Secure Outsourcing of Matrix Determinant Computation under the Malicious Cloud

Computing the determinant of large matrix is a time-consuming task, which is appearing more and more widely in science and engineering problems in the era of big data. Fortunately, cloud computing can provide large storage and computation resources, and thus, act as an ideal platform to complete computation outsourced from resource-constrained devices. However, cloud computing also causes security issues. For example, the curious cloud may spy on user privacy through outsourced data. The malicious cloud violating computing scripts, as well as cloud hardware failure, will lead to incorrect results. Therefore, we propose a secure outsourcing algorithm to compute the determinant of large matrix under the malicious cloud mode in this paper. The algorithm protects the privacy of the original matrix by applying row/column permutation and other transformations to the matrix. To resist malicious cheating on the computation tasks, a new verification method is utilized in our algorithm. Unlike previous algorithms that require multiple rounds of verification, our verification requires only one round without trading off the cheating detectability, which greatly reduces the local computation burden. Both theoretical and experimental analysis demonstrate that our algorithm achieves a better efficiency on local users than previous ones on various dimensions of matrices, without sacrificing the security requirements in terms of privacy protection and cheating detectability.

Exponential multivalued forbidden configurations

The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$. Comment: 12 pages; v3: formatted for DMTCS; v2: Corollary 3.2 added, typos fixed, some proofs clarified

Guiro: User-Guided Matrix Reordering Preprint Behrisch2019

Matrix representations are one of the main established and empirically proven to be effective visualization techniques for relational (or network) data. However, matrices —similar to node-link diagrams— are most effective if their layout reveals the underlying data topology. Given the many developed algorithms, a practical problem arises: “Which matrix reordering algorithm should I choose for my dataset at hand?” To make matters worse, different reordering algorithms applied to the same dataset may let significantly different visual matrix patterns emerge. This leads to the question of trustworthiness and explainability of these fully automated, often heuristic, black-box processes. We present GUIRO, a Visual Analytics system that helps novices, network analysts, and algorithm designers to open the black-box. Users can investigate the usefulness and expressiveness of 70 accessible matrix reordering algorithms. For network analysts, we introduce a novel model space representation and two interaction techniques for a user-guided reordering of rows or columns, and especially groups thereof (submatrix reordering). These novel techniques contribute to the understanding of the global and local dataset topology. We support algorithm designers by giving them access to 16 reordering quality metrics and visual exploration means for comparing reordering implementations on a row/column permutation level. We evaluated GUIRO in a guided explorative user study with 12 subjects, a case study demonstrating its usefulness in a real-world scenario, and through an expert study gathering feedback on our design decisions. We found that our proposed methods help even inexperienced users to understand matrix patterns and allow a user-guided steering of reordering algorithms. GUIRO helps to increase the transparency of matrix reordering algorithms, thus helping a broad range of users to get a better insight into the complex reordering process, in turn supporting data and reordering algorithm insights.

Forbidden Families of Minimal Quadratic and Cubic Configurations

A matrix is simple if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a configuration, denoted $F\prec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $\mathcal{F}$ be a family of matrices. We define the extremal function $\text{forb}(m, \mathcal{F}) = \max\{|A|\colon A \text{ is an }m-\text{rowed simple matrix and has no configuration } F\in\mathcal{F}\}$. We consider pairs $\mathcal{F}=\{F_1,F_2\}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $\text{forb}(m, F_i)$ has greater asymptotic growth than $\text{forb}(m, \mathcal{F})$, extending research started by Anstee and Koch.

Forbidden Berge Hypergraphs

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a Berge hypergraph if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $\|A\|$ denote the number of columns in $A$, we define the extremal function $\mathrm{Bh}(m,{ F})=\max\{\|A\|\,:\, A \hbox{ }m\hbox{-rowed simple matrix and no Berge hypergraph }F\}$. We determine the asymptotics of $\mathrm{Bh}(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determining the maximum number of copies of $K_r$ in a $m$-vertex graph that has no $K_{s,t}$ subgraph, a problem studied by Alon and Shikhelman.

Large forbidden configurations and design theory

Let forb(m, F) denote the maximum number of columns possible in a (0, 1)-matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F. We consider cases where the configuration F has a number of columns that grows with m. For a k × l matrix G, define s · G to be the concatenation of s copies of G. In a number of cases we determine forb(m, mα · G) is Θ(mk+α). Results of Keevash on the existence of designs provide constructions that can be used to give asymptotic lower bounds. An induction idea of Anstee and Lu is useful in obtaining upper bounds.

Construction of non-isomorphic covering arrays

A covering array CA([Formula: see text]) of strength [Formula: see text] and order [Formula: see text] is an [Formula: see text] array over [Formula: see text] with the property that every [Formula: see text] subarray covers all members of [Formula: see text] at least once. When the value of [Formula: see text] is the minimum possible it is named as the covering array number (CAN) i.e. [Formula: see text]. Two CAs are isomorphic if one of them can be derived from the other by a combination of a row permutation, a column permutation, and a symbol permutation in a subset of columns. Isomorphic CAs have equivalent coverage properties, and can be considered as the same CA; the truly distinct CAs are those which are non-isomorphic among them. An interesting and hard problem is to construct all the non-isomorphic CAs that exist for a particular combination of the parameters [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We constructed the non-isomorphic CAs for 70 combinations of values of the parameters [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the results allow us to determine CAN(3,13,2) =16, CAN(3,14,2) =16, CAN(3,15,2) =17, CAN(3,16,2) =17, and CAN(2,10,3) =14. The exact lower bound for these covering arrays numbers had not been determined before either by computational search or by algebraic analysis.

A Survey of Forbidden Configuration Results

Let $F$ be a $k\times \ell$ (0,1)-matrix. We say a (0,1)-matrix $A$ has $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a trace and in the language of hypergraphs a configuration is a subhypergraph.Let $F$ be a given $k\times \ell$ (0,1)-matrix. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. The matrix $F$ need not be simple. We define $\hbox{forb}(m,F)$ as the maximum number of columns of any simple $m$-rowed matrix $A$ which do not contain $F$ as a configuration. Thus if $A$ is an $m\times n$ simple matrix which has no submatrix which is a row and column permutation of $F$ then $n\le\hbox{forb}(m,F)$. Or alternatively if $A$ is an $m\times (\hbox{forb}(m,F)+1)$ simple matrix then $A$ has a submatrix which is a row and column permutation of $F$. We call $F$ a forbidden configuration. The fundamental result is due to Sauer, Perles and Shelah, Vapnik and Chervonenkis. For $K_k$ denoting the $k\times 2^k$ submatrix of all (0,1)-columns on $k$ rows, then $\hbox{forb}(m,K_k)=\binom{m}{k-1}+\binom{m}{k-2}+\cdots \binom{m}{0}$. We seek asymptotic results for $\hbox{forb}(m,F)$ for a fixed $F$ and as $m$ tends to infinity . A conjecture of Anstee and Sali predicts the asymptotically best constructions from which to derive the asymptotics of $\hbox{forb}(m,F)$. The conjecture has helped guide the research and has been verified for $k\times \ell$ $F$ with $k=1,2,3$ and for simple $F$ with $k=4$ as well as other cases including $\ell=1,2$. We also seek exact values for $\hbox{forb}(m,F)$.

Genetic Algorithms Applied to Problems of Forbidden Configurations

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say a (0,1)-matrix $A$ avoids $F$ (as a configuration) if there is no submatrix of $A$ which is a row and column permutation of $F$. Let $\|{A}\|$ denote the number of columns of $A$. We define $\mathrm{forb}(m,F)=\max\{\|{A}\|\ : A$ is an $m$-rowed simple matrix that avoids $F \}$. Define an extremal matrix as an $m$-rowed simple matrix $A$ with that avoids $F$ and $\|{A}\|=\mathrm{forb}(m,F)$. We describe the use of Local Search Algorithms (in particular a Genetic Algorithm) for finding extremal matrices. We apply this technique to two forbidden configurations in turn, obtaining a guess for the structure of an $m\times\mathrm{forb}(m,F)$ simple matrix avoiding $F$ and then proving the guess is indeed correct. The Genetic Algorithm was also helpful in finding the proof.

Small forbidden configurations V: Exact bounds for 4 × 2 cases

The present paper continues the work begun by Anstee, Ferguson, Griggs, Kamoosi and Sali on small forbidden configurations. We define a matrix to besimpleif it is a (0, 1)-matrix with no repeated columns. LetFbe ak× (0, 1)-matrix (the forbidden configuration). AssumeAis anm×nsimple matrix which has no submatrix which is a row and column permutation ofF. We define forb (m, F) as the largestn, which would depend onmandF, so that such anAexists.DefineFabcdas the (a+b+c+d) × 2 matrix consisting ofarows of [11],brows of [10],crows of [01] anddrows of [00]. With the exception ofF2110, we compute forb (m; Fabcd) for all 4 × 2Fabcd. A number of cases follow easily from previous results and general observations. A number follow by clever inductions based on a single column such as forb (m; F1111) = 4m− 4 and forb (m; F1210) = forb (m; F1201) = forb (m; F0310) = (2m)+m+ 2 (proofs are different). A different idea proves forb (m; F0220) = (2m) + 2m− 1 with the forbidden configuration being related to a result of Kleitman. Our results suggest that determining forb (m; F2110) is heavily related to designs and we offer some constructions of matrices avoidingF2110using existing designs.