Path Enumeration Research Articles

Markovian traffic equilibrium assignment based on network generalized extreme value model

This study establishes Markovian traffic equilibrium assignment based on the network generalized extreme value (NGEV) model, which we call NGEV equilibrium assignment. The use of the NGEV model for route choice modeling has recently been proposed, and it enables capturing the path correlation without explicit path enumeration. However, the theoretical properties of the model in traffic assignment have yet to be investigated in the literature, which has limited the practical applicability of the NGEV model in the traffic assignment field. This study addresses the research gap by providing the theoretical developments necessary for the NGEV equilibrium assignment. We first show that the NGEV assignment can be formulated and solved under the same path algebra as the traditional Markovian traffic assignment models. Moreover, we present the equivalent optimization formulations to the NGEV equilibrium assignment. The formulations allow us to derive both primal and dual types of efficient solution algorithms. In particular, the dual algorithm is based on the accelerated gradient method that is for the first time applied in the traffic assignment. The numerical experiments showed the excellent convergence and complementary relationship of the proposed primal and dual algorithms.

An Automatic Planning-Based Attack Path Discovery Approach from IT to OT Networks

With the convergence of IT and OT networks, more opportunities can be found to destroy physical processes by cyberattacks. Discovering attack paths plays a vital role in describing possible sequences of exploitation. Automated planning that is an important branch of artificial intelligence (AI) is introduced into the attack graph modeling. However, while adopting the modeling method for large-scale IT and OT networks, it is difficult to meet urgent demands, such as scattered data management, scalability, and automation. To that end, an automatic planning-based attack path discovery approach is proposed in this paper. At first, information of the attacking knowledge and network topology is formally represented in a standardized planning domain definition language (PDDL), integrated into a graph data model. Subsequently, device reachability graph partitioning algorithm is introduced to obtain subgraphs that are small enough and of limited size, which facilitates the discovery of attack paths through the AI planner as soon as possible. In order to further cope with scalability problems, a multithreading manner is used to execute the attack path enumeration for each subgraph. Finally, an automatic workflow with the assistance of a graph database is provided for constructing the PDDL problem file for each subgraph and traversal query in an interactive way. A case study is presented to demonstrate effectiveness of attack path discovery and efficiency with the increase in number of devices.

Gaussian Asymptotics of Jack Measures on Partitions From Weighted Enumeration of Ribbon Paths

Abstract In this paper, we determine two asymptotic results for Jack measures $M(v^{\textrm {out}}, v^{\textrm {in}})$, a measure on partitions defined by two specializations $v^{\textrm {out}}, v^{\textrm {in}}$ of Jack polynomials proposed by Borodin–Olshanski in [10]. Assuming $v^{\textrm {out}} = v^{\textrm {in}}$, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to vanishing, fixed, and diverging values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call “ribbon paths,” show for arbitrary $v^{\textrm {out}}, v^{\textrm {in}}$ that certain Jack measure joint cumulants ${\kappa _n}$ are weighted sums of connected ribbon paths on $n$ sites with $n-1+g$ pairings, and derive our two results from the contributions of $(n,g)=(1,0)$ and $(2,0)$, respectively. Our analysis makes use of Nazarov–Sklyanin’s spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack–Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy–Dołęga.

Distributed hop-constrained s-t simple path enumeration at billion scale

Hop-constrained s-t simple path (HC-s-t path) enumeration is a fundamental problem in graph analysis and has received considerable attention recently. Straightforward distributed solutions are inefficient and suffer from poor scalabiltiy when addressing this problem in billion-scale graphs due to the disability of pruning fruitless exploration or huge memory consumption. Motivated by this, in this paper, we aim to devise an efficient and scalable distributed algorithm to enumerate the HC-s-t paths in billion-scale graphs. We first propose a new hybrid search paradigm tailored for HC-s-t path enumeration. Based on the new search paradigm, we devise a distributed enumeration algorithm following the divide-and-conquer strategy. The algorithm can not only prune fruitless exploration, but also well bound the memory consumption with high parallelism. We also devise an effective workload balance mechanism that is automatically triggered by the idle machines to handle skewed workloads. Moreover, we explore the bidirectional search strategy to further improve enumeration efficiency. The experiment results demonstrate the efficiency of our proposed algorithm.

APPLICATIONS OF TRAVELING SALESMAN PROBLEM ON THE OPTIMAL SIGHTSEEING ORDERS OF MACAO WORLD HERITAGE SITES WITH REAL TIME OR DISTANCE VALUES BETWEEN EVERY PAIR OF SITES

The optimal route of sightseeing orders for visiting every Macao World Heritage Site at exactly once was calculated with Simulated Annealing and Metropolis Algorithm (SAMA) after considering real required time or traveling distance between pairs of sites by either driving a car, taking a bus, or on foot. We found out that, with the optimal tour path, it took roughly 78 minutes for driving a car, 115 minutes on foot, while 117 minutes for taking a bus. On the other hand, the optimal total distance for driving a car would be 13.918 km while for pedestrians to walk, 7.844 km. These results probably mean that there is large space for the improvement on public transportation in this city. Comparison of computation time demanded between the brute- force enumeration of all possible paths and SAMA was also presented, together with animation of the processes for the algorithm to find out the optimal route. It is expected that computation time is astronomically increasing for the brute-force enumeration with more number of sites, while it only takes SAMA much less order of magnitude in time to calculate the optimal solution for larger number of sites. Several optimal options of routes were also provided in each transportation method. However, it is possible that in some types of transportation there could be only one optimal route having no circular or mirrored duplicates.

Path-integral treatment of quantum bouncers

The one-sided bouncer (1SB) and the symmetric bouncer (SB) involve a one-dimensional particle in a piecewise linear potential. For such problems, the time-dependent quantum mechanical propagator cannot be found in closed form. The semiclassical Feynman path integral (PI) is a very appealing approach, as it approximates the propagator by a closed-form expression (a sum over a finite number of classical paths). In this paper we solve the classical path enumeration problem. We obtain closed-form expressions for the initial velocity, bounce times, focal times, action, van Vleck determinant, and Morse index for each classical path. We calculate the propagator within the semiclassical approximation. The numerical results agree with eigenfunction expansion results away from caustics. We derive mappings between the 1SB and SB, which explains why each bounce of the 1SB increases the Morse index by 2 and results in a phase change of π. We interpret the semiclassical Feynman PI to obtain visualizations of matter wave propagation based on interference between classical paths, in analogy with the traditional visualization of light wave propagation as interference between classical ray paths.

Universal singular exponents in catalytic variable equations

Catalytic equations appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of maps of size n in various planar map classes grows asymptotically like c⋅n−5/2γn, for suitable positive constants c and γ. Essentially we have to distinguish between linear catalytic equations (where the subexponential growth is n−3/2) and non-linear catalytic equations (where we have n−5/2 as in planar maps). Furthermore we provide a quite general central limit theorem for parameters that can be encoded by catalytic functional equations, even when they are not positive.

Fast diffraction pathfinding for dynamic sound propagation

In the context of geometric acoustic simulation, one of the more perceptually important yet difficult to simulate acoustic effects is diffraction, a phenomenon that allows sound to propagate around obstructions and corners. A significant bottleneck in real-time simulation of diffraction is the enumeration of high-order diffraction propagation paths in scenes with complex geometry (e.g. highly tessellated surfaces). To this end, we present a dynamic geometric diffraction approach that consists of an extensive mesh preprocessing pipeline and complementary runtime algorithm. The preprocessing module identifies a small subset of edges that are important for diffraction using a novel silhouette edge detection heuristic. It also extends these edges with planar diffraction geometry and precomputes a graph data structure encoding the visibility between the edges. The runtime module uses bidirectional path tracing against the diffraction geometry to probabilistically explore potential paths between sources and listeners, then evaluates the intensities for these paths using the Uniform Theory of Diffraction. It uses the edge visibility graph and the A* pathfinding algorithm to robustly and efficiently find additional high-order diffraction paths. We demonstrate how this technique can simulate 10th-order diffraction up to 568 times faster than the previous state of the art, and can efficiently handle large scenes with both high geometric complexity and high numbers of sources.

Alternative method of counting the number of efficient paths in a transportation network

The number of efficient paths between an origin-destination (O-D) pair provides the route diversity degree of possibly used paths in a transportation network and has many important applications. The existing counting method was based on the Bell loading method for Logit model to determine the number of efficient paths between any two nodes without path enumeration. However, this method has a high time-complexity and requires many unnecessary computations, which significantly hinder its use in large-scale networks. Inspired by the Dial loading method for Logit model, this paper develops a more computationally attractive method to count not only the number of efficient paths, but also the number of efficient paths using each link/node, and the total/average cost of these paths between each O-D pair. Besides circumventing path enumeration, the proposed method has a much lower time-complexity. Numerical examples are then provided to demonstrate the validity and efficiency of the proposed method.

A convex programming approach for ridesharing user equilibrium under fixed driver/rider demand

With the proliferation of smartphone-based ridesharing apps around the world, traffic assignment with ridesharing is drawing increasing attention in recent years. A number of ridesharing user equilibrium (RUE) models have been proposed, but most of them are formulated as path-based mixed complementarity problems based on presumed ridesharing price and inconvenience functions, thus are inconvenient to implement in reality. In this study, by redefining the set of feasible driver trajectories and the market equilibrium conditions for ridesharing, we propose an alternative approach to modeling the RUE when the driver- and rider-demand for each OD pair are fixed and given. We show that the resulting RUE conditions can be equivalently transformed into a convex programming problem, and the existence and uniqueness of RUE link flows are guaranteed under mild conditions. The structure of the model is similar to the classic Beckmann’s formulation, except for the additional ridesharing demand-supply constraints. So a dual subgradient algorithm with averaging is proposed to solve the problem, and the dual sub-problem can be solved by the Frank-Wolfe method. The algorithm effectively avoids path enumeration, therefore is implementable on large networks. The impact of problem size on the computational efficiency of the algorithm is theoretically analyzed and numerically demonstrated.

Enumeration of Lattice paths with infinite types of steps and the Chung-Feller property

In this paper we consider the lattice paths that go from the origin to a point on the right half of the plane with step set S={Ui=(1+i,1−i)|i≥0}⋃{Vi=(i,−i)|i≥1} such that the step Ui is assigned with weight ui and the step Vi is assigned with weight vi for i≥1 except that the step U0=(1,1) is assigned with weight 1. Let G(n,k) be the set of all lattice paths ending at the point (n,2k−n) and Gn,k=|G(n,k)|, and let U(n,k) (resp. V(n,k)) be the set of all lattice paths (resp. nonnegative lattice paths) ending at (2n−k,k) and Un,k=|U(n,k)| (resp. Vn,k=|V(n,k)|). We will show that (Gn,k)n,k∈N, (Un,k)n,k∈N and (Vn,k)n,k∈N are all Riordan arrays. Correlations between these Riordan arrays are studied. Consequently, a new Chung-Feller type property is obtained, and the bijective proof is provided. We also list numerous interesting examples.

Efficient Hop-constrained s-t Simple Path Enumeration

Graph is a ubiquitous structure representing entities and their relationships applied in many areas such as social networks, web graphs, and biological networks. One of the fundamental tasks in graph analytics is to investigate the relations between two vertices (e.g., users, items, and entities) such as how a vertex A influences another vertex B, or to what extent A and B are similar to each other, based on the graph topology structure. For this purpose, we study the problem of hop-constrained s-t simple path enumeration in this paper, which aims to list all simple paths from a source vertex s to a target vertex t with hop-constraint k. We first propose a polynomial delay algorithm, namely BC-DFS, based on a barrier-based pruning technique. Then, a join-oriented algorithm, namely JOIN, is designed to further enhance the query response time. On the theoretical side, BC-DFS is a polynomial delay algorithm with O(km) time per output where m is the number of edges in the graph. This time complexity is similar to the best known theoretical result for the polynomial delay algorithms of this problem. On the practical side, our comprehensive experiments on 15 real-life networks demonstrate the superior performance of the BC-DFS algorithm compared to the state-of-the-art techniques. It is also reported that the JOIN algorithm can further significantly enhance the query response time. In this paper, we also study the hop-constrained path enumeration problem with diversity constraint and propose a block-oriented algorithm, namely SCB. To further speed up the computation, hybrid lower bounds based on reverse shortest-path tree are also developed, namely SCB+. The experiments show our proposed methods significantly improve the query time and scalability comparing with baselines.

An elementary approach to solve recursions relative to the enumeration of S-Motzkin paths

S-Motzkin paths (bijective to ternary trees) and partial version of them are calculated using only elementary methods from linear algebra.

On some problems about ternary paths: a linear algebra approach

Ternary paths consist of an upstep of one unit, a downstep of two units, never go below the x-axis, and return to the x-axis. This paper addresses the enumeration of partial ternary paths, ending at a given level i, reading the path either from left-to-right or from right-to-left. Since the paths are not symmetric with respect to left vs. right, as classical Dyck paths, this leads to different results. The right-to-left enumeration is quite challenging, but leads at the end to very satisfying results. The methods are elementary (solving systems of linear equations). In this way, several conjectures left open in Naiomi Cameron’s Ph.D. thesis could be successfully settled.

A computationally efficient metric for identification of critical links in large transportation networks

Maintaining safety and resilience in transportation system requires identification of critical components. The existing transportation link criticality metrics typically require multiple traffic assignments to perform a full scan of the network. Link removals create computational burden and can cause network disconnectivity which makes it problematic to run traffic assignment. The authors previously introduced Link Criticality Index (LCI) that identifies the criticality ranking within a single User Equilibrium (UE) traffic assignment using Frank-Wolfe (FW) algorithm. While LCI is shown to provide balanced rankings with respect to connectivity and flow conditions, its computational efficiency diminishes for larger networks due to the need for path enumeration. This paper formulates an adjusted LCI to make it computationally efficient for larger networks. Adjusted LCI utilizes 1) viable routes instead of the complete list of enumerated paths; and, 2) a path-based traffic assignment algorithm (Gradient Projection) to replace FW. The consistency of rankings between original/adjusted LCIs, and other measures from the literature are analyzed and compared for three networks, and through an experimental setup. The proposed method is also applied on a large-scale transportation network, i.e., Chicago network. The results show that both modifications are effective and yield faster results without compromising LCI's advantages.

Enumeration of S-Motzkin paths from left to right and from right to left: a kernel method approach

Abstract The area of S-Motzkin paths (bijective to ternary trees) is calculated using the kernel method by enumerating these (partial) paths with fixed end-point resp. starting point.

A Matrix Approach for Analyzing Signal Flow Graph

Mason’s gain formula can grow factorially because of growth in the enumeration of paths in a directed graph. Each of the (n − 2)! permutation of the intermediate vertices includes a path between input and output nodes. This paper presents a novel method for analyzing the loop gain of a signal flow graph based on the transform matrix approach. This approach only requires matrix determinant operations to determine the transfer function with complexity O(n3) in the worst case, therefore rendering it more efficient than Mason’s gain formula. We derive the transfer function of the signal flow graph to the ratio of different cofactor matrices of the augmented matrix. By using the cofactor expansion, we then obtain a correspondence between the topological operation of deleting a vertex from a signal flow graph and the algebraic operation of eliminating a variable from the set of equations. A set of loops sharing the same backward edges, referred to as a loop group, is used to simplify the loop enumeration. Two examples of feedback networks demonstrate the intuitive approach to obtain the transfer function for both numerical and computer-aided symbolic analysis, which yields the same results as Mason’s gain formula. The transfer matrix offers an excellent physical insight, because it enables visualization of the signal flow.

Mixed-Integer Linear Programming Models for Multi-Robot Non-Adversarial Search

In this letter, we consider the Multi-Robot Efficient Search Path Planning (MESPP) problem, where a team of robots is deployed in a graph-represented environment to capture a moving target within a given deadline. We prove this problem to be NP-hard, and present the first set of Mixed-Integer Linear Programming (MILP) models to tackle the MESPP problem. Our models are the first to encompass multiple searchers, arbitrary capture ranges, and false negatives simultaneously. While state-of-the-art algorithms for MESPP are based on simple path enumeration, the adoption of MILP as a planning paradigm allows to leverage the powerful techniques of modern solvers, yielding better computational performance and, as a consequence, longer planning horizons. The models are designed for computing optimal solutions offline, but can be easily adapted for a distributed online approach. Our simulations show that it is possible to achieve 98% decrease in computational time relative to the previous state-of-the-art. We also show that the distributed approach performs nearly as well as the centralized, within 6% in the settings studied in this letter, with the advantage of requiring significant less time - an important consideration in practical search missions.

Optimal lane expansion model for a battery electric vehicle transportation network considering range anxiety and demand uncertainty

This paper investigates a novel design problem involving an optimal government lane expansion scheme for a battery electric vehicle (BEV) transportation network. A lane expansion model is established considering the BEV charging time, driver range anxiety and uncertain transportation demand. This model aims to minimize the total travel time (i.e., sum of the driving time and charging time) of all drivers in the transportation network and optimize the lane expansion scheme (i.e., the number and location of extended lanes in the network) under the established investment ceiling. To address demand uncertainty, an improved adjustable robust optimization method is further proposed to relax the model by introducing two control parameters. Based on the framework of the active set algorithm, a local optimal solution algorithm is designed to effectively solve the abovementioned model. Column generation is embedded in the above algorithm to avoid path enumeration. Sensitivity analyses are conducted for different control parameters and government investment scales. The results show that the model and algorithm we proposed can provide a theoretical basis for the government to improve the traffic efficiency of the transportation network and achieve the goal of sustainable transport.

Modelling the order of scoring in team sports

Abstract This paper considers sports matches in which two teams compete to score more points within a set amount of time (e.g. football, ice hockey). We focus on the order in which the competing teams score during the match (order of scoring). This type of order of scoring problem has not been addressed previously, and doing so here gives new insights into sports matches. For example, our analysis can deal with a situation that spectators find matches that involve comebacks particularly exciting. To describe such problems mathematically, we formulate the probabilities of (i) the favourite team leading throughout the match and (ii) the favourite team falling behind the opposing team but then making a comeback. These probabilities are derived using an independent Poisson model and lattice path enumeration, the latter of which involves the well-known ballot theorem. The independence assumption allows lattice path enumeration to be applied directly to the Poisson model and various scoring patterns to be addressed. We confirm that the values obtained from the proposed models agree well with actual sports data from football, futsal and ice hockey.