Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions

A random geometric graph (RGG) is defined by placing n points uniformly at random in [0,n ^1/d ] ^d , and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω(n) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a Euclidean distance of $\omega (\frac{\log n}{r^{d-1}} )$ , their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is Θ(n ^1/d /r) w.h.p. We also prove that the condition on the Euclidean distance above is essentially tight. We also analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that w.h.p. this algorithm informs every node in the largest connected component of an RGG within Θ(n ^1/d /r+logn) rounds.     

Online computation with advice

We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice. The advice is a function, defined by the online algorithm, of the whole request sequence. The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b=0 corresponds to the classical online model, and b=⌈log∣A∣⌉, where A is the algorithm’s action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones.In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the k-server problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1≤b≤Θ(logn), where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio Ω(log(n)/b) and we present a deterministic online algorithm for MTS with competitive ratio O(log(n)/b). For the k-server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k^O(1/b) for any choice of Θ(1)≤b≤logk.     

Computing the subset partial order for dense families of sets

We give an algorithm to compute the subset partial order (called the subset graph) for a family F of sets containing k sets with N elements in total and domain size n. Our algorithm requires O(nk²/logk) time and space on a Pointer Machine. When F is dense, i.e. N=Θ(nk), the algorithm requires O(N²/log²N) time and space. We give a construction for a dense family whose subset graph is of size Θ(N²/log²N), indicating the optimality of our algorithm for dense families. The subset graph can be dynamically maintained when F undergoes set insertions and deletions in O(nk/logk) time per update (that is sub-linear in N for the case of dense families). If we assume words of b?k bits, allow bits to be packed in words, and use bitwise operations, the above running time and space requirements can be reduced by a factor of blog(k/b+1)/logk and b²log(k/b+1)/logk respectively.     

Transportation Distances on the Circle

This paper is devoted to the study of the Monge-Kantorovich theory of optimal mass transport, in the special case of one-dimensional and circular distributions. More precisely, we study the Monge-Kantorovich problem between discrete distributions on the unit circle S ¹, in the case where the ground distance between two points x and y is defined as h(d(x,y)), where d is the geodesic distance on the circle and h a convex and increasing function. This study complements previous results in the literature, holding only for a ground distance equal to the geodesic distance d. We first prove that computing a Monge-Kantorovich distance between two given sets of pairwise different points boils down to cut the circle at a well chosen point and to compute the same distance on the real line. This result is then used to obtain a dissimilarity measure between 1-D and circular discrete histograms. In a last part, a study is conducted to compare the advantages and drawbacks of transportation distances relying on convex or concave cost functions, and of the classical L ¹ distance. Simple retrieval experiments based on the hue component of color images are shown to illustrate the interest of circular distances. The framework is eventually applied to the problem of color transfer between images.     

Boolean-width of graphs

We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods-Boolean sums of neighborhoods-across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an n-vertex graph given with a decomposition tree of boolean-width k, we solve Maximum Weight Independent Set in time O(n²k2^2k) and Minimum Weight Dominating Set in time O(n²+nk2^3k). With an additional n² factor in the runtime, we can also count all independent sets and dominating sets of each cardinality.Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of GF(2)-sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Θ(n²) vertices has boolean-width Θ(logn) and rank-width, clique-width, tree-width, and branch-width Θ(n).     

Signatures versus histograms: Definitions, distances and algorithms

The aim of this paper is to present a new method to compare histograms. The main advantage is that there is an important time-complexity reduction with respect to the methods presented before. This reduction is statistically and analytically demonstrated in the paper.The distances between histograms that we presented are defined on a structure called signature, which is a lossless representation of histograms. Moreover, the type of the elements of the sets that the histograms represent are ordinal, nominal and modulo.We show that the computational cost of these distances is O(z^′) for the ordinal and nominal types and O(z^′2) for the modulo one, z^′ being the number of non-empty bins of the histograms. The computational cost of the algorithms presented in the literature depends on the number of bins of the histograms. In most of the applications, the obtained histograms are sparse; then considering only the non-empty bins drastically decreases the time consumption of the comparison.The distances and algorithms presented in this paper are experimentally validated on the comparison of images obtained from public databases and positioning of mobile robots through the recognition of indoor scenes (captured in a learning stage).     

Analysis of scalable data-privatization threading algorithms for hybrid MPI/OpenMP parallelization of molecular dynamics

We propose and analyze threading algorithms for hybrid MPI/OpenMP parallelization of a molecular-dynamics simulation, which are scalable on large multicore clusters. Two data-privatization thread scheduling algorithms via nucleation-growth allocation are introduced: (1) compact-volume allocation scheduling (CVAS); and (2) breadth-first allocation scheduling (BFAS). The algorithms combine fine-grain dynamic load balancing and minimal memory-footprint data privatization threading. We show that the computational costs of CVAS and BFAS are bounded by Θ(n ^5/3 p ^?2/3) and Θ(n), respectively, for p threads working on n particles on a multicore compute node. Memory consumption per node of both algorithms scales as O(n+n ^2/3 p ^1/3), but CVAS has smaller prefactors due to a geometric effect. Based on these analyses, we derive the selection criterion between the two algorithms in terms of the granularity, n/p. We observe that memory consumption is reduced by 75 % for p=16 and n=8,192 compared to a naïve data privatization, while maintaining thread imbalance below 5 %. We obtain a strong-scaling speedup of 14.4 with 16-way threading on a four quad-core AMD Opteron node. In addition, our MPI/OpenMP code achieves 2.58× and 2.16× speedups over the MPI-only implementation on 32,768 cores of BlueGene/P for 0.84 and 1.68 million particle systems, respectively.     

Learning large-alphabet and analog circuits with value injection queries

We consider the problem of learning an acyclic discrete circuit with n wires, fan-in bounded by k and alphabet size s using value injection queries. For the class of transitively reduced circuits, we develop the Distinguishing Paths Algorithm, that learns such a circuit using (ns) ^O(k) value injection queries and time polynomial in the number of queries. We describe a generalization of the algorithm to the class of circuits with shortcut width bounded by b that uses (ns) ^O(k+b) value injection queries. Both algorithms use value injection queries that fix only O(kd) wires, where d is the depth of the target circuit. We give a reduction showing that without such restrictions on the topology of the circuit, the learning problem may be computationally intractable when s=n ^Θ(1), even for circuits of depth O(log?n). We then apply our large-alphabet learning algorithms to the problem of approximate learning of analog circuits whose gate functions satisfy a Lipschitz condition. Finally, we consider models in which behavioral equivalence queries are also available, and extend and improve the learning algorithms of (Angluin in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, pp. 584–593, 2006) to handle general classes of gate functions that are polynomial time learnable from counterexamples.     

Block sensitivity of minterm-transitive functions

Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown that these functions, if nonconstant, must have high complexity according to various measures.In a recent work of this type, Sun (2007) [9] gave lower bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f)=Ω(N^1/3). He also showed that there exists such a function for which bs(f)=O(N^3/7lnN). His example belongs to a subclass of transitively invariant functions called “minterm-transitive” functions, defined by Chakraborty (2005) [3].We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy bs(f)=Ω(N^3/7). Thus, Sun’s example has nearly minimal block sensitivity for this subclass. Second, we improve Sun’s example: we exhibit a minterm-transitive function for which bs(f)=O(N^3/7ln^1/7N).     

Connectivity-preserving transformations of binary images

A binary image I is B_a, W_b-connected, where a, b ∈ {4, 8}, if its foreground is a-connected and its background is b-connected. We consider a local modification of a B_a, w_b-connected image I in which a black pixel can be interchanged with an adjacent white pixel provided that this preserves the connectivity of both the foreground and the background of I. We have shown that for any (a, b) ∈ {(4, 8), (8, 4), (8, 8)}, any two B_a, w_b-connected images I and J each with n black pixels differ by a sequence of Θ(n²) interchanges. We have also shown that any two B₄, W₄-connected images I and J each with n black pixels differ by a sequence of O(n⁴) interchanges.     

Practical constructive schemes for deterministic shared-memory access

We present three explicit schemes for distributingM variables amongN memory modules, whereM=Θ(N ^1.5),M = Θ(N ²), andM=Θ(N ³), respectively. Each variable is replicated into a constant number of copies stored in distinct modules. We show thatN processors, directly accessing the memories through a complete interconnection, can read/write any set ofN variables in worst-case timeO (N ^1/3),O(N ^1/2), andO(N ^2/3), respectively for the three schemes. The access times for the last two schemes are optimal with respect to the particular redundancy values used by such schemes. The address computation can be carried out efficiently by each processor without recourse to a complete memory map and requiring onlyO(1) internal storage.     

An optimal algorithm for constructing the weighted voronoi diagram in the plane

Let S denote a set of n points in the plane such that each point p has assigned a positive weight w(p) which expresses its capability to influence its neighbourhood. In this sense, the weighted distance of an arbitrary point x from p is given by d_e(x,p)/w(p) where d_e denotes the Euclidean distance function. The weighted Voronoi diagram for S is a subdivision of the plane such that each point p in S is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of S.An algorithm which constructs the weighted Voronoi diagram for S in O(n²) time is outlined in this paper. The method is optimal as the diagram can consist of Θ(n²) faces, edges and vertices.     

Approximate motion planning and the complexity of the boundary of the union of simple geometric figures

We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have a running time of Ω(n ²), wheren is the number of obstacle corners. We introduce thetightness of a motion-planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with a running time ofO((a/b · 1/?_crit + 1)n(logn)²), wherea ≥b are the lengths of the sides of a rectangle and ?_crit is the tightness of the problem. We show further that the complexity (= number of vertices) of the boundary ofn bow ties (see Figure 1) isO(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented.     

A linear time algorithm for finding an optimal degree-bounded subtree of an edge-weighted tree

Given an edge-weighted tree T=(V(T),E(T)) and its subtree T^′, for any v∈V(T), the distance d(v,T^′) is defined as the minimum weighted distance from v to any vertex in T^′. The distance d(T,T^′) is defined as the sum of all distances of the form d(v,T^′) where v∈V(T). We give an algorithm to find a T^′ that minimizes d(T,T^′) and for all w∈V(T^′), the degree degT^′(w) of w is not more than a prespecified bound db(w)(0?db(w)?degT(w)) at w. The worst-case running time of our algorithm is in O(|V(T)|).     

Birth and growth of multicyclic components in random hypergraphs

Define an ?-component to be a connected b-uniform hypergraph with k edges and k(b−1)−? vertices. In this paper, we investigate the growth of size and complexity of connected components of a random hypergraph process. We prove that the expected number of creations of ?-components during a random hypergraph process tends to 1 as b is fixed and ? tends to infinity with the total number of vertices n while remaining ?=o(n^1/3). We also show that the expected number of vertices that ever belong to an ?-component is ∼12^1/3?^1/3n^2/3(b−1)^−1/3. We prove that the expected number of times hypertrees are swallowed by ?-components is ∼2^1/33^−1/3n^1/3?^−1/3(b−1)^−5/3. It follows that with high probability the largest ?-component during the process is of size of order O(?^1/3n^2/3(b−1)^−1/3). Our results give insight into the size of giant components inside the phase transition of random hypergraphs and generalize previous results about graphs.     

Dynamic Matrix Rank with Partial Lookahead

We consider the problem of maintaining information about the rank of a matrix M under changes to its entries. For an n×n matrix M, we show an amortized upper bound of O(n ^ω?1) arithmetic operations per change for this problem, where ω<2.373 is the exponent for matrix multiplication, under the assumption that there is a lookahead of up to Θ(n) locations. That is, we know up to the next Θ(n) locations (i ₁,j ₁),(i ₂,j ₂),…?, whose entries are going to change, in advance; however we do not know the new entries in these locations in advance. We get the new entries in these locations in a dynamic manner. The dynamic matrix rank problem was first studied by Frandsen and Frandsen who showed an upper bound of O(n ^1.575) and a lower bound of Ω(n) for this problem and later Sankowski showed an upper bound of O(n ^1.495) for this problem when allowing randomization and a small probability of error. These algorithms do not assume any lookahead. For the dynamic matrix rank problem with lookahead, Sankowski and Mucha showed a randomized algorithm (with a small probability of error) that is more efficient than these algorithms.     

Delay efficient scheduling via redundant constraints in multihop networks

We consider the problem of delay-efficient scheduling in general multihop networks. While the class of max-weight type algorithms are known to be throughput optimal for this problem, they typically incur undesired delay performance. In this paper, we propose the Delay-Efficient SCheduling algorithm (DESC). DESC is built upon the idea of accelerating queues (AQ), which are virtual queues that quickly propagate the traffic arrival information along the routing paths. DESC is motivated by the use of redundant constraints to accelerate convergence in the classic optimization context. We show that DESC is throughput-optimal. The delay bound of DESC can be better than previous bounds of the max-weight type algorithms which did not use such traffic information. We also show that under DESC, the service rates allocated to the flows converge quickly to their target values and the average total “network service lag” is small. In particular, when there are O(1) flows and the rate vector is of Θ(1) distance away from the boundary of the capacity region, the average total “service lag” only grows linearly in the network size.     

Log-logarithmic worst-case range queries are possible in space Θ(N)

Let S denote a set of N records whose keys are distinct nonnegative integers less than some initially specified bound M. This paper introduces a new data structure, called the y-fast trie, which uses Θ(N) space and Θ(log log M) time for range queries on a random access machine. We will also define a simpler but less efficient structure, called the x-fast trie.     

DNF are teachable in the average case

We study the average number of well-chosen labeled examples that are required for a helpful teacher to uniquely specify a target function within a concept class. This “average teaching dimension” has been studied in learning theory and combinatorics and is an attractive alternative to the “worst-case” teaching dimension of Goldman and Kearns which is exponential for many interesting concept classes. Recently Balbach showed that the classes of 1-decision lists and 2-term DNF each have linear average teaching dimension. As our main result, we extend Balbach’s teaching result for 2-term DNF by showing that for any 1≤s≤2 ^Θ(n), the well-studied concept classes of at-most-s-term DNF and at-most-s-term monotone DNF each have average teaching dimension O(ns). The proofs use detailed analyses of the combinatorial structure of “most” DNF formulas and monotone DNF formulas. We also establish asymptotic separations between the worst-case and average teaching dimension for various other interesting Boolean concept classes such as juntas and sparse GF ₂ polynomials.     

A new critical theorem for adaptive nonlinear stabilization

It is fairly well known that there are fundamental differences between adaptive control of continuous-time and discrete-time nonlinear systems. In fact, even for the seemingly simple single-input single-output control system y_t+1=θ₁f(y_t)+u_t+w_t+1 with a scalar unknown parameter θ₁ and noise disturbance {w_t}, and with a known function f(⋅) having possible nonlinear growth rate characterized by |f(x)|=Θ(|x|^b) with b≥1, the necessary and sufficient condition for the system to be globally stabilizable by adaptive feedback is b<4. This was first found and proved by Guo (1997) for the Gaussian white noise case, and then proved by Li and Xie (2006) for the bounded noise case. Subsequently, a number of other types of “critical values” and “impossibility theorems” on the maximum capability of adaptive feedback were also found, mainly for systems with known control parameter as in the above model. In this paper, we will study the above basic model again but with additional unknown control parameter θ₂, i.e., u_t is replaced by θ₂u_t in the above model. Interestingly, it turns out that the system is globally stabilizable if and only ifb<3. This is a new critical theorem for adaptive nonlinear stabilization, which has meaningful implications for the control of more general uncertain nonlinear systems.