Space-Efficient DFS and Applications: Simpler, Leaner, Faster

The problem of space-efficient depth-first search (DFS) is reconsidered. A particularly simple and fast algorithm is presented that, on a directed or undirected input graph $G=(V,E)$ with $n$ vertices and $m$ edges, carries out a DFS in $O(n+m)$ time with $n+\sum_{v\in V_{\ge 3}}\lceil{\log_2(d_v-1)}\rceil +O(\log n)\le n+m+O(\log n)$ bits of working memory, where $d_v$ is the (total) degree of $v$, for each $v\in V$, and $V_{\ge 3}=\{v\in V\mid d_v\ge 3\}$. A slightly more complicated variant of the algorithm works in the same time with at most $n+({4/5})m+O(\log n)$ bits. It is also shown that a DFS can be carried out in a graph with $n$ vertices and $m$ edges in $O(n+m\log^*\! n)$ time with $O(n)$ bits or in $O(n+m)$ time with either $O(n\log\log(4+{m/n}))$ bits or, for arbitrary integer $k\ge 1$, $O(n\log^{(k)}\! n)$ bits. These results among them subsume or improve most earlier results on space-efficient DFS. Some of the new time and space bounds are shown to extend to applications of DFS such as the computation of cut vertices, bridges, biconnected components and 2-edge-connected components in undirected graphs

Simple 2^f-Color Choice Dictionaries

A c-color choice dictionary of size n in N is a fundamental data structure in the development of space-efficient algorithms that stores the colors of n elements and that supports operations to get and change the color of an element as well as an operation choice that returns an arbitrary element of that color. For an integer f>0 and a constant c=2^f, we present a word-RAM algorithm for a c-color choice dictionary of size n that supports all operations above in constant time and uses only nf+1 bits, which is optimal if all operations have to run in o(n/w) time where w is the word size. In addition, we extend our choice dictionary by an operation union without using more space

ESTIMATION AND CONTROL OF NONLINEAR SYSTEMS: MODEL-BASED AND MODEL-FREE APPROACHES

State estimation and subsequent controller design for a general nonlinear system is an important problem that have been studied over the past decades. Many applications, e.g., atmospheric and oceanic sampling or lift control of an airfoil, display strongly nonlinear dynamics with very high dimensionality. Some of these applications use smaller underwater or aerial sensing platforms with insufficient on-board computation power to use a Monte-Carlo approach of particle filters. Hence, they need a computationally efficient filtering method for state-estimation without a severe penalty on the performance. On the other hand, the difficulty of obtaining a reliable model of the underlying system, e.g., a high-dimensional fluid dynamical environment or vehicle flow in a complex traffic network, calls for the design of a data-driven estimation and controller when abundant measurements are present from a variety of sensors. This dissertation places these problems in two broad categories: model-based and model-free estimation and output feedback. In the first part of the dissertation, a semi-parametric method with Gaussian mixture model (GMM) is used to approximate the unknown density of states. Then a Kalman filter and its nonlinear variants are employed to propagate and update each Gaussian mode with a Bayesian update rule. The linear observation model permits a Kalman filter covariance update for each Gaussian mode. The estimation error is shown to be stochastically bounded and this is illustrated numerically. The estimate is used in an observer-based feedback control to stabilize a general closed-loop system. A transferoperator- based approach is then proposed for the motion update for Bayesian filtering of a nonlinear system. A finite-dimensional approximation of the Perron-Frobenius (PF) operator yields a method called constrained Ulam dynamic mode decomposition (CUDMD). This algorithm is applied for output feedback of a pitching airfoil in unsteady flow. For the second part, an echo-state network (ESN) based approach equipped with an ensemble Kalman filter is proposed for data-driven estimation of a nonlinear system from a time series. A random reservoir of recurrent neural connections with the echo-state property (ESP) is trained from a time-series data. It is then used as a model-predictor for an ensemble Kalman filter for sparse estimation. The proposed data-driven estimation method is applied to predict the traffic flow from a set of mobility data of the UMD campus. A data-driven model-identification and controller design is also developed for control-affine nonlinear systems that are ubiquitous in several aerospace applications. We seek to find an approximate linear/bilinear representation of these nonlinear systems from data using the extended dynamic mode decomposition algorithm (EDMD) and apply Liealgebraic methods to analyze the controllability and design a controller. The proposed method utilizes the Koopman canonical transform (KCT) to approximate the dynamics into a bilinear system (Koopman bilinear form) under certain assumptions. The accuracy of this approximation is then analytically justified with the universal approximation property of the Koopman eigenfunctions. The resulting bilinear system is then subjected to controllability analysis using the Myhill semigroup and Lie algebraic structures, and a fixed endpoint optimal controller is designed using the Pontryagin’s principle