Empty domain

Template:Distinguish In computer science, the longest common substring problem is to find the longest string (or strings) that is a substring (or are substrings) of two or more strings.

Example

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".

  ABABC
    |||
   BABCA
    |||
    ABCBA

Problem definition

Given two strings, $S$ of length $m$ and $T$ of length $n$ , find the longest strings which are substrings of both $S$ and $T$ .

A generalisation is the k-common substring problem. Given the set of strings $S=\{S_{1},...,S_{K}\}$ , where $|S_{i}|=n_{i}$ and $\Sigma n_{i}=N$ . Find for each $2\leq k\leq K$ , the longest strings which occur as substrings of at least $k$ strings.

Algorithms

One can find the lengths and starting positions of the longest common substrings of $S$ and $T$ in $\Theta (n+m)$ with the help of a generalised suffix tree. Finding them by dynamic programming costs $\Theta (nm)$ . The solutions to the generalised problem take $\Theta (n_{1}+...+n_{K})$ space and $\Theta (n_{1}$ ·...· $n_{K})$ time with dynamic programming and take $\Theta (N*K)$ time with generalized suffix tree.

Suffix tree

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.

Building the suffix tree takes $\Theta (N)$ 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 $\Theta (NK)$ time. If the suffix tree is prepared for constant time lowest common ancestor retrieval, it can be solved in $\Theta (N)$ time.^[1]

Dynamic programming

First find the longest common suffix for all pairs of prefixes of the strings. The longest common suffix is

${\mathit {LCSuff}}(S_{1..p},T_{1..q})={\begin{cases}{\mathit {LCSuff}}(S_{1..p-1},T_{1..q-1})+1&\mathrm {if} \;S[p]=T[q]\\0&\mathrm {otherwise} .\end{cases}}$

For the example strings "ABAB" and "BABA":

	A	B	A	B
	0	0	0	0
B	0	1	0	1
A	1	0	2	0
B	0	2	0	3
A	1	0	3	0

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".

{\mathit {LCSubstr}}(S,T)=\max _{1\leq i\leq m,1\leq j\leq n}{\mathit {LCSuff}}(S_{1..i},T_{1..j})\;

This can be extended to more than two strings by adding more dimensions to the table.

Pseudocode

DTZ's auction group in Singapore auctions all types of residential, workplace and retail properties, retailers, homes, accommodations, boarding houses, industrial buildings and development websites. Auctions are at the moment held as soon as a month.

Whitehaven @ Pasir Panjang – A boutique improvement nicely nestled peacefully in serene Pasir Panjang personal estate presenting a hundred and twenty rare freehold private apartments tastefully designed by the famend Ong & Ong Architect. Only a short drive away from Science Park and NUS Campus, Jade Residences, a recent Freehold condominium which offers high quality lifestyle with wonderful facilities and conveniences proper at its door steps. Its fashionable linear architectural fashion promotes peace and tranquility living nestled within the D19 personal housing enclave. Rising workplace sector leads real estate market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline International Market Perspectives - 1st Quarter 2014

There are a lot of websites out there stating to be one of the best seek for propertycondominiumhouse, and likewise some ways to discover a low cost propertycondominiumhouse. Owning a propertycondominiumhouse in Singapore is the dream of virtually all individuals in Singapore, It is likely one of the large choice we make in a lifetime. Even if you happen to're new to Property listing singapore funding, we are right here that will help you in making the best resolution to purchase a propertycondominiumhouse at the least expensive value.

Jun 18 ROCHESTER in MIXED USE IMPROVEMENT $1338000 / 1br - 861ft² - (THE ROCHESTER CLOSE TO NORTH BUONA VISTA RD) pic real property - by broker Jun 18 MIXED USE IMPROVEMENT @ ROCHESTER @ ROCHESTER PK $1880000 / 1br - 1281ft² - (ROCHESTER CLOSE TO NORTH BUONA VISTA) pic real estate - by broker Tue 17 Jun Jun 17 Sunny Artwork Deco Gem Near Seashore-Super Deal!!! $103600 / 2br - 980ft² - (Ventnor) pic actual estate - by owner Jun 17 Freehold semi-d for rent (Jalan Rebana ) $7000000 / 5909ft² - (Jalan Rebana ) actual property - by dealer Jun sixteen Ascent @ 456 in D12 (456 Balestier Highway,Singapore) pic real property - by proprietor Jun 16 RETAIL SHOP AT SIM LIM SQUARE FOR SALE, IT MALL, ROCHOR, BUGIS MRT $2000000 / 506ft² - (ROCHOR, BUGIS MRT) pic real estate - by dealer HDB Scheme Any DBSS BTO

In case you are eligible to purchase landed houses (open solely to Singapore residents) it is without doubt one of the best property investment choices. Landed housing varieties solely a small fraction of available residential property in Singapore, due to shortage of land right here. In the long term it should hold its worth and appreciate as the supply is small. In truth, landed housing costs have risen the most, having doubled within the last eight years or so. However he got here back the following day with two suitcases full of money. Typically we've got to clarify to such folks that there are rules and paperwork in Singapore and you can't just buy a home like that,' she said. For conveyancing matters there shall be a recommendedLondon Regulation agency familiar with Singapore London propertyinvestors to symbolize you

Sales transaction volumes have been expected to hit four,000 units for 2012, close to the mixed EC gross sales volume in 2010 and 2011, in accordance with Savills Singapore. Nevertheless the last quarter was weak. In Q4 2012, sales transactions were 22.8% down q-q to 7,931 units, in line with the URA. The quarterly sales discount was felt throughout the board. When the sale just starts, I am not in a hurry to buy. It's completely different from a private sale open for privileged clients for one day solely. Orchard / Holland (D09-10) House For Sale The Tembusu is a singular large freehold land outdoors the central area. Designed by multiple award-profitable architects Arc Studio Architecture + Urbanism, the event is targeted for launch in mid 2013. Post your Property Condos Close to MRT

The following pseudocode finds the set of longest common substrings between two strings with dynamic programming:

function LCSubstr(S[1..m], T[1..n])
    L := array(1..m, 1..n)
    z := 0
    ret := {}
    for i := 1..m
        for j := 1..n
            if S[i] == T[j]
                if i == 1 or j == 1
                    L[i,j] := 1
                else
                    L[i,j] := L[i-1,j-1] + 1
                if L[i,j] > z
                    z := L[i,j]
                    ret := {S[i-z+1..i]}
                elif L[i,j] == z
                    ret := ret ∪ {S[i-z+1..i]}
            else L[i,j]=0;
    return ret

This algorithm runs in $O(nm)$ time. The variable z is used to hold the length of the longest common substring found so far. The set ret is used to hold the set of strings which are of length z. The set ret can be saved efficiently by just storing the index i, which is the last character of the longest common substring (of size z) instead of S[i-z+1..z]. Thus all the longest common substrings would be, for each i in ret, S[(ret[i]-z)..(ret[i])].

The following tricks can be used to reduce the memory usage of an implementation:

Keep only the last and current row of the DP table to save memory ( $O(\min(m,n))$ instead of $O(nm)$ )
Store only non-zero values in the rows. This can be done using hash tables instead of arrays. This is useful for large alphabets.

References

↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

[Gus97-1] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[1]

Empty domain

Contents

Example

Problem definition

Algorithms

Suffix tree

Dynamic programming

Pseudocode

See also

References

External links

Navigation menu

Empty domain

Example

Problem definition

Algorithms

Suffix tree

Dynamic programming

Pseudocode

See also

References

External links

Navigation menu

Search