Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe streaming $k$-mismatch problem
118
0
0.0
(
0
)
Added by
Raphael Clifford
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We consider the streaming complexity of a fundamental task in approximate pattern matching: the $k$-mismatch problem. It asks to compute Hamming distances between a pattern of length $n$ and all length-$n$ substrings of a text for which the Hamming distance does not exceed a given threshold $k$. In our problem formulation, we report not only the Hamming distance but also, on demand, the full emph{mismatch information}, that is the list of mismatched pairs of symbols and their indices. The twin challenges of streaming pattern matching derive from the need both to achieve small working space and also to guarantee that every arriving input symbol is processed quickly. We present a streaming algorithm for the $k$-mismatch problem which uses $O(klog{n}logfrac{n}{k})$ bits of space and spends ourcomplexity time on each symbol of the input stream, which consists of the pattern followed by the text. The running time almost matches the classic offline solution and the space usage is within a logarithmic factor of optimal. Our new algorithm therefore effectively resolves and also extends an open problem first posed in FOCS09. En route to this solution, we also give a deterministic $O( k (log frac{n}{k} + log |Sigma|) )$-bit encoding of all the alignments with Hamming distance at most $k$ of a length-$n$ pattern within a text of length $O(n)$. This secondary result provides an optimal solution to a natural communication complexity problem which may be of independent interest.
rate research
Read More
We revisit the $k$-mismatch problem in the streaming model on a pattern of length $m$ and a streaming text of length $n$, both over a size-$sigma$ alphabet. The current state-of-the-art algorithm for the streaming $k$-mismatch problem, by Clifford et al. [SODA 2019], uses $tilde O(k)$ space and $tilde Obig(sqrt kbig)$ worst-case time per character. The space complexity is known to be (unconditionally) optimal, and the worst-case time per character matches a conditional lower bound. However, there is a gap between the total time cost of the algorithm, which is $tilde O(nsqrt k)$, and the fastest known offline algorithm, which costs $tilde Obig(n + minbig(frac{nk}{sqrt m},sigma nbig)big)$ time. Moreover, it is not known whether improvements over the $tilde O(nsqrt k)$ total time are possible when using more than $O(k)$ space. We address these gaps by designing a randomized streaming algorithm for the $k$-mismatch problem that, given an integer parameter $kle s le m$, uses $tilde O(s)$ space and costs $tilde Obig(n+minbig(frac {nk^2}m,frac{nk}{sqrt s},frac{sigma nm}sbig)big)$ total time. For $s=m$, the total runtime becomes $tilde Obig(n + minbig(frac{nk}{sqrt m},sigma nbig)big)$, which matches the time cost of the fastest offline algorithm. Moreover, the worst-case time cost per character is still $tilde Obig(sqrt kbig)$.
The shift distance $mathsf{sh}(S_1,S_2)$ between two strings $S_1$ and $S_2$ of the same length is defined as the minimum Hamming distance between $S_1$ and any rotation (cyclic shift) of $S_2$. We study the problem of sketching the shift distance, which is the following communication complexity problem: Strings $S_1$ and $S_2$ of length $n$ are given to two identical players (encoders), who independently compute sketches (summaries) $mathtt{sk}(S_1)$ and $mathtt{sk}(S_2)$, respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) $mathsf{sh}(S_1,S_2)$ with high probability. This paper primarily focuses on the more general $k$-mismatch version of the problem, where the decoder is allowed to declare a failure if $mathsf{sh}(S_1,S_2)>k$, where $k$ is a parameter known to all parties. Andoni et al. (STOC13) introduced exact circular $k$-mismatch sketches of size $widetilde{O}(k+D(n))$, where $D(n)$ is the number of divisors of $n$. Andoni et al. also showed that their sketch size is optimal in the class of linear homomorphic sketches. We circumvent this lower bound by designing a (non-linear) exact circular $k$-mismatch sketch of size $widetilde{O}(k)$; this size matches communication-complexity lower bounds. We also design $(1pm varepsilon)$-approximate circular $k$-mismatch sketch of size $widetilde{O}(min(varepsilon^{-2}sqrt{k}, varepsilon^{-1.5}sqrt{n}))$, which improves upon an $widetilde{O}(varepsilon^{-2}sqrt{n})$-size sketch of Crouch and McGregor (APPROX11).
The Densest $k$-Subgraph (D$k$S) problem, and its corresponding minimization problem Smallest $p$-Edge Subgraph (S$p$ES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both D$k$S and S$p$ES from graphs to hypergraphs. We consider the Densest $k$-Subhypergraph problem (given a hypergraph $(V, E)$, find a subset $Wsubseteq V$ of $k$ vertices so as to maximize the number of hyperedges contained in $W$) and define the Minimum $p$-Union problem (given a hypergraph, choose $p$ of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an $O(n^{4(4-sqrt{3})/13 + epsilon}) leq O(n^{0.697831+epsilon})$-approximation (for arbitrary constant $epsilon > 0$) for Densest $k$-Subhypergraph and an $tilde O(n^{2/5})$-approximation for Minimum $p$-Union. We also give an $O(sqrt{m})$-approximation for Minimum $p$-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.
In this paper, we introduce and study the Non-Uniform k-Center problem (NUkC). Given a finite metric space $(X,d)$ and a collection of balls of radii ${r_1geq cdots ge r_k}$, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation $alpha$, such that the union of balls of radius $alphacdot r_i$ around the $i$th center covers all the points in $X$. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds. The NUkC problem generalizes the classic $k$-center problem when all the $k$ radii are the same (which can be assumed to be $1$ after scaling). It also generalizes the $k$-center with outliers (kCwO) problem when there are $k$ balls of radius $1$ and $ell$ balls of radius $0$. There are $2$-approximation and $3$-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years. We first observe that no $O(1)$-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an $(O(1),O(1))$-bi-criteria approximation result: we give an $O(1)$-approximation to the optimal dilation, however, we may open $Theta(1)$ centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal $2$-approximation to the kCwO problem improving upon the long-standing $3$-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community.
In this paper we initiate the study of the heterogeneous capacitated $k$-center problem: given a metric space $X = (F cup C, d)$, and a collection of capacities. The goal is to open each capacity at a unique facility location in $F$, and also to assign clients to facilities so that the number of clients assigned to any facility is at most the capacity installed; the objective is then to minimize the maximum distance between a client and its assigned facility. If all the capacities $c_i$s are identical, the problem becomes the well-studied uniform capacitated $k$-center problem for which constant-factor approximations are known. The additional choice of determining which capacity should be installed in which location makes our problem considerably different from this problem, as well the non-uniform generalizations studied thus far in literature. In fact, one of our contributions is in relating the heterogeneous problem to special-cases of the classical Santa Claus problem. Using this connection, and by designing new algorithms for these special cases, we get the following results: (a)A quasi-polynomial time $O(log n/epsilon)$-approximation where every capacity is violated by $1+varepsilon$, (b) A polynomial time $O(1)$-approximation where every capacity is violated by an $O(log n)$ factor. We get improved results for the {em soft-capacities} version where we can place multiple facilities in the same location.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
