Efficient approximation algorithms for the Hamming center problem

We study Hamming versions of two classical clustering problems. The Hamming radius p-clustering problem (HRC) for a set S of k binary strings, each of length n, is to find p binary strings of length n that minimize the maximum Hamming distance between a string in S and the closest of the p strings; this minimum value is termed the p-radius of S and is denoted by ϱ The related Hamming diameter p-clustering problem (HDC) is to split S into p groups so that the maximum of the Hamming group diameters is minimized; this latter value is called the p-diameter of S. First, we provide an integer programming formulation of HRC which yields exact solutions in polynomial time whenever k and p are constant. We also observe that HDC admits straightforward polynomial-time solutions when k = O(log n) or p = 2. Next, by reduction from the corresponding geometric p-clustering problems in the plane under the L1 metric, we show that neither HRC nor HDC can be approximated within any constant factor smaller than two unless P=NP. We also prove that for any Ɛ > 0 it is NP-hard to split S into at most pk1/7-Ɛ clusters whose Hamming diameter doesn't exceed the p-diameter. Furthermore, we note that by adapting Gonzalez' farthest-point clustering algorithm [6], HRC and HDC can be approximated within a factor of two in time O(pkn). Next, we describe a 2O(pϱ/Ɛ)kO(p/Ɛ)n2-time (1+ Ɛ)- approximation algorithm for HRC. In particular, it runs in polynomial time when p = O(1) and ϱ = O(log(k+n)): Finally, we show how to find in O((n/Ɛ + kn log n + k2 log n)(2ϱk)2/Ɛ) time a set L of O(p log k) strings of length n such that for each string in S there is at least one string in L within distance (1 + Ɛ)ϱ, for any constant 0 < Ɛ < 1.

Approximation algorithms for Hamming clustering problems

Given a dictionary ${\mathcal W}$ consisting of n binary strings of length m each, a d-query asks if there exists a string in ${\mathcal W}$ within Hamming distance d of a given binary query string q. The problem was posed by Minsky and Papert in 1969 as a challenge to data structure design. There is a tradeoff between time and space in solving the problem of answering a d-query. Recently developed time-efficient methods for text indexing with errors can be used to answer a d-query in O(m) time. However, these methods use O(nlog d n) (or more) additional space which is not practical for large databases. We present a method for the problem assuming the standard RAM model of computation. We process the dictionary to construct an edge-labelled tree with distinct labels to siblings, and with bounded branching factor and height. Storing the resulting tree does not require asymptotically more space than the size of an ordinary trie that stores the given dictionary. We present an algorithm for the d-query problem that takes O(m(3 log4/3 n – 1) d (log2 n) d + 1) time, and uses only O(m) additional space. We also generalize the results for the case of the problem when a larger alphabet, or edit distance are used. We achieve $O(m(2|\Sigma|-1)^{d}(log_{(2|\Sigma|-1)}{\it n} -1) ^{d}(log_{2}n)^{d+1})$ time complexity for the problem when Hamming distance is used. The time complexity increases by a factor of $O(d(2|\Sigma|-1)^d(log_{2}n)^{d})$ when we use edit distance. The algorithms are efficient when the approximate dictionary look-up involves long words defined over small alphabets. The algorithm can be modified such that it allows for words of different lengths as well as different lengths of query strings.

We study the fine-grained complexity of the famous k-center problem in the metric induced by a graph with n vertices and m edges. The problem is NP-hard to approximate within a factor strictly better than 2, and several 2-approximation algorithms are known. Two of the most well-known approaches for the 2-approximation are (1) finding a maximal distance-r independent set (where the minimum pairwise distance is greater than r) and (2) Gonzalez's algorithm that iteratively adds the center farthest from the currently chosen centers. For the approach based on distance-r independent sets, Thorup [SIAM J. Comput. '05] already gave a nearly linear time algorithm. While Thorup's algorithm is not complicated, it still requires tools such as an approximate oracle for neighborhood size by Cohen [J. Comput. Syst. Sci. '97]. Our main result is a nearly straightforward algorithm that improves the running time by an O (log n) factor. It results in a (2 + ε)-approximation for k-center in O((m + n log n) log n log(n/ε)) time. For Gonzalez's algorithm [Theor. Comput. Sci. 85], we show that the simple Õ(mk)-time implementation is nearly optimal if we insist on an exact implementation. On the other hand, we show that a (1 + ε)-approximate version of the algorithm is efficiently implementable, leading to a (2 + ε)-approximation algorithm running in time O((m + n log n) log2(n)/ε). We also show that, unlike in the algorithm based on distance-r independent sets, the dependency on 1/ε in the running time is essentially optimal for (1 + ε)-approximate Gonzalez's.

We propose a technique of extraction repeatable binary string from fingerprint images. We extract binary features from topological relations between minutiae points. For an arbitrary minutiae point we trace neighboring ridges until we encounter event: minutiae or projection of minutiae. Then we encode these events. Thus we obtain 50-100 bit descriptions for each minutiae point. In order to extract longer binary string, we propose two techniques. First, we select 2^l binary strings (centers) of the same length as topological descriptor. Then for each minutiae descriptor we select k nearest centers. Selected subset of centers is represented by 2^l bit long binary string. The self-aligned technique has significant errors (FRR=30% at FAR=0.1%). The second tehnique requires public helper. We concatenate descriptions of 5-8 minutiae points. Data that are necessary to find and order minutiae subset is stored in a public helper. Thus we extracted 384-512 bit binary string with approximately 20% erroneous bits. Errors are corrected using two-layer BCH-major voting codes. Experiments on FVC2002DB1 dataset and private dataset (100 different fingers, with 3 samples per finger) show that 20-40 bit error-free binary string can be reproduced from genuine fingerprint with 90% success rate.

We here study max hamming xsat, i.e., the problem of finding two xsat models at maximum Hamming distance. By using a recent xsat solver as an auxiliary function, an O(2n) time algorithm can be constructed, where n is the number of variables. This upper time bound can be further improved to O(1.8348n) by introducing a new kind of branching, more directly suited for finding models at maximum Hamming distance. The techniques presented here are likely to be of practical use as well as of theoretical value, proving that there are non-trivial algorithms for maximum Hamming distance problems.

In this paper we gather several improvements in the field of exact and approximate exponential-time algorithms for the Bandwidth problem. For graphs with treewidth t we present a O(n O(t) 2n) exact algorithm. Moreover for the same class of graphs we introduce a subexponential constant-approximation scheme – for any α> 0 there exists a (1 + α)-approximation algorithm running in \(O(\exp(c(t + \sqrt{n/\alpha})\log n))\) time where c is a universal constant. These results seem interesting since Unger has proved that Bandwidth does not belong to APX even when the input graph is a tree (assuming P ≠ NP). So somewhat surprisingly, despite Unger’s result it turns out that not only a subexponential constant approximation is possible but also a subexponential approximation scheme exists. Furthermore, for any positive integer r, we present a (4r − 1)-approximation algorithm that solves Bandwidth for an arbitrary input graph in \(O^*(2^{n\over r})\) time and polynomial space. Finally we improve the currently best known exact algorithm for arbitrary graphs with a O(4.473n) time and space algorithm.In the algorithms for the small treewidth we develop a technique based on the Fast Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Björklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O *(2n) time and space, what matches the best known results.KeywordsFast Fourier TransformApproximation AlgorithmChromatic NumberExact AlgorithmInput GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

De Bruijn sequences are circular strings of length $2^n$ whose length n substrings are the binary strings of length n. Our focus is on creating circular strings of length $\binom{n}{w}$ for the binary strings of length n with weight (number of 1s) equal to w. In this case, each fixed-weight string can be encoded by its first $n{-}1$ bits since the final bit is redundant. For this reason, we construct circular strings of length $\binom{n-1}{w}+\binom{n-1}{w-1}$ whose length $n{-}1$ substrings are the binary strings of length $n{-}1$ with weight w or $w{-}1$. Our construction is reminiscent of the construction for the lexicographically least de Bruijn sequence, except the underlying algorithm is applied to cool-lex order instead of lexicographic order. The construction can be efficiently implemented so that successive blocks of n bits are generated in constant amortized time while using $O(n \log n)$-space. This article's results were also used to create de Bruijn sequences for binary strings of length n with a specified maximum weight.

Measuring the similarity or difference of two strings is a fundamental task in computer science and engineering. Among the various measures, Hamming distance and Levenshtein distance have been widely utilized. In this paper, we present neural networks for computing these distances, employing the Rectified Linear Unit function as the activation function. We show that the number of units in the networks is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(n)</i> for the Hamming distance between two strings of length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(mn)</i> for the Levenshtein distance between two strings of lengths <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>. These complexities align with those of traditional algorithms. While the depth of the latter network is large (O(m+n)), we show that the depth can be significantly reduced at the expense of employing a larger number of units. Furthermore, we show that the networks can be modified for learning a string from a set of strings along with their respective distances to the target string. We performed computational experiments to evaluate the potential usufulness of our proposed networks. The results suggest that these networks can efficiently compute distances in a time proportional to the network size, and can proficiently learn a string when utilizing the Hamming distance.

On the bottleneck tree alignment problems

A new class of block codes with specified length and maximum Hamming distance between the codewords is given. It is also shown that the proposed codes have the largest number of words when the maximum Hamming distance specified is 1, 2, or 3.

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

This paper proposes an efficient binary representation of delta Quantization Parameter (QP) for High Efficiency Video Coding (HEVC). Video encoders adapt QPs of coding blocks for visual quality optimization and rate control. Although they send only delta QPs (dQPs) obtained by causal prediction, the side information overhead is expensive. Therefore the HEVC design necessitates an efficient dQP coding. The proposed scheme converts a dQP to a binary string in which the first and second bins indicate the significance and sign of the dQP respectively and the rest represents the magnitude minus 1. Furthermore, it detects and truncates redundant bins in the binary strings by using the sign and an admissible dQP range. Thus it reduces the length of dQP binary strings and improves dQP coding efficiency. Simulation results using HEVC reference software demonstrate that the proposed scheme improves the dQP coding efficiency by 6% while reducing its bin rates by 25%.

We study the following all-or-nothing multicommodity flow problem in planar graphs. Input: A graph G with n vertices and k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Find: A largest subset W of {1, ...., k such that for every i in W, we can send one unit of flow between si and ti. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri et al. (STOC'04) give a poly-logarithmic factor approximation algorithm and show that a natural LP-relaxation has a poly-logarithmic integrality gap. This result is in contrast with the integrality gap Ω(√n) for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an O(1)-approximation algorithm and show that the integrality gap is O(1). In particular, in polynomial time, we can find an index set W with |W| = Ω(OPT) and eight si-ti paths for each i in W such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the recent result by S'eguin-Charbonneau and Shepherd (FOCS'11) who give an O(1)-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).

We study the NP-hard problem of learning a decision tree (DT) of smallest depth or size from data. We provide the first parameterized complexity analysis of the problem and draw a detailed parameterized complexity map for the natural parameters: size or depth of the DT, maximum domain size of all features, and the maximum Hamming distance between any two examples. Our main result shows that learning DTs of smallest depth or size is fixed-parameter tractable (FPT) parameterized by the combination of all three of these parameters. We contrast this FPT-result by various hardness results that underline the algorithmic significance of the considered parameters.