Theoretical Computer Science最新文献

Clustering is a basic technology in data mining, and similarity measurement plays a crucial role in it. The existing clustering algorithms, especially those for social networks, pay more attention to users' properties while ignoring the global measurement across social relationships. In this paper, a new clustering algorithm is proposed, which not only considers the distance of users' properties but also considers users' social influence. Social influence can be further divided into mutual influence and self influence. With mutual influence, we can deal with users' interests and measure their similarities by introducing areas and activities, thus better weighing the influence between them in an indirect way. Separately, we formulate a new propagation model, PR-Threshold++, by merging the PageRank algorithm and Linear Threshold model, to model the self influence. Based on that, we design a novel similarity by exploiting users' distance, mutual influence, and self influence. Finally, we adjust K-medoids according to our similarity and use real-world datasets to evaluate their performance in intensive simulations.

Two crucial metrics used to evaluate the fault tolerance of interconnection networks are connectivity and diagnosability. By improving the connectivity and diagnosability of the interconnection network, its fault tolerance can be enhanced. In this paper, we focus on determining the g-extra connectivity $(0\le g\le 10)$ of the divide-and-swap cube $DS{C}_n$, as well as its diagnosability based on the pessimistic diagnosis strategy and g-extra precise diagnosis strategy, under the PMC and MM* models. The research analysis suggests that compared with some other connectivity and diagnosability of $DS{C}_n$, such as classical connectivity, structure connectivity, super connectivity, and classical diagnosability, the extra connectivity/diagnosability and pessimistic diagnosability of $DS{C}_n$ enable it to have a higher fault tolerance. Moreover, we propose two $O(N{\log }_{2}N)$ effective diagnosis algorithms of $DS{C}_n$: the g-extra diagnosis algorithm (EX-Diagnosis${}_{DS{C}_n}$) and the pessimistic diagnosis algorithm (PE-Diagnosis${}_{DS{C}_n}$), where the EX-Diagnosis${}_{DS{C}_n}$ algorithm can accurately diagnose the state of all processors in $DS{C}_n$.

Partial key exposure attacks pose a significant threat to RSA-type cryptosystems. These attacks factorize the RSA modulus by utilizing partial knowledge of the decryption exponent, which is typically revealed by side-channel attacks, cold boot attacks, etc. Such partial information is often located in non-consecutive blocks. However, the majority of the proposed attacks on Prime Power RSA have only considered a single unexposed block. Meanwhile, related attacks are incapable of being expanded to multiple unexposed blocks or achieving optimal results.

In this paper, we propose partial key exposure attacks on Prime Power RSA modulus $N=p^r{q}^l$ with n unknown blocks, where $n\ge 2$. We reduce this extended attack to solving multivariate linear modular equations and apply lattice-based approaches, including Herrmann-May's method (ASIACRYPT'08), Takayasu-Kunihiro's method (ACISP'13), and Lu-Zhang-Peng-Lin's method (ASIACRYPT'15), to solve them. Furthermore, we improve Lu et al.'s method by adding helpful polynomials and removing unhelpful polynomials to construct a better lattice basis. We also extend Lu et al.'s method by introducing a new parameter to make the lattice basis construction more flexible. Our improved and extended methods can be used for attacks when $l=1$ and $l\ge 1$, respectively. These new attacks require less partial information than previous methods. For example, in the case where $n=2$, we reduce the amount of partial information needed from 80.7% to 77.8% when $r=2,l=1$, and from 64.0% to 44.9% when $r=3,l=2$.

We prove a complexity trichotomy theorem for a class of partition functions over k-regular graphs, where the signature is complex valued and not necessarily symmetric. In details, we establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: For every parameter setting in $C$ for the spin system, the partition function is either (1) computable in polynomial time for every graph, or (2) #P-hard for general graphs but computable in polynomial time for planar graphs, or (3) #P-hard even for planar graphs.

We introduce the Observation Route Problem (ORP) defined as follows: Given a set of n pairwise disjoint obstacles (regions) in the plane, find a shortest tour (route) such that an observer walking along this tour can see (observe) each obstacle from some point of the tour. The observer does not need to see the entire boundary of an obstacle. The tour is not allowed to intersect the interior of any region (i.e., the regions are obstacles and therefore out of bounds). The problem exhibits similarity to both the Traveling Salesman Problem with Neighborhoods (TSPN) and the External Watchman Route Problem (EWRP). We distinguish two variants: the range of visibility is either limited to a bounding rectangle, or unlimited. We obtain the following results:

(I) Given a family of n disjoint convex bodies in the plane, computing a shortest observation route does not admit a $(c\log n)$-approximation unless $P=\mathrm{NP}$ for an absolute constant $c>0$. (This holds for both limited and unlimited vision.)

(II) Given a family of disjoint convex bodies in the plane, computing a shortest external watchman route is $\mathrm{NP}$-hard. (This holds for both limited and unlimited vision; and even for families of axis-aligned squares.)

(III) Given a family of n disjoint fat convex polygons in the plane, an observation tour whose length is at most $O(\log n)$ times the optimal can be computed in polynomial time. (This holds for limited vision.)

(IV) For every $n\ge 5$, there exists a convex polygon with n sides and all angles obtuse such that its perimeter is not a shortest external watchman route. This refutes a conjecture by Absar and Whitesides (2006).

In this paper, we present lower bounds and algorithmic constructions of union-intersection-bounded families of sets. The lower bound is established using the Lovász Local Lemma. This bound matches the best known bound for the size of union-intersection-bounded families of sets. We then use the variable framework for the Lovász Local Lemma, to discuss an algorithm that outputs explicit constructions that attain the lower bound. The algorithm has polynomial complexity in the number of points in the family.

A weak dominance drawing Γ of a DAG $G=(V,E)$ is a d-dimensional drawing such that $D\left(u\right)<D\left(v\right)$ for every dimension D of Γ if there is a directed path from a vertex u to a vertex v in G, where $D\left(w\right)$ is the coordinate of vertex $w\in V$ in dimension D of Γ. If $D\left(u\right)<D\left(v\right)$ for every dimension D of Γ, but there is no path from u to v, we have a falsely implied path (fip). Minimizing the number of fips is an important theoretical and practical problem. Computing 2-dimensional weak dominance drawings with minimum number of fips is NP-hard. We show that this problem is FPT parameterized by the dimension d and the modular width mw. A key ingredient of our proof is the Compaction Lemma, where we show an interesting property of any weak dominance drawing of G with the minimum number of fips. This FPT result in weak dominance, which is interesting by itself because the fip-minimization problem is NP-hard, is used to prove our main contributions. Computing the dominance dimension of G, that is, the minimum number of dimensions d for which G has a d-dimensional dominance drawing (a weak dominance drawing with 0 fips), is a well-known NP-hard problem. We show that the dominance dimension of G is bounded by $\frac{mw}{2}$ (or mw, if $mw<4$) and that computing the dominance dimension of G is an FPT problem with parameter mw. As far as we know, this the first FPT-algorithm to compute the dominance dimension of a DAG.

A dynamic flow network consists of a directed graph, where nodes called sources represent locations of evacuees, and nodes called sinks represent locations of evacuation facilities. Each source and each sink are given supply representing the number of evacuees and demand representing the maximum number of acceptable evacuees, respectively. Each edge is given capacity and transit time. Here, the capacity of an edge bounds the rate at which evacuees can enter the edge per unit time, and the transit time represents the time which evacuees take to travel across the edge. The evacuation completion time is the minimum time at which each evacuee can arrive at one of the evacuation facilities. Given a dynamic flow network without sinks, once sinks are located on some nodes or edges, the evacuation completion time for this sink location is determined. We then consider the problem of locating sinks to minimize the evacuation completion time, called the sink location problem. The problems have been given polynomial-time algorithms only for limited networks such as paths [1], [2], [3], cycles [1], and trees [4], [5], [6], but no polynomial-time algorithms are known for more complex network classes. In this paper, we prove that the 1-sink location problem can be solved in polynomial-time when an input network is a grid with uniform edge capacity and transit time.

We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset $S=\{a_1,a_2,\dots ,a_n\}$ of integers $a_i\in Z^{⁎}$ (where $Z^{⁎}$ denotes the set of non-zero integers) such that ${\sum }_{i=1}^n{a}_i=0$, find a maximum cardinality partition $\{S_1,S_2,\dots ,S_k\}$ of $S$ such that, for every $1\le i\le k$, ${\sum }_{a_j\in S_i}{a}_j=0$. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in $S$ and (iii) the largest integer in $S$.

In 2023, Zhang et al. proposed a novel diagnostic parameter, namely the cyclic diagnosability and explored the cyclic diagnosability of $Q_n$. In this paper, the cyclic diagnosability of $B{H}_n$ is determined under the PMC model and the $M{M}^{⁎}$ model.