Physarum Solver Research Articles

Dynamic remodeling model based on chemotaxis of slime molds.

Social infrastructure networks, essential for daily life and economic activities, encompass utilities such as water, electricity, roads, and telecommunications. Dynamic remodeling of these systems is crucial for responding to continuous changes, unexpected events, and increased demand. This study proposes a new dynamic remodeling model inspired by biological mechanisms, focusing on a model based on the chemotaxis of slime molds. Slime molds adapt spontaneously to environmental changes by remodeling through the growth and degeneration of tubes. This capability can be applied to optimizing and dynamic remodeling social infrastructure networks. This study elucidated the chemotactic response characteristics of slime molds using biological experiments. The mold's response was observed by considering changes in the concentration of chemicals as environmental changes, confirming that slime molds adapt to environmental changes by shortening their periodic cycles. Subsequently, based on this dynamic response, we propose a new dynamic model (oscillated Physarum solver, O-PS) that extends the existing Physarum solver (PS). Numerical simulations demonstrated that the O-PS possesses rapid and efficient path-remodeling capabilities. In particular, within a simplified maze network, the O-PS was confirmed to have the same shortest-path searching ability as the PS, while being capable of faster remodeling. This study offers a new approach for optimizing and dynamically remodeling social infrastructure networks by mimicking biological mechanisms, enabling the rapid identification of solutions considering multiple objectives under complex constraints. Furthermore, the variation in convergence speed with oscillation frequency in the O-PS suggests flexibility in responding to environmental changes. Further research is required to develop more effective remodeling strategies.

PANDA: A physarum-inspired algorithm to solve the multi-objective discrete network design problem

The Multi-Objective Discrete Network Design Problem (MO-DNDP) describes the optimisation of a transport network across multiple metrics. Due to the impractical processing times of exact solutions, heuristics have been harnessed to solve this problem. However, the scale of the networks, the set of candidate links and the metrics assessed have been limited. This research proposes a MO-DNDP that encapsulates 8 metrics across three real networks used as benchmarks scaling from 17 km2 to 41,850 km2 to encapsulate the complexity of optimising transport networks at different scales, traffic demands and constraint densities. To solve this more comprehensive MO-DNDP, this paper proposes the Physarum-inspired algorithm PANDA that improves upon the classical Physarum Solver algorithm by harnessing the attractant sensing and merging capabilities of the biological Physarum Polycephalum slime mould. PANDA is a first attempt at solving the complex MO-DNDP using the Physarum foraging model. PANDA consistently scored better than the classical Solver in the land use rating by circumventing landforms, built-up areas, and environmentally sensitive regions. Additionally, PANDA constructed networks that were on average 30 % better than the existing benchmark networks and 32 % better than the Physarum Solver networks. PANDA improved upon the benchmarks by constructing lower capacity roads that were optimised for the actual traffic demand between each origin–destination pair instead of shared arterials.

The capacity constraint physarum solver

Physarum polycephalum inspired algorithm (PPA) has attracted immense attention recently. By modeling real-world problems into a graph and adopting proper adaption equation, PPA could solve system optimization or user equilibrium problems. However, few PPA-based algorithms can tackle problems with capacity constraints, lacking the ability to control link flows in the PPA. To tackle this problem, a novel framework, the capacitated PPA (CPPA), is proposed to allow capacity constraints toward link flow in the PPA. To demonstrate the validity of the CPPA, we have developed three applications of the CPPA, i.e., the CPPA for the maximum-flow problem (CPPA-MF), the CPPA for the minimum-cost–maximum-flow problem (CPPA-MCMF), and the CPPA for the link-capacitated traffic assignment problem (CPPA-CTAP). In the experiments, the CPPA-MF shows competitive performance among four baseline algorithms; the CPPA-MCMF is approximately 50% faster than the baseline algorithms; the CPPA-CTAP reduce the oversaturated degree from approximately 40% to 1.5% compared to the original algorithm. The experimental results prove the validation of the CPPA framework. The CPPA is also robust since it have been successfully applied in three different scenarios. The proposed CPPA extends the PPA’s application to capacity-constrain problems and might shed some lights on its future applications in complex real-world problems.

A Parallel Fully Dynamic Iterative Bio-Inspired Shortest Path Algorithm

We propose a fully dynamic bio-inspired parallel algorithm for solving the shortest path problem on dynamically changing graphs based on Physarum Solver, which is an amoeba shortest path algorithm. Although many sequential algorithms exist for solving dynamic shortest path problem, there are only few parallel algorithms, most of which identify the set of vertices affected by the dynamic changes. However, when the graph size becomes large, the process of determining affected vertices is time-consuming. In fact, when the percentage of changing edges is large, most of the studies in the literature are infeasible. The proposed algorithm is able to identify affected vertices and reconstruct them spontaneously in parallel. Moreover, it is designed to be suitable for dynamically changing graphs since it uses the information arising in earlier iterations. Thus, it computes effectively dynamic shortest path even if percentage of changing edges is large. At each iteration of the proposed algorithm, a sparse linear system of equations needs to be solved, which is the most time-consuming and challenging step of the algorithm especially when the problem size is large. We propose a parallel preconditioned iterative method for solving those sparse linear systems. The proposed preconditioner is specifically designed based on the properties of the coefficient matrix of those linear systems, and the effectiveness of the proposed preconditioner is compared against other state-of-the-art preconditioners on dynamic graphs. The proposed algorithm is evaluated using three different large graph models representing diverse real-life applications on a parallel multicore cluster. The parallel scalability of the proposed algorithm is compared against $${\varDelta }$$ -stepping, which is the most representative parallel implementation of Dijkstra’s algorithm in the Parallel Boost Graph Library.

Adaptive Load Balancing Ad Hoc Routing Scheme Inspired by True Slime Mold

Engineering neo-biomimetics, i.e. imitation models based on body structures and behavior of living organisms, relied upon to solve complex problems, have been studied in various fields. In distributed networks, such as ad-hoc networks and wireless sensor networks, the behavior of a variety of true slime molds which are capable of constructing multipath flow networks based on the amount of body, has been studied. Ad hoc networks only consist of mobile terminals (nodes) that can relay packets along an established route. However, link relations and the available bandwidth of the nodes change dynamically due to the mobility of nodes. In addition, the speed of communication between nodes also varies due to node positions and their communication-related quality. Thus, practical use of ad-hoc networks still remains an issue, because it is difficult to establish stable routes under such environments. This study aims to propose an adaptive load balancing routing technique that adaptively diversifies the transmission paths based on the available bandwidth, residual battery life, and the data transmission volume, by applying a mathematical model of slime mold routing, known as the physarum solver. We confirm the effectiveness of its adaptive behavior in dynamic environments using computer simulations.

Dynamic Physarum Solver: a bio-inspired shortest path method of dynamically changing graphs

In dynamic graphs, edge weights of the graph change with time and solving the shortest path problem in such graphs is an important real-world problem. The studies in the literature require excessive computational time for computing the dynamic shortest path since determining changing edge weights is difficult especially when the graph size becomes large. In this paper, we propose a dynamic bio-inspired algorithm for finding the dynamic shortest path for large graphs based on Physarum Solver, which is a shortest path algorithm for static graphs. The proposed method is evaluated using three different large graph models representing diverse real-life applications. The effect of changing edge weights on the solution time is evaluated for each graph model separately and compared against ∆-stepping, which is the most representative implementation of Dijkstra’s algorithm. Experimental results show that the proposed method easily adapts edge weight changes and computes the dynamic shortest path efficiently.

An Accelerated Physarum Solver for Network Optimization.

As a novel computational paradigm, Physarum solver has received increasing attention from the researchers in tackling a plethora of network optimization problems. However, the convergence of Physarum solver is grounded by solving a system of linear equations iteratively, which often leads to low computational performance. Two factors have been highlighted along the process: 1) high time complexity in solving the system of linear equations and 2) extensive iterations required for convergence. Thus, Physarum solver has been largely restricted by its unsatisfactory computational performance. In this paper, we aim to address these two issues by developing two enhancement strategies: 1) pruning inactive nodes and 2) terminating Physarum solver in advance. First, extensive nodes and edges become and stay inactive after a few iterations in identifying the shortest path. Removing these inactive nodes and edges significantly decreases the graph size, thereby reducing computational complexity. Second, we define a transition phase for edges. All of the paths experiencing such a transition phase are dynamically aggregated to form a set of near-optimal paths among which the optimal path is included. Depth-first search is then leveraged to identify the optimal path from the near-optimal paths set. Earlier termination of Physarum solver saves considerable iterations while guaranteeing the optimality of the found solution. Empirically, 20 randomly generated sparse and complete graphs with network sizes ranging from 50 to 2000 as well as two real-world traffic networks are used to compare the performance of accelerated Physarum solver to the other two state-of-the-art algorithms.

A parallel bio-inspried shortest path algorithm

Physarum polycephalum is an amoeba-like organism and is able to find the shortest path in a labyrinth. Inspired by P. polycephalum, recently, a mathematical model and an algorithm (Physarum Solver) was developed. There are, however, only sequential implementations of this algorithm. In this paper, a fast and efficient parallel Physarum Solver is proposed. The proposed algorithm requires the solution of linear systems whose coefficient matrix is a symmetric M-matrix. The solution of the linear system is the most time consuming step of the Physarum Solver which is classically handled by direct solvers without taking advantage of the fact that the coefficient matrix is an M-matrix. However, direct solvers are infeasible for solving large real-world problems. In the proposed parallel Physarum Solver, an effective parallel iterative linear solver with a parallel preconditioner for M-matrices is used. The parallel scalability, solution time, and accuracy of the proposed algorithm are presented and compared to a state-of-the-art parallel implementation of $$\varDelta $$ -stepping shortest path algorithm in the Parallel Boost Graph Library. Our implementation exhibits a remarkable parallel speedup with comparable accuracy for synthetic and real world applications.

A Bio-Inspired Approach to Traffic Network Equilibrium Assignment Problem.

Finding an equilibrium state of the traffic assignment plays a significant role in the design of transportation networks. We adapt the path finding mathematical model of slime mold Physarum polycephalum to solve the traffic equilibrium assignment problem. We make three contributions in this paper. First, we propose a generalized Physarum model to solve the shortest path problem in directed and asymmetric graphs. Second, we extend it further to resolve the network design problem with multiple source nodes and sink nodes. At last, we demonstrate that the Physarum solver converges to the user-optimized (Wardrop) equilibrium by dynamically updating the costs of links in the network. In addition, convergence of the developed algorithm is proved. Numerical examples are used to demonstrate the efficiency of the proposed algorithm. The superiority of the proposed algorithm is demonstrated in comparison with several other algorithms, including the Frank-Wolfe algorithm, conjugate Frank-Wolfe algorithm, biconjugate Frank-Wolfe algorithm, and gradient projection algorithm.

Physarum solver: a bio-inspired method for sustainable supply chain network design problem

A supplier of products and services aims to minimize the capacity investment cost and the operational cost incurred by unwanted byproducts, e.g. carbon dioxide emission. In this paper, we consider a sustainable supply chain network design problem, where the capacity and the product flow along each link are design variables. We formulate it as a multi-criteria optimization problem. A bio-inspired algorithm is developed to tackle this problem. We illustrate how to design a sustainable supply chain network in three steps. First, we develop a generalized model inspired by the foraging behaviour of slime mould Physarum polycephalum to handle the network optimization with multiple sinks. Second, we propose a strategy to update the link cost iteratively, thus making the Physarum model to converge to a user equilibrium. Third, we perform an equivalent operation to transform a system optimum problem into a corresponding user equilibrium problem so that it is solvable in the Physarum model. The efficiency of the proposed algorithm is illustrated with numerical examples.

An intelligent physarum solver for supply chain network design under profit maximization and oligopolistic competition

We propose an efficient bio-inspired algorithm for design of optimal supply chain networks in a competitive oligopoly markets. The firms compete in manufacture, storage and distribution of a product to several markets. Each firm aims at maximisation of its own profit by optimising the design capacity and product flow in the supply chain. We model the supply chain network as a multi-layer graph of manufacturing nodes, distribution nodes and storage centres. To optimise the network, we adopt the mechanisms of a foraging behaviour of slime mould Physarum polycephalym. First, we extend the original Physarum model to deal with networks with multiple sources and sinks. Second, we develop a novel method to solve the user equilibrium (UE) problem by exploiting the adaptivity of the Physarum model: we update the link costs according to the product flow. Third, we refer to an equivalent transformation between system optimum problem and UE problem to determine the optimal product flows and design capacities of a supply chain. At last, we present an approach to update the amount of product supplied by each firm. By comparing our solutions with that in Nagurney (2010b) on several numerical examples, we demonstrate the efficiency and practicality of the proposed method.

A Bio-Inspired Algorithm for Route Selection in Wireless Sensor Networks

How to determine the optimal communication path in wireless sensor networks (WSNs) is a fundamental problem. In this letter, we formulate the optimal communication path problem and convert it into the shortest path tree (SPT) problem by considering an external base station and sensors as the root node and leaf nodes, respectively. Inspired by a path-finding mathematical model Physarum solver, a novel bio-inspired algorithm is proposed to solve the SPT problem in WSNs. Experimental results demonstrate that the proposed algorithm also has an advantage of adaptivity and performs better than Physarum solver in dynamic small WSNs.

Physarum Learner: A bio-inspired way of learning structure from data

A novel Score-based Physarum Learner algorithm for learning Bayesian Network structure from data is introduced and shown to outperform common score based structure learning algorithms for some benchmark data sets. The Score-based Physarum Learner first initializes a fully connected Physarum-Maze with random conductances. In each Physarum Solver iteration, the source and sink nodes are changed randomly, and the conductances are updated. Connections exceeding a predefined conductance threshold are considered as Bayesian Network edges, and the score of the connected nodes are examined in both directions. A positive or negative feedback is given to the edge conductance based on the calculated scores. Due to randomness in selecting connections for evaluation, an ensemble of Score-based Physarum Learner is used to build the final Bayesian Network structure.

An amoeboid algorithm for solving linear transportation problem

Transportation Problem (TP) is one of the basic operational research problems, which plays an important role in many practical applications. In this paper, a bio-inspired mathematical model is proposed to handle the Linear Transportation Problem (LTP) in directed networks by modifying the original amoeba model Physarum Solver. Several examples are used to prove that the provided model can effectively solve Balanced Transportation Problem (BTP), Unbalanced Transportation Problem (UTP), especially the Generalized Transportation Problem (GTP), in a nondiscrete way.

Physarum solver: A biologically inspired method of road-network navigation

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.

1P495 Solving the shortest path problem by Physarum solver - Modeling of the Adaptive Network of True Slime Mold(24. Mathematical biology,Poster Session,Abstract,Meeting Program of EABS & BSJ 2006)

1P495 Solving the shortest path problem by Physarum solver - Modeling of the Adaptive Network of True Slime Mold(24. Mathematical biology,Poster Session,Abstract,Meeting Program of EABS & BSJ 2006)