Two Plane-Probing Algorithms for the Computation of the Normal Vector to a Digital Plane

Digital planes are sets of integer points located between two parallel planes. We present a new algorithm that computes the normal vector of a digital plane given only a predicate “is a point x in the digital plane or not”. In opposition to classical recognition algorithm, this algorithm decides on-the-fly which points to test in order to output at the end the exact surface characteristics of the plane. We present two variants: the H-algorithm, which is purely local, and the R-algorithm which probes further along rays coming out from the local neighborhood tested by the H-algorithm. Both algorithms are shown to output the correct normal to the digital planes if the starting point is a lower leaning point. The worst-case time complexity is in \(O(\omega )\) for the H-algorithm and \(O(\omega \log \omega )\) for the R-algorithm, where \(\omega \) is the arithmetic thickness of the digital plane. In practice, the H-algorithm often outputs a reduced basis of the digital plane while the R-algorithm always returns a reduced basis. Both variants perform much better than the theoretical bound, with an average behavior close to \(O(\log \omega )\). Finally, we show how this algorithm can be used to analyze the geometry of arbitrary digital surfaces, by computing normals and identifying convex, concave or saddle parts of the surface. This paper is an extension of Lachaud et al. (Proceedings of 19th IAPR international conference discrete geometry for computer imagery (DGCI’2016), Nantes, France. Springer, Cham, 2016).     

A fault tolerant peer-to-peer spatial data structure

In recent years, many layered indexing techniques over distributed hash table (DHT)-based peer-to-peer (P2P) systems have been proposed to realize distributed range search. In this paper, we present a fault tolerant constant degree dynamic Distributed Spatial Data Structure called DSDS that supports orthogonal range search on a set of N d-dimensional points published on n nodes. We describe a total order binary relation algorithm to publish points among supernodes and determine supernode keys. A non-redundant rainbow skip graph is used to coordinate message passing among nodes. The worst case orthogonal range search cost in a d-dimensional DSDS with n nodes is \(O\left (\log n+m+\frac {K}{B}\right )\) messages, where m is the number of nodes intersecting the query, K is the number of points reported in range, and B is the number of points that can fit in one message. A complete backup copy of data points stored in other nodes provides redundancy for our DSDS. This redundancy permits answering a range search query in the case of failure of a single node. For single node failure, the DSDS routing system can be recovered to a fully functional state at a cost of O(log n) messages. Backup sets in DSDS nodes are used to first process a query in the most efficient dimension, and then used to process a query containing the data in a failed node in d-dimensional space. The DSDS search algorithm can process queries in d-dimensional space and still tolerate failure of one node. Search cost in the worst case with a failed node increases to \(O\left (d\log n+dm+\frac {K}{B}\right )\) messages for d dimensions.     

On fractional faces of the metric polytope

The integrality recognition problem is considered on a sequence M _{n, k} of nested relaxations of a Boolean quadric polytope, including the rooted semimetric M _n and metric M _{n, 3} polytopes. The constraints of the metric polytope cut off all faces of the rooted semimetric polytope that contain only fractional vertices. This makes it possible to solve the integrality recognition problem on M _n in polynomial time. To solve the integrality recognition problem on the metric polytope, we consider the possibility of cutting off all fractional faces of M _{n, 3} by a certain relaxation M _{n, k} . The coordinates of points of the metric polytope are represented in homogeneous form as a three-dimensional block matrix. We show that in studying the question of cutting off the fractional faces of the metric polytope, it is sufficient to consider only constraints in the form of triangle inequalities.     

A new strategy for generating shortest addition sequences

An addition sequence problem is given a set of numbers X = {n ₁, n ₂, . . . , n _m }, what is the minimal number of additions needed to compute all m numbers starting from 1? This problem is NP-complete. In this paper, we present a branch and bound algorithm to generate an addition sequence with a minimal number of elements for a set X by using a new strategy. Then we improve the generation by generalizing some results on addition chains (m = 1) to addition sequences and finding what we will call a presumed upper bound for each n _j , 1 ≤ j ≤ m, in the search tree.     

An Improved Approximation Algorithm for the Most Points Covering Problem

In this paper we present a polynomial time approximation scheme for the most points covering problem. In the most points covering problem, n points in R ², r>0, and an integer m>0 are given and the goal is to cover the maximum number of points with m disks with radius r. The dual of the most points covering problem is the partial covering problem in which n points in R ² are given, and we try to cover at least p≤n points of these n points with the minimum number of disks. Both these problems are NP-hard. To solve the most points covering problem, we use the solution of the partial covering problem to obtain an upper bound for the problem and then we generate a valid solution for the most points covering problem by a careful modification of the partial covering solution. We first present an improved approximation algorithm for the partial covering problem which has a better running time than the previous algorithm for this problem. Using this algorithm, we attain a \((1 - \frac{{2\varepsilon }}{{1 +\varepsilon }})\)-approximation algorithm for the most points covering problem. The running time of our algorithm is \(O((1+\varepsilon )mn+\epsilon^{-1}n^{4\sqrt{2}\epsilon^{-1}+2}) \) which is polynomial with respect to both m and n, whereas the previously known algorithm for this problem runs in \(O(n \log n +n\epsilon^{-6m+6} \log (\frac{1}{\epsilon}))\) which is exponential regarding m.     

Bimagic Vertex Labelings

The notion of the equivalence of vertex labelings on a given graph is introduced. The equivalence of three bimagic labelings for regular graphs is proved. A particular solution is obtained for the problem of the existence of a 1-vertex bimagic vertex labeling of multipartite graphs, namely, for graphs isomorphic with K_{n, n, m}. It is proved that the sequence of bi-regular graphs K_n(ij)?=?((K_n???1???M)?+?K₁)???(u_nu_i)???(u_nu_j) admits 1-vertex bimagic vertex labeling, where u_i, u_j is any pair of non-adjacent vertices in the graph K_n???1???M, u_n is a vertex of K₁, M is perfect matching of the complete graph K_n???1. It is established that if an r-regular graph G of order n is distance magic, then graph G + G has a 1-vertex bimagic vertex labeling with magic constants (n?+?1)(n?+?r)/2?+?n² and (n?+?1)(n?+?r)/2?+?nr. Two new types of graphs that do not admit 1-vertex bimagic vertex labelings are defined.     

On Metric Dimension of Nonbinary Hamming Spaces

For q-ary Hamming spaces we address the problem of the minimum number of points such that any point of the space is uniquely determined by its (Hamming) distances to them. It is conjectured that for a fixed q and growing dimension n of the Hamming space this number asymptotically behaves as 2n/ log _q n. We prove this conjecture for q = 3 and q = 4; for q = 2 its validity has been known for half a century.     

Chain sequences with ordered enclosing

To enhance the efficiency of numerical control machines for cutting of sheet material, a method of searching for the cutter trajectory as a path with ordered enclosing in the flat graph G = (V, E) corresponding to the cutting design is proposed. It is shown that this path can be represented as a sequence of no more than n/2 + 1 chains with ordered enclosing, where n is the number of odd-order vertices in G. An algorithm for finding the sought sequence of chains, with a complexity of O(|E|) is developed.     

Property-Preserving Data Reconstruction

We initiate a new line of investigation into online property-preserving data reconstruction. Consider a dataset which is assumed to satisfy various (known) structural properties; e.g., it may consist of sorted numbers, or points on a manifold, or vectors in a polyhedral cone, or codewords from an error-correcting code. Because of noise and errors, however, an (unknown) fraction of the data is deemed unsound, i.e., in violation with the expected structural properties. Can one still query into the dataset in an online fashion and be provided data that is always sound? In other words, can one design a filter which, when given a query to any item I in the dataset, returns a sound item J that, although not necessarily in the dataset, differs from I as infrequently as possible. No preprocessing should be allowed and queries should be answered online.We consider the case of a monotone function. Specifically, the dataset encodes a function f:{1,…,n}?? R that is at (unknown) distance ε from monotone, meaning that f can—and must—be modified at ε n places to become monotone.Our main result is a randomized filter that can answer any query in O(log?² nlog? log?n) time while modifying the function f at only O(ε n) places. The amortized time over n function evaluations is O(log?n). The filter works as stated with probability arbitrarily close to 1. We provide an alternative filter with O(log?n) worst case query time and O(ε nlog?n) function modifications. For reconstructing d-dimensional monotone functions of the form f:{1,…,n} ^d ? ? R, we present a filter that takes (2 ^O(d)(log?n)^4d?2log?log?n) time per query and modifies at most O(ε n ^d ) function values (for constant d).     

Negation-Limited Complexity of Parity and Inverters

In negation-limited complexity, one considers circuits with a limited number of NOT gates, being motivated by the gap in our understanding of monotone versus general circuit complexity, and hoping to better understand the power of NOT gates. We give improved lower bounds for the size (the number of AND/OR/NOT) of negation-limited circuits computing Parity and for the size of negation-limited inverters. An inverter is a circuit with inputs x ₁,…,x _n and outputs ¬ x ₁,…,¬ x _n . We show that: (a) for n=2 ^r ?1, circuits computing Parity with r?1 NOT gates have size at least 6n?log?₂(n+1)?O(1), and (b) for n=2 ^r ?1, inverters with r NOT gates have size at least 8n?log?₂(n+1)?O(1). We derive our bounds above by considering the minimum size of a circuit with at most r NOT gates that computes Parity for sorted inputs x ₁≥???≥x _n . For an arbitrary r, we completely determine the minimum size. It is 2n?r?2 for odd n and 2n?r?1 for even n for ?log?₂(n+1)??1≤r≤n/2, and it is ?3n/2??1 for r≥n/2. We also determine the minimum size of an inverter for sorted inputs with at most r NOT gates. It is 4n?3r for ?log?₂(n+1)?≤r≤n. In particular, the negation-limited inverter for sorted inputs due to Fischer, which is a core component in all the known constructions of negation-limited inverters, is shown to have the minimum possible size. Our fairly simple lower bound proofs use gate elimination arguments in a somewhat novel way.     

Star reduction among minimal length addition chains

An addition chain for a natural number n is a sequence \({1=a_0 < a_1 < \cdots < a_r=n}\) of numbers such that for each 0 < i ≤ r, a _i  = a _j  + a _k for some 0 ≤ k ≤ j < i. The minimal length of an addition chain for n is denoted by ?(n). If j = i ? 1, then step i is called a star step. We show that there is a minimal length addition chain for n such that the last four steps are stars. Then we conjecture that there is a minimal length addition chain for n such that the last \({\lfloor\frac{\ell(n)}{2}\rfloor}\)-steps are stars. We verify that the conjecture is true for all numbers up to 2¹⁸. An application of the result and the conjecture to generate a minimal length addition chain reduce the average CPU time by 23–29% and 38–58% respectively, and memory storage by 16–18% and 26–45% respectively for m-bit numbers with 14 ≤ m ≤ 22.     

On the Advice Complexity of the k-server Problem Under Sparse Metrics

We consider the k-Server problem under the advice model of computation when the underlying metric space is sparse. On one side, we introduce Θ(1)-competitive algorithms for a wide range of sparse graphs. These algorithms require advice of (almost) linear size. We show that for graphs of size N and treewidth α, there is an online algorithm that receives O (n(log α + log log N))^* bits of advice and optimally serves any sequence of length n. We also prove that if a graph admits a system of μ collective tree (q, r)-spanners, then there is a (q + r)-competitive algorithm which requires O (n(log μ + log log N)) bits of advice. Among other results, this gives a 3-competitive algorithm for planar graphs, when provided with O (n log log N) bits of advice. On the other side, we prove that advice of size Ω(n) is required to obtain a 1-competitive algorithm for sequences of length n even for the 2-server problem on a path metric of size N ≥ 3. Through another lower bound argument, we show that at least \(\frac {n}{2}(\log \alpha - 1.22)\) bits of advice is required to obtain an optimal solution for metric spaces of treewidth α, where 4 ≤ α < 2k.     

Efficient and secure multi secret sharing schemes based on boolean XOR and arithmetic modulo

Multi Secret Sharing (MSS) scheme is an efficient method of transmitting more than one secret securely. In (n, n)-MSS scheme n secrets are used to create n shares and for reconstruction, all n shares are required. In state of the art schemes n secrets are used to construct n or n + 1 shares, but one can recover partial secret information from less than n shares. There is a need to develop an efficient and secure (n, n)-MSS scheme so that the threshold property can be satisfied. In this paper, we propose three different (n, n)-MSS schemes. In the first and second schemes, Boolean XOR is used and in the third scheme, we used Modular Arithmetic. For quantitative analysis, Similarity metrics, Structural, and Differential measures are considered. A proposed scheme using Modular Arithmetic performs better compared to Boolean XOR. The proposed (n, n)-MSS schemes outperform the existing techniques in terms of security, time complexity, and randomness of shares.     

On conditional diagnosability and reliability of the BC networks

An n-dimensional bijective connection network (in brief, BC network), denoted by X _n , is an n-regular graph with 2 ⁿ nodes and n2 ^n?1 edges. Hypercubes, crossed cubes, twisted cubes, and Möbius cubes all belong to the class of BC networks (Fan and He in Chin. J. Comput. 26(1):84–90, [2003]). We prove that the super connectivity of X _n is 2n?2 for n≥3 and the conditional diagnosability of X _n is 4n?7 for n≥5. As a corollary of this result, we obtain the super connectivity and conditional diagnosability of the hypercubes, twisted cubes, crossed cubes, and Möbius cubes.     

The hierarchical Petersen network: a new interconnection network with fixed degree

Network cost and fixed-degree characteristic for the graph are important factors to evaluate interconnection networks. In this paper, we propose hierarchical Petersen network (HPN) that is constructed in recursive and hierarchical structure based on a Petersen graph as a basic module. The degree of HPN(n) is 5, and HPN(n) has \(10^n\) nodes and \(2.5 \times 10^n\) edges. And we analyze its basic topological properties, routing algorithm, diameter, spanning tree, broadcasting algorithm and embedding. From the analysis, we prove that the diameter and network cost of HPN(n) are \(3\log _{10}N-1\) and \(15 \log _{10}N-1\), respectively, and it contains a spanning tree with the degree of 4. In addition, we propose link-disjoint one-to-all broadcasting algorithm and show that HPN(n) can be embedded into FP\(_k\) with expansion 1, dilation 2k and congestion 4. For most of the fixed-degree networks proposed, network cost and diameter require \(O(\sqrt{N})\) and the degree of the graph requires O(N). However, HPN(n) requires O(1) for the degree and \(O(\log _{10}N)\) for both diameter and network cost. As a result, the suggested interconnection network in this paper is superior to current fixed-degree and hierarchical networks in terms of network cost, diameter and the degree of the graph.     

A novel weighting scheme for random k-SAT关于随机 k-SAT 的新加权方法

Consider a random k-conjunctive normal form F_k(n, rn) with n variables and rn clauses. We prove that if the probability that the formula F_k(n, rn) is satisfiable tends to 0 as n→∞, then r ? 2.83, 8.09, 18.91, 40.81, and 84.87, for k = 3, 4, 5, 6, and 7, respectively.     

Tilings of nonorientable surfaces by Steiner triple systems

A Steiner triple system of order n (for short, STS(n)) is a system of three-element blocks (triples) of elements of an n-set such that each unordered pair of elements occurs in precisely one triple. Assign to each triple (i,j,k) ? STS(n) a topological triangle with vertices i, j, and k. Gluing together like sides of the triangles that correspond to a pair of disjoint STS(n) of a special form yields a black-and-white tiling of some closed surface. For each n ≡ 3 (mod 6) we prove that there exist nonisomorphic tilings of nonorientable surfaces by pairs of Steiner triple systems of order n. We also show that for half of the values n ≡ 1 (mod 6) there are nonisomorphic tilings of nonorientable closed surfaces.     

On (<Emphasis Type="Italic">a</Emphasis>, <Emphasis Type="Italic">d</Emphasis> )-Distance Anti-Magic and 1-Vertex Bimagic Vertex Labelings of Certain Types of Graphs

The results for the corona P _n ?°?P₁ are generalized, which make it possible to state that P _n ?°?P₁ is not an ( a, d)-distance antimagic graph for arbitrary values of a and d. A condition for the existence of an ( a, d)-distance antimagic labeling of a hypercube Q _n is obtained. Functional dependencies are found that generate this type of labeling for Q _n . It is proved by the method of mathematical induction that Q _n is a (2 ⁿ ?+?n???1,?n???2) -distance antimagic graph. Three types of graphs are defined that do not allow a 1-vertex bimagic vertex labeling. A relation between a distance magic labeling of a regular graph G and a 1-vertex bimagic vertex labeling of G?∪?G is established.     

Guaranteed strategies for escaping encirclement

This paper considers a conflict situation on the plane as follows. A fast evader E has to break out the encirclement of slow pursuers P _{j1,...,j n} = {P _j1,..., P _jn }, n ≥ 3, with a miss distance not smaller than r ≥ 0. First, we estimate the minimum guaranteed miss distance from E to a pursuer P _a , a ∈ {j ₁,..., j _n }, when the former moves along a given straight line. Then the obtained results are used to calculate the guaranteed estimates to a group of two pursuers P _b,c = {P _b , P _c }, b, c ∈ {j ₁,..., j _n }, b ≠ c, when E maneuvers by crossing the rectilinear segment P _b P _c , and the state passes to the domain of the game space where E applies a strategy under which the miss distance to any of the pursuers is not decreased. In addition, we describe an approach to the games with a group of pursuers P _{j1,... jn} , n ≥ 3, in which E seeks to break out the encirclement by passing between two pursuers P _b and P _c , entering the domain of the game space where E can increase the miss distance to all pursuers by straight motion. By comparing the guaranteed miss distances with r for all alternatives b, c ∈ {j ₁,..., j _n }, b ≠ c, and a ? {b, c}, it is possible to choose the best alternative and also to extract the histories of the game in which the designed evasion strategies guarantee a safe break out from the encirclement.     

Mutually independent Hamiltonian cycles in dual-cubes

The hypercube family Q _n is one of the most well-known interconnection networks in parallel computers. With Q _n , dual-cube networks, denoted by DC _n , was introduced and shown to be a (n+1)-regular, vertex symmetric graph with some fault-tolerant Hamiltonian properties. In addition, DC _n ’s are shown to be superior to Q _n ’s in many aspects. In this article, we will prove that the n-dimensional dual-cube DC _n contains n+1 mutually independent Hamiltonian cycles for n≥2. More specifically, let v _i ∈V(DC _n ) for 0≤i≤|V(DC _n )|?1 and let \(\langle v_{0},v_{1},\ldots ,v_{|V(\mathit{DC}_{n})|-1},v_{0}\rangle\) be a Hamiltonian cycle of DC _n . We prove that DC _n contains n+1 Hamiltonian cycles of the form \(\langle v_{0},v_{1}^{k},\ldots,v_{|V(\mathit{DC}_{n})|-1}^{k},v_{0}\rangle\) for 0≤k≤n, in which v _i ^k ≠v _i ^k′ whenever k≠k′. The result is optimal since each vertex of DC _n has only n+1 neighbors.