Clique Numbers of Random Subgraphs of Some Distance Graphs

We consider a class of graphs G(n, r, s) = (V (n, r),E(n, r, s)) defined as follows:

$$V(n,r) = \{ x = ({x_{1,}},{x_2}...{x_n}):{x_i} \in \{ 0,1\} ,{x_{1,}} + {x_2} + ... + {x_n} = r\} ,E(n,r,s) = \{ \{ x,y\} :(x,y) = s\} $$

where (x, y) is the Euclidean scalar product. We study random subgraphs G(G(n, r, s), p) with edges independently chosen from the set E(n, r, s) with probability p each. We find nontrivial lower and upper bounds on the clique number of such graphs.

     

Observational constraints on a hyperbolic potential in brane-world inflation

We focus on the large field of a hyperbolic potential form, which is characterized by a parameter f, in the framework of the brane-world inflation in Randall-Sundrum-II model. From the observed form of the power spectrum P _R (k), the parameter f should be of order 0.1m _p to 0.001m _p , the brane tension must be in the range λ ~ (1?10)×10⁵⁷ GeV⁴, and the energy scale is around V₀ ^1/4 ~ 10¹⁵ GeV. We find that the inflationary parameters (n _s , r, and dn _s /d(ln k) depend only on the number of e-folds N. The compatibility of these parameters with the last Planck measurements is realized with large values of N.     

On resol vability of Steine r systems S( v = 2m, 4, 3) of rank r ≤ v − m + 1 o ve r $$\mathbb{F}_2 $$

Two new constructions of Steiner quadruple systems S(v, 4, 3) are given. Both preserve resolvability of the original Steiner system and make it possible to control the rank of the resulting system. It is proved that any Steiner system S(v = 2 ^m , 4, 3) of rank r ≤ v ? m + 1 over F₂ is resolvable and that all systems of this rank can be constructed in this way. Thus, we find the number of all different Steiner systems of rank r = v ? m + 1.     

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.     

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.     

On ℤ<Subscript>4</Subscript>-linear codes with the parameters of Reed-Muller codes

For any pair of integers r and m, 0 ≤ r ≤ m, we construct a class of quaternary linear codes whose binary images under the Gray map are codes with the parameters of the classical rth-order Reed-Muller code RM(r, m).     

On the <Emphasis Type="Italic">k</Emphasis>-accessibility of cores of <Emphasis Type="Italic">TU</Emphasis>-cooperative games

This paper proposes a strengthening of the author’s core-accessibility theorem for balanced TU-cooperative games. The obtained strengthening relaxes the influence of the nontransitivity of classical domination αv on the quality of the sequential improvement of dominated imputations in a game v. More specifically, we establish the k-accessibility of the core C(α _v ) of any balanced TU-cooperative game v for all natural numbers k: for each dominated imputation x, there exists a converging sequence of imputations x₀, x₁,..., such that x₀ = x, lim x _r ∈ C(α _v ) and x_r?m is dominated by any successive imputation x _r with m ∈ [1, k] and r ≥ m. For showing that the TU-property is essential to provide the k-accessibility of the core, we give an example of an NTU-cooperative game G with a ”black hole” representing a nonempty closed subset B ? G(N) of dominated imputations that contains all the α _G -monotonic sequential improvement trajectories originating at any point x ∈ B.     

Tracking frequent items over distributed probabilistic data

Tracking frequent items (also called heavy hitters) is one of the most fundamental queries in real-time data due to its wide applications, such as logistics monitoring, association rule based analysis, etc. Recently, with the growing popularity of Internet of Things (IoT) and pervasive computing, a large amount of real-time data is usually collected from multiple sources in a distributed environment. Unfortunately, data collected from each source is often uncertain due to various factors: imprecise reading, data integration from multiple sources (or versions), transmission errors, etc. In addition, due to network delay and limited by the economic budget associated with large-scale data communication over a distributed network, an essential problem is to track the global frequent items from all distributed uncertain data sites with the minimum communication cost. In this paper, we focus on the problem of tracking distributed probabilistic frequent items (TDPF). Specifically, given k distributed sites S = {S ₁, … , S _k }, each of which is associated with an uncertain database \(\mathcal {D}_{i}\) of size n _i , a centralized server (or called a coordinator) H, a minimum support ratio r, and a probabilistic threshold t, we are required to find a set of items with minimum communication cost, each item X of which satisfies P r(s u p(X) ≥ r × N) > t, where s u p(X) is a random variable to describe the support of X and \(N={\sum }_{i=1}^{k}n_{i}\). In order to reduce the communication cost, we propose a local threshold-based deterministic algorithm and a sketch-based sampling approximate algorithm, respectively. The effectiveness and efficiency of the proposed algorithms are verified with extensive experiments on both real and synthetic uncertain datasets.     

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.     

Optimal Path Embedding in the Exchanged Crossed Cube

The (s + t + 1)-dimensional exchanged crossed cube, denoted as ECQ(s, t), combines the strong points of the exchanged hypercube and the crossed cube. It has been proven that ECQ(s, t) has more attractive properties than other variations of the fundamental hypercube in terms of fewer edges, lower cost factor and smaller diameter. In this paper, we study the embedding of paths of distinct lengths between any two different vertices in ECQ(s, t). We prove the result in ECQ(s, t): if s ≥ 3, t ≥ 3, for any two different vertices, all paths whose lengths are between \( \max \left\{9,\left\lceil \frac{s+1}{2}\right\rceil +\left\lceil \frac{t+1}{2}\right\rceil +4\right\} \) and 2 ^s+t+1 ? 1 can be embedded between the two vertices with dilation 1. Note that the diameter of ECQ(s, t) is \( \left\lceil \frac{s+1}{2}\right\rceil +\left\lceil \frac{t+1}{2}\right\rceil +2 \). The obtained result is optimal in the sense that the dilations of path embeddings are all 1. The result reveals the fact that ECQ(s, t) preserves the path embedding capability to a large extent, while it only has about one half edges of CQ _n .     

Asymptotic bounds for the rate of colored superimposed codes

We introduce the notion of a q-ary (r ₀, r ₁, ..., r _q?1) superimposed code. We obtain upper and lower asymptotic bounds for the rate of these (colored) codes.     

Almost cover-free codes

We say that an s-subset of codewords of a code X is (s, l)-bad if X contains l other codewords such that the conjunction of these l words is covered by the disjunction of the words of the s-subset. Otherwise, an s-subset of codewords of X is said to be (s, l)-bad. A binary code X is called a disjunctive (s, l) cover-free (CF) code if X does not contain (s, l)-bad subsets. We consider a probabilistic generalization of (s, l) CF codes: we say that a binary code is an (s, l) almost cover-free (ACF) code if almost all s-subsets of its codewords are (s, l)-good. The most interesting result is the proof of a lower and an upper bound for the capacity of (s, l) ACF codes; the ratio of these bounds tends as s→∞ to the limit value log₂ e/(le).     

Multidimensional analogs of the kendall equation for priority queueing systems: computational aspects

An analogy between celebrated Kendall equation for busy periods in the system M|GI|1 and analytical results for busy periods in the priority systemsM _r |GI _r |1 is drawn. These results can be viewed as generalizations of the functional Kendall equation. The methodology and algorithms of numerical solution of recurrent functional equations which appear in the analysis of such queueing systems are developed. The efficiency of the algorithms is achieved by acceleration of the numerical procedure of solving the classical Kendall equation. An algorithm of calculation of the system workload coefficient calculation is given.     

Sleeping on the Job: Energy-Efficient and Robust Broadcast for Radio Networks

We address the problem of minimizing power consumption when broadcasting a message from one node to all the other nodes in a radio network. To enable power savings for such a problem, we introduce a compelling new data streaming problem which we call the Bad Santa problem. Our results on this problem apply for any situation where: (1) a node can listen to a set of n nodes, out of which at least half are non-faulty and know the correct message; and (2) each of these n nodes sends according to some predetermined schedule which assigns each of them its own unique time slot. In this situation, we show that in order to receive the correct message with probability 1, it is necessary and sufficient for the listening node to listen to a \(\Theta(\sqrt{n})\) expected number of time slots. Moreover, if we allow for repetitions of transmissions so that each sending node sends the message O(log?^? n) times (i.e. in O(log?^? n) rounds each consisting of the n time slots), then listening to O(log?^? n) expected number of time slots suffices. We show that this is near optimal.We describe an application of our result to the popular grid model for a radio network. Each node in the network is located on a point in a two dimensional grid, and whenever a node sends a message m, all awake nodes within L _∞ distance r receive m. In this model, up to \(t<\frac{r}{2}(2r+1)\) nodes within any 2r+1 by 2r+1 square in the grid can suffer Byzantine faults. Moreover, we assume that the nodes that suffer Byzantine faults are chosen and controlled by an adversary that knows everything except for the random bits of each non-faulty node. This type of adversary models worst-case behavior due to malicious attacks on the network; mobile nodes moving around in the network; or static nodes losing power or ceasing to function. Let n=r(2r+1). We show how to solve the broadcast problem in this model with each node sending and receiving an expected \(O(n\log^{2}{|m|}+\sqrt{n}|m|)\) bits where |m| is the number of bits in m, and, after broadcasting a fingerprint of m, each node is awake only an expected \(O(\sqrt{n})\) time slots. Moreover, for t≤(1?ε)(r/2)(2r+1), for any constant ε>0, we can achieve an even better energy savings. In particular, if we allow each node to send O(log?^? n) times, we achieve reliable broadcast with each node sending O(nlog?²|m|+(log?^? n)|m|) bits and receiving an expected O(nlog?²|m|+(log?^? n)|m|) bits and, after broadcasting a fingerprint of m, each node is awake for only an expected O(log?^? n) time slots. Our results compare favorably with previous protocols that required each node to send Θ(|m|) bits, receive Θ(n|m|) bits and be awake for Θ(n) time slots.     

Bounded Unpopularity Matchings

We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching M is popular if there is no matching M′ such that more people prefer M′ to M than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied by Abraham et al. (SIAM J. Comput. 37(4):1030–1045, 2007). If there is no popular matching, a reasonable substitute is a matching whose unpopularity is bounded. We consider two measures of unpopularity—unpopularity factor denoted by u(M) and unpopularity margin denoted by g(M). McCutchen recently showed that computing a matching M with the minimum value of u(M) or g(M) is NP-hard, and that if G does not admit a popular matching, then we have u(M)≥2 for all matchings M in G.Here we show that a matching M that achieves u(M)=2 can be computed in \(O(m\sqrt{n})\) time (where m is the number of edges in G and n is the number of nodes) provided a certain graph H admits a matching that matches all people. We also describe a sequence of graphs: H=H ₂,H ₃,…,H _k such that if H _k admits a matching that matches all people, then we can compute in \(O(km\sqrt{n})\) time a matching M such that u(M)≤k?1 and \(g(M)\le n(1-\frac{2}{k})\). Simulation results suggest that our algorithm finds a matching with low unpopularity in random instances.     

The weight-constrained maximum-density subtree problem and related problems in trees

Given a tree T=(V,E) of n nodes such that each node v is associated with a value-weight pair (val _v ,w _v ), where value val _v is a real number and weight w _v is a non-negative integer, the density of T is defined as \(\frac{\sum_{v\in V}{\mathit{val}}_{v}}{\sum_{v\in V}w_{v}}\). A subtree of T is a connected subgraph (V′,E′) of T, where V′?V and E′?E. Given two integers w _min? and w _max?, the weight-constrained maximum-density subtree problem on T is to find a maximum-density subtree T′=(V′,E′) satisfying w _min?≤∑_v∈V′ w _v ≤w _max?. In this paper, we first present an O(w _max? n)-time algorithm to find a weight-constrained maximum-density path in a tree T, and then present an O(w _max? ² n)-time algorithm to find a weight-constrained maximum-density subtree in T. Finally, given a node subset S?V, we also present an O(w _max? ² n)-time algorithm to find a weight-constrained maximum-density subtree in T which covers all the nodes in S.     

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.     

<Emphasis Type="Italic">α</Emphasis>-Systems of differential inclusions and their unification

This paper introduces α-systems of differential inclusions on a bounded time interval [t₀, ?] and defines α-weakly invariant sets in [t₀, ?] × ?_n, where ?_n is a phase space of the differential inclusions. We study the problems connected with bringing the motions (trajectories) of the differential inclusions from an α-system to a given compact set M ? ?_n at the moment ? (the approach problems). The issues of extracting the solvability set W ? [t₀, ?] × ?ⁿ in the problem of bringing the motions of an α-system to M and the issues of calculating the maximal α-weakly invariant set W^c ? [t₀, ?] × ?ⁿ are also discussed. The notion of the quasi-Hamiltonian of an α-system (α-Hamiltonian) is proposed, which seems important for the problems of bringing the motions of the α-system to M.     

Reverse Furthest Neighbors Query in Road Networks

Given a road network G = (V,E), where V (E) denotes the set of vertices(edges) in G, a set of points of interest P and a query point q residing in G, the reverse furthest neighbors (Rfn _R ) query in road networks fetches a set of points p ∈ P that take q as their furthest neighbor compared with all points in P ∪ {q}. This is the monochromatic Rfn _R (Mrfn _R ) query. Another interesting version of Rfn _R query is the bichromatic reverse furthest neighbor (Brfn _R ) query. Given two sets of points P and Q, and a query point q ∈ Q, a Brfn _R query fetches a set of points p ∈ P that take q as their furthest neighbor compared with all points in Q. This paper presents efficient algorithms for both Mrfn _R and Brfn _R queries, which utilize landmarks and partitioning-based techniques. Experiments on real datasets confirm the efficiency and scalability of proposed algorithms.     

MDS codes in Doob graphs

The Doob graph D(m, n), where m > 0, is a Cartesian product of m copies of the Shrikhande graph and n copies of the complete graph K ₄ on four vertices. The Doob graph D(m, n) is a distance-regular graph with the same parameters as the Hamming graph H(2m + n, 4). We give a characterization of MDS codes in Doob graphs D(m, n) with code distance at least 3. Up to equivalence, there are m ³/36+7m ²/24+11m/12+1?(m mod 2)/8?(m mod 3)/9 MDS codes with code distance 2m + n in D(m, n), two codes with distance 3 in each of D(2, 0) and D(2, 1) and with distance 4 in D(2, 1), and one code with distance 3 in each of D(1, 2) and D(1, 3) and with distance 4 in each of D(1, 3) and D(2, 2).