This paper introduces a new family of models of intensional Martin-L "of type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and on cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.
Spitters [Section 3] SpittersB:cubstt uses a somewhat different formulation of Moore path in the cubical topos @cite_2 . His notion is the reflexive-transitive closure of the usual path types given by the bounded interval. For better properties and to get a closer relationship with our version, one would like to quotient these cubical Spitters-Moore'' path objects up to degenerate paths; but the undecidability of degeneracy seems to stop one being able to do that while retaining the (uniform) Kan-fibrancy of such path objects. Here we can side-step such issues, since notably our models manage to avoid using a notion of Kan fibrancy at all.
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This article presents three characterizations of the weak factorization systems on finitely complete categories that interpret intensional dependent type theory with Sigma-, Pi-, and Id-types. The first characterization is that the weak factorization system (L,R) has the properties that L is stable under pullback along R and that all maps to a terminal object are in R. We call such weak factorization systems type-theoretic. The second is that the weak factorization system has an Id-presentation: roughly, it is generated by Id-types in the empty context. The third is that the weak factorization system (L, R) is generated by a Moore relation system, a generalization of the notion of Moore paths. We say an algorithm on n by n matrices with entries in [-M, M] (or n-node graphs with edge weights from [-M, M]) is truly sub cubic if it runs in O(n^ 3- ( M)) time for some > 0. We define a notion of sub cubic reducibility, and show that many important problems on graphs and matrices solvable in O(n^3) time are equivalent under sub cubic reductions. Namely, the following weighted problems either all have truly sub cubic algorithms, or none of them do: - The all-pairs shortest paths problem (APSP). - Detecting if a weighted graph has a triangle of negative total edge weight. - Listing up to n^ 2.99 negative triangles in an edge-weighted graph. - Finding a minimum weight cycle in a graph of non-negative edge weights. - The replacement paths problem in an edge-weighted digraph. - Finding the second shortest simple path between two nodes in an edge-weighted digraph. - Checking whether a given matrix defines a metric. - Verifying the correctness of a matrix product over the ( , +)-semiring. Therefore, if APSP cannot be solved in n^ 3- time for any > 0, then many other problems also need essentially cubic time. In fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure. These equivalences simplify prior work on sub cubic algorithms for all-pairs path problems, since it now suffices to give appropriate sub cubic triangle detection algorithms. Other consequences of our work are new combinatorial approaches to Boolean matrix multiplication over the (OR, AND)-semiring (abbreviated as BMM). We show that practical advances in triangle detection would imply practical BMM algorithms, among other results. Building on our techniques, we give two new BMM algorithms: a derandomization of the recent combinatorial BMM algorithm of Bansal and Williams (FOCS'09), and an improved quantum algorithm for BMM. A distance sensitivity oracle is a data structure which, given two nodes s and t in a directed edge-weighted graph G and an edge e, returns the shortest length of an s-t path not containing e, a so called replacement path for the triple (s, t, e). Such oracles are used to quickly recover from edge failures. In this paper we consider the case of integer weights in the interval [-M, M], and present the first distance sensitivity oracle that achieves simultaneously sub cubic preprocessing time and sub linear query time. More precisely, for a given parameter [0, 1], our oracle has preprocessing time O (Mn^ + 1 2 +Mn^ + (4- ) ) and query time O (n^ 1- ). Here This paper presents a novel framework for the recognition of objects based on their silhouettes. The main idea is to measure the distance between two shapes as the minimum extent of deformation necessary for one shape to match the other. Since the space of deformations is very high-dimensional, three steps are taken to make the search practical: 1) define an equivalence class for shapes based on shock-graph topology, 2) define an equivalence class for deformation paths based on shock-graph transitions, and 3) avoid complexity-increasing deformation paths by moving toward shock-graph degeneracy. Despite these steps, which tremendously reduce the search requirement, there still remain numerous deformation paths to consider. To that end, we employ an edit-distance algorithm for shock graphs that finds the optimal deformation path in polynomial time. The proposed approach gives intuitive correspondences for a variety of shapes and is robust in the presence of a wide range of visual transformations. The recognition rates on two distinct databases of 99 and 216 shapes each indicate highly successful within category matches (100 percent in top three matches), which render the framework potentially usable in a range of shape-based recognition applications. External 2-dimensional searching is a fundamental problem with many applications in relational, object-oriented, spatial, and temporal databases. For example, interval intersection can be reduced to 2-sided, 2-dimensional searching and indexing class hierarchies of objects to 3-sided, 2-dimensional searching. Path caching is a new technique that can be used to transform a number of time space efficient data structures for internal 2-dimensional searching (such as segment trees, interval trees, and priority search trees) into I O efficient external ones. Let n be the size of the database, B the page size, and t the output size of a query. Using path caching, we provide the first data structure with optimal I O query time @math for 2-sided, 2-dimensional searching. Furthermore, we show that path caching requires a small space overhead @math and is simple enough to admit dynamic updates in optimal @math amortized time. We also extend this data structure to handle 3-sided, 2-dimensional searching with optimal I O query-time, at the expense of slightly higher storage and update overheads. Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the state-of-the-art algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s (by Floyd and Warshall). Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs. We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the all-pairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions to fast matrix multiplication. For maximum bottleneck paths and minimum nondecreasing paths we are the first to break the cubic barrier, obtaining truly subcubic strongly polynomial algorithms. We also consider a nonalgebraic, combinatorial approach, which is considered more efficient in practice compared to methods based on fast matrix multiplication. We present a combinatorial data structure that maintains a matrix so that products with given sparse vectors can be computed efficiently. This allows us to obtain good running times for path problems in unweighted sparse graphs. This thesis also gives algorithms for some single source path problems. We obtain the first linear time algorithm for the single source minimum nondecreasing paths problem. We give some extensions to this, including an algorithm to find cheapest minimum nondecreasing paths. Besides finding optimal paths, we consider the related problem of finding optimal cycles. In particular, we focus on the problem of finding in a weighted graph a triangle of maximum weight sum. We obtain the first truly subcubic algorithm for finding a maximum weight triangle in a node-weighted graph. We also present algorithms for the edge-weighted case. These algorithms immediately imply good algorithms for finding maximum weight k-cliques, or arbitrary maximum weight pattern subgraphs of fixed size. This paper presents an analytical method for planning an efficient tool-path in machining free-form surfaces on 3-axis milling machines. This new approach uses a nonconstant offset of the previous tool-path, which guarantees the cutter moving in an unmachined area of the part surface and without redundant machining. The method comprises three steps : (1) the calculation of the tool-path interval, (2) the conversion from the path interval to the parametric interval, and (3) the synthesis of efficient tool-path planning. The rectilinear shortest path problem can be stated as follows: given a set of m non-intersecting simple polygonal obstacles in the plane, find a shortest L"1-metric (rectilinear) path from a point s to a point t that avoids all the obstacles. The path can touch an obstacle but does not cross it. This paper presents an algorithm with time complexity O(n+m(lgn)^3^ ^2), which is close to the known lower bound of @W(n+mlgm) for finding such a path. Here, n is the number of vertices of all the obstacles together. A point-disjoint path cover of a directed graph is a collection of point-disjoint paths (some paths possibly having zero length) which covers all the points. A path cover which minimizes the number of paths corresponds to an optimal sequence of the steps of a computer program for efficient coding and documentation. The minimization problem for the general directed graph is hard in the sense of being NP-complete. In the case of cycle-free digraphs, however, the problem is polynomial, for it is shown that it can be reduced to the maximum-matching problem. A heuristic given here for finding a near optimal path cover for the general case is based upon applying the maximum-matching algorithm to the subgraphs of an interval decomposition. Authentication using a path of trusted intermediaries, each able to authenticate the next in the path, is a well-known technique for authenticating channels in a large distributed system. In this paper, we explore the use of multiple paths to redundantly authenticate a channel and focus on two notions of path independence-disjoint paths and connective paths-that seem to increase assurance in the authentication. We give evidence that there are no efficient algorithms for locating maximum sets of paths with these independence properties and propose several approximation algorithms for these problems. We also describe a service we have deployed, called PathServer, that makes use of our algorithms to find such sets of paths to support authentication in PGP applications. A path or a polygonal domain is C-oriented if the orientations of its edges belong to a set of C given orientations; this is a generalization of the notable rectilinear case ( C = 2 ). We study exact and approximation algorithms for minimum-link C-oriented paths and paths with unrestricted orientations, both in C-oriented and in general domains.Our two main algorithms are as follows:A subquadratic-time algorithm with a non-trivial approximation guarantee for general (unrestricted-orientation) minimum-link paths in general domains.An algorithm to find a minimum-link C-oriented path in a C-oriented domain. Our algorithm is simpler and more time-space efficient than the prior algorithm.We also obtain several related results:3SUM-hardness of determining the link distance with unrestricted orientations (even in a rectilinear domain).An optimal algorithm for finding a minimum-link rectilinear path in a rectilinear domain. The algorithm and its analysis are simpler than the existing ones.An extension of our methods to find a C-oriented minimum-link path in a general (not necessarily C-oriented) domain.A more efficient algorithm to compute a 2-approximate C-oriented minimum-link path.A notion of "robust" paths. We show how minimum-link C-oriented paths approximate the robust paths with unrestricted orientations to within an additive error of 1. In this paper, we study the structure and computational complexity of optimal multi-robot path planning problems on graphs. Our results encompass three formulations of the discrete multi-robot path planning problem, including a variant that allows synchronous rotations of robots along fully occupied, disjoint cycles on the graph. Allowing rotation of robots provides a more natural model for multi-robot path planning because robots can communicate. Our optimality objectives are to minimize the total arrival time, the makespan (last arrival time), and the total distance. On the structure side, we show that, in general, these objectives demonstrate a pairwise Pareto optimal structure and cannot be simultaneously optimized. On the computational complexity side, we extend previous work and show that, regardless of the underlying multi-robot path planning problem, these objectives are all intractable to compute. In particular, our NP-hardness proof for the time optimal versions, based on a minimal and direct reduction from the 3-satisfiability problem, shows that these problems remain NP-hard even when there are only two groups of robots (i.e. robots within each group are interchangeable). This paper presents a new geometrical formulation of the manipulation task planning problem in robotics. The problem is shown to be a constrained instance of the coordinated motion planning problem for multiple moving bodies. The constraints are related to the placement and the motion of objects, and can be expressed geometrically. We give a general paradigm for building Manipulation Task Planners based on the proposed formulation. A manipulation task appears as a path in the configuration space of the robot and all movable objects. A manipulation path is a sequence of constrained paths: transit-paths , where the robot moves "alone'', and transfer-paths , where the robot holds an object. The approach consists in building a Manipulation Graph that models the connectivity between certain regions in the global configuration space by transit-paths and transfer-paths. This approach is then applied to the case of a finite number of object placements and grasps. The nodes in the Manipulation Graph correspond to well identified configurations and the edges correspond to paths built from a series of configuration space slices. An implemented system is presented and discussed. Recent techniques for inferring business relationships between ASs [1,2] have yielded maps that have extremely few invalid BGP paths in the terminology of Gao[3]. However, some relationships inferred by these newer algorithms are incorrect, leading to the deduction of unrealistic AS hierarchies. We investigate this problem and discover what causes it. Having obtained such insight, we generalize the problem of AS relationship inference as a multiobjective optimization problem with node-degree-based corrections to the original objective function of minimizing the number of invalid paths. We solve the generalized version of the problem using the semidefinite programming relaxation of the MAX2SAT problem. Keeping the number of invalid paths small, we obtain a more veracious solution than that yielded by recent heuristics. We study Kan extensions in three weakenings of the Eilenberg-Moore dou- ble category associated to a double monad, that was introduced by Grandis and Par e. To be precise, given a normal oplax double monad T on a double categoryK, we consider the double categories consisting of pseudo T -algebras, 'weak' vertical T -morphisms, hor- izontal T -morphisms and T -cells, where 'weak' means either 'lax', 'colax' or 'pseudo'. Denoting these double categories by Algw(T ), where w = l, c or ps accordingly, our main result gives, in each of these cases, conditions ensuring that (pointwise) Kan extensions can be lifted along the forgetful double functor Algw(T ) ! K. As an application we recover and generalise a result by Getzler, on the lifting of pointwise left Kan extensions along symmetric monoidal enriched functors. As an application of Getzler's result we prove, in suitable symmetric monoidal categories, the existence of bicommutative Hopf monoids that are freely generated by cocommutative comonoids. Constrained path problems have to do with finding paths in graphs subject to constraints. We present a constraint programming approach for solving the Ordered disjoint-paths problem (ODP), i.e., the Disjoint-paths problem where the pairs are associated with ordering constraints. In our approach, we reduce ODP to the Ordered simple path with mandatory nodes problem (OSPMN), i.e., the problem of finding a simple path containing a set of mandatory nodes in a given order. The reduction of the problem is motivated by the fact that we have an appropriate way of dealing with OSPMN based on DomReachability, a propagator that implements a generalized reachability constraint on a directed graph based on the concept of graph variables. The DomReachability constraint has three arguments: (1) a flow graph, i.e., a directed graph with a source node; (2) the dominance relation graph on nodes and edges of the flow graph; and (3) the transitive closure of the flow graph. Our experimental evaluation of DomReachability shows that it provides strong pruning, obtaining solutions with very little search. Furthermore, we show that DomReachability is also useful for defining a good labeling strategy. These experimental results give evidence that DomReachability is a useful primitive for solving constrained path problems over directed graphs. Routing policies are typically partitioned into a few classes that capture the most common practices in use today[1]. Unfortunately, it is known that the reality of routing policies[2] and peering relationships is far more complex than those few classes[1,3]. We take the next step of searching for the appropriate granularity at which policies should be modeled. For this purpose, we study how and where to configure per-prefix policies in an AS-level model of the Internet, such that the selected paths in the model are consistent with those observed in BGP data from multiple vantage points. By comparing business relationships with per-prefix filters, we investigate the role and limitations of business relationships as a model for policies. We observe that popular locations for filtering correspond to valleys where no path should be propagated according to inferred business relationships. This result reinforces the validity of the valley-free property used for business relationships inference. However, given the sometimes large path diversity ASs have, business relationships do not contain enough information to decide which path will be chosen as the best. To model how individual ASs choose their best paths, we introduce a new abstraction: next-hop atoms. Next-hop atoms capture the different sets of neighboring ASs an AS uses for its best routes. We show that a large fraction of next-hop atoms correspond to per-neighbor path choices. A non-negligible fraction of path choices, however, correspond to hot-potato routing and tie-breaking within the BGP decision process, very detailed aspects of Internet routing. Similarity search is an important function in many applications, which usually focuses on measuring the similarity between objects with the same type. However, in many scenarios, we need to measure the relatedness between objects with different types. With the surge of study on heterogeneous networks, the relevance measure on objects with different types becomes increasingly important. In this paper, we study the relevance search problem in heterogeneous networks, where the task is to measure the relatedness of heterogeneous objects (including objects with the same type or different types). A novel measure HeteSim is proposed, which has the following attributes: (1) a uniform measure: it can measure the relatedness of objects with the same or different types in a uniform framework; (2) a path-constrained measure: the relatedness of object pairs are defined based on the search path that connects two objects through following a sequence of node types; (3) a semi-metric measure: HeteSim has some good properties (e.g., self-maximum and symmetric), which are crucial to many data mining tasks. Moreover, we analyze the computation characteristics of HeteSim and propose the corresponding quick computation strategies. Empirical studies show that HeteSim can effectively and efficiently evaluate the relatedness of heterogeneous objects. Motion planning is a key problem in robotics that is concerned with finding a path that satisfies a goal specification subject to constraints. In its simplest form, the solution to this problem consists of finding a path connecting two states, and the only constraint is to avoid collisions. Even for this version of the motion planning problem, there is no efficient solution for the general case [1]. The addition of differential constraints on robot motion or more general goal specifications makes motion planning even harder. Given its complexity, most planning algorithms forego completeness and optimality for slightly weaker notions such as resolution completeness, probabilistic completeness [2], and asymptotic optimality. Bisimulation and bisimilarity are fundamental notions in comparing state-based systems. Their extensions to a variety of systems have been actively pursued in recent years, a notable direction being quantitative extensions. In this paper we present an abstract categorical framework for such extended (bi)simulation notions. We use coalgebras as system models and fibrations for organizing predicates—following the seminal work by Hermida and Jacobs—but our focus is on the structural aspect of fibrational frameworks. Specifically we use morphisms of fibrations as well as canonical liftings of functors via Kan extensions. We apply this categorical framework by deriving some known properties of the Hausdorff pseudometric and approximate bisimulation in control theory.