Topological Stability of Kinetic kk-Centers

We study the $k$-center problem in a kinetic setting: given a set of continuously moving points $P$ in the plane, determine a set of $k$ (moving) disks that cover $P$ at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of $k$-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution---the topological stability ratio---considering various metrics and various optimization criteria. For $k = 2$ we provide tight bounds, and for small $k > 2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant $k$

Efficient Multi-Robot Coverage of a Known Environment

This paper addresses the complete area coverage problem of a known environment by multiple-robots. Complete area coverage is the problem of moving an end-effector over all available space while avoiding existing obstacles. In such tasks, using multiple robots can increase the efficiency of the area coverage in terms of minimizing the operational time and increase the robustness in the face of robot attrition. Unfortunately, the problem of finding an optimal solution for such an area coverage problem with multiple robots is known to be NP-complete. In this paper we present two approximation heuristics for solving the multi-robot coverage problem. The first solution presented is a direct extension of an efficient single robot area coverage algorithm, based on an exact cellular decomposition. The second algorithm is a greedy approach that divides the area into equal regions and applies an efficient single-robot coverage algorithm to each region. We present experimental results for two algorithms. Results indicate that our approaches provide good coverage distribution between robots and minimize the workload per robot, meanwhile ensuring complete coverage of the area.Comment: In proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 201

Optimization of Wireless Sensor Network Lifetime by Deploying Relay Sensors

Topology control in wireless sensor networks helps to lower node energy consumption by reducing transmission power and by confining interference, collisions and consequently retransmissions. Decrease in node energy consumption implies probability of increasing network lifetime. In this paper, first we analyze popular topology control algorithms used for optimizing the power consumption in the wireless sensor network and later propose a novel technique wherein power consumption is traded with additional relay nodes. We introduce relay nodes to make the network connected without increasing the transmit power. The relay node decreases the transmit power required while it may increase end-to-end delay.  We design and analyze an algorithm that place an almost minimum number of relay nodes required to make network connected. We have implemented greedy version of this algorithm and demonstrated in simulation that it produces a high quality link. We use InterAvg, InterMax (no of nodes that can offer interference) MinMax, and MinTotal as metrics to analyze and compare various algorithms. Matlab and NS-2 are used for simulation purpose. Keywords: Energy saving, sensor networks, Interference, network connectivity, topology control

Coverage & cooperation: Completing complex tasks as quickly as possible using teams of robots

As the robotics industry grows and robots enter our homes and public spaces, they are increasingly expected to work in cooperation with each other. My thesis focuses on multirobot planning, specifically in the context of coverage robots, such as robotic lawnmowers and vacuum cleaners. Two problems unique to multirobot teams are task allocation and search. I present a task allocation algorithm which balances the workload amongst all robots in the team with the objective of minimizing the overall mission time. I also present a search algorithm which robots can use to find lost teammates. It uses a probabilistic belief of a target robot’s position to create a planning tree and then searches by following the best path in the tree. For robust multirobot coverage, I use both the task allocation and search algorithms. First the coverage region is divided into a set of small coverage tasks which minimize the number of turns the robots will need to take. These tasks are then allocated to individual robots. During the mission, robots replan with nearby robots to rebalance the workload and, once a robot has finished its tasks, it searches for teammates to help them finish their tasks faster

Universal Algorithms for Clustering Problems

This paper presents universal algorithms for clustering problems, including the widely studied $k$-median, $k$-means, and $k$-center objectives. The input is a metric space containing all potential client locations. The algorithm must select $k$ cluster centers such that they are a good solution for any subset of clients that actually realize. Specifically, we aim for low regret, defined as the maximum over all subsets of the difference between the cost of the algorithm's solution and that of an optimal solution. A universal algorithm's solution $SOL$ for a clustering problem is said to be an $(\alpha, \beta)$-approximation if for all subsets of clients $C'$, it satisfies $SOL(C') \leq \alpha \cdot OPT(C') + \beta \cdot MR$, where $OPT(C')$ is the cost of the optimal solution for clients $C'$ and $MR$ is the minimum regret achievable by any solution. Our main results are universal algorithms for the standard clustering objectives of $k$-median, $k$-means, and $k$-center that achieve $(O(1), O(1))$-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other $\ell_p$-objectives and the setting where some subset of the clients are fixed. We also give hardness results showing that $(\alpha, \beta)$-approximation is NP-hard if $\alpha$ or $\beta$ is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, $(O(1), O(1))$-approximation is the strongest type of guarantee obtainable for universal clustering.Comment: Appeared in ICALP 2021, Track A. Fixed mismatch between paper title and arXiv titl

CUSTOMER SEGMENTATION ANALYSIS OF CANNABIS RETAIL DATA: A MACHINE LEARNING APPROACH

As the legal cannabis industry emerges from its nascent stages, there is increasing motivation for retailers to look for data or strategies that can help them segment or describe their customers in a succinct, but informative manner. While many cannabis operators view the state-mandated traceability as a necessary burden, it provides a goldmine for internal customer analysis. Traditionally, segmentation analysis focuses on demographic or RFM (recency-frequency-monetary) segmentation. Yet, neither of these methods has the capacity to provide insight into a customer’s purchasing behavior. With the help of 4Front Ventures, a battle-tested multinational cannabis operator, this report focuses on segmenting customers using cannabis-specific data (such as flower and concentrate consumption) and machine learning methods (K-Means and Agglomerative Hierarchical Clustering) to generate newfound ways to explore a dispensary’s consumer base. The findings are that there are roughly five or six clusters of customers with each cluster having unique purchasing traits that define them. Although the results are meaningful, this report could benefit with exploring more clustering algorithms, comparing results across dispensaries within the same state, or investigating segmentations in other state markets