Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges

Computing \emph{all best swap edges} (ABSE) of a spanning tree $T$ of a given $n$-vertex and $m$-edge undirected and weighted graph $G$ means to select, for each edge $e$ of $T$, a corresponding non-tree edge $f$, in such a way that the tree obtained by replacing $e$ with $f$ enjoys some optimality criterion (which is naturally defined according to some objective function originally addressed by $T$). Solving efficiently an ABSE problem is by now a classic algorithmic issue, since it conveys a very successful way of coping with a (transient) \emph{edge failure} in tree-based communication networks: just replace the failing edge with its respective swap edge, so as that the connectivity is promptly reestablished by minimizing the rerouting and set-up costs. In this paper, we solve the ABSE problem for the case in which $T$ is a \emph{single-source shortest-path tree} of $G$, and our two selected swap criteria aim to minimize either the \emph{maximum} or the \emph{average stretch} in the swap tree of all the paths emanating from the source. Having these criteria in mind, the obtained structures can then be reviewed as \emph{edge-fault-tolerant single-source spanners}. For them, we propose two efficient algorithms running in $O(m n +n^2 \log n)$ and $O(m n \log \alpha(m,n))$ time, respectively, and we show that the guaranteed (either maximum or average, respectively) stretch factor is equal to 3, and this is tight. Moreover, for the maximum stretch, we also propose an almost linear $O(m \log \alpha(m,n))$ time algorithm computing a set of \emph{good} swap edges, each of which will guarantee a relative approximation factor on the maximum stretch of $3/2$ (tight) as opposed to that provided by the corresponding BSE. Surprisingly, no previous results were known for these two very natural swap problems.Comment: 15 pages, 4 figures, SIROCCO 201

Functional Extensionality for Refinement Types

Refinement type checkers are a powerful way to reason about functional programs. For example, one can prove properties of a slow, specification implementation, porting the proofs to an optimized implementation that behaves the same. Without functional extensionality, proofs must relate functions that are fully applied. When data itself has a higher-order representation, fully applied proofs face serious impediments! When working with first-order data, fully applied proofs lead to noisome duplication when using higher-order functions. While dependent type theories are typically consistent with functional extensionality axioms, refinement type systems with semantic subtyping treat naive phrasings of functional extensionality inconsistently, leading to unsoundness. We demonstrate this unsoundness and develop a new approach to equality in Liquid Haskell: we define a propositional equality in a library we call PEq. Using PEq avoids the unsoundness while still proving useful equalities at higher types; we demonstrate its use in several case studies. We validate PEq by building a small model and developing its metatheory. Additionally, we prove metaproperties of PEq inside Liquid Haskell itself using an unnamed folklore technique, which we dub `classy induction'

Collaborative Appearance-Based Place Recognition and Improving Place Recognition Using Detection of Dynamic Objects

This dissertation makes contributions to the problem of Long-Term Appearance-Based Place Recognition. We present a framework for place recognition in a collaborative scheme and a method to reduce the impact of dynamic objects on place representations. We demonstrate our findings using a state-of-the-art place recognition approach. We begin in Part I by describing the general problem of place recognition and its importance in applications where accurate localization is crucial. We discuss feature detection and description and also explain the functioning of several place recognition frameworks. In Part II, we present a novel framework for collaboration between agents from a pure appearance-based place recognition perspective. Using this framework, multiple agents can efficiently share partial or complete knowledge about places and benefit from their teamwork. This collaborative framework allows agents with limited storage and memory capacity to become useful in environment exploration tasks (for instance, by enabling remote recognition); includes procedures to manage an agent’s memory load and distributes knowledge of places across agents; allows the reuse of knowledge from one agent to another; and increases the tolerance for failure of individual agents. Part II also defines metrics which allow us to measure the performance of a system that uses the collaborative framework. Finally, in Part III, we present an innovative method to improve the recognition of places in environments densely populated by dynamic objects. We demonstrate that we can improve the recognition performance in these environments by incorporating high- level information from dynamic objects. Tests conducted using a synthetic dataset show the benefits of our approach. The proposed method allows the system to significantly improve the recognition performance in the photo-realistic dataset while reducing storage requirements, resulting in up to 23.7 percent less storage space than the state-of-the-art approach that we have extended; smaller representations also reduced the time required to match places. In Part III, we also formulate the concept of a valid place representation and determine the quality of the observation based on dynamic objects present in the agent’s view. Of course, recognition systems that are sensitive to dynamic objects incur additional computational costs to recognize those objects. We show that this additional cost is outweighed by the benefits that incorporating dynamic object detection in the place recognition pipeline. Our findings can be used in many applications, including applications for navigation, e.g. assisting visually impaired individuals with navigating indoors, or autonomous vehicles