The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
D Grigoriev - One of the best experts on this subject based on the ideXlab platform.
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Polytime Algorithm for the Shortest Path in a Homotopy Class amidst Semi-Algebraic Obstacles in the Plane
1998Co-Authors: D Grigoriev, A SlissenkoAbstract:Given a set of semi-algebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct the shortest path between these points in a given Homotopy Class. This path is unique and has some canonical form. We use the representation of Homotopy Classes in a way that is as general as the Classical one. It consists in representing generators of a free group which describes the Classes of Homotopy by disjoint cuts GS97] homeomorphic to rays. We show that given such a system of generators and a word representing a Homotopy Class, one can contruct the shortest path of this Class in time polynomial in the size of the word and in the size of the representation of the obstacles and the cuts. The Homotopy Class may also be represented by a path, then the polynomial complexity will depend on the size of the representation of this path. As a technical notion we introduce one particular system of cuts, which we call an extremity basis, that proves to be especially convenient for algorithmic purposes. The considered problem is motivated by robot motion planning and by theoretical questions arising in shortest path approximations in higher dimensions.
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computing minimum link path in a Homotopy Class amidst semi algebraic obstacles in the plane
Lecture Notes in Computer Science, 1997Co-Authors: D Grigoriev, A SlissenkoAbstract:Given a set of semi-algebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct a polygonal path between these points which has the minimum number of segments (links) and the minimum ‘total turn’, that is the sum of absolute values of angles of turns of the consecutive polygon links. We describe an algorithm that solves the problem spending polynomial time to construct one segment of the minimum-link and minimum-turn polygon if to use a modification of real RAMs which permits to handle the solutions of algebraic equations. It is known that the number of segments in such a minimum-link polygon can be exponential as function of the length of the input data or even of the degree of polynomials representing the semi-algebraic set. In fact, we describe how to construct a minimum-link-turn path for a given Class of Homotopy(whose shortest path has no self-intersections), and provide a rigorous and rather a universal way of reasoning about Homotopy Classes in contexts related to algorithms. It was previously shown by Heintz-Krick-Slissenko-Solerno that a shortest path in the situation under consideration is semi-algebraic, and an extended real RAM that is able to compute integrals of algebraic functions can find it in polytime.
Francis Lazarus - One of the best experts on this subject based on the ideXlab platform.
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Optimal pants decompositions and shortest homotopic cycles on an orientable surface
Journal of the ACM, 2007Co-Authors: Éric Colin De Verdière, Francis LazarusAbstract:We consider the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface. For this purpose, we use a pants decomposition of the surface: a set of disjoint simple cycles that cut the surface into pairs of pants (spheres with three holes). We solve this problem in a framework where the cycles are closed walks on the vertex-edge graph of a combinatorial surface that may overlap but do not cross. We give an algorithm that transforms an input pants decomposition into another homotopic pants decomposition that is optimal: each cycle is as short as possible in its Homotopy Class. As a consequence, finding a shortest cycle homotopic to a given simple cycle amounts to extending the cycle into a pants decomposition and to optimizing it: the resulting pants decomposition contains the desired cycle. We describe two algorithms for extending a cycle to a pants decomposition. All algorithms in this article are polynomial, assuming uniformity of the weights of the vertex-edge graph of the surface.
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optimal system of loops on an orientable surface
Discrete and Computational Geometry, 2005Co-Authors: Éric Colin De Verdière, Francis LazarusAbstract:Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a {\it system of loops} for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its Homotopy Class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.
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Optimal pants decompositions and shortest homotopic cycles on an orientable surface
Lecture Notes in Computer Science, 2003Co-Authors: Éric Colin De Verdière, Francis LazarusAbstract:A pants decomposition of a compact orientable surface M is a set of disjoint simple cycles which cuts M into pairs of pants, i.e., spheres with three boundaries. Assuming M is a polyhedral surface, with weighted vertex-edge graph G, we consider combinatorial pants decompositions: the cycles are closed walks in G that may overlap but do not cross. We give an algorithm which, given a pants decomposition, computes a homotopic pants decomposition in which each cycle is a shortest cycle in its Homotopy Class. In particular, the resulting decomposition is optimal (as short as possible among all homotopic pants decompositions), and any optimal pants decomposition is made of shortest homotopic cycles. Qur algorithm is polynomial in the complexity of the input and in the longest-to-shortest edge ratio of G. The same algorithm can be applied, given a simple cycle C, to compute a shortest cycle homotopic to C which is itself simple.
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optimal system of loops on an orientable surface
Foundations of Computer Science, 2002Co-Authors: Éric Colin De Verdière, Francis LazarusAbstract:Every compact orientable boundaryless surface /spl Mscr/ can be cut along simple loops with a common point /spl upsi//sub 0/, pairwise disjoint except at /spl upsi//sub 0/, so that the resulting surface is a topological disk; such a set of loops is called a fundamental system of loops for /spl Mscr/. The resulting disk is a polygon in which the edges are pairwise identified on the surface; it is called a polygonal schema Assuming that /spl Mscr/ is triangulated, and that each edge has a given length, we are interested in a shortest (or optimal) system homotopic to a given one, drawn on the vertex-edge graph of /spl Mscr/. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its Homotopy Class. We give a polynomial (under some reasonable assumptions) algorithm to build such a system. As a byproduct, we get a polynomial algorithm to compute a shortest simple loop homotopic to a given simple loop.
Maxim Likhachev - One of the best experts on this subject based on the ideXlab platform.
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effective footstep planning using Homotopy Class guidance
Artificial Intelligence, 2020Co-Authors: Vinitha Ranganeni, Sahit Chintalapudi, Oren Salzman, Maxim LikhachevAbstract:Abstract Planning the motion for humanoid robots is a computationally-complex task due to the high dimensionality of the system. Thus, a common approach is to first plan in the low-dimensional space induced by the robot's feet—a task referred to as footstep planning. This low-dimensional plan is then used to guide the full motion of the robot. One approach that has proven successful in footstep planning is using search-based planners such as A* and its many variants. To do so, these search-based planners have to be endowed with effective heuristics to efficiently guide them through the search space. However, designing effective heuristics is a time-consuming task that requires the user to have good domain knowledge. Thus, our goal is to be able to effectively plan the footstep motions taken by a humanoid robot while obviating the burden on the user to carefully design local-minima free heuristics. To this end, we propose to use user-defined Homotopy Classes in the workspace that are intuitive to define. These Homotopy Classes are used to automatically generate heuristic functions that efficiently guide the footstep planner. Additionally, we present an extension to Homotopy Classes such that they are applicable to complex multi-level environments. We compare our approach for footstep planning with a standard approach that uses a heuristic common to footstep planning. In simple scenarios, the performance of both algorithms is comparable. However, in more complex scenarios our approach allows for a speedup in planning of several orders of magnitude when compared to the standard approach.
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effective footstep planning for humanoids using Homotopy Class guidance
International Conference on Automated Planning and Scheduling, 2017Co-Authors: Vinitha Ranganeni, Oren Salzman, Maxim LikhachevAbstract:Planning the motion for humanoid robots is a computationally-complex task due to the high dimensionality of the system. Thus, a common approach is to first plan in the low-dimensional space induced by the robot's feet--a task referred to as footstep planning. This low-dimensional plan is then used to guide the full motion of the robot. One approach that has proven successful in footstep planning is using search-based planners such as A* and its many variants. To do so, these search-based planners have to be endowed with effective heuristics to efficiently guide them through the search space. However, designing effective heuristics is a time-consuming task that requires the user to have good domain knowledge. Thus, our goal is to be able to effectively plan the footstep motions taken by a humanoid robot while obviating the burden on the user to carefully design local-minima free heuristics. To this end, we propose to use user-defined Homotopy Classes in the workspace that are intuitive to define. These Homotopy Classes are used to automatically generate heuristic functions that efficiently guide the footstep planner. We compare our approach for footstep planning with a standard approach that uses a heuristic common to footstep planning. In simple scenarios, the performance of both algorithms is comparable. However, in more complex scenarios our approach allows for a speedup in planning of several orders of magnitude when compared to the standard approach.
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A Homotopy-like Class Invariant for Sub-manifolds of Punctured Euclidean Spaces
arXiv: Differential Geometry, 2011Co-Authors: Subhrajit Bhattacharya, Maxim Likhachev, Vijay KumarAbstract:We consider the $D$-dimensional Euclidean space, $\mathbb{R}^D$, with certain $(D-N)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{singularity manifolds} and represent by $\widetilde{\mathcal{S}}$) removed from it. We define and investigate the problem of finding a Homotopy-like Class invariant ($\chi$-Homotopy) for certain $(N-1)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{candidate manifolds} and represent by $\omega$) of $\mathbb{R}^D \setminus \widetilde{\mathcal{S}}$, with special emphasis on computational aspects of the problem. We determine a differential $(N-1)$-form, $\psi_{\widetilde{\mathcal{S}}}$, such that $\chi_{\widetilde{\mathcal{S}}}(\omega) = \int_\omega \psi_{\widetilde{\mathcal{S}}}$ is a Class invariant for such candidate manifolds. We show that the formula agrees with formulae from Cauchy integral theorem and Residue theorem of complex analysis (when $D=2,N=2$), Biot-Savart law and Ampere's law of theory of electromagnetism (when $D=3,N=2$), and the Gauss divergence theorem (when $D=3,N=3$), and discover that the underlying equivalence relation suggested by each of these well-known theorems is the $\chi$-Homotopy of sub-manifolds of these low dimensional punctured Euclidean spaces. We describe numerical techniques for computing $\psi_{\widetilde{\mathcal{S}}}$ and its integral on $\omega$, and give numerical validations of the proposed theory for a problem in a 5-dimensional Euclidean space. We also discuss a specific application from \emph{robot path planning problem}, when N=2, and describe a method for computing least cost paths with Homotopy Class constraints using \emph{graph search techniques}.
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search based path planning with Homotopy Class constraints
Annual Symposium on Combinatorial Search, 2010Co-Authors: Subhrajit Bhattacharya, Vijay Kumar, Maxim LikhachevAbstract:Goal-directed path planning is one of the basic and widely studied problems in the field of mobile robotics. Homotopy Classes of trajectories, arising due to the presence of obstacles, are defined as sets of trajectories that can be transformed into each other by gradual bending and stretching without colliding with obstacles. The problem of finding least-cost paths restricted to a specific Homotopy Class or finding least-cost paths that do not belong to certain Homotopy Classes arises frequently in such applications as predicting paths for dynamic entities and computing heuristics for path planning with dynamic constraints. In the present work, we develop a compact way of representing Homotopy Classes and propose an efficient method of graph search-based optimal path planning with constraints on Homotopy Classes. The method is based on representing the environment of the robot as a complex plane and making use of the Cauchy Integral Theorem. We prove optimality of the method and show its efficiency experimentally.
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search based path planning with Homotopy Class constraints in 3d
National Conference on Artificial Intelligence, 2010Co-Authors: Subhrajit Bhattacharya, Maxim Likhachev, Vijay KumarAbstract:Homotopy Classes of trajectories, arising due to the presence of obstacles, are defined as sets of trajectories that can be transformed into each other by gradual bending and stretching without colliding with obstacles. The problem of exploring/finding the different Homotopy Classes in an environment and the problem of finding least-cost paths restricted to a specific Homotopy Class (or not belonging to certain Homotopy Classes) arises frequently in such applications as predicting paths for unpredictable entities and deployment of multiple agents for efficient exploration of an environment. In (Bhattacharya, Kumar, and Likhachev 2010) we have shown how Homotopy Classes of trajectories on a two-dimensional plane with obstacles can be Classified and identified using the Cauchy Integral Theorem and the Residue Theorem from Complex Analysis. In more recent work (Bhattacharya, Likhachev, and Kumar 2011) we extended this representation to three-dimensional spaces by exploiting certain laws from the Theory of Electromagnetism (Biot-Savart law and Ampere's Law) for representing and identifying Homotopy Classes in three dimensions in an efficient way. Using such a representation, we showed that Homotopy Class constraints can be seamlessly weaved into graph search techniques for determining optimal path constrained to certain Homotopy Classes or forbidden from others, as well as for exploring different Homotopy Classes in an environment.
A Slissenko - One of the best experts on this subject based on the ideXlab platform.
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Polytime Algorithm for the Shortest Path in a Homotopy Class amidst Semi-Algebraic Obstacles in the Plane
1998Co-Authors: D Grigoriev, A SlissenkoAbstract:Given a set of semi-algebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct the shortest path between these points in a given Homotopy Class. This path is unique and has some canonical form. We use the representation of Homotopy Classes in a way that is as general as the Classical one. It consists in representing generators of a free group which describes the Classes of Homotopy by disjoint cuts GS97] homeomorphic to rays. We show that given such a system of generators and a word representing a Homotopy Class, one can contruct the shortest path of this Class in time polynomial in the size of the word and in the size of the representation of the obstacles and the cuts. The Homotopy Class may also be represented by a path, then the polynomial complexity will depend on the size of the representation of this path. As a technical notion we introduce one particular system of cuts, which we call an extremity basis, that proves to be especially convenient for algorithmic purposes. The considered problem is motivated by robot motion planning and by theoretical questions arising in shortest path approximations in higher dimensions.
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computing minimum link path in a Homotopy Class amidst semi algebraic obstacles in the plane
Lecture Notes in Computer Science, 1997Co-Authors: D Grigoriev, A SlissenkoAbstract:Given a set of semi-algebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct a polygonal path between these points which has the minimum number of segments (links) and the minimum ‘total turn’, that is the sum of absolute values of angles of turns of the consecutive polygon links. We describe an algorithm that solves the problem spending polynomial time to construct one segment of the minimum-link and minimum-turn polygon if to use a modification of real RAMs which permits to handle the solutions of algebraic equations. It is known that the number of segments in such a minimum-link polygon can be exponential as function of the length of the input data or even of the degree of polynomials representing the semi-algebraic set. In fact, we describe how to construct a minimum-link-turn path for a given Class of Homotopy(whose shortest path has no self-intersections), and provide a rigorous and rather a universal way of reasoning about Homotopy Classes in contexts related to algorithms. It was previously shown by Heintz-Krick-Slissenko-Solerno that a shortest path in the situation under consideration is semi-algebraic, and an extended real RAM that is able to compute integrals of algebraic functions can find it in polytime.
Éric Colin De Verdière - One of the best experts on this subject based on the ideXlab platform.
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Optimal pants decompositions and shortest homotopic cycles on an orientable surface
Journal of the ACM, 2007Co-Authors: Éric Colin De Verdière, Francis LazarusAbstract:We consider the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface. For this purpose, we use a pants decomposition of the surface: a set of disjoint simple cycles that cut the surface into pairs of pants (spheres with three holes). We solve this problem in a framework where the cycles are closed walks on the vertex-edge graph of a combinatorial surface that may overlap but do not cross. We give an algorithm that transforms an input pants decomposition into another homotopic pants decomposition that is optimal: each cycle is as short as possible in its Homotopy Class. As a consequence, finding a shortest cycle homotopic to a given simple cycle amounts to extending the cycle into a pants decomposition and to optimizing it: the resulting pants decomposition contains the desired cycle. We describe two algorithms for extending a cycle to a pants decomposition. All algorithms in this article are polynomial, assuming uniformity of the weights of the vertex-edge graph of the surface.
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optimal system of loops on an orientable surface
Discrete and Computational Geometry, 2005Co-Authors: Éric Colin De Verdière, Francis LazarusAbstract:Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a {\it system of loops} for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its Homotopy Class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.
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Optimal pants decompositions and shortest homotopic cycles on an orientable surface
Lecture Notes in Computer Science, 2003Co-Authors: Éric Colin De Verdière, Francis LazarusAbstract:A pants decomposition of a compact orientable surface M is a set of disjoint simple cycles which cuts M into pairs of pants, i.e., spheres with three boundaries. Assuming M is a polyhedral surface, with weighted vertex-edge graph G, we consider combinatorial pants decompositions: the cycles are closed walks in G that may overlap but do not cross. We give an algorithm which, given a pants decomposition, computes a homotopic pants decomposition in which each cycle is a shortest cycle in its Homotopy Class. In particular, the resulting decomposition is optimal (as short as possible among all homotopic pants decompositions), and any optimal pants decomposition is made of shortest homotopic cycles. Qur algorithm is polynomial in the complexity of the input and in the longest-to-shortest edge ratio of G. The same algorithm can be applied, given a simple cycle C, to compute a shortest cycle homotopic to C which is itself simple.
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optimal system of loops on an orientable surface
Foundations of Computer Science, 2002Co-Authors: Éric Colin De Verdière, Francis LazarusAbstract:Every compact orientable boundaryless surface /spl Mscr/ can be cut along simple loops with a common point /spl upsi//sub 0/, pairwise disjoint except at /spl upsi//sub 0/, so that the resulting surface is a topological disk; such a set of loops is called a fundamental system of loops for /spl Mscr/. The resulting disk is a polygon in which the edges are pairwise identified on the surface; it is called a polygonal schema Assuming that /spl Mscr/ is triangulated, and that each edge has a given length, we are interested in a shortest (or optimal) system homotopic to a given one, drawn on the vertex-edge graph of /spl Mscr/. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its Homotopy Class. We give a polynomial (under some reasonable assumptions) algorithm to build such a system. As a byproduct, we get a polynomial algorithm to compute a shortest simple loop homotopic to a given simple loop.