25
votes
How does editing software (like Microsoft word or Gmail) pick the 2nd string to compare in Levenshtein distance?
Yes, the entire dictionary is compared against each word. This can be fast by using a trie and an algorithm similar to Levenshtein's.
I have built my own spelling corrector that checks words against a ...
- 451
19
votes
How does editing software (like Microsoft word or Gmail) pick the 2nd string to compare in Levenshtein distance?
Companies with search engines (e.g. Microsoft or Google) don't always directly search for the string with the smallest Levenshtein distance. They have a huge database of search queries, from which ...
- 21.7k
16
votes
Accepted
Longest common substring in linear time
Let $m$ and $n$ be the lengths of two given strings,
Linear time assuming the size of the alphabet is constant.
Yes, the longest common substring of two given strings can be found in $O(m+n)$ time, ...
- 38.6k
12
votes
How to check if two strings are permutations of each other using O(1) additional space?
Denote the arrays by $A,B$, and suppose they are of length $n$.
Suppose first that the values in each array are distinct. Here is an algorithm that uses $O(1)$ space:
Compute the minimum values of ...
- 275k
11
votes
Accepted
Assigning a unique representation to equivalent circular queues
Enumerate all possible rotations of the queue. Take the lexicographically first of them. Use this as your representative. If you want a short index into a hash table, take the hash of that. Then ...
D.W.♦
- 156k
10
votes
Accepted
From Guido's essays, how does this function avoid quadratic behavior in a string concatenation algorithm?
Let's assume that adding two strings of lengths $a,b$ takes time $a+b$. Consider the following strategy to convert a list of $n$ characters into a list:
Read the list in chunks of $k$, convert them ...
- 275k
10
votes
Accepted
Count number of non-contiguous occurrences in string
A dynamic programming algorithm in $\mathcal{O}(|S| |T|)$ should do the trick.
Let's denote $S = s_1…s_m$ and $T = t_1…t_n$. For $0\leqslant i \leqslant m$, $0\leqslant j \leqslant n$, let $N(i, j)$ ...
- 12.5k
10
votes
Accepted
Can we prove that any set of finite strings without substrings is finite?
Assuming that by substring you mean consecutive substring, take
$$ L= \{10^n1 : n \ge 0\}. $$
On the other hand, if you refer to nonconsecutive substrings, or subsequence ordering, then Higman's Lemma ...
- 275k
8
votes
Accepted
Find all substrings that fit the mask with asterisks
Yes, there is a more efficient algorithm. Your algorithm can take exponential time.
You can check whether there exists any match in $O(nm)$ time, where $n$ is the length of text and $m$ is the ...
D.W.♦
- 156k
8
votes
How does editing software (like Microsoft word or Gmail) pick the 2nd string to compare in Levenshtein distance?
I just tried and found that the spelling checker on my phone finds a perfectly fine replacement for “gekki wirkd”. Look at your keyboard, and it is obvious.
A good spelling checker does much better ...
- 28.4k
8
votes
Accepted
What is the name of the following binary encoding?
Your encoding is not self-terminating, which makes it somewhat less useful than encodings such as universal codes.
Given an integer $n \geq 0$, write $n+2$ in binary without leading zeroes, and remove ...
- 275k
7
votes
Have I invented a new data structure?
I've never seen this data structure before. However, it doesn't seem like a good choice for storing a set of words, for most purposes. I see three significant disadvantages:
Performance. Looking up ...
D.W.♦
- 156k
7
votes
Accepted
How to check if two strings are permutations of each other using O(1) additional space?
The naive approach would be building histograms of both strings and checking whether they are the same. Since we are not allowed to store such a data structure (whose size would be linear to the size ...
- 607
7
votes
Accepted
Finding the smallest string that contains a given set of substrings
This problem is called shortest superstring problem. John Gallant, David Maier and James Astorer proved it is NP-hard in 19791.
Given two strings $A$ and $B$, let $|A|$ denote the length of $A$, and ...
- 7,405
7
votes
Find all n bit numbers with k ones and unique under circular shift
Binary strings considered up to rotation are known as necklaces. You are interested in enumerating binary necklaces with given density. You can find one solution in Wang and Savage, A Gray Code for ...
- 275k
6
votes
Accepted
Find all pairs of strings in a set with Levenshtein distance < d
There's a "trick" you can use that might potentially speed up your algorithm a little: shingling. No guarantees that it'll necessarily help in your particular case, though.
Lemma. If the edit ...
D.W.♦
- 156k
6
votes
How to speed up process of finding duplicates/similar items in a large amount of strings?
Edit distance is definitely not the way to proceed. The standard approach toward near-identical document deduplication is to compute hashes of shingles. Here's one way to do it:
Compute a set of k-...
- 448
6
votes
Is there a data structure for efficiently searching a string that contains a given substring?
You could append all of the $n$ strings together, and add an arbitrary character '\$' not in the pattern to separate them. Then you could apply the Z algorithm on your original pattern and this new ...
- 280
6
votes
Accepted
What is an efficient data structure for prefix matching?
A trie is asymptotically optimal for this. No data structure can achieve better asymptotic running time.
If you care about constant factors, the only way to know what will be optimal is to try ...
D.W.♦
- 156k
6
votes
Accepted
How many "compressible" strings are there?
A simple counting argument shows that the number of strings of length $N$ such that $K(S) \leq M$ is at most $2^{M+1}$.
Conversely, considering the program $\Pi$ that gets an integer $r$ and a string ...
- 275k
5
votes
Accepted
Longest Repeated (Scattered) Subsequence in a String
The special case of $k = n/2$ is the same problem as this CST.SE question How hard is unshuffling a string? asks.
Buss and Soltys proved NP-completeness of this problem [1] by reducing 3-Partition ...
- 2,205
5
votes
Complexity of a naive algorithm for finding the longest Fibonacci substring
Say that $F(n)$ occurs at some position if the substring starting at that position is compatible with either $F(n)$ or its complementation. How close can occurrences of $F(n)$ be? Take as an example $...
- 275k
5
votes
Accepted
What do you call a function from symbols of alphabet to languages?
It looks like it refers to a substitution, but that the given definition is incomplete.
Anyway, in formal language theory, people often use monoid morphisms between free monoids. To avoid overloading ...
- 6,014
5
votes
Accepted
Space complexity of string indices: O(1) or O(log|S|)?
These are correct (unless you explicitly specify a non-standard model of computing):
$O(1)$ space,
$O(1)$ words of space,
$O(\log|S|)$ bits of space.
- 1,996
5
votes
String matching algorithm - check if a string matches a pattern
There are various algorithms for pattern matching in string.
Exact String matching algorithms
Brute Force
Rabin Karp
Boyer-Moree
KMP
Aho Corasick etc.
Approximate String Matching Algorithm
...
- 372
5
votes
Accepted
Counting substrings with a given number of different characters in O(N)
You can solve this in $O(n)$ time using two (well, three) pointers that both move leftward.
Let $S$ be the string. We'll let $i$ range from $n$ down to $1$, and for each value of $i$, we're going to ...
D.W.♦
- 156k
5
votes
Accepted
Complexity of Block edit distance with Swapping only
When one operation is exactly "removing a block and inserting it between two other positions", the problem of computing the string distance is known as Transposition Distance. It is NP-hard even if ...
5
votes
Accepted
Algorithm: given very large file of strings, find lines containing substring
You can construct a suffix tree for that very large file and once generated the same can be used for querying.
For Suffix tree generation use Suffix Array approach, there are many algorithm to ...
- 119
5
votes
I'm looking for an algorithm to find unknown patterns in a string
Construct the suffix tree of your string, which takes time linear in the length of the string (assuming a finite alphabet). Every inner node represents a repeat, their respective descendant leaves ...
- 72k
5
votes
Accepted
Maximum number of different substrings in big string
Here is a general solution for an alphabet of size $d \geq 3$ and a string of length $n$.
Every string of length $n$ has $n-\ell+1$ substrings of length $\ell$. Hence the number of different ...
- 275k
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