Quantum Algorithms for Matrix Products over Semirings

In this paper we construct quantum algorithms for matrix products over several algebraic structures called semirings, including the (max,min)-matrix product, the distance matrix product and the Boolean matrix product. In particular, we obtain the following results. We construct a quantum algorithm computing the product of two n x n matrices over the (max,min) semiring with time complexity O(n^{2.473}). In comparison, the best known classical algorithm for the same problem, by Duan and Pettie, has complexity O(n^{2.687}). As an application, we obtain a O(n^{2.473})-time quantum algorithm for computing the all-pairs bottleneck paths of a graph with n vertices, while classically the best upper bound for this task is O(n^{2.687}), again by Duan and Pettie. We construct a quantum algorithm computing the L most significant bits of each entry of the distance product of two n x n matrices in time O(2^{0.64L} n^{2.46}). In comparison, prior to the present work, the best known classical algorithm for the same problem, by Vassilevska and Williams and Yuster, had complexity O(2^{L}n^{2.69}). Our techniques lead to further improvements for classical algorithms as well, reducing the classical complexity to O(2^{0.96L}n^{2.69}), which gives a sublinear dependency on 2^L. The above two algorithms are the first quantum algorithms that perform better than the $\tilde O(n^{5/2})$-time straightforward quantum algorithm based on quantum search for matrix multiplication over these semirings. We also consider the Boolean semiring, and construct a quantum algorithm computing the product of two n x n Boolean matrices that outperforms the best known classical algorithms for sparse matrices. For instance, if the input matrices have O(n^{1.686...}) non-zero entries, then our algorithm has time complexity O(n^{2.277}), while the best classical algorithm has complexity O(n^{2.373}).Comment: 19 page

Deterministic Sampling and Range Counting in Geometric Data Streams

We present memory-efficient deterministic algorithms for constructing epsilon-nets and epsilon-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.Comment: 12 pages, 1 figur

Zone Diagrams in Euclidean Spaces and in Other Normed Spaces

Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.Comment: Title page + 16 pages, 20 figure

Rigid ball-polyhedra in Euclidean 3-space

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron one can assign an inner dihedral angle and say that the given ball-polyhedron is locally rigid with respect to its inner dihedral angles if the vertex-edge-face structure of the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron up to congruence locally. The main result of this paper is a Cauchy-type rigidity theorem for ball-polyhedra stating that any simple and standard ball-polyhedron is locally rigid with respect to its inner dihedral angles.Comment: 11 pages, 2 figure

On Compact Routing for the Internet

While there exist compact routing schemes designed for grids, trees, and Internet-like topologies that offer routing tables of sizes that scale logarithmically with the network size, we demonstrate in this paper that in view of recent results in compact routing research, such logarithmic scaling on Internet-like topologies is fundamentally impossible in the presence of topology dynamics or topology-independent (flat) addressing. We use analytic arguments to show that the number of routing control messages per topology change cannot scale better than linearly on Internet-like topologies. We also employ simulations to confirm that logarithmic routing table size scaling gets broken by topology-independent addressing, a cornerstone of popular locator-identifier split proposals aiming at improving routing scaling in the presence of network topology dynamics or host mobility. These pessimistic findings lead us to the conclusion that a fundamental re-examination of assumptions behind routing models and abstractions is needed in order to find a routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

Statistical mechanics of budget-constrained auctions

Finding the optimal assignment in budget-constrained auctions is a combinatorial optimization problem with many important applications, a notable example being the sale of advertisement space by search engines (in this context the problem is often referred to as the off-line AdWords problem). Based on the cavity method of statistical mechanics, we introduce a message passing algorithm that is capable of solving efficiently random instances of the problem extracted from a natural distribution, and we derive from its properties the phase diagram of the problem. As the control parameter (average value of the budgets) is varied, we find two phase transitions delimiting a region in which long-range correlations arise.Comment: Minor revisio

The Supremum Norm of the Discrepancy Function: Recent Results and Connections

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.Comment: 15 pages, 3 Figures. Reports on talks presented by the authors at the 10th international conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Sydney Australia, February 2011. v2: Comments of the referee are incorporate

Kožna dekontaminacija živčanoga bojnog otrova sarina s apsorpcijskim pripravkom u uvjetima in vivo

Our Institute’s nuclear, biological, and chemical defense research team continuously investigates and develops preparations for skin decontamination against nerve agents. In this in vivo study, we evaluated skin decontamination efficacy against sarin by a synthetic preparation called Mineral Cationic Carrier (MCC®) with known ion exchange, absorption efficacy and bioactive potential. Mice were treated with increasing doses of sarin applied on their skin, and MCC® was administered immediately after contamination. The results showed that decontamination with MCC® could achieve therapeutic efficacy corresponding to 3 x LD50 of percutaneous sarin and call for further research.Istraživački tim NBKO (nuklearno-biološko-kemijske obrane) radi na pronalasku i razvoju pripravka za dekontaminaciju kože od živčanih bojnih otrova. Cilj ovog istraživanja bio je ispitati dekontaminacijska svojstva (adsorpcijska i/ili kemisorpcijska) pripravka MCC® rabeći živčani bojni otrov sarin kao kožni kontaminant u uvjetima in vivo. MCC® je sintetski pripravak koji je biokemijski aktivan i ima ionskoizmjenjivačka i adsorpcijska svojstva. Istraživanje u uvjetima in vivo napravljeno je na miševima aplikacijom rastućih doza sarina na kožu životinje. Pripravak MCC® uporabljen je kao kožni dekontaminant neposredno nakon kožne kontaminacije sarinom. Istraživanja su pokazala da pripravak MCC® posjeduje adsorpcijska svojstva, ujedno važna za dekontaminaciju živčanih bojnih otrova. Eksperimenti u uvjetima in vivo na miševima (NOD-soj) pokazali su da se dekontaminacijom pripravkom MCC® može postići terapijski učinak od 3 LD50 (perkutano, sarin)

Optimal Reachability for Weighted Timed Games

Weighted timed automata are timed automata annotated with costs on locations and transitions. The optimal game-reachability problem for these automata is to find the best-cost strategy of supplying the inputs so as to ensure reachability of a target set within a specified number of iterations. The only known complexity bound for this problem is a doubly-exponential upper bound. We establish a singly-exponential upper bound and show that there exist automata with exponentially many states in a single region with pair-wise distinct optimal strategies