531 research outputs found
Expected length of the longest common subsequence for large alphabets
We consider the length L of the longest common subsequence of two randomly
uniformly and independently chosen n character words over a k-ary alphabet.
Subadditivity arguments yield that the expected value of L, when normalized by
n, converges to a constant C_k. We prove a conjecture of Sankoff and Mainville
from the early 80's claiming that C_k\sqrt{k} goes to 2 as k goes to infinity.Comment: 14 pages, 1 figure, LaTe
On a Speculated Relation Between Chv\'atal-Sankoff Constants of Several Sequences
It is well known that, when normalized by n, the expected length of a longest
common subsequence of d sequences of length n over an alphabet of size sigma
converges to a constant gamma_{sigma,d}. We disprove a speculation by Steele
regarding a possible relation between gamma_{2,d} and gamma_{2,2}. In order to
do that we also obtain new lower bounds for gamma_{sigma,d}, when both sigma
and d are small integers.Comment: 13 pages. To appear in Combinatorics, Probability and Computin
Multivariate Fine-Grained Complexity of Longest Common Subsequence
We revisit the classic combinatorial pattern matching problem of finding a
longest common subsequence (LCS). For strings and of length , a
textbook algorithm solves LCS in time , but although much effort has
been spent, no -time algorithm is known. Recent work
indeed shows that such an algorithm would refute the Strong Exponential Time
Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann,
K\"unnemann FOCS'15].
Despite the quadratic-time barrier, for over 40 years an enduring scientific
interest continued to produce fast algorithms for LCS and its variations.
Particular attention was put into identifying and exploiting input parameters
that yield strongly subquadratic time algorithms for special cases of interest,
e.g., differential file comparison. This line of research was successfully
pursued until 1990, at which time significant improvements came to a halt. In
this paper, using the lens of fine-grained complexity, our goal is to (1)
justify the lack of further improvements and (2) determine whether some special
cases of LCS admit faster algorithms than currently known.
To this end, we provide a systematic study of the multivariate complexity of
LCS, taking into account all parameters previously discussed in the literature:
the input size , the length of the shorter string
, the length of an LCS of and , the numbers of
deletions and , the alphabet size, as well as
the numbers of matching pairs and dominant pairs . For any class of
instances defined by fixing each parameter individually to a polynomial in
terms of the input size, we prove a SETH-based lower bound matching one of
three known algorithms. Specifically, we determine the optimal running time for
LCS under SETH as .
[...]Comment: Presented at SODA'18. Full Version. 66 page
Near-Linear Time Insertion-Deletion Codes and (1+)-Approximating Edit Distance via Indexing
We introduce fast-decodable indexing schemes for edit distance which can be
used to speed up edit distance computations to near-linear time if one of the
strings is indexed by an indexing string . In particular, for every length
and every , one can in near linear time construct a string
with , such that, indexing
any string , symbol-by-symbol, with results in a string where for which edit
distance computations are easy, i.e., one can compute a
-approximation of the edit distance between and any other
string in time.
Our indexing schemes can be used to improve the decoding complexity of
state-of-the-art error correcting codes for insertions and deletions. In
particular, they lead to near-linear time decoding algorithms for the
insertion-deletion codes of [Haeupler, Shahrasbi; STOC `17] and faster decoding
algorithms for list-decodable insertion-deletion codes of [Haeupler, Shahrasbi,
Sudan; ICALP `18]. Interestingly, the latter codes are a crucial ingredient in
the construction of fast-decodable indexing schemes
A Central Limit Theorem for the Length of the Longest Common Subsequences in Random Words
Let and be two independent sequences of
independent identically distributed random variables taking their values in a
common finite alphabet and having the same law. Let be the length of the
longest common subsequences of the two random words and
. Under a lower bound assumption on the order of its variance,
is shown to satisfy a central limit theorem. This is in contrast to the
limiting distribution of the length of the longest common subsequences in two
independent uniform random permutations of , which is shown to
be the Tracy-Widom distribution.Comment: Some corrections, typos corrected and improvement
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