Given k strings each of length at most n, computing the shortest common supersequence of them is a well-known NP-hard problem (when k is unbounded). On the other hand, when k=2, such a shortest common supersequence can be computed in O(n^2) time using dynamic programming as a textbook example. In this paper, we consider the problem of computing a minimal common supersequence and enumerating all minimal common supersequences for k=2 input strings. Our results are summarized as follows. A minimal common supersequence of k=2 input strings can be computed in O(n) time. (The method also works when k is a constant). All minimal common supersequences between two input strings can be enumerated with a data structure of O(n^2) space and an O(n) time delay, and the data structure can be constructed in O(n^3) time.

In this paper we initiate the study of computing a maximal (not necessarily maximum) repeating pattern in a single input string, where the corresponding problems have been studied (e.g., a maximal common subsequence) only in two or more input strings by Hirota and Sakai starting 2019. Given an input string S of length n, we can compute a maximal square subsequence of S in O(nlog n) time, greatly improving the O(n^2) bound for computing the longest square subsequence of S. For a maximal k-repeating subsequence, our bound is O(f(k)nlog n), where f(k) is a computable function such that f(k) < k· 4^k. This greatly improves the O(n^2k-1) bound for computing a longest k-repeating subsequence of S, for k≥ 3. Both results hold for the constrained case, i.e., when the solution must contain a subsequence X of S, though with higher running times.

Reciprocal translocation is an important genome rearrangement operation that involves swapping non-empty terminal segments between two chromosomes. Given two multi-chromosomal genomes, A and B, the problem of sorting by reciprocal translocations aims to find the shortest sequence of reciprocal translocations that can transform A into B. This problem can be solved in polynomial time, and, as far as we know, the best time complexity is O(n^3/2√(log n)) . In this paper, by leveraging the results on the fully dynamic graph connectivity problem, we improve the time complexity to O(n(N^2log n+log ^2n)/loglog n) , where n represents the number of genes and N represents the number of chromosomes. In real evolutionary scenarios, the number of chromosomes involved in rearrangement events is typically small. Consequently, when N is treated as a constant or a polynomial function of log n , our algorithm achieves a time complexity better than O(n^3/2√(log n)) .

In this paper, we investigate the twin bridges problem in polygons, whose counterpart on trees has been studied recently. Given two disjoint simple polygons P and Q, each with at most n vertices, the problem is to find two vertices p_1 and p_2 in P (and q_1 and q_2 in Q) to build two bridges p_1q_1 and p_2q_2 such that the constrained diameter of the resulting geometric structure (e.g., a polygon with a hole when the two bridges do not intersect), i.e., the maximum of the shortest distance between two vertices in P, Q, or just in one of them, through at least one of the bridges, is minimized. The main results are summarized as follows: (1) If P and Q are arbitrary polygons, the problem can be solved in O(n^6log n) time. (2) These results hold even if the bridges between p_i and q_i , i∈{1,2} , are geodesic. (3) We show the general problem is NP-hard: given m disjoint polygons P_1,...,P_m , add two bridges between P_i and P_i+1 , i∈ [m-1] , such that the constrained diameter of the resulting geometric structure (ideally, a polygon with multiple holes, but could be more complex if some of the bridges intersect) is minimized.

We consider the parametric shortest paths problem in a linearly interpolated graph. Given two positively-weighted directed graphs G_0=(V,E,ω_0) and G_1=(V,E,ω_1), the linearly interpolated graph is the family of graphs (1-λ)G_0+λG_1, parameterized by λ∈ [0,1]. The problem is to compute all distinct parametric shortest paths. We compute a data structure in Θ(k|E|log |V|) time, where k is the number of distinct parametric shortest paths over all λ∈ [0,1] that exist for a nontrivial interval of parameters, each corresponding to a linear function in a maximal sub-interval of [0,1]. Using this data structure, a shortest path query takes Θ(log k) time.

The study of complex proteoforms with mutations and post-translational modifications has gained increasing attention with the advancement of mass spectrometry (MS)-based techniques. Achieving high proteoform sequence coverage by MS is essential for accurately characterizing these complex proteoforms. Extensive efforts have been made to increase the proteoform sequence coverage using deep bottom-up and top-down MS strategies. In this study, we evaluated top-down and middle-down MS approaches for enhancing proteoform sequence coverage using three proteins: ubiquitin, myoglobin, and carbonic anhydrase II. In the top-down MS approach, we applied in-source fragmentation (ISF) to generate pseudo-MS3 spectra, thereby improving sequence coverage. For the middle-down MS strategy, we performed short-duration enzymatic digestions to produce longer peptides that preserve more proteoform sequence information. Our experimental results demonstrated that ISF and partial digestion significantly increased the sequence coverage of the proteins, achieving coverage greater than 90%.

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.

In this article, we initiate the study on some problems related to multiple protein scaffold filling, with or without references. The objective is to maximize the sum of Blosum62 scores of the filled sequences when no reference is given, or to maximize the Blosum62 score between the filled sequence and a reference. We present the following results: (1) given n scaffolds generated from the top-down tandem mass spectra, finding k scaffolds whose corresponding contents can be used to fill into a target sequence (or a sequence whose Blosum62 score with a reference is maximized) takes Ω(nk-ε) time, unless the Strong Exponential Time Hypothesis (SETH) fails. (2) Given two or more protein scaffolds and the corresponding multisets of amino acids to be filled accordingly, the corresponding optimization problem can be solved in polynomial time with dynamic programming. (3) Due to the high (and impractical) running times of these algorithms, we implement several heuristic algorithms for the special cases when three scaffolds are given. The corresponding empirical results are quite promising, from the testing of 12 datasets spanning five different kinds of proteins: antibody, myoglobin, mitochondrial respiratory-chain, calmodulin, and thioredoxin.

A 1.5D imprecise terrain is an x-monotone polyline with fixed x-coordinates, the y-coordinate of each vertex is not fixed but is constrained to be in a given vertical interval. A 2.5D imprecise terrain is a triangulation with fixed x and y-coordinates, but the z-coordinate of each vertex is constrained to a given vertical interval. Given an imprecise terrain with n intervals, the optimistic shortest watchtower problem asks for a terrain T realized by a precise point in each vertical interval such that the height of the shortest vertical line segment whose lower endpoint lies on T and upper endpoint sees the entire terrain is minimized. In this paper, we present a linear time algorithm to solve the 1.5D optimistic shortest watchtower problem exactly. For the discrete version of the 2.5D case (where the watchtower must be placed on a vertex of T), and we give an additive approximation scheme running in O(OPT/εn^3) time, achieving a solution within an additive error of ε from the optimal solution value OPT.

In this paper, we initiate the study on some problems related to multiple protein scaffold filling, with or without references. The objective is to maximize the sum of the Blosum62 scores of the filled sequences when no reference is given, or to maximize the Blosum62 score between the filled sequence and a reference. We present the following results: (1) given n scaffolds generated from the top-down tandem mass spectra, finding k scaffolds whose corresponding contents can be used to fill into a target sequence (or, a sequence whose Blosum62 score with a reference is maximized) takes (n^k-ε) time, unless the SETH (Strong Exponential Time Hypothesis) fails. (2) given two or more protein scaffolds and the corresponding multisets of amino acids to be filled accordingly, the corresponding optimization problem can be solved in polynomial time with dynamic programming. (3) Due to the high (and impractical) running times of the algorithms in (2), we implement several heuristic algorithms for the special cases when three scaffolds are given, and the corresponding empirical results are quite promising—although we also find that for the greedy algorithm more biological information needs to be incorporated to generate biologically meaningful results.

In de novo protein sequencing, we often could only obtain an incomplete protein sequence, namely a scaffold, from top-down and bottom-up tandem mass spectrometry. While most sections of proteins can be inferred from their homologous sequences, some specific section of proteins is always missing and it is hard to predict the missing amino acids in the gaps of the scaffolds. Thus, we only focus on predicting the gaps based on a probabilistic algorithm and a machine learning model instead predicting the complete protein sequence using generative AI models in this paper. We study two versions of the protein scaffold filling problem with known gap size and known gap mass, respectively. For the known size gaps version, we develop several machine learning models based on random forest, k-nearest neighbors, decision tree, and fully connected neural network. For the known gap mass problem, we design a probabilistic algorithm to predict the missing amino acids in the gaps. The experimental results on both real and simulation data show that our proposed algorithms show promising results of 100% and close to 100% accuracy, respectively.

The protein scaffold filling problem remains a significant challenge in computational proteomics, which is critical for accurate protein function prediction and drug design. Despite recent advancements, current sequencing methods often yield incomplete protein sequences, referred to as scaffolds, which require precise filling for further analysis. This paper presents a web-based application, implemented using the Django framework, adopting our previously developed machine learning and deep learning techniques for protein scaffold filling. The platform allows users to try our pre-trained models or train models on their datasets for new scaffolds. This system provides a versatile tool for researchers in computational proteomics, enhancing the efficiency of protein sequence prediction. The developed web application can be accessed through https://psf.ncat.edu/.

We formulate low-level malware detection using algorithms based on feature matching as Order-based Malware Detection with Critical Instructions (General-OMDCI): given a pattern in the form of a sequence M of colored blocks, where each block contains a critical character (representing a unique sequence of critical instructions potentially associated with malware but without certainty), and a program A, represented as a sequence of n colored blocks with critical characters, the goal is to find two subsequences, M' of M and A' of A, with blocks matching in color and whose critical characters form a permutation of each other. When M is a permutation in both colors and critical characters the problem is called OMDCI. If we additionally require M'=M, then the problem is called OMDCI+; if in this case d=|M| is used as a parameter, then the OMDCI+ problem is easily shown to be FPT. Our main (negative) results are on the cases when |M| is arbitrary and are summarized as follows: OMDCI+ is NP-complete, which implies OMDCI is also NP-complete. For the special case of OMDCI, deciding if the optimal solution has length 0 (i.e., deciding if no part of M appears in A) is co-NP-hard. As a result, the OMDCI problem does not admit an FPT algorithm unless P=co-NP. In summary, our results imply that using algorithms based on feature matching to identify malware or determine the absence of malware in a given low-level program are both hard.

Motivated by the recent trend of increasing number of e-cars and hybrid cars, we investigate the problem of building a red-black spanner for a mixed-charging vehicular network. In such a network, we have two kinds of gas/charging stations: electric (black) and the traditional gas (red) stations. Our requirement is that one cannot connect two gas stations directly in the spanner (i.e., no red-red edge), and the goal is to build a linear-size spanner with a bounded stretch factor under this requirement. (In 2-d, it can be shown that a spanner with an optimal stretch factor could have a quadratic size and if one is constrained to build the spanner purely from a given road network then it is impossible to obtain a bounded stretch factor.) Our main results are summarized as follows.1.In 1-d, a linear-size red-black spanner is built to satisfy the ‘no red-red edge’ requirement which achieves the optimal stretch factor.2.In 2-d and under the L2 metric, we build a linear-size red-black spanner satisfying the ‘no red-red edge’ requirement which achieves a stretch factor of 1.998.3.In 2-d and under the L1 metric, a linear-size red-black spanner is built to satisfy the ‘no red-red edge’ requirement which achieves a stretch factor of 3.613.

De novo sequencing is an important research field in proteomics, aiming to directly infer the amino acid sequence of unknown polypeptides solely based on tandem mass spectrometry (MS/MS) without relying on databases. Traditional peptide sequence prediction methods often rely on manual feature extraction and statistical models, which have certain limitations. In recent years, end-to-end models based on deep learning have significantly improved prediction accuracy. However, most existing models focus on the global Transformer architecture, with insufficient characterization of local peak cluster details. Additionally, the presence of considerable noise in mass spectrometry data and the large differences in amino acid sequences between different species leave room for improvement in accuracy on species-specific datasets. The key innovation of DyCoNovo is the introduction of a lightweight dynamic convolution network module on the basis of CasaNovo, providing spectral representations that combine global context and local details for de novo peptide sequencing research. Meanwhile, a phased contrastive learning strategy is adopted, which enhances the model's generalization ability across different species datasets and under low signal-to-noise ratio data, as well as improves training stability. On a standard benchmark covering nine species and approximately 1.5 million spectra, compared with CasaNovo, DyCoNovo achieves an average increase of 13.9 % in amino acid accuracy and 19 % in peptide accuracy. Compared with ContraNovo, the latest model adopting contrastive learning strategies, DyCoNovo shows an average increase of 2.4 % in amino acid accuracy and 2.1 % in peptide accuracy. DyCoNovo demonstrates more significant advantages in low signal-to-noise ratio data, verifying the effectiveness of dynamic convolution in robust modeling of local peak shapes. This model effectively improves the accuracy of de novo peptide sequencing.

De novo peptide sequencing infers amino acid sequences directly from tandem mass spectra without relying on protein databases. This paper present RT-GCTnovo, a model that integrates multi-scale feature extraction, gated fusion, and dynamic decoding to improve sequencing accuracy. The model employs Transformer encoders and convolutional neural networks to capture global and local spectral features, enhanced by retention time information as a new constraint. A sigmoid gating mechanism adaptively fuses these features, and a semiautoregressive decoder with dynamic mass masking iteratively refines predictions. Experimental results on multispecies datasets show that RT-GCTnovo outperforms DeepNovo, Casanovo, and TSARseqNovo, achieving 57.7 % peptide-level accuracy and 79.8% amino acid-level accuracy.

De novo peptide sequencing is important for proteomics to predict unknown peptide sequences based on mass spectrometry data. Traditional methods for peptide sequence prediction have certain limitations because they often rely on handcrafted feature extraction and statistical models. To enhance prediction accuracy and rationality, a novel de novo peptide prediction model based on Transformer and Graph Convolutional Networks (GCN) is designed. In this model BERT is firstly used to extract peptide embedding features. These extracted features are integrated with physicochemical characteristics such as precursor mass, mass-to-charge ratio (m/z), and intensity to improve global feature representation. A model of the graph structural relationships between peptides using GCN can capture complex interactions to further enhance the capability of representation. Finally, the autoregressive decoder of a GPT-2 generation module is used to generate peptide sequences. And some strategies such as Top-k sampling, Top-p sampling, and temperature control are incorporated to ensure the rationality and diversity of the generated sequences. Experimental results demonstrate that the proposed model achieves an average improvement of 5.29

In this paper, we start with a variation of the star cover problem called the Two-Squirrel problem. Given a set $P$ of $2n$ points in the plane, and two sites $c_1$ and $c_2$, compute two $n$-stars $S_1$ and $S_2$ centered at $c_1$ and $c_2$ respectively such that the maximum weight of $S_1$ and $S_2$ is minimized. This problem is strongly NP-hard by a reduction from Equal-size Set-Partition with Rationals. Then we consider two variations of the Two-Squirrel problem, namely the Two-MST and Two-TSP problem, which are both NP-hard. The NP-hardness for the latter is obvious while the former needs a non-trivial reduction from Equal-size Set-Partition with Rationals. In terms of approximation algorithms, for Two-MST and Two-TSP we give factor 3.6402 and $4+\varepsilon$ approximations respectively. Finally, we also show some interesting polynomial-time solvable cases for Two-MST.

The analysis of genomes evolving by inversions has led to a famous problem of sorting a permutation by reversals. On the other hand, during the evolution of many genomes, the involvement of repeated segments is unavoidable, which has inspired new interests on the model of sorting by signed/unsigned symmetric reversals. Recently, the signed version of this problem has been investigated. In this paper, we focus on the problem of Sorting by Unsigned Symmetric Reversals, which requires a series of unsigned symmetric reversals to transform a chromosome.A into chromosome.B, where chromosomes are represented by a sequence of unsigned symbols (possibly with repetitions). Our main result is on the special case where each symbol appears at most twice in a chromosome. We give a sufficient and necessary condition to answer the decision problem, and design an algorithm with approximation factor 1.5 for the corresponding optimization problem, which runs in O(n(4)) time.

It is widely recognized that structural variation represents a significant source of genetic variation. As a well-known type of structural variation, reversion is studied drastically by both biologists and computer scientists. Recently, scientists have found that repetitive sequences always appear at the ends of the segment where the reversal occurred on the chromosome, which has inspired new interests in models for sorting unsigned chromosomes by symmetric reversals (abbreviated MUSR). The problem of MUSR asks for a minimum number of unsigned symmetric reversals to transform a chromosome S into another chromosome T, and requires symmetric reversals to be performed on the segment flanked by the same letter. In this paper, we show that MUSR is NP-hard through an intricate reduction from the MAX-(3,B2)-SAT problem. Moreover, we provide an innovative depiction of the optimal solution and then develop an improved approximation algorithm that ensures the approximation factor of 2 ln3+7/8 (approximately 1.15) and a time complexity of O(n(2)).