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| {{Distinguish|longest common subsequence problem}}
| | In any given year, about seven percent - between 13 million and 14 million people - will experience a depressive disorder,. This feeling of fear can turn out to be severe and get in the way of accomplishing work, school, and other everyday activities. Don't fill your body up with refined sugars and unhealthy food, a diet that is balanced and nutritional is what you require. After all, we learned in school anything that changes the way the mind or body works is a drug. As a result, treating the anxiety with medication or therapy may alleviate the insomnia as well. <br><br>Although this disorder reacts to everyone differently, it can be very frustrating to live your life with it. Neurotransmitters, such as norepinephrine, dopamine, and serotonin are thought to be the culprits, and when released in the brain can cause of a variety of mental illnesses. re probably already familiar with some of the more common strategies. Amazing minds, and incredibly fascinating and useful information. Another way to find out if your dog has separation anxiety is by asking your neighbor is he or she heard the dog barking excessively. <br><br>both stress and anxiety during a punching or kicking exercise. There is now a treatment available that can help to cure anxiety. Due to this type of noise, technicians will give you MRI ear plugs and also offer choice of music. If this has been happening to you lately, you are likely having panic attacks. It is very important to have someone to talk to especially when there is something bothering on your mind to somehow ease the burden's that is inside you. <br><br>- Mental 'muddiness' (inability to focus or concentrate). They have faith on the actions taken and believe that such actions will work out in the future, thus not focusing much on an uncontrollable event that might happen in future. Whatever the case, a little small talk prior to the interview can go a long way toward establishing who will be willing to fight for you in the selection process. Keeping up with reading is one of the best things any parent can do to keep their child's mind fresh and ready to learn. Lastly, people will demand for the root causes of all these anxiety conditions. <br><br>What should be important to you is not the type of anxiety you suffer from but how soon it can be cured. We have reviewed e - Books, video programs, audio programs, music relaxation and natural herbal remedies and have found none come close to the same effictiveness of Panic Away. Encourage your teen to approach the counter and order alone. Before you hit the sheets at night, you should think about all of the great stuff that is occurring in your life right now. Just like humans when we were little and sought the comfort and shelter of our mother or father during a storm or anxious moment.<br><br>If you have any concerns relating to where and ways to use [http://www.stress-lavoro-correlato.info/sitemap/ treatments for anxiety], you could contact us at the web site. |
| In [[computer science]], the '''longest common substring problem''' is to find the longest [[string_(computer_science)|string]] (or strings) that is a [[substring]] (or are substrings) of two or more strings. | |
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| ==Example==
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| The longest common substring of the strings "ABABC", "BABCA" and "ABCBA" is string "ABC" of length 3. Other common substrings are "AB", "BC" and "BA".
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| ABABC
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| BABCA
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| ABCBA
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| ==Problem definition==
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| Given two strings, <math>S</math> of length <math>m</math> and <math>T</math> of length <math>n</math>, find the longest strings which are substrings of both <math>S</math> and <math>T</math>.
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| A generalisation is the '''k-common substring problem'''. Given the set of strings <math>S = \{S_1, ..., S_K\}</math>, where <math>|S_i|=n_i</math> and <math>\Sigma n_i = N</math>. Find for each <math>2 \leq k \leq K</math>, the longest strings which occur as substrings of at least <math>k</math> strings.
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| ==Algorithms==
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| One can find the lengths and starting positions of the longest common substrings of <math>S</math> and <math>T</math> in <math>\Theta(n+m)</math> with the help of a [[generalised suffix tree]]. Finding them by [[dynamic programming]] costs <math>\Theta(nm)</math>. The solutions to the generalised problem take <math>\Theta(n_1 + ... + n_K)</math> space and <math>\Theta(n_1</math>·...·<math>n_K)</math> time with [[dynamic programming]] and take <math>\Theta(N * K)</math> time with [[generalized suffix tree]].
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| ===Suffix tree===
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| [[Image:Suffix tree ABAB BABA ABBA.svg|thumb|400px|right|[[Generalised suffix tree]] for the strings "ABAB", "BABA" and "ABBA", numbered 0, 1 and 2.]]
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| The longest common substrings of a set of strings can be found by building a [[generalised suffix tree]] for the strings, and then finding the deepest internal nodes which have leaf nodes from all the strings in the subtree below it. The figure on the right is the suffix tree for the strings "ABAB", "BABA" and "ABBA", padded with unique string terminators, to become "ABAB$0", "BABA$1" and "ABBA$2". The nodes representing "A", "B", "AB" and "BA" all have descendant leaves from all of the strings, numbered 0, 1 and 2.
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| Building the suffix tree takes <math>\Theta(N)</math> time (if the size of the alphabet is constant). If the tree is traversed from the bottom up with a bit vector telling which strings are seen below each node, the k-common substring problem can be solved in <math>\Theta(NK)</math> time. If the suffix tree is prepared for constant time [[lowest common ancestor]] retrieval, it can be solved in <math>\Theta(N)</math> time.<ref name="Gus97">{{cite book | last = Gusfield | first = Dan | origyear = 1997 | year = 1999 | title = Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology | publisher = Cambridge University Press | location = USA | isbn = 0-521-58519-8}}</ref>
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| ===[[Dynamic programming]]===
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| First find the longest common [[substring#Suffix|suffix]] for all pairs of [[substring#Prefix|prefixes]] of the strings. The longest common suffix is
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| <math> | |
| \mathit{LCSuff}(S_{1..p}, T_{1..q}) =
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| \begin{cases}
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| \mathit{LCSuff}(S_{1..p-1}, T_{1..q-1}) + 1 & \mathrm{if } \; S[p] = T[q] \\
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| 0 & \mathrm{otherwise}.
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| \end{cases}
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| </math> | |
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| For the example strings "ABAB" and "BABA":
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| {| class="wikitable" style="text-align:center"
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| |-
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| ! width="15" | !! width="15" | !! width="15" |A !! width="15" |B !! width="15" |A !! width="15" |B
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| |-
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| | || 0 || 0 || 0 || 0 || 0
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| |-
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| ! B
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| | 0 || 0 || style="color:red" |1 || 0 || 1
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| |-
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| ! A
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| | 0 || style="color:red" |1 || 0 || style="color:red" |2 || 0
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| |-
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| ! B
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| | 0 || 0 || style="color:red" |2 || 0 || style="color:red" |3
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| |-
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| ! A
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| | 0 || 1 || 0 || style="color:red" | 3 || 0
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| |}
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| The maximal of these longest common suffixes of possible prefixes must be the longest common substrings of ''S'' and ''T''. These are shown on diagonals, in red, in the table. For this example, the longest common substrings are "BAB" and "ABA".
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| :<math>
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| \mathit{LCSubstr}(S, T) = \max_{1 \leq i \leq m, 1 \leq j \leq n} \mathit{LCSuff}(S_{1..i}, T_{1..j}) \;
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| </math>
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| This can be extended to more than two strings by adding more dimensions to the table.
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| ==Pseudocode==
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| {{Wikibooks|Algorithm implementation|Strings/Longest common substring|Longest common substring}}
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| The following pseudocode finds the set of longest common substrings between two strings with dynamic programming:
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| '''function''' LCSubstr(S[1..m], T[1..n])
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| L := '''array'''(1..m, 1..n)
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| z := 0
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| ret := {}
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| '''for''' i := 1..m
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| '''for''' j := 1..n
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| '''if''' S[i] == T[j]
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| '''if''' i == 1 or j == 1
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| L[i,j] := 1
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| '''else'''
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| L[i,j] := L[i-1,j-1] + 1
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| '''if''' L[i,j] > z
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| z := L[i,j]
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| ret := {S[i-z+1..i]}
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| '''elif''' L[i,j] == z
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| ret := ret ∪ {S[i-z+1..i]}
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| '''else''' L[i,j]=0;
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| '''return''' ret
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| This algorithm runs in <math>O(n m)</math> time. The variable <code>z</code> is used to hold the length of the longest common substring found so far. The set <code>ret</code> is used to hold the set of strings which are of length <code>z</code>. The set <code>ret</code> can be saved efficiently by just storing the index <code>i</code>, which is the last character of the longest common substring (of size z) instead of <code>S[i-z+1..z]</code>. Thus all the longest common substrings would be, for each i in <code>ret</code>, <code>S[(ret[i]-z)..(ret[i])]</code>.
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| The following tricks can be used to reduce the memory usage of an implementation:
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| * Keep only the last and current row of the DP table to save memory (<math>O(\min(m, n))</math> instead of <math>O(n m)</math>)
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| * Store only non-zero values in the rows. This can be done using hash tables instead of arrays. This is useful for large alphabets.
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| ==See also==
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| *[[Longest palindromic substring]]
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| ==References==
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| <references/> | |
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| ==External links==
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| *[http://nist.gov/dads/HTML/longestCommonSubstring.html Dictionary of Algorithms and Data Structures: longest common substring]
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| *[http://metacpan.org/module/String::LCSS_XS Perl/XS implementation of the dynamic programming algorithm]
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| *[http://metacpan.org/module/Tree::Suffix Perl/XS implementation of the suffix tree algorithm]
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| *[http://en.wikibooks.org/wiki/Algorithm_implementation/Strings/Longest_common_substring Dynamic programming implementations in various languages on wikibooks]
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| *[http://www.emanueleferonato.com/2010/12/01/solving-the-longest-common-substring-problem-with-as3/ working AS3 implementation of the dynamic programming algorithm]
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| [[Category:Problems on strings]]
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| [[Category:Dynamic programming]]
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In any given year, about seven percent - between 13 million and 14 million people - will experience a depressive disorder,. This feeling of fear can turn out to be severe and get in the way of accomplishing work, school, and other everyday activities. Don't fill your body up with refined sugars and unhealthy food, a diet that is balanced and nutritional is what you require. After all, we learned in school anything that changes the way the mind or body works is a drug. As a result, treating the anxiety with medication or therapy may alleviate the insomnia as well.
Although this disorder reacts to everyone differently, it can be very frustrating to live your life with it. Neurotransmitters, such as norepinephrine, dopamine, and serotonin are thought to be the culprits, and when released in the brain can cause of a variety of mental illnesses. re probably already familiar with some of the more common strategies. Amazing minds, and incredibly fascinating and useful information. Another way to find out if your dog has separation anxiety is by asking your neighbor is he or she heard the dog barking excessively.
both stress and anxiety during a punching or kicking exercise. There is now a treatment available that can help to cure anxiety. Due to this type of noise, technicians will give you MRI ear plugs and also offer choice of music. If this has been happening to you lately, you are likely having panic attacks. It is very important to have someone to talk to especially when there is something bothering on your mind to somehow ease the burden's that is inside you.
- Mental 'muddiness' (inability to focus or concentrate). They have faith on the actions taken and believe that such actions will work out in the future, thus not focusing much on an uncontrollable event that might happen in future. Whatever the case, a little small talk prior to the interview can go a long way toward establishing who will be willing to fight for you in the selection process. Keeping up with reading is one of the best things any parent can do to keep their child's mind fresh and ready to learn. Lastly, people will demand for the root causes of all these anxiety conditions.
What should be important to you is not the type of anxiety you suffer from but how soon it can be cured. We have reviewed e - Books, video programs, audio programs, music relaxation and natural herbal remedies and have found none come close to the same effictiveness of Panic Away. Encourage your teen to approach the counter and order alone. Before you hit the sheets at night, you should think about all of the great stuff that is occurring in your life right now. Just like humans when we were little and sought the comfort and shelter of our mother or father during a storm or anxious moment.
If you have any concerns relating to where and ways to use treatments for anxiety, you could contact us at the web site.