The directed Physarum dynamics is known to solve positive linear programs: minimize @math subject to @math and @math for a positive cost vector @math . The directed Physarum dynamics evolves a positive vector @math according to the dynamics @math . Here @math is the solution to @math that minimizes the "energy" @math . In this paper, we study the non-uniform directed dynamics @math , where @math is a positive diagonal matrix. The non-uniform dynamics is more complex than the uniform dynamics (with @math being the identity matrix), as it allows each component of @math to react with different speed to the differences between @math and @math . Our contribution is to show that the non-uniform directed dynamics solves positive linear programs.
is a slime mold that apparently is able to solve shortest path problems. Nakagaki, Yamada, and T ' o th @cite_0 report about the following experiment; see Figure . They built a maze, covered it by pieces of Physarum (the slime can be cut into pieces which will reunite if brought into vicinity), and then fed the slime with oatmeal at two locations. After a few hours the slime retracted to a path following the shortest path in the maze connecting the food sources. The authors report that they repeated the experiment with different mazes; in all experiments, Physarum retracted to the shortest path.
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Physarum Polycephalum is a slime mold that apparently is able to solve shortest path problems. A mathematical model has been proposed by biologists to describe the feedback mechanism used by the slime mold to adapt its tubular channels while foraging two food sources s0 and s1. We prove that, under this model, the mass of the mold will eventually converge to the shortest s0-s1 path of the network that the mold lies on, independently of the structure of the network or of the initial mass distribution. This matches the experimental observations by the biologists and can be seen as an example of a "natural algorithm", that is, an algorithm developed by evolution over millions of years. Physarum polycephalum is a unicellular, multinucleate slime mold in the Mycetozoa group that exhibits remarkable capabilities, including the ability to construct efficient networks when foraging food [1]. The Physarum cell grows, as long as nutrition is abundant. When nutrition is limited, Physarum forms a network of interconnected veins; the veins are gel-like tubes in which the cytoplasm flows. It has been experimentally observed that, with two food sources, Physarum's tubular network often retracts to the shortest path between the food sources. Tero, Kobayashi and Nakagaki [2] proposed a model for the dynamics of the Physarum, which in the computer simulations converged to the shortest path on any initial network. We analytically prove that, under this model, the mass of the mold has to eventually converge to the shortest path in the initial network between the two food sources, independently of the structure of the initial network or of the initial mass distribution. This presentation is based on the work in [3,4] and is supported by the Italian Flagship Initiative "InterOmics" (PB.P05). References [1] Toshiyuki Nakagaki, Hiroyasu Yamada, Agota Toth: Maze-solving by an amoeboid organism. Nature 407:470 (2000) [2] Atsushi Tero, Ryo Kobayashi, Toshiyuki Nakagaki: A mathematical model for adaptive transport network in path finding by true slime mold. Journal of Theoretical Biology 244: 553-564 (2007) [3] Vincenzo Bonifaci, Kurt Mehlhorn, Girish Varma: Physarum can compute shortest paths. Journal of Theoretical Biology 309: 121-133 (2012) [4] Vincenzo Bonifaci: Physarum can compute shortest paths: A short proof. Information Processing Letters 113(1-2): 4-7 (2013) We describe here a mathematical model of the adaptive dynamics of a transport network of the true slime mold Physarum polycephalum, an amoeboid organism that exhibits path-finding behavior in a maze. This organism possesses a network of tubular elements, by means of which nutrients and signals circulate through the plasmodium. When the organism is put in a maze, the network changes its shape to connect two exits by the shortest path. This process of path-finding is attributed to an underlying physiological mechanism: a tube thickens as the flux through it increases. The experimental evidence for this is, however, only qualitative. We constructed a mathematical model of the general form of the tube dynamics. Our model contains a key parameter corresponding to the extent of the feedback regulation between the thickness of a tube and the flux through it. We demonstrate the dependence of the behavior of the model on this parameter. When two food sources are presented to the slime mold Physarum in the dark, a thick tube for absorbing nutrients is formed that connects the food sources through the shortest route. When the light-avoiding organism is partially illuminated, however, the tube connecting the food sources follows a different route. Defining risk as the experimentally measurable rate of light-avoiding movement, the minimum-risk path is exhibited by the organism, determined by integrating along the path. A model for an adaptive-tube network is presented that is in good agreement with the experimental observations. Physarum polycephalum (true slime mold) has recently emerged as a fascinating example of biological computation through morphogenesis. Despite being a single cell organism, experiments have observed that through its growth process, the Physarum is able to solve various minimum cost flow problems. This paper analyzes a mathematical model of the Physarum growth dynamics. We show how to encode general linear programming (LP) problems as instances of the Physarum. We prove that under the growth dynamics, the Physarum is guaranteed to converge to the optimal solution of the LP. We further derive an efficient discrete algorithm based on the Physarum model, and experimentally verify its performance on assignment problems. The plasmodium of the slime mould Physarum polycephalum is a large amoeba-like cell consisting of a dendritic network of tube-like structures (pseudopodia). It changes its shape as it crawls over a plain agar gel and, if food is placed at two different points, it will put out pseudopodia that connect the two food sources. Here we show that this simple organism has the ability to find the minimum-length solution between two points in a labyrinth. We have proposed a mathematical model for the adaptive dynamics of the transport network in an amoeba-like organism, the true slime mold Physarum polycephalum. The model is based on physiological observations of this species, but can also be used for path-finding in the complicated networks of mazes and road maps. In this paper, we describe the physiological basis and the formulation of the model, as well as the results of simulations of some complicated networks. The path-finding method used by Physarum is a good example of cellular computation. In this paper we study dynamics inspired by Physarum polycephalum (a slime mold) for solving linear programs [NTY00, IJNT11, JZ12]. These dynamics are arrived at by a local and mechanistic interpretation of the inner workings of the slime mold and a global optimization perspective has been lacking even in the simplest of instances. Our first result is an interpretation of the dynamics as an optimization process. We show that Physarum dynamics can be seen as a steepest-descent type algorithm on a certain Riemannian manifold. Moreover, we prove that the trajectories of Physarum are in fact paths of optimizers to a parametrized family of convex programs, in which the objective is a linear cost function regularized by an entropy barrier. Subsequently, we rigorously establish several important properties of solution curves of Physarum. We prove global existence of such solutions and show that they have limits, being optimal solutions of the underlying LP. Finally, we show that the discretization of the Physarum dynamics is efficient for a class of linear programs, which include unimodular constraint matrices. Thus, together, our results shed some light on how nature might be solving instances of perhaps the most complex problem in P: linear programming. In this paper we study dynamics inspired by Physarum polycephalum (a slime mold) for solving linear programs [NTY00, IJNT11, JZ12]. These dynamics are arrived at by a local and mechanistic interpretation of the inner workings of the slime mold and a global optimization perspective has been lacking even in the simplest of instances. Our first result is an interpretation of the dynamics as an optimization process. We show that Physarum dynamics can be seen as a steepest-descent type algorithm on a certain Riemannian manifold. Moreover, we prove that the trajectories of Physarum are in fact paths of optimizers to a parametrized family of convex programs, in which the objective is a linear cost function regularized by an entropy barrier. Subsequently, we rigorously establish several important properties of solution curves of Physarum. We prove global existence of such solutions and show that they have limits, being optimal solutions of the underlying LP. Finally, we show that the discretization of the Physarum dynamics is efficient for a class of linear programs, which include unimodular constraint matrices. Thus, together, our results shed some light on how nature might be solving instances of perhaps the most complex problem in P: linear programming. Many biological systems are composed of unreliable components which self-organize efficiently into systems that can tackle complex problems. One such example is the true slimemold Physarum polycephalum which is an amoeba-like organism that seeks food sources and efficiently distributes nutrients throughout its cell body. The distribution of nutrients is accomplished by a self-assembled resource distribution network of small tubes with varying diameter which can evolve with changing environmental conditions without any global control. In this paper, we use a phenomenological model for the tube evolution in slime mold and map it to a path formation protocol for wireless sensor networks. By selecting certain evolution parameters in the protocol, the network may evolve toward single paths connecting data sources to a data sink. In other parameter regimes, the protocol may evolve toward multiple redundant paths. We present detailed analysis of a small model network. A thorough understanding of the simple network leads to design insights into appropriate parameter selection. We also validate the design via simulation of large-scale realistic wireless sensor networks using the QualNet network simulator. The present study investigated the trail-following behavior of the subterranean termite Coptotermes gestroi (Wasmann Rhinotermitidae) under laboratory conditions. The results showed that workers were the first to initiate the exploration to the food source. When food was discovered they returned to the nest laying a trail for recruiting nestmates to the food source. In this situation, workers always traveled significantly faster when returning from the arenas. Both workers and soldiers were recruited to the food source; however, the soldier worker proportion was higher during the first phase of the recruitment. When no food was available, the number of recruited nestmates and the speed on their way back to the nest were significantly lower. The results also showed that scout foragers always laid trail pheromones when entering into unknown territories, and that chemical signals found in the food could induce workers of C. gestroi to increase their travel speed. We consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it. We seek algorithms that achieve a bounded ratio of the worst-case distance traversed in order to see all visible points of the environment (thus creating a map), divided by the optimum distance needed to verify the map, if we had it in the beginning. The situation is complicated by the fact that the latter off-line problem (the problem of optimally verifying a map) is NP-hard. Although we show that there is no such “competitive” algorithm for general obstacle courses, we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it. We restrict ourselves to the rectilinear case, where each side of the obstacles and the room is parallel to one of the coordinates, and the robot must also move either parallel or perpendicular to the sides. (In a subsequent paper, we will discuss the extension to polygons of general shapes.) We also discuss the off-line problem for simple rectilinear polygons and find an optimal solution (in the L 1 metric) in polynomial time, in the case where the entry and the exit are different points. Given a robot R, a set S of obstacles, and points p and q, the Shortest Path Problem is to find the shortest path for R to move from p to q without crashing into any of the obstacles. We show that if the problem is restricted to a disc-shaped robot in the plane with nonintersecting polygons as obstacles then the shortest path can be found in time O(n 2 log n) where n is the number of edges that make up the polygonal obstacles. This matches the best time currently known for the simpler problem of finding the shortest path in the plane for a point robot. We present a practical algorithm for finding minimum-length paths between points in the Euclidean plane with (not necessarily convex) polygonal obstacles. Prior to this work, the best known algorithm for finding the shortest path between two points in the plane required O(n 2 log n) time and O (n 2 ) space, where n denotes the number of obstacle edges. Assuming that a triangulation or a Voronoi diagram for the obstacle space is provided with the input (if is not, either one can be precomputed in O ( n log n) time), we present an O(kn) time algorithm, where k denotes the number of “islands” (connected components) in the obstacle space. The algorithm uses only O(n) space and, given a source point s , produces an O(n) size data structure such that the distance between s and any other point x in the plane ( x ) is not necessarily an obstacle vertex or a point on an obstacle edge) can be computed in O (1) time. The algorithm can also be used to compute shortest paths for the movement of a disk (so that optimal movement for arbitrary objects can be computed to the accuracy of enclosing them with the smallest possible disk). We consider the problem of navigating through an unknownenvironment in which the obstacles are disjoint oriented rectanglesenclosed in an n x n square room. The task of navigatingalgorithm is to reach the center of the room starting from one of thecorners. While there always exists a path of length n , the best previously knownnavigating algorithm finds paths of length n 20 1nn . We give an efficient deterministicalgorithm which finds a path of length O ( n ln n ); this algorithm uses tactileinformation only. Moreover, we prove that any deterministic algorithmcan be forced to traverse a distance ofO( n ln n ), even if it uses visualinformation. 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. Using insights from biological processes could help to design new optimization techniques for long-standing computational problems. This paper exploits a cellular computing model in the slime mold physarum polycephalum to solve the Steiner tree problem which is an important NP-hard problem in various applications, especially in network design. Inspired by the path-finding and network formation capability of physarum, we develop a new optimization algorithm, named as the physarum optimization, with low complexity and high parallelism. To validate and evaluate our proposed models and algorithm, we further apply the physarum optimization to the minimal exposure problem which is a fundamental problem corresponding to the worst-case coverage in wireless sensor networks. Complexity analysis and simulation results show that our proposed algorithm could achieve good performance with low complexity. Moreover, the core mechanism of our physarum optimization also may provide a useful starting point to develop some practical distributed algorithms for network design. We present new techniques for establishing lower bounds in robot motion planning problems. Our scheme is based on path encoding and uses homotopy equivalence classes of paths to encode state. We first apply the method to the shortest path problem in 3 dimensions. The problem is to find the shortest path under an Lp metric (e.g. a euclidean metric) between two points amid polyhedral obstacles. Although this problem has been extensively studied, there were no previously known lower bounds. We show that there may be exponentially many shortest path classes in single-source multiple-destination problems, and that the single-source single-destination problem is NP-hard. We use a similar proof technique to show that two dimensional dynamic motion planning with bounded velocity is NP-hard. Finally we extend the technique to compliant motion planning with uncertainty in control. Specifically, we consider a point in 3 dimensions which is commanded to move in a straight line, but whose actual motion may differ from the commanded motion, possibly involving sliding against obstacles. Given that the point initially lies in some start region, the problem of finding a sequence of commanded velocities which is guaranteed to move the point to the goal is shown to be non-deterministic exponential time hard, making it the first provably intractable problem in robotics. Multiple objects in the plane are called pseudodisks if the boundaries of any two of them intersect transversely at most twice. Given a set of @math (possibly intersecting) convex pseudodisks of @math complexity each and two points @math and @math in the plane, we present an efficient algorithm for computing a shortest @math -to- @math path avoiding the pseudodisks. After the union of the pseudodisks is computed, which can be done in @math randomized time or @math deterministic time, our algorithm runs in @math deterministic time, where @math is the size of the extended visibility graph of the union of the pseudodisks. Note that @math in the worst case. In over two decades, the previously best algorithms for this problem have not improved on the bound of @math time, even when all the pseudodisks are pairwise disjoint disks. Our technique is also applicable to a motion planning problem of finding a shortest path to translate a convex object in the plane from one location to anot... We propose a humanoid robot navigation planning framework that reuses previous experience to decrease planning time. The framework is intended for navigating complex unstructured environments using both palm and foot contacts. In a complex environment, discrete-search-based contact space planners trade-off between high branching factor and action flexibility. Although approaches such as weighted A∗, ARA∗ and ANA∗ could speed up the search by compromising on optimality, they can be very slow when the heuristic is inaccurate. In the proposed framework, an experience-retrieval module is added in parallel to ANA∗. This module collects previously-generated motion plans and clusters them based on contact pose similarity to form a motion plan library. To retrieve an appropriate plan from the library for a given environment, the framework uses a distance between the contact poses in the plan and environment surfaces. Candidate plans are then modified with local trajectory optimization until a plan fitting the query environment is found. Our experiments show that the proposed framework outperforms planning-from-scratch in success rate in unstructured environments by at least 28 and can navigate difficult environments such as rubble and narrow corridors.