Single-cell sequencing (SCS) enables the study of tumor evolution at the resolution of a single cell. SCS data can be represented as a binary matrix, where the $ij$-th entry indicates whether cell $i$ has mutation $j$. There is a simple characterization of when the data is compatible with a perfect phylogeny based on the absence of a special "conflict" submatrix. In practice, SCS data are noisy, which raises the natural question of the minimum number of entries that must be flipped in the data matrix to make it conflict-free and thus compatible with a perfect phylogeny. Furthermore, the likelihood of a false positive is several orders of magnitude smaller than that of a false negative rate. We consider a variation of the minimum-flip problem parameterized by the number of false positives. Restricting the false positive rate to a small range, often multiple optimal solutions can arise. While previous work has focused on reconstructing a single optimal phylogenetic tree, we are interested in the relations that are present among all optimal solutions; we call such relations essential. In this work, we propose an efficient algorithm based on integer linear programming to determine the essential relation on the cells given an SCS data matrix. We test our tool, ${\sf EssentCell}$, on several data sets and discuss the results found.

Transposition is a well-known genome rearrangement event that switches two consecutive sub-strings on a string. Since a transposition makes changes to a string, the genome here is just a string. The problem of transforming one string into the other by a sequence of transposition operations has attracted a lot of attention. However, it has been reported that genome rearrangement events are often associated with repeated sub-strings. In particular, a transposition operation is most likely associated with three identical repeated sub-strings. A transposition operation on two consecutive sub-strings $x$ and $y$ switches the two sub-strings and transforms the whole string $zxyw$ into the other string $zyxw$, where $z$ and $w$ represents the two sub-strings on the left and right of $xy$, respectively. When repeated sub-strings are considered, the two consecutive sub-strings $x$ and $y$ are flanked with three identical repeated sub-strings $R$ and the flanked transposition transforms the whole string $zRxRyRw$ into $zRyRxRw$. For a flanked transposition operation, the neighbors of $x$ and $y$ remain the same before and after the transposition. In this paper, we investigate the problem of transforming one string into the other by a number of flanked transpositions. First, we present a necessary and sufficient condition to determine if a string can be transformed into the other by a sequence of flanked transpositions. We then design a decision algorithm with running time $\mathcal {O}(n)$ to test if such a condition holds. We also show that transforming one string into the other by using minimum number of flanked transpositions is NP-hard. A string $\pi$ of $n$ letters is simple if the $n-1$ consecutive pairs of letters are distinct. We present an $\mathcal {O}(n^{2})$ approximation algorithm with ratio 2 for the optimization version of the special case, where both input strings are simple.

Abstract Motivated by computing duplication patterns in sequences, a new problem called the longest letter-duplicated subsequence (LLDS) is proposed. Given a sequence S of length n, a letter-duplicated subsequence is a subsequence of S in the form of $$x_1^{d_1}x_2^{d_2}\ldots x_k^{d_k}$$ x 1 d 1 x 2 d 2 … x k d k with $$x_i\in \Sigma $$ x i ∈ Σ , $$x_j\ne x_{j+1}$$ x j ≠ x j + 1 and $$d_i\ge 2$$ d i ≥ 2 for all i in [k] and j in $$[k-1]$$ [ k - 1 ] . A linear time algorithm for computing a longest letter-duplicated subsequence (LLDS) of S can be easily obtained. In this paper, we focus on two variants of this problem: (1) ‘all-appearance’ version, i.e., all letters in $$\Sigma $$ Σ must appear in the solution, and (2) the weighted version. For the former, we obtain dichotomous results: We prove that, when each letter appears in S at least 4 times, the problem and a relaxed version on feasibility testing (FT) are both NP-hard. The reduction is from $$(3^+,1,2^-)$$ ( 3 + , 1 , 2 - ) -SAT, where all 3-clauses (i.e., containing 3 lals) are monotone (i.e., containing only positive literals) and all 2-clauses contain only negative literals. We then show that when each letter appears in S at most 3 times, then the problem admits an O(n) time algorithm. Finally, we consider the weighted version, where the weight of a block $$x_i^{d_i} (d_i\ge 2)$$ x i d i ( d i ≥ 2 ) could be any positive function which might not grow with $$d_i$$ d i . We give a non-trivial $$O(n^2)$$ O ( n 2 ) time dynamic programming algorithm for this version, i.e., computing an LD-subsequence of S whose weight is maximized.

Rearrangement sorting problems impact profoundly in measuring genome similarities and tracing historic scenarios of species. However, recent studies on genome rearrangement mechanisms disclosed a statistically significant evidence, repeats are situated at the ends of rearrangement relevant segments and stay unchanged before and after rearrangements.To reflect the principle behind this evidence, we propose flanked block-interchange, an operation on strings that exchanges two substrings flanked by identical left and right symbols in a string. The flanked block-interchange distance problem is formulated as finding a shortest sequence of flanked block-interchanges to transform a string into the other. We propose a sufficient and necessary condition for deciding whether two strings can be transformed into each other by flanked block-interchanges. This condition is linear time verifiable. Under this condition for two strings, we present a [Formula: see text]-approximation algorithm for the flanked block-interchange distance problem where each symbol occurs at most k times in a string and a polynomial algorithm for this problem where each symbol occurs at most twice in a string. We show that the problem of flanked block-interchange distance is NP-hard at last.

Given a sequence $S$ of length $n$, a letter-duplicated subsequence is a subsequence of $S$ in the form of $x_1^{d_1}x_2^{d_2}\cdots x_k^{d_k}$ with $x_i\inΣ$, $x_j\neq x_{j+1}$ and $d_i\geq 2$ for all $i$ in $[k]$ and $j$ in $[k-1]$. A linear time algorithm for computing the longest letter-duplicated subsequence (LLDS) of $S$ can be easily obtained. In this paper, we focus on two variants of this problem. We first consider the constrained version when $Σ$ is unbounded, each letter appears in $S$ at least 6 times and all the letters in $Σ$ must appear in the solution. We show that the problem is NP-hard (a further twist indicates that the problem does not admit any polynomial time approximation). The reduction is from possibly the simplest version of SAT that is NP-complete, $(\leq 2,1,\leq 3)$-SAT, where each variable appears at most twice positively and exact once negatively, and each clause contains at most three literals and some clauses must contain exactly two literals. (We hope that this technique will serve as a general tool to help us proving the NP-hardness for some more tricky sequence problems involving only one sequence -- much harder than with at least two input sequences, which we apply successfully at the end of the paper on some extra variations of the LLDS problem.) We then show that when each letter appears in $S$ at most 3 times, then the problem admits a factor $1.5-O(\frac{1}{n})$ approximation. Finally, we consider the weighted version, where the weight of a block $x_i^{d_i} (d_i\geq 2)$ could be any positive function which might not grow with $d_i$. We give a non-trivial $O(n^2)$ time dynamic programming algorithm for this version, i.e., computing an LD-subsequence of $S$ whose weight is maximized.

We are on the verge of a post-genomics era in which whole protein sequencing will be quickly carried out. Protein se-quencing plays an important role in identifying protein functions, analyzing protein-protein interactions, and characterizing post-translational modifications, etc. The protein sequencing problem is to determine the complete sequence of amino acids in proteins. De novo protein sequencing using top-down and bottom-up tandem mass spectrometry suffers from the problem of producing only partial sequences of target proteins, namely scaffold. In this paper, we explore the possibility of using deep learning techniques to perform the task of predicting amino acids in partially sequenced proteins by two phases. First, our methods involve querying the NCBI Protein Blast server to find closest matching homologous sequences to a scaffold as a training dataset. Second, we train several deep learning models based on a convolutional neural network and long short term memory to predict missing amino acids in the scaffold in the forward and reverse directions. We comprehensively evaluate our proposed methods on an alemtuzumab dataset and our results show that the proposed methods achieve high sequence coverage and high sequence accuracy with 100 % on the the light chain of alemtuzumab scaffold data.

In computational biology, tandem duplication is an important biological phenomenon which can occur either at the genome or at the DNA level. A tandem duplication takes a copy of a genome segment and inserts it right after the segment---this can be represented as the string operation $AXB \Rightarrow AXXB$. Tandem exon duplications have been found in many species such as human, fly, and worm and have been largely studied in computational biology. The tandem duplication (TD) distance problem we investigate in this paper is defined as follows: given two strings $S$ and $T$ over the same alphabet $\Sigma$, compute the smallest sequence of TDs required to convert $S$ to $T$. The natural question of whether the TD distance can be computed in polynomial time was posed in 2004 by Leupold et al. and had remained open, despite the fact that TDs have received much attention ever since. In this paper, we focus on the special case when all characters of $S$ are distinct. This is known as the exemplar TD distance, which is of special relevance in bioinformatics. We first prove that this problem is NP-hard when the alphabet size is unbounded, settling the 16-year-old open problem. We then show how to adapt the proof to $|\Sigma|=4$, hence proving the NP-hardness of the TD problem for any $|\Sigma|\geq 4$. One of the tools we develop for the reduction is a new problem called Cost-Effective Subgraph, for which we obtain W[1]-hardness results that might be of independent interest. We finally show that computing the exemplar TD distance between $S$ and $T$ is fixed-parameter tractable. Our results open the door to many other questions, and we conclude with several open problems.

Recently, due to the genomic sequence analysis in several types of cancer, genomic data based on copy number profiles (CNP for short) are getting more and more popular. A CNP is a vector where each component is a non-negative integer representing the number of copies of a specific segment of interest. The motivation is that in the late stage of certain types of cancer, the genomes are progressing rapidly by segmental duplications and deletions, and hence obtaining the exact sequences becomes difficult. Instead, the number of copies of important segments can be predicted from expression analysis and carries important biological information. Therefore, significant research has recently been devoted to the analysis of genomic data represented as CNP’s. In this paper, we present two streams of results. The first is the negative results on two open problems regarding the computational complexity of the Minimum Copy Number Generation (MCNG) problem posed by Qingge et al. in 2018. The Minimum Copy Number Generation (MCNG) is defined as follows: given a string S in which each character represents a gene or segment, and a CNP C, compute a string T from S, with the minimum number of segmental duplications and deletions, such that cnp(T)=C. It was shown by Qingge et al. that the problem is NP-hard if the duplications are tandem and they left the open question of whether the problem remains NP-hard if arbitrary duplications and/or deletions are used. We answer this question affirmatively in this paper; in fact, we prove that it is NP-hard to even obtain a constant factor approximation. This is achieved through a general-purpose lemma on set-cover reductions that require an exact cover in one direction, but not the other, which might be of independent interest. We also prove that the corresponding parameterized version is W[1]-hard, answering another open question by Qingge et al. The other result is positive and is based on a new (and more general) problem regarding CNP’s. The Copy Number Profile Conforming (CNPC) problem is formally defined as follows: given two CNP’s C₁ and C₂, compute two strings S₁ and S₂ with cnp(S₁)=C₁ and cnp(S₂)=C₂ such that the distance between S₁ and S₂, d(S₁,S₂), is minimized. Here, d(S₁,S₂) is a very general term, which means it could be any genome rearrangement distance (like reversal, transposition, and tandem duplication, etc). We make the first step by showing that if d(S₁,S₂) is measured by the breakpoint distance then the problem is polynomially solvable. We expect that this will trigger some related research along the line in the near future.

In computational biology, tandem duplication is an important biological phenomenon which can occur either at the genome or at the DNA level. A tandem duplication takes a copy of a genome segment and inserts it right after the segment - this can be represented as the string operation AXB ⇒ AXXB. Tandem exon duplications have been found in many species such as human, fly or worm, and have been largely studied in computational biology. The Tandem Duplication (TD) distance problem we investigate in this paper is defined as follows: given two strings S and T over the same alphabet, compute the smallest sequence of tandem duplications required to convert S to T. The natural question of whether the TD distance can be computed in polynomial time was posed in 2004 by Leupold et al. and had remained open, despite the fact that tandem duplications have received much attention ever since. In this paper, we prove that this problem is NP-hard, settling the 16-year old open problem. We further show that this hardness holds even if all characters of S are distinct. This is known as the exemplar TD distance, which is of special relevance in bioinformatics. One of the tools we develop for the reduction is a new problem called the Cost-Effective Subgraph, for which we obtain W[1]-hardness results that might be of independent interest. We finally show that computing the exemplar TD distance between S and T is fixed-parameter tractable. Our results open the door to many other questions, and we conclude with several open problems.

The one-sided Exemplar Adjacency Number (EAN) is a known problem for computing the exemplar similarity between a generic linear genome${\mathcal G}$with gene duplications and an exemplar genome$H$(over the same set of$n$gene families). In this problem, we need to compute an exemplar genome$G$, which is a permutation obtained from${\mathcal G}$, such that the number of common adjacencies between$G$and$H$is maximized. Unfortunately, the problem is not only NP-hard but also NP-hard to approximate. In this paper, we approach the problem by relaxing the constraint such that a sub-permutation$G^{+}$obtained from${\mathcal G}$does not have to include all the gene families, but still needs to have a length at least$k$. Hence$G^{+}$is called apseudo-exemplargenome. Then, a slightly more general problem (One-sided EAN+) is defined: compute a pseudo-exemplar genome$G^{+}$from${\mathcal G}$such that the number of common adjacencies between$H$and$G^{+}$is maximized. Certainly One-sided EAN+ contains One-sided EAN as a special case; moreover, it presents some flexibility in designing algorithms. First, we relax and formulate the One-sided EAN+ problem as the maximum independent set (MIS) on a colored interval graph and hence reduce the appearance of each gene to at most two times. We show that this new relaxation is still NP-complete, though a simple factor-2 approximation algorithm can be designed; moreover, we also prove that the problem cannot be approximated within$2-\varepsilon$by a local search technique. We then show that this relaxed version is fixed-parameter tractable (FPT). Second, to ensure that each gene appears in$G^+$at most once, we use integer linear programming (ILP) to solve this problem. Finally, we implement our algorithm and compare it with the up-to-date software GREDU, with simulated signed and unsigned genomes. It turns out that our algorithm is more stable and can process genomes of length up to 12,000 for signed genomes (while GREDU can falter on such a large signed genome and it cannot handle unsigned genomes at all).

The Maximal Strip Recovery problem (MSR) and its complementary (CMSR) are well-studied NP-hard problems in computational genomics. The input of these dual problems are two signed permutations. The goal is to delete some gene markers from both permutations, such that, in the remaining permutations, each gene marker has at least one common neighbor. Equivalently, the resulting permutations could be partitioned into common strips of length at least two. Then MSR is to maximize the number of remaining genes, while the objective of CMSR is to delete the minimum number of gene markers. In this paper, we present a new approximation algorithm for the Complementary Maximal Strip Recovery (CMSR) problem. Our approximation factor is 2, improving the currently best 7/3-approximation algorithm. Although the improvement on the factor is not huge, the analysis is greatly simplified by a compensating method, commonly referred to as the non-oblivious local search technique. In such a method a substitution may not always increase the value of the current solution (it sometimes may even decrease the solution value), though it always improves the value of another function seemingly unrelated to the objective function.

Massive trust assessment (MTA) in an Online Social Network (OSN), i.e., computing the trustworthiness of all users in the network, is crucial in various OSN-related applications. Existing solutions are either too slow or inaccurate in addressing the MTA problem. We propose the OpinionWalk algorithm that accurately and efficiently conducts MTA in an OSN. OpinionWalk models trust by the Dirichlet distribution and uses a matrix to represent the direct trust relations among users. From the perspective of a user, other users' trustworthiness are stored in a column vector that is iteratively updated when the algorithm “walks” through the network, in a breadth-first search manner. We identify the overlapping subproblems property in MTA and prove OpinionWalk is a more efficient solution. The accuracy and execution time of OpinionWalk are evaluated and compared to benchmark algorithms including EigenTrust, TrustRank, MoleTrust, TidalTrust and AssessTrust, using two real-world datasets (Advogato and Pretty Good Privacy). Experimental results indicate that OpinionWalk is an efficient and accurate solution to MTA, compared to previous algorithms.

The genomic scaffold filling problem has attracted a lot of attention recently. The problem is on filling an incomplete sequence (scaffold) I into I', with respect to a complete reference genome G, such that the number of adjacencies between G and I' is maximized. The problem is NP-complete and APX-hard, and admits a 1.2-approximation. However, the sequence input I is not quite practical and does not fit most of the real datasets (where a scaffold is more often given as a list of contigs). In this paper, we revisit the genomic scaffold filling problem by considering this important case when, (1) a scaffold S is given, the missing genes X = c(G) - c(S) can only be inserted in between the contigs, and the objective is to maximize the number of adjacencies between G and the filled S' and (2) a scaffold S is given, a subset of the missing genes X' subset X = c(G) - c(S) can only be inserted in between the contigs, and the objective is still to maximize the number of adjacencies between G and the filled S''. For problem (1), we present a simple NP-completeness proof, we then present a factor-2 greedy approximation algorithm, and finally we show that the problem is FPT when each gene appears at most d times in G. For problem (2), we prove that the problem is W[1]-hard and then we present a factor-2 FPT-approximation for the case when each gene appears at most d times in G.

The Chain Pair Simplification problem (CPS) was posed by Bereg et al. who were motivated by the problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones. In this problem, given two polygonal chains of lengths n and m, the goal is to simplify both of them simultaneously, so that the lengths of the resulting simplifications as well as the discrete Frechet distance between them are bounded. When the vertices of the simplifications are arbitrary (i.e., not necessarily from the original chains), the problem is called General CPS (GCPS). In this paper we consider for the first time the complexity of GCPS under both the discrete Frechet distance (GCPS-3F) and the Hausdorff distance (GCPS-2H). (In the former version, the quality of the two simplifications is measured by the discrete Fr'echet distance, and in the latter version it is measured by the Hausdorff distance.) We prove that GCPS-3F is polynomially solvable, by presenting an widetilde-O((n+m)^6 min{n,m}) time algorithm for the corresponding minimization problem. We also present an O((n+m)^4) 2-approximation algorithm for the problem. On the other hand, we show that GCPS-2H is NP-complete, and present an approximation algorithm for the problem.

We consider the emerging problem of comparing the similarity between (unlabeled) pedigrees. More specifically, we focus on the simplest pedigrees, namely, the 2-generation pedigrees. We show that the isomorphism testing for two 2-generation pedigrees is GI-hard. If the 2-generation pedigrees are monogamous (i.e., each individual at level-1 can mate with exactly one partner) then the isomorphism testing problem can be solved in polynomial time. We then consider the problem by relaxing it into an NP-complete decomposition problem which can be formulated as the Minimum Common Integer Pair Partition (MCIPP) problem, which we show to be FPT by exploiting a property of the optimal solution. While there is still some difficulty to overcome, this lays down a solid foundation for this research.

While extending the lifetime of a wireless sensor network (WSN) with full coverage has been extensively studied, it was found recently that the lifetime of a WSN can be prolonged significantly if partial covers are used instead. In this paper, we formally define the problem of extending the lifetime of a WSN using partial covers. (Throughout this paper, we assume that each point of the given target region is covered at least k times by the input sensors.) We first present a centralized algorithm using an optimal subroutine which computes the densest strip with width 2r, where r is the minimum sensing radius of all sensors. By using a known 1-D algorithm, we can cover the center of the strip with k full covers and the remaining subproblems can be solved recursively to have the eventual k partial covers. We then introduce a distributed algorithm without any assumption on the coordinates and directions of sensors, as long as each sensor knows the presence of other sensors within its sensing region. Finally, we present some experimental results comparing the performance of these algorithms with the previous homological partial cover solution. In all small instances the results generated by our algorithms are significantly better than those generated by the homological method. For two larger instances (a larger domain with n around 1000), the homological method cannot finish while both of our algorithms generate promising results.

Scaffold filling is a new combinatorial optimization problem in genome sequencing. The one-sided scaffold filling problem can be described as given an incomplete genome I and a complete (reference) genome G, fill the missing genes into I such that the number of common (string) adjacencies between the resulting genome I' and G is maximized. This problem is NP-complete for genome with duplicated genes and the best known approximation factor is 1.33, which uses a greedy strategy. In this paper, we prove a better lower bound of the optimal solution, and devise a new algorithm by exploiting the maximum matching method and a local improvement technique, which improves the approximation factor to 1.25. For genome with gene repetitions, this is the only known NP-complete problem which admits an approximation with a small constant factor (less than 1.5).

For protein structure alignment and comparison, a lot of work has been done using RMSD as the distance measure, which has drawbacks under certain circumstances. Thus, the discrete Fréchet distance was recently applied to the problem of protein (backbone) structure alignment and comparison with promising results. For this problem, visualization is also important because protein chain backbones can have as many as 500-600 $(\alpha)$-carbon atoms, which constitute the vertices in the comparison. Even with an excellent alignment, the similarity of two polygonal chains can be difficult to visualize unless the chains are nearly identical. Thus, the chain pair simplification problem (CPS-3F) was proposed in 2008 to simultaneously simplify both chains with respect to each other under the discrete Fréchet distance. The complexity of CPS-3F is unknown, so heuristic methods have been developed. Here, we define a variation of CPS-3F, called the constrained CPS-3F problem ($({\rm CPS\hbox{-}3F}^+)$), and prove that it is polynomially solvable by presenting a dynamic programming solution, which we then prove is a factor-2 approximation for CPS-3F. We then compare the $({\rm CPS\hbox{-}3F}^+)$ solutions with previous empirical results, and further demonstrate some of the benefits of the simplified comparisons. Chain pair simplification based on the Hausdorff distance (CPS-2H) is known to be NP-complete, and here we prove that the constrained version ($(\rm CPS\hbox{-}2H^+)$) is also NP-complete. Finally, we discuss future work and implications along with a software library implementation, named the Fréchet-based Protein Alignment & Comparison Toolkit (FPACT).

Motivated by the trend of genome sequencing without completing the sequence of the whole genomes, a problem on filling an incomplete multichromosomal genome (or scaffold) I with respect to a complete target genome G was studied. The objective is to minimize the resulting genomic distance between I' and G, where I' is the corresponding filled scaffold. We call this problem the onesided scaffold filling problem. In this paper, we conduct a systematic study for the scaffold filling problem under the breakpoint distance and its variants, for both unichromosomal and multichromosomal genomes (with and without gene repetitions). When the input genome contains no gene repetition (i.e., is a fragment of a permutation), we show that the two-sided scaffold filling problem (i.e., G is also incomplete) is polynomially solvable for unichromosomal genomes under the breakpoint distance and for multichromosomal genomes under the genomic (or DCJ--Double-Cut-and-Join) distance. However, when the input genome contains some repeated genes, even the one-sided scaffold filling problem becomes NP-complete when the similarity measure is the maximum number of adjacencies between two sequences. For this problem, we also present efficient constant-factor approximation algorithms: factor-2 for the general case and factor 1.33 for the one-sided case.

In this paper, we focus on a roadside directional sensor network where sensors with directional Field Of Views (FOVs) are placed along a roadway. We study the problem of wakeup scheduling, with the objective of maximizing network lifetime under the constraint that full coverage and network connectivity are maintained at all times. First, we present centralized polynomial time algorithms to optimally solve the problems of scheduling sensor nodes with fixed sensing orientations. Moreover, an effective heuristic algorithm is proposed to solve the problem of scheduling sensor nodes with variable sensing orientations. In addition, we also present distributed algorithms for both fixed and variable cases. Simulation results based on a roadway in the Yellowstone National Park have been presented to show the performance of the proposed algorithms.

MOTIVATION: The double cut and join operation (abbreviated as DCJ) has been extensively used for genomic rearrangement. Although the DCJ distance between signed genomes with both linear and circular (uni- and multi-) chromosomes is well studied, the only known result for the NP-complete unsigned DCJ distance problem is an approximation algorithm for unsigned linear unichromosomal genomes. In this article, we study the problem of computing the DCJ distance on two unsigned linear multichromosomal genomes (abbreviated as UDCJ). RESULTS: We devise a 1.5-approximation algorithm for UDCJ by exploiting the distance formula for signed genomes. In addition, we show that UDCJ admits a weak kernel of size 2k and hence an FPT algorithm running in O(2(2k)n) time.

Given a polyhedral terrain T with n vertices, the two-watchtower problem for T calls for finding two vertical segments, called watchtowers, of smallest common height, whose bottom endpoints (bases) lie on T, and whose top endpoints guard T, in the sense that each point on T is visible from at least one of them. In this paper we present the following results for the two-watchtower problem in R2 and R3: (1) We show that the discrete two-watchtowers problem in R2, where the bases are constrained to lie at vertices of T, can be solved in O(n2 log4n) time, significantly improving previous solutions. The algorithm works, without increasing its asymptotic running time, even if, one of the towers is allowed to be placed anywhere on T. (2) We show that the continuous two-watchtower problem in R2, where the bases can lie anywhere on T, can be solved in O(n3α(n)log3n) time, again significantly improving previous results. (3) Still in R2, we show that the continuous version of the problem of guarding a finite set P ⊂ T of m points by two watchtowers of smallest height can be solved in O(mn log4n) time. (4) The discrete version of the two-watchtower problem in R3 can be solved in O(n11/3 polylog(n)) time; this is the first nontrivial result for this problem in R3.

Abstract In this article we report our research on building FEATURES—an intelligent web search engine that is able to perform real‐time adaptive feature (i.e., keyword) and document learning. Not only does FEATURES learn from the user's document relevance feedback, but it also automatically extracts and suggests indexing keywords relevant to a search query and learns from the user's keyword relevance feedback so that it is able to speed up its search process and to enhance its search performance. We design two efficient and mutual‐benefiting learning algorithms that work concurrently, one for feature learning and the other for document learning. FEATURES employs these algorithms together with an internal index database and a real‐time meta‐searcher to perform adaptive real‐time learning to find desired documents with as little relevance feedback from the user as possible. The architecture and performance of FEATURES are also discussed.

We investigate the applicability of on-line learning algorithms to the real-world problem of Web search. Consider that Web documents are indexed using n Boolean features. We first present a practically efficient online learning algorithm TW2 to search for Web documents represented by a disjunction of at most k relevant features. We then design and implement WebSail, a real-time adaptive Web search learner, with TW2 as its learning component. WebSail learns from the user's relevance feedback in real-time and helps the user to search for the desired Web documents. The architecture and performance of WebSail are also discussed.

We introduce the unoriented $\Theta$-maximum as a new criterion for describing the shape of a set of planar points. We present efficient algorithms for computing the unoriented $\Theta$-maximum of a set of planar points. We also propose a simple linear expected time algorithm for computing the unoriented $\Theta$-maximum of a set of planar points when $\Theta=\pi/2$.

Article Free Access Share on A polynomial time solution for labeling a rectilinear map Authors: Chung Keung Poon Dept. of Computer Science, City University of Hong Kong Dept. of Computer Science, City University of Hong KongView Profile , Binhai Zhu Dept. of Computer Science, City University of Hong Kong Dept. of Computer Science, City University of Hong KongView Profile , Franis Chin Dept. of Computer Science, University of Hong Kong Dept. of Computer Science, University of Hong KongView Profile Authors Info & Claims SCG '97: Proceedings of the thirteenth annual symposium on Computational geometryAugust 1997Pages 451–453https://doi.org/10.1145/262839.263079Published:01 August 1997Publication History 9citation277DownloadsMetricsTotal Citations9Total Downloads277Last 12 Months25Last 6 weeks6 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 Publisher SiteeReaderPDF

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This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point simply by walking through the triangulation, after selecting a good starting point by random sampling. The analysis generalizes and extends a recent result of d = 2 dimensions by proving this procedure to take expected time close to O(n{sup 1/(d+1)}) for point location in Delaunay triangulations of n random points in d = 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice.