103 research outputs found
Why is it hard to beat for Longest Common Weakly Increasing Subsequence?
The Longest Common Weakly Increasing Subsequence problem (LCWIS) is a variant
of the classic Longest Common Subsequence problem (LCS). Both problems can be
solved with simple quadratic time algorithms. A recent line of research led to
a number of matching conditional lower bounds for LCS and other related
problems. However, the status of LCWIS remained open.
In this paper we show that LCWIS cannot be solved in strongly subquadratic
time unless the Strong Exponential Time Hypothesis (SETH) is false.
The ideas which we developed can also be used to obtain a lower bound based
on a safer assumption of NC-SETH, i.e. a version of SETH which talks about NC
circuits instead of less expressive CNF formulas
A Simple Algorithm for Approximating the Text-To-Pattern Hamming Distance
The algorithmic task of computing the Hamming distance between a given pattern of length m and each location in a text of length n, both over a general alphabet Sigma, is one of the most fundamental algorithmic tasks in string algorithms. The fastest known runtime for exact computation is tilde O(nsqrt m). We recently introduced a complicated randomized algorithm for obtaining a (1 +/- eps) approximation for each location in the text in O( (n/eps) log(1/eps) log n log m log |Sigma|) total time, breaking a barrier that stood for 22 years. In this paper, we introduce an elementary and simple randomized algorithm that takes O((n/eps) log n log m) time
Coloring Graphs having Few Colorings over Path Decompositions
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no
time algorithm for
deciding if an -vertex graph with pathwidth
admits a proper vertex coloring with colors unless the Strong Exponential
Time Hypothesis (SETH) is false. We show here that nevertheless, when
, where is the maximum degree in the
graph , there is a better algorithm, at least when there are few colorings.
We present a Monte Carlo algorithm that given a graph along with a path
decomposition of with pathwidth runs in time, that
distinguishes between -colorable graphs having at most proper
-colorings and non--colorable graphs. We also show how to obtain a
-coloring in the same asymptotic running time. Our algorithm avoids
violating SETH for one since high degree vertices still cost too much and the
mentioned hardness construction uses a lot of them.
We exploit a new variation of the famous Alon--Tarsi theorem that has an
algorithmic advantage over the original form. The original theorem shows a
graph has an orientation with outdegree less than at every vertex, with a
different number of odd and even Eulerian subgraphs only if the graph is
-colorable, but there is no known way of efficiently finding such an
orientation. Our new form shows that if we instead count another difference of
even and odd subgraphs meeting modular degree constraints at every vertex
picked uniformly at random, we have a fair chance of getting a non-zero value
if the graph has few -colorings. Yet every non--colorable graph gives a
zero difference, so a random set of constraints stands a good chance of being
useful for separating the two cases.Comment: Strengthened result from uniquely -colorable graphs to graphs with
few -colorings. Also improved running tim
Edit Distance Cannot Be Computed in Strongly Subquadratic Time (Unless SETH is False)
© 2018 Society for Industrial and Applied Mathematics. The edit distance (a.k.a. the Levenshtein distance) between two strings is defined as the minimum number of insertions, deletions, or substitutions of symbols needed to transform one string into another. The problem of computing the edit distance between two strings is a classical computational task, with a well-known algorithm based on dynamic programming. Unfortunately, all known algorithms for this problem run in nearly quadratic time. In this paper we provide evidence that the near-quadratic running time bounds known for the problem of computing edit distance might be tight. Specifically, we show that if the edit distance can be computed in time O(n2−δ) for some constant δ > 0, then the satisfiability of conjunctive normal form formulas with N variables and M clauses can be solved in time MO(1)2(1−)N for a constant > 0. The latter result would violate the strong exponential time hypothesis, which postulates that such algorithms do not exist
Edit Distance Cannot Be Computed in Strongly Subquadratic Time (unless SETH is false)
The edit distance (a.k.a. the Levenshtein distance) between two strings is defined as the minimum number of insertions, deletions or substitutions of symbols needed to transform one string into another. The problem of computing the edit distance between two strings is a classical computational task, with a well-known algorithm based on dynamic programming. Unfortunately, all known algorithms for this problem run in nearly quadratic time.
In this paper we provide evidence that the near-quadratic running time bounds known for the problem of computing edit distance might be {tight}. Specifically, we show that, if the edit distance can be computed in time O(n[superscript 2-δ]) for some constant δ>0, then the satisfiability of conjunctive normal form formulas with N variables and M clauses can be solved in time M[superscript O(1)] 2[superscript (1-ε)N] for a constant ε>0. The latter result would violate the Strong Exponential Time Hypothesis, which postulates that such algorithms do not existNational Science Foundation (U.S.)IBM (PhD Felllowship)Center for Massive Data Algorithmics (MADALGO)Simons Foundation (Investigator Award
Coloring Graphs Having Few Colorings Over Path Decompositions
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them.
We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases
On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
Conditional lower bounds for dynamic graph problems has received a great deal
of attention in recent years. While many results are now known for the
fully-dynamic case and such bounds often imply worst-case bounds for the
partially dynamic setting, it seems much more difficult to prove amortized
bounds for incremental and decremental algorithms. In this paper we consider
partially dynamic versions of three classic problems in graph theory. Based on
popular conjectures we show that:
-- No algorithm with amortized update time exists for
incremental or decremental maximum cardinality bipartite matching. This
significantly improves on the bound for sparse graphs
of Henzinger et al. [STOC'15] and bound of Kopelowitz,
Pettie and Porat. Our linear bound also appears more natural. In addition, the
result we present separates the node-addition model from the edge insertion
model, as an algorithm with total update time exists for the
former by Bosek et al. [FOCS'14].
-- No algorithm with amortized update time exists for
incremental or decremental maximum flow in directed and weighted sparse graphs.
No such lower bound was known for partially dynamic maximum flow previously.
Furthermore no algorithm with amortized update time
exists for directed and unweighted graphs or undirected and weighted graphs.
-- No algorithm with amortized update time exists
for incremental or decremental -approximating the diameter
of an unweighted graph. We also show a slightly stronger bound if node
additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit
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