intuitive and then mathematically rigorous analysis of the algorithm. It is concluded that the expected running time of this algorithm is O(n) and the worst case (when the Randomized-Partition algorithm chooses bad pivots) is O(n^{2}).

At the last minutes of the lecture a worst-case linear time order statistics algorithm is presented. This algorithm was invented by five scientists - M. Blum, R. W. Floyd, V. Pratt, R. Rivest and R. Tarjan. I found the original publication of this algorithm by these gentlemen: Time Bounds for Selection.

You're welcome to watch lecture six:

Lecture six notes:

Topics covered in lecture six:

- [00:30] The problem of order statistics.
- [01:50] Naive algorithm for finding k-th smallest element.
- [04:30] Randomized divide and conquer algorithm.
- [11:20] Example of algorithm run on array (6, 10, 13, 5, 8, 3, 2, 11) to find 7-th smallest element.
- [15:30] Intuitive analysis of Randomized-Select algorithm.
- [20:30] Mathematically rigorous analysis of expected running time.
- [43:55] Worst-case linear time order statistics algorithm.

Have fun! The next post will be all about hashing!

PS. This course is taught from the CLRS book (also called "Introduction to Algorithms"):

**7**Comments August 25, 2008

# MIT's Introduction to Algorithms, Lectures 4 and 5: Sorting

This is the third post in an article series about MIT's lecture course "**Introduction to Algorithms**." In this post I will review lectures four and five, which are on the topic of sorting.

The previous post covered a lecture on "Divide and Conquer" algorithm design technique and its applications.

Lecture four is devoted entirely to a single sorting algorithm which uses this technique. The algorithm I am talking about is the "**Quicksort**" algorithm. The quicksort algorithm was invented by Charles Hoare in 1962 and it is the most widely used sorting algorithm in practice.

I wrote about quicksort before in "Three Beautiful Quicksorts" post, where its running time was analyzed experimentally. This lecture does it theoretically.

Lecture five talks about theoretical running time limits of sorting using comparisons and then discusses two linear time sorting algorithms -- Counting sort and Radix sort.

## Lecture 4: Quicksort

The lecture starts by giving the divide and conquer description of the algorithm:

**Divide**: Partition the array to be sorted into two subarrays around pivot x, such that elements in the lower subarray <= x, and elements in the upper subarray >= x.**Conquer**: Recursively sort lower and upper subarrays using quicksort.**Combine**: Since the subarrays are sorted in place, no work is needed to combine them. The entire array is now sorted!

The main algorithm can be written in a few lines of code:

Quicksort(A, p, r) // sorts list A[p..r] if p < r q = Partition(A, p, r) Quicksort(A, p, q-1) Quicksort(A, q+1, r)

The key in this algorithm is the Partition subroutine:

Partition(A, p, q) // partitions elements in array A[p..q] in-place around element x = A[p], // so that A[p..i] <= x and A[i] == x and A[i+1..q] >= x x = A[p] i = p for j = p+1 to q if A[j] <= x i = i+1 swap(A[i], A[j]) swap(A[p], A[i]) return i

The lecture then proceeds with the analysis of Quicksort's worst-case running time. It is concluded that if the array to be sorted is already sorted or already reverse sorted, the running time is O(n^{2}), which is no better than Insertion sort (seen in lecture one)! What happens in the worst case is that all the elements get partitioned to one side of the chosen pivot. For example, given an array (1, 2, 3, 4), the partition algorithm chooses 1 as the pivot, then all the elements stay where they were, and no partitioning actually happens. To overcome this problem **Randomized Quicksort** algorithm is introduced.

The main idea of randomized quicksort hides in the Partition subroutine. Instead of partitioning around the first element, it partition around a random element in the array!

Nice things about randomized quicksort are:

- Its running time is independent of initial element order.
- No assumptions need to be made about the statistical distribution of input.
- No specific input elicits the worst case behavior.
- The worst case is determined only by the output of a random-number generator.

Randomized-Partition(A, p, q) <strong>swap(A[p], A[rand(p,q)])</strong> // the only thing changed in the original Partition subroutine! x = A[p] i = p for j = p+1 to q if A[j] <= x i = i+1 swap(A[i], A[j]) swap(A[p], A[i]) return i

The rest of the lecture is very math-heavy and uses indicator random variables to conclude that the **expected running time** of randomized quicksort is O(n·lg(n)).

In practice quicksort is 3 or more times faster than merge sort.

Please see Three Beautiful Quicksorts post for more information about the version of **industrial-strength quicksort**.

You're welcome to watch lecture four:

Topics covered in lecture four:

- [00:35] Introduction to quicksort.
- [02:30] Divide and conquer approach to quicksort.
- [05:20] Key idea - linear time (Θ(n)) partitioning subroutine.
- [07:50] Structure of partitioning algorithm.
- [11:40] Example of partition algorithm run on array (6, 10, 13, 5, 8, 3, 2, 11).
- [16:00] Pseudocode of quicksort algorithm.
- [19:00] Analysis of quicksort algorithm.
- [20:25] Worst case analysis of quicksort.
- [24:15] Recursion tree for worst case.
- [28:55] Best case analysis of quicksort, if partition splits elements in half.
- [28:55] Analysis of quicksort, if partition splits elements in a proportion 1/10 : 9/10.
- [33:33] Recursion tree of this analysis.
- [04:30] Analysis of quicksort, if partition alternates between lucky, unlucky, lucky, unlucky, ...
- [46:50] Randomized quicksort.
- [51:10] Analysis of randomized quicksort.

Lecture four notes:

## Lecture 5: Lower Sorting Bounds and Linear Sorting

The lecture starts with a question -- **How fast can we sort?** Erik Demaine (professor) answers and says that it depends on the model of what you can do with the elements.

The previous lectures introduced several algorithms that can sort n numbers in O(n·lg(n)) time. Merge sort achieves this upper bound in the worst case; quicksort achieves it on average. These algorithms share an interesting property: the sorted order they determine is based only on comparisons between the input elements. Such sorting algorithms are called **Comparison Sorts**, which is a model for sorting.

It turns out that any comparison sort algorithm can be translated into something that is called a **Decision Tree**. A decision tree is a full binary tree that represents the comparisons between elements that are performed by a particular sorting algorithm operating on an input of a given size.

Erik uses decision trees to derive the lower bound for running time of comparison based sorting algorithms. The result is that no comparison-based sort can do better than O(n·lg(n)).

The lecture continues with bursting outside of comparison model and looks at sorting in linear time using no comparisons.

The first linear time algorithm covered in the lecture is **Counting Sort**. The basic idea of counting sort is to determine, for each input element x, the number of elements less than x. This information can be used to place element x directly into its position in the output array. For example, if there are 17 elements less than x, then x belongs in output position 18.

The second linear time algorithm is **Radix Sort**, which sorts a list of numbers by examining each digit at a given position separately.

Erik ends the lecture by analyzing correctness and running time of radix sort.

Video of lecture five:

Topics covered in lecture five:

- [00:30] How fast can we sort?
- [02:27] Review of running times of quicksort, heapsort, merge sort and insertion sort.
- [04:50] Comparison sorting (model for sorting).
- [06:50] Decision trees.
- [09:35] General description of decision trees.
- [14:25] Decision trees model comparison sorts.
- [20:00] Lower bound on decision tree sorting.
- [31:35] Sorting in linear time.
- [32:30] Counting sort.
- [38:05] Example of counting sort run on an array (4, 1, 3, 4, 3)
- [50:00] Radix sort.
- [56:10] Example of radix sort run on array (329, 457, 657, 839, 546, 720, 355).
- [01:00:30] Correctness of radix sort.
- [01:04:25] Analysis of radix sort.

Lecture five notes:

Have fun sorting! The next post will be about order statistics (given an array find n-th smallest element)!

PS. This course is taught from the CLRS book (also called "Introduction to Algorithms"):