Finding a Box Representation for a Graph in $O(n^2\Delta^2\ln n)$ time

An axis-parallel box in $b$-dimensional space is a Cartesian product $R_1 \times R_2 \times \cdots \times R_b$ where $R_i$ (for $1 \leq i \leq b$) is a closed interval of the form $[a_i, b_i]$ on the real line. For a graph $G$, its boxicity is the minimum dimension $b$, such that $G$ is representable as the intersection graph of (axis-parallel) boxes in $b$-dimensional space. The concept of boxicity finds application in various areas of research like ecology, operation research etc. Chandran, Francis and Sivadasan gave an $O(\Delta n^2 \ln^2 n)$ randomized algorithm to construct a box representation for any graph $G$ on $n$ vertices in $\lceil (\Delta + 2)\ln n \rceil$ dimensions, where $\Delta$ is the maximum degree of the graph. They also came up with a deterministic algorithm that runs in $O(n^4 \Delta )$ time. Here, we present an $O(n^2 \Delta^2 \ln n)$ deterministic algorithm that constructs the box representation for any graph in $\lceil (\Delta + 2)\ln n \rceil$ dimensions.

Multi-agent search using?sensors with heterogeneous capabilities

In this thesis we address the problem of multi-agent search. We formulate two deploy and search strategies based on optimal deployment of agents in search space so as to maximize the search effectiveness in a single step. We show that a variation of centroidal Voronoi configuration is the optimal deployment. When the agents have sensors with different capabilities, the problem will be heterogeneous in nature. We introduce a new concept namely, generalized Voronoi partition in order to formulate and solve the heterogeneous multi-agent search problem. We address a few theoretical issues such as optimality of deployment, convergence and spatial distributedness of the control law and the search strategies. Simulation experiments are carried out to compare performances of the proposed strategies with a few simple search strategies.

Search on a Hypercubic Lattice using Quantum Random Walk

We investigate the spatial search problem on the two-dimensional square lattice, using the Dirac evolution operator discretized according to the staggered lattice fermion formalism. d=2 is the critical dimension for the spatial search problem, where infrared divergence of the evolution operator leads to logarithmic factors in the scaling behavior. As a result, the construction used in our accompanying article [ A. Patel and M. A. Rahaman Phys. Rev. A 82 032330 (2010)] provides an O(√NlnN) algorithm, which is not optimal. The scaling behavior can be improved to O(√NlnN) by cleverly controlling the massless Dirac evolution operator by an ancilla qubit, as proposed by Tulsi Phys. Rev. A 78 012310 (2008). We reinterpret the ancilla control as introduction of an effective mass at the marked vertex, and optimize the proportionality constants of the scaling behavior of the algorithm by numerically tuning the parameters.

Bidding Dynamics of Rational Advertisers in Sponsored Search Auctions on the Web

In this paper, we address a key problem faced by advertisers in sponsored search auctions on the web: how much to bid, given the bids of the other advertisers, so as to maximize individual payoffs? Assuming the generalized second price auction as the auction mechanism, we formulate this problem in the framework of an infinite horizon alternative-move game of advertiser bidding behavior. For a sponsored search auction involving two advertisers, we characterize all the pure strategy and mixed strategy Nash equilibria. We also prove that the bid prices will lead to a Nash equilibrium, if the advertisers follow a myopic best response bidding strategy. Following this, we investigate the bidding behavior of the advertisers if they use Q-learning. We discover empirically an interesting trend that the Q-values converge even if both the advertisers learn simultaneously.

Multi-agent Search using Voronoi partitions

This paper addresses the problem of multiagent search in an unknown environment. The agents are autonomous in nature and are equipped with necessary sensors to carry out the search operation. The uncertainty, or lack of information about the search area is known a priori as a probability density function. The agents are deployed in an optimal way so as to maximize the one step uncertainty reduction. The agents continue to deploy themselves and reduce uncertainty till the uncertainty density is reduced over the search space below a minimum acceptable level. It has been shown, using LaSalle’s invariance principle, that a distributed control law which moves each of the agents towards the centroid of its Voronoi partition, modified by the sensor range leads to single step optimal deployment. This principle is now used to devise search trajectories for the agents. The simulations were carried out in 2D space with saturation on speeds of the agents. The results show that the control strategy per step indeed moves the agents to the respective centroid and the algorithm reduces the uncertainty distribution to the required level within a few steps.

An algebraic formulation of exact force-, moment-isotropy in spatial parallel manipulators

In this paper, we present an algebraic method to study and design spatial parallel manipulators that demonstrate isotropy in the force and moment distributions.We use the force and moment transformation matrices separately,and derive conditions for their isotropy individually as well as in combination. The isotropy conditions are derived in closed-form in terms of the invariants of the quadratic forms associated with these matrices. The formulation has been applied to a class of Stewart platform manipulators. We obtain multi-parameter families of isotropic manipulator analytically. In addition to computing the isotropic configurations of an existing manipulator,we demonstrate a procedure for designing the manipulator for isotropy at a given configuration.

Self Assessment-Based Decision Making for Multiagent Cooperative Search

This paper addresses a search problem with multiple limited capability search agents in a partially connected dynamical networked environment under different information structures. A self assessment-based decision-making scheme for multiple agents is proposed that uses a modified negotiation scheme with low communication overheads. The scheme has attractive features of fast decision-making and scalability to large number of agents without increasing the complexity of the algorithm. Two models of the self assessment schemes are developed to study the effect of increase in information exchange during decision-making. Some analytical results on the maximum number of self assessment cycles, effect of increasing communication range, completeness of the algorithm, lower bound and upper bound on the search time are also obtained. The performance of the various self assessment schemes in terms of total uncertainty reduction in the search region, using different information structures is studied. It is shown that the communication requirement for self assessment scheme is almost half of the negotiation schemes and its performance is close to the optimal solution. Comparisons with different sequential search schemes are also carried out. Note to Practitioners-In the futuristic military and civilian applications such as search and rescue, surveillance, patrol, oil spill, etc., a swarm of UAVs can be deployed to carry out the mission for information collection. These UAVs have limited sensor and communication ranges. In order to enhance the performance of the mission and to complete the mission quickly, cooperation between UAVs is important. Designing cooperative search strategies for multiple UAVs with these constraints is a difficult task. Apart from this, another requirement in the hostile territory is to minimize communication while making decisions. This adds further complexity to the decision-making algorithms. In this paper, a self-assessment-based decision-making scheme, for multiple UAVs performing a search mission, is proposed. The agents make their decisions based on the information acquired through their sensors and by cooperation with neighbors. The complexity of the decision-making scheme is very low. It can arrive at decisions fast with low communication overheads, while accommodating various information structures used for increasing the fidelity of the uncertainty maps. Theoretical results proving completeness of the algorithm and the lower and upper bounds on the search time are also provided.

Exploiting Programmable Network Interfaces for Parallel Query Execution in Workstation Clusters

Workstation clusters equipped with high performance interconnect having programmable network processors facilitate interesting opportunities to enhance the performance of parallel application run on them. In this paper, we propose schemes where certain application level processing in parallel database query execution is performed on the network processor. We evaluate the performance of TPC-H queries executing on a high end cluster where all tuple processing is done on the host processor, using a timed Petri net model, and find that tuple processing costs on the host processor dominate the execution time. These results are validated using a small cluster. We therefore propose 4 schemes where certain tuple processing activity is offloaded to the network processor. The first 2 schemes offload the tuple splitting activity - computation to identify the node on which to process the tuples, resulting in an execution time speedup of 1.09 relative to the base scheme, but with I/O bus becoming the bottleneck resource. In the 3rd scheme in addition to offloading tuple processing activity, the disk and network interface are combined to avoid the I/O bus bottleneck, which results in speedups up to 1.16, but with high host processor utilization. Our 4th scheme where the network processor also performs apart of join operation along with the host processor, gives a speedup of 1.47 along with balanced system resource utilizations. Further we observe that the proposed schemes perform equally well even in a scaled architecture i.e., when the number of processors is increased from 2 to 64

Quantum Algorithms with Fixed Points: The Case of Database Search

The standard quantum search algorithm lacks a feature, enjoyed by many classical algorithms, of having a fixed-point, i.e. a monotonic convergence towards the solution. Here we present two variations of the quantum search algorithm, which get around this limitation. The first replaces selective inversions in the algorithm by selective phase shifts of $\frac{\pi}{3}$. The second controls the selective inversion operations using two ancilla qubits, and irreversible measurement operations on the ancilla qubits drive the starting state towards the target state. Using $q$ oracle queries, these variations reduce the probability of finding a non-target state from $\epsilon$ to $\epsilon^{2q+1}$, which is asymptotically optimal. Similar ideas can lead to robust quantum algorithms, and provide conceptually new schemes for error correction.