|
|
| Line 1: |
Line 1: |
| In [[coding theory]], the '''Sardinas–Patterson algorithm''' is a classical algorithm for determining in [[polynomial time]] whether a given [[variable-length code]] is uniquely decodable, named after August Albert Sardinas and George W. Patterson, who published it in 1953.{{sfnp|Sardinas|Patterson|1953}} The algorithm carries out a systematic search for a string which admits two different decompositions into codewords. As [[Donald Knuth|Knuth]] reports, the algorithm was rediscovered about ten years later in 1963 by [[Robert Floyd|Floyd]], despite the fact that it was at the time already well known in coding theory.<ref>Knuth (2003), p. 2</ref>
| |
|
| |
|
| ==Idea of the algorithm==
| |
|
| |
|
| Consider the code <math>\{\, a \mapsto 1, b \mapsto 011, c\mapsto 01110, d\mapsto 1110, e\mapsto 10011\,\}</math>. This code, which is based on an example by Berstel,<ref>Berstel et al. (2009), Example 2.3.1 p. 63</ref> is an example of a code which is not uniquely decodable, since the string
| | The main advantage of using the blog is that anyone can use the Word - Press blog and customize the elements in the theme regardless to limited knowledge about internet and website development. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. One really cool features about this amazing and free wp plugin is that the code it generates is completely portable. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. Understanding how Word - Press works can be a challenge, but it is not too difficult when you learn more about it. <br><br>Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. If you have any concerns concerning wherever and how to use [http://zpib.com/wordpress_dropbox_backup_589562 wordpress dropbox backup], you can get hold of us at our own web-page. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. It sorts the results of a search according to category, tags and comments. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer. <br><br>Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. Enough automated blog posts plus a system keeps you and your clients happy. There are plenty of tables that are attached to this particular database. <br><br>The disadvantage is it requires a considerable amount of time to set every thing up. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. Thus it is difficult to outrank any one of these because of their different usages. If you choose a blog then people will be able to post articles on your site and people will be able to make comments on your posts (unless you turn comments off). If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>Under Settings —> Reading, determine if posts or a static page will be your home page, and if your home page is a static page, what page will contain blog posts. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. Word - Press is the most popular personal publishing platform which was launched in 2003. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account. |
| | |
| :011101110011
| |
| | |
| can be interpreted as the sequence of codewords
| |
| | |
| :01110 – 1110 – 011,
| |
| | |
| but also as the sequence of codewords
| |
| | |
| :011 – 1 – 011 – 10011.
| |
| | |
| Two possible decodings of this encoded string are thus given by ''cdb'' and ''babe''.
| |
| | |
| In general, a codeword can be found by the following idea: In the first round, we choose two codewords <math>x_1</math> and <math>y_1</math> such that <math>x_1</math> is a [[Prefix (computer science)|prefix]] of <math>y_1</math>, that is,
| |
| <math>x_1w = y_1</math> for some "dangling suffix" <math>w</math>. If one tries first <math>x_1=011</math> and <math>y_1=01110</math>, the dangling [[Suffix (computer science)|suffix]] is <math>w = 10</math>. If we manage to find two sequences <math>x_2,\ldots,x_p</math> and <math>y_2,\ldots,y_q</math> of codewords such that
| |
| <math>x_2\cdots x_p = wy_2\cdots y_q</math>, then we are finished: For then the string <math>x = x_1x_2\cdots x_p</math> can alternatively be decomposed as <math>y_1y_2\cdots y_q</math>, and we have found the desired string having at least two different decompositions into codewords. In the second round, we try out two different approaches: the first trial is to look for a codeword that has ''w'' as prefix. Then we obtain a new dangling suffix ''w''', with which we can continue our search. If we eventually encounter a dangling suffix that is itself a codeword (or the [[empty word]]), then the search will terminate, as we know there exists a string with two decompositions. The second trial is to seek for a codeword that is itself a prefix of ''w''. In our example, we have <math>w = 10</math>, and the sequence ''1'' is a codeword. We can thus also continue with ''w'=0'' as the new dangling suffix.
| |
| | |
| ==Precise description of the algorithm==
| |
| | |
| The algorithm is described most conveniently using quotients of [[formal language]]s. In general, for two sets of strings ''D'' and ''N'', the (left) quotient <math>N^{-1}D</math> is defined as the residual words obtained from ''D'' by removing some prefix in ''N''. Formally, <math>N^{-1}D = \{\,y \mid xy\in D ~\textrm{ and }~ x \in N \,\}</math>. Now let <math>C</math> denote the (finite) set of codewords in the given code.
| |
| | |
| The algorithm proceeds in rounds, where we maintain in each round not only one dangling suffix as described above, but the (finite) set of all potential dangling suffixes. Starting with round <math>i=1</math>, the set of potential dangling suffixes will be denoted by <math>S_i</math>. The sets <math>S_i</math> are defined [[recursive definition|inductively]] as follows:
| |
| | |
| <math>S_1 = C^{-1}C \setminus \{\varepsilon\}</math>. Here, the symbol <math>\varepsilon</math> denotes the [[empty word]].
| |
| | |
| <math>S_{i+1} = C^{-1}S_i\cup S_i^{-1}C</math>, for all <math>i\ge 1</math>.
| |
| | |
| The algorithm computes the sets <math>S_i</math> in increasing order of <math>i</math>. As soon as one of the <math>S_i</math> contains a word from ''C'' or the empty word, then the algorithm terminates and answers that the given code is not uniquely decodable. Otherwise, once a set <math>S_i</math>
| |
| equals a previously encountered set <math>S_j</math> with <math>j<i</math>, then the algorithm would enter in principle an endless loop. Instead of continuing endlessly, it answers that the given code is uniquely decodable.
| |
| | |
| ==Termination and correctness of the algorithm==
| |
| Since all sets <math>S_i</math> are sets of suffixes of a finite set of codewords, there are only finitely many different candidates for <math>S_i</math>. Since visiting one of the sets for the second time will cause the algorithm to stop, the algorithm cannot continue endlessly and thus must always [[terminate]]. More precisely, the total number of dangling suffixes that the algorithm considers is at most equal to the total of the lengths of the codewords in the input, so the algorithm runs in [[polynomial time]] as a function of this input length. By using a [[suffix tree]] to speed the comparison between each dangling suffix and the codewords, the time for the algorithm can be bounded by O(''nk''), where ''n'' is the total length of the codewords and ''k'' is the number of codewords.{{sfnp|Rodeh|1982}} The algorithm can also be implemented to run on a [[nondeterministic turing machine]] that uses only [[NL (complexity)|logarithmic space]]; the problem of testing unique decipherability is [[NL-complete]], so this space bound is optimal.<ref>{{harvtxt|Rytter|1986}} proves that the complementary problem, of testing for the existence of a string with two decodings, is NL-complete, and therefore that unique decipherability is co-NL-complete. The equivalence of NL-completeness and co-NL-completeness follows from the [[Immerman–Szelepcsényi theorem]].</ref>
| |
| | |
| A proof that the algorithm is [[correctness (computer science)|correct]], i.e. that it always gives the correct answer, is found in the textbooks by [[Arto Salomaa|Salomaa]]<ref>Salomaa (1981)</ref> and by Berstel et al.<ref>Berstel et al. (2009), Chapter 2.3</ref>
| |
| | |
| ==See also==
| |
| | |
| *[[Kraft's inequality]] in some cases provides a quick way to exclude the possibility that a given code is uniquely decodable.
| |
| *[[Prefix code]]s and [[block code]]s are important classes of codes which are uniquely decodable by definition.
| |
| *[[Timeline of information theory]]
| |
| | |
| ==Notes==
| |
| | |
| <references/>
| |
| | |
| ==References==
| |
| * {{cite book | last1=Berstel | first1=Jean | last2=Perrin | first2=Dominique | last3=Reutenauer | first3=Christophe | title=Codes and automata | series=Encyclopedia of Mathematics and its Applications | volume=129 | location=Cambridge | publisher=Cambridge University Press | year=2010 | isbn=978-0-521-88831-8 | url=http://www-igm.univ-mlv.fr/~berstel/LivreCodes/Codes.html | zbl=1187.94001 }}
| |
| * {{cite book | last1=Berstel | first1=Jean | last2=Reutenauer | first2=Christophe | title=Noncommutative rational series with applications | series=Encyclopedia of Mathematics and Its Applications | volume=137 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2011 | isbn=978-0-521-19022-0 | zbl=1250.68007 }}
| |
| * {{cite journal | first=Donald E. | last=Knuth | authorlink=Donald Knuth | title=Robert W Floyd, In Memoriam | journal=SIGACT News | volume=34 | number=4 | pages=3–13 | date=December 2003 }}
| |
| *{{cite journal | ref = harv
| |
| | last = Rodeh | first = M.
| |
| | doi = 10.1109/TIT.1982.1056535
| |
| | issue = 4
| |
| | journal = IEEE Transactions on Information Theory
| |
| | pages = 648–651
| |
| | title = A fast test for unique decipherability based on suffix trees (Corresp.)
| |
| | volume = 28
| |
| | year = 1982}}.
| |
| *{{cite journal | ref = harv
| |
| | last = Rytter | first = Wojciech
| |
| | doi = 10.1016/0020-0190(86)90121-3
| |
| | issue = 1
| |
| | journal = Information Processing Letters
| |
| | mr = 853618
| |
| | pages = 1–3
| |
| | title = The space complexity of the unique decipherability problem
| |
| | volume = 23
| |
| | year = 1986}}.
| |
| * {{cite book | first=Arto | last=Salomaa | authorlink=Arto Salomaa | title=Jewels of Formal Language Theory | publisher=Pitman Publishing | isbn=0-273-08522-0 | year=1981 | zbl=0487.68064 }}
| |
| *{{cite conference|first1=August Albert|last1=Sardinas|first2=George W.|last2=Patterson|contribution=A necessary and sufficient condition for the unique decomposition of coded messages|title=Convention Record of the [[Institute of Radio Engineers|I.R.E.]], 1953 National Convention, Part 8: Information Theory|pages=104–108|year=1953}}.
| |
| | |
| ;Further reading
| |
| | |
| * [[Robert G. Gallager]]: ''Information Theory and Reliable Communication.'' Wiley, 1968
| |
| | |
| {{DEFAULTSORT:Sardinas-Patterson algorithm}}
| |
| [[Category:Algorithms]]
| |
| [[Category:Coding theory]]
| |
| [[Category:Data compression]]
| |
The main advantage of using the blog is that anyone can use the Word - Press blog and customize the elements in the theme regardless to limited knowledge about internet and website development. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. One really cool features about this amazing and free wp plugin is that the code it generates is completely portable. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. Understanding how Word - Press works can be a challenge, but it is not too difficult when you learn more about it.
Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. If you have any concerns concerning wherever and how to use wordpress dropbox backup, you can get hold of us at our own web-page. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. It sorts the results of a search according to category, tags and comments. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer.
Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. Enough automated blog posts plus a system keeps you and your clients happy. There are plenty of tables that are attached to this particular database.
The disadvantage is it requires a considerable amount of time to set every thing up. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. Thus it is difficult to outrank any one of these because of their different usages. If you choose a blog then people will be able to post articles on your site and people will be able to make comments on your posts (unless you turn comments off). If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL.
Under Settings —> Reading, determine if posts or a static page will be your home page, and if your home page is a static page, what page will contain blog posts. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. Word - Press is the most popular personal publishing platform which was launched in 2003. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account.