On the Internal Topological Structure of Plane Regions

The study of topological information of spatial objects has for a long time been a focus of research in disciplines like computational geometry, spatial reasoning, cognitive science, and robotics. While the majority of these researches emphasised the topological relations between spatial objects, this work studies the internal topological structure of bounded plane regions, which could consist of multiple pieces and/or have holes and islands to any finite level. The insufficiency of simple regions (regions homeomorphic to closed disks) to cope with the variety and complexity of spatial entities and phenomena has been widely acknowledged. Another significant drawback of simple regions is that they are not closed under set operations union, intersection, and difference. This paper considers bounded semi-algebraic regions, which are closed under set operations and can closely approximate most plane regions arising in practice.Comment: A short version appeared in KR-10. Several results have been rephrased and omitted proofs are given here. (Sanjiang Li. A Layered Graph Representation for Complex Regions, in Proceedings of the 12th International Conference on the Principles of Knowledge Representation and Reasoning (KR-10), pages 581-583, Toronto, Canada, May 9-13, 2010

Semi-dynamic shortest-path tree algorithms for directed graphs with arbitrary weights

Given a directed graph $G$ with arbitrary real-valued weights, the single source shortest-path problem (SSSP) asks for, given a source $s$ in $G$, finding a shortest path from $s$ to each vertex $v$ in $G$. A classical SSSP algorithm detects a negative cycle of $G$ or constructs a shortest-path tree (SPT) rooted at $s$ in $O(mn)$ time, where $m,n$ are the numbers of edges and vertices in $G$ respectively. In many practical applications, new constraints come from time to time and we need to update the SPT frequently. Given an SPT $T$ of $G$, suppose the weight on a certain edge is modified. We show by rigorous proof that the well-known {\sf Ball-String} algorithm for positively weighted graphs can be adapted to solve the dynamic SPT problem for directed graphs with arbitrary weights. Let $n_0$ be the number of vertices that are affected (i.e., vertices that have different distances from $s$ or different parents in the input and output SPTs) and $m_0$ the number of edges incident to an affected vertex. The adapted algorithms terminate in $O(m_0+n_0 \log n_0)$ time, either detecting a negative cycle (only in the decremental case) or constructing a new SPT $T'$ for the updated graph. We show by an example that the output SPT $T'$ may have more than necessary edge changes to $T$. To remedy this, we give a general method for transforming $T'$ into an SPT with minimal edge changes in time $O(n_0)$ provided that $G$ has no cycles with zero length.Comment: 27 pages, 3 figure

Multi-agent coordination using nearest neighbor rules: revisiting the Vicsek model

Recently, Jadbabaie, Lin, and Morse (IEEE TAC, 48(6)2003:988-1001) offered a mathematical analysis of the discrete time model of groups of mobile autonomous agents raised by Vicsek et al. in 1995. In their paper, Jadbabaie et al. showed that all agents shall move in the same heading, provided that these agents are periodically linked together. This paper sharpens this result by showing that coordination will be reached under a very weak condition that requires all agents are finally linked together. This condition is also strictly weaker than the one Jadbabaie et al. desired.Comment: 11 pages, linguistic mistakes corrected, title modifie

Relational reasoning in the region connection calculus

This paper is mainly concerned with the relation-algebraical aspects of the well-known Region Connection Calculus (RCC). We show that the contact relation algebra (CRA) of certain RCC model is not atomic complete and hence infinite. So in general an extensional composition table for the RCC cannot be obtained by simply refining the RCC8 relations. After having shown that each RCC model is a consistent model of the RCC11 CT, we give an exhaustive investigation about extensional interpretation of the RCC11 CT. More important, we show the complemented closed disk algebra is a representation for the relation algebra determined by the RCC11 table. The domain of this algebra contains two classes of regions, the closed disks and closures of their complements in the real plane.Comment: Latex2e, 35 pages, 2 figure

Let's Play Mahjong!

Mahjong is a very popular tile-based game commonly played by four players. Each player begins with a hand of 13 tiles and, in turn, players draw and discard (i.e., change) tiles until they complete a legal hand using a 14th tile. In this paper, we initiate a mathematical and AI study of the Mahjong game and try to answer two fundamental questions: how bad is a hand of 14 tiles? and which tile should I discard? We define and characterise the notion of deficiency and present an optimal policy to discard a tile in order to increase the chance of completing a legal hand within $k$ tile changes for each $k\geq 1$.Comment: 20 pages, 1 figur

Reasoning with Topological and Directional Spatial Information

Current research on qualitative spatial representation and reasoning mainly focuses on one single aspect of space. In real world applications, however, multiple spatial aspects are often involved simultaneously. This paper investigates problems arising in reasoning with combined topological and directional information. We use the RCC8 algebra and the Rectangle Algebra (RA) for expressing topological and directional information respectively. We give examples to show that the bipath-consistency algorithm BIPATH is incomplete for solving even basic RCC8 and RA constraints. If topological constraints are taken from some maximal tractable subclasses of RCC8, and directional constraints are taken from a subalgebra, termed DIR49, of RA, then we show that BIPATH is able to separate topological constraints from directional ones. This means, given a set of hybrid topological and directional constraints from the above subclasses of RCC8 and RA, we can transfer the joint satisfaction problem in polynomial time to two independent satisfaction problems in RCC8 and RA. For general RA constraints, we give a method to compute solutions that satisfy all topological constraints and approximately satisfy each RA constraint to any prescribed precision

Reasoning about Cardinal Directions between Extended Objects: The Hardness Result

The cardinal direction calculus (CDC) proposed by Goyal and Egenhofer is a very expressive qualitative calculus for directional information of extended objects. Early work has shown that consistency checking of complete networks of basic CDC constraints is tractable while reasoning with the CDC in general is NP-hard. This paper shows, however, if allowing some constraints unspecified, then consistency checking of possibly incomplete networks of basic CDC constraints is already intractable. This draws a sharp boundary between the tractable and intractable subclasses of the CDC. The result is achieved by a reduction from the well-known 3-SAT problem.Comment: 24 pages, 24 figure

On Quotients of Formal Power Series

Quotient is a basic operation of formal languages, which plays a key role in the construction of minimal deterministic finite automata (DFA) and the universal automata. In this paper, we extend this operation to formal power series and systemically investigate its implications in the study of weighted automata. In particular, we define two quotient operations for formal power series that coincide when calculated by a word. We term the first operation as (left or right) \emph{quotient}, and the second as (left or right) \emph{residual}. To support the definitions of quotients and residuals, the underlying semiring is restricted to complete semirings or complete c-semirings. Algebraical properties that are similar to the classical case are obtained in the formal power series case. Moreover, we show closure properties, under quotients and residuals, of regular series and weighted context-free series are similar as in formal languages. Using these operations, we define for each formal power series $A$ two weighted automata ${\cal M}_A$ and ${\cal U}_A$. Both weighted automata accepts $A$, and ${\cal M}_A$ is the minimal deterministic weighted automaton of $A$. The universality of ${\cal U}_A$ is justified and, in particular, we show that ${\cal M}_A$ is a sub-automaton of ${\cal U}_A$. Last but not least, an effective method to construct the universal automaton is also presented in this paper.Comment: 48 pages, 3 figures, 30 conference

Exploring Directional Path-Consistency for Solving Constraint Networks

Among the local consistency techniques used for solving constraint networks, path-consistency (PC) has received a great deal of attention. However, enforcing PC is computationally expensive and sometimes even unnecessary. Directional path-consistency (DPC) is a weaker notion of PC that considers a given variable ordering and can thus be enforced more efficiently than PC. This paper shows that DPC (the DPC enforcing algorithm of Dechter and Pearl) decides the constraint satisfaction problem (CSP) of a constraint language if it is complete and has the variable elimination property (VEP). However, we also show that no complete VEP constraint language can have a domain with more than 2 values. We then present a simple variant of the DPC algorithm, called DPC*, and show that the CSP of a constraint language can be decided by DPC* if it is closed under a majority operation. In fact, DPC* is sufficient for guaranteeing backtrack-free search for such constraint networks. Examples of majority-closed constraint classes include the classes of connected row-convex (CRC) constraints and tree-preserving constraints, which have found applications in various domains, such as scene labeling, temporal reasoning, geometric reasoning, and logical filtering. Our experimental evaluations show that DPC* significantly outperforms the state-of-the-art algorithms for solving majority-closed constraints

Multiagent Simple Temporal Problem: The Arc-Consistency Approach

The Simple Temporal Problem (STP) is a fundamental temporal reasoning problem and has recently been extended to the Multiagent Simple Temporal Problem (MaSTP). In this paper we present a novel approach that is based on enforcing arc-consistency (AC) on the input (multiagent) simple temporal network. We show that the AC-based approach is sufficient for solving both the STP and MaSTP and provide efficient algorithms for them. As our AC-based approach does not impose new constraints between agents, it does not violate the privacy of the agents and is superior to the state-of-the-art approach to MaSTP. Empirical evaluations on diverse benchmark datasets also show that our AC-based algorithms for STP and MaSTP are significantly more efficient than existing approaches.Comment: Accepted by The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18