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- Creators: School of Mathematical and Statistical Sciences
- Creators: Richa, Andréa W.
- Member of: Theses and Dissertations
- Status: Published
Optimal foraging theory provides a suite of tools that model the best way that an animal will <br/>structure its searching and processing decisions in uncertain environments. It has been <br/>successful characterizing real patterns of animal decision making, thereby providing insights<br/>into why animals behave the way they do. However, it does not speak to how animals make<br/>decisions that tend to be adaptive. Using simulation studies, prior work has shown empirically<br/>that a simple decision-making heuristic tends to produce prey-choice behaviors that, on <br/>average, match the predicted behaviors of optimal foraging theory. That heuristic chooses<br/>to spend time processing an encountered prey item if that prey item's marginal rate of<br/>caloric gain (in calories per unit of processing time) is greater than the forager's<br/>current long-term rate of accumulated caloric gain (in calories per unit of total searching<br/>and processing time). Although this heuristic may seem intuitive, a rigorous mathematical<br/>argument for why it tends to produce the theorized optimal foraging theory behavior has<br/>not been developed. In this thesis, an analytical argument is given for why this<br/>simple decision-making heuristic is expected to realize the optimal performance<br/>predicted by optimal foraging theory. This theoretical guarantee not only provides support<br/>for why such a heuristic might be favored by natural selection, but it also provides<br/>support for why such a heuristic might a reliable tool for decision-making in autonomous<br/>engineered agents moving through theatres of uncertain rewards. Ultimately, this simple<br/>decision-making heuristic may provide a recipe for reinforcement learning in small robots<br/>with little computational capabilities.
Over the years, advances in research have continued to decrease the size of computers from the size of<br/>a room to a small device that could fit in one’s palm. However, if an application does not require extensive<br/>computation power nor accessories such as a screen, the corresponding machine could be microscopic,<br/>only a few nanometers big. Researchers at MIT have successfully created Syncells, which are micro-<br/>scale robots with limited computation power and memory that can communicate locally to achieve<br/>complex collective tasks. In order to control these Syncells for a desired outcome, they must each run a<br/>simple distributed algorithm. As they are only capable of local communication, Syncells cannot receive<br/>commands from a control center, so their algorithms cannot be centralized. In this work, we created a<br/>distributed algorithm that each Syncell can execute so that the system of Syncells is able to find and<br/>converge to a specific target within the environment. The most direct applications of this problem are in<br/>medicine. Such a system could be used as a safer alternative to invasive surgery or could be used to treat<br/>internal bleeding or tumors. We tested and analyzed our algorithm through simulation and visualization<br/>in Python. Overall, our algorithm successfully caused the system of particles to converge on a specific<br/>target present within the environment.
The main contributions of this thesis are three-fold: First, a bi-criteria approximation algorithm is presented for this all-or-nothing multicommodity flow (ANF) problem. This algorithm is the first to achieve a constant approximation of the maximum throughput with an edge capacity violation ratio that is at most logarithmic in n, with high probability. The approach used is based on a version of randomized rounding that keeps splittable flows, rather than approximating those via a non-splittable path for each commodity: This allows it to work for arbitrary directed edge-capacitated graphs, unlike most of the prior work on the ANF problem. The algorithm also works if a weighted throughput is considered, where the benefit gained by fully satisfying the demand for commodity i is determined by a given weight w_i>0. Second, a derandomization of the algorithm is presented that maintains the same approximation bounds, using novel pessimistic estimators for Bernstein's inequality. In addition, it is shown how the framework can be adapted to achieve a polylogarithmic fraction of the maximum throughput while maintaining a constant edge capacity violation, if the network capacity is large enough. Lastly, one important aspect of the randomized and derandomized algorithms is their simplicity, which lends to efficient implementations in practice. The implementations of both randomized rounding and derandomized algorithms for the ANF problem are presented and show their efficiency in practice.
}}=\tau$. This research will focus on improving approximations on the lower bound of $\tau$. Toward this end we will examine algorithmic enumeration, and series analysis for self-avoiding polygons.