最短路径的求解算法

文章提出了一种求最短路径的算法,该算法能高效地求出一个顶点到其它各顶点的所有最短路径。用C语言设计了相应的程序验证了此算法。     

Finding the k Shortest Paths in Parallel

A concurrent-read exclusive-write PRAM algorithm is developed to find the k shortest paths between pairs of vertices in an edge-weighted directed graph. Repetitions of vertices along the paths are allowed. The algorithm computes an implicit representation of the k shortest paths to a given destination vertex from every vertex of a graph with n vertices and m edges, using O(m+nk log ² k) work and O( log^3k log ^*k+ log n( log log k+ log ^*n)) time, assuming that a shortest path tree rooted at the destination is pre-computed. The paths themselves can be extracted from the implicit representation in O( log k + log n) time, and O(n log n +L) work, where L is the total length of the output. Received July 2, 1997; revised June 18, 1998.     

求最短路径的新算法

本文提出了一种求最短路径的新算法,并用C语言设计相应的程序验证了此算法。实验表明,该算法能高效地求出一个顶点到其它各项点的所有最短路径。     

求图中顶点之间所有最短路径的一种实用算法

提出了求一个顶点到另一个顶点的所有最短路径的一个算法，此算法中设计了一些独特的数据结构。在算法运行的整个过程中，求一个有效顶点（后面定义）到终点的所有最短路径的过程（入栈、出栈等操作）实际只进行一遍，用C语言编制的相应程序验证了这个算法的可靠性和实用性。     

Finding Reliable Shortest Paths in Road Networks Under Uncertainty

The aim of this study is to investigate the solution algorithm for solving the problem of determining reliable shortest paths in road networks with stochastic travel times. The availability of reliable shortest paths enables travelers, in the face of travel time uncertainty, to plan their trips with a pre-specified on-time arrival probability. In this study, the reliable shortest path between origin and destination nodes is determined using a multiple-criteria shortest path approach when link travel times follow normal distributions. The dominance conditions involved in such problems are established, thereby reducing the number of generated non-dominated paths during the search processes. Two solution algorithms, multi-criteria label-setting and A* algorithms, are proposed and their complexities analyzed. Computational results using large scale networks are presented. Numerical examples using data from a real-world advanced traveller information system is also given to illustrate the applicability of the solution algorithms in practice.     

所有最短路径的求解算法

本文提出了一种求所有最短路径的算法,能高效地求出一个顶点到其它各顶点的所有最短路径。此外,我们用C语言设计的相应程序验证了此算法。     

A Two-Criterion Lexicographic Algorithm for Finding All Shortest Paths in Networks

An algorithm for finding all shortest paths in an undirected network is considered. The following two criteria are used: the minimum number of arcs in a path and a minimum path length. The algorithm is analyzed for complexity, and it is empirically shown that, with increasing the network density, its computational efficiency becomes higher than that of the Floyd algorithm adequately modified to find the shortest path with the use of a two-step criterion.     

Parallel Mesh Algorithms for Grid Graph Shortest Paths with Application to Separation of Touching Chromosomes

The grid graph shortest path problem has many applications. In this paper, we present practical mesh algorithms using a local cost-reducing operation for various forms of the grid graph shortest path problem. The algorithms are very simple and can easily mark the vertices on shortest paths between any two vertices. The time complexity of the algorithm is proportional to the maximum length of the shortest paths with a very small multiplicative constant. Also in this paper, we discuss the application of the parallel algorithms in automatic chromosome analysis to intelligently split touching chromosomes. We identify local features useful for finding a potential path to separate touching chromosomes. We then define a distance measure based on the local features and find the best splitting path to cut touching chromosomes. The splitting algorithm only uses local information and is highly parallel.     

每对结点间最短路径VC++实现

最短路径问题应用广泛、实现多样。动态规划法能够有效计算出每对节点间的最短路径。本文利用VC++实现了动态规划法的每对节点间最短路径问题。     

On Dynamic Shortest Paths Problems

We obtain the following results related to dynamic versions of the shortest-paths problem:

基于带标记的并发可达标识图的关键路径的求解方法

将AOE 网转换成有色时延Petri网模型,在模型转换过程中同时计算出各位置所对应的事件的最早开始时间,给出了模拟AOE 网的有色时延Petri网模型的带标记的并发可达标识图的构建算法;利用并发可达标识图中的标记序列直接得到关键路径并计算出完成所有活动所需的最短时间。实例与仿真实验结果表明,当AOE网中平均存在3个以上的并发活动时,所提方法执行效率优于传统的求解关键路径的算法,并发活动越多,所提算法效率越高。     

用（）(t)的广义连接图求有障碍时的最短路径

在有障碍时求两点间的最短路径是VLSI设计、机器人设计等领域中的基本问题,连接图是研究此问题的基本工具.现有算法构造的最好的连接图GF是基于自由区的概念而设计的,其顶数和边数分别为O(t)和O(tlogt),其中t为障碍的极边数.提出了广义自由区和极大正规划分的概念,在此基础上得到广义连接图GG,用来表征广义自由区之间的邻接情况,其顶数和边数均为()(t),且具有平面图的性质.同时还提出了基于扫描线的极大正规划分构造算法,其时间复杂度为O(tlogt);并提出规范路径的概念,以及采用"不改向"启发式策略的A*算法在广义连接图GG中寻找两点间的最短路径,算法的时间复杂度由基于GF的现有算法的O(tlogt)降低到()(t).     

On Bounded Leg Shortest Paths Problems

Let V be a set of points in a d-dimensional l _p -metric space. Let s,t∈V and let L be a real number. An L-bounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set has length of at most L. The minimal path among all these paths is the L-bounded leg shortest path from s to t. In the s–t Bounded Leg Shortest Path (stBLSP) problem we are given two points s and t and a real number L, and are required to compute an L-bounded leg shortest path from s to t. In the All-Pairs Bounded Leg Shortest Path (apBLSP) problem we are required to build a data structure that, given any two query points from V and a real number L, outputs the length of the L-bounded leg shortest path (a distance query) or the path itself (a path query). In this paper we obtain the following results:     

Shortest Paths in Time-Dependent FIFO Networks

In this paper, we study the time-dependent shortest paths problem for two types of time-dependent FIFO networks. First, we consider networks where the availability of links, given by a set of disjoint time intervals for each link, changes over time. Here, each interval is assigned a non-negative real value which represents the travel time on the link during the corresponding interval. The resulting shortest path problem is the time-dependent shortest path problem for availability intervals ( $\mathcal{TDSP}_{\mathrm{int}}$ ), which asks to compute all shortest paths to any (or all) destination node(s) d for all possible start times at a given source node s. Second, we study time-dependent networks where the cost of using a link is given by a non-decreasing piece-wise linear function of a real-valued argument. Here, each piece-wise linear function represents the travel time on the link based on the time when the link is used. The resulting shortest paths problem is the time-dependent shortest path problem for piece-wise linear functions ( $\mathcal{TDSP}_{\mathrm{lin}}$ ) which asks to compute, for a given source node s and destination d, the shortest paths from s to d, for all possible starting times. We present an algorithm for the $\mathcal{TDSP}_{\mathrm{lin}}$ problem that runs in time O((F _d +γ)(|E|+|V|log?|V|)) where F _d is the output size (i.e., number of linear pieces needed to represent the earliest arrival time function to d) and γ is the input size (i.e., number of linear pieces needed to represent the local earliest arrival time functions for all links in the network). We then solve the $\mathcal{TDSP}_{\mathrm{int}}$ problem in O(λ(|E|+|V|log?|V|)) time by reducing it to an instance of the $\mathcal{TDSP}_{\mathrm{lin}}$ problem. Here, λ denotes the total number of availability intervals in the entire network. Both methods improve significantly on the previously known algorithms.     

Improved Algorithms for Dynamic Shortest Paths

We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time. A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting tradeoff between preprocessing, query, and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to n -vertex digraphs of genus O(n ^1-ε ) for any ε>0 . Received September 24, 1996; revised May 13, 1998, and January 20, 1999.     

Sketching Clothoid Splines Using Shortest Paths

Clothoid splines are gaining popularity as a curve representation due to their intrinsically pleasing curvature, which varies piecewise linearly over arc length. However, constructing them from hand‐drawn strokes remains difficult. Building on recent results, we describe a novel algorithm for approximating a sketched stroke with a fair (i.e., visually pleasing) clothoid spline. Fairness depends on proper segmentation of the stroke into curve primitives — lines, arcs, and clothoids. Our main idea is to cast the segmentation as a shortest path problem on a carefully constructed weighted graph. The nodes in our graph correspond to a vastly overcomplete set of curve primitives that are fit to every subsegment of the sketch, and edges correspond to transitions of a specified degree of continuity between curve primitives. The shortest path in the graph corresponds to a desirable segmentation of the input curve. Once the segmentation is found, the primitives are fit to the curve using non‐linear constrained optimization. We demonstrate that the curves produced by our method have good curvature profiles, while staying close to the user sketch.     

Computing Approximate Shortest Paths on Convex Polytopes

The algorithms for computing a shortest path on a polyhedral surface are slow, complicated, and numerically unstable. We have developed and implemented a robust and efficient algorithm for computing approximate shortest paths on a convex polyhedral surface. Given a convex polyhedral surface P in \reals ³ , two points s, t ∈ P , and a parameter \eps > 0 , it computes a path between s and t on P whose length is at most (1+\eps) times the length of the shortest path between those points. It constructs in time O(n/\sqrt \eps ) a graph of size O(1/\eps ⁴ ) , computes a shortest path on this graph, and projects the path onto the surface in O(n/\eps) time, where n is the number of vertices of P . In the postprocessing step we have added a heuristic that considerably improves the quality of the resulting path. Received July 25, 2000; revised June 6, 2001.     

A Branch-Checking Algorithm for All-Pairs Shortest Paths

We formulate and study an algorithm for all-pairs shortest paths in a network with $n $ nodes and $m $ arcs of positive length. Using the dynamic programming principle of optimality of subpaths the algorithm avoids redundant updates of distance labels. A shortest $v$--$w$ path, say $\langle v, r_{1} , r_{2} , \ldots , r_{k } = w \rangle$ with $k $ arcs ($k \geq 1$), is only then combined with an arc $(w,t) \in A$ to update the distance label of pair $v$--$t$, if $(w,t) $ is present on the shortest $r_{\ell } $--$ t$ path for each node $r_{\ell}$ $(\ell=k- 1 , k- 2, \ldots, 1) $. The algorithm extracts shortest paths in order of length from a data structure and builds two shortest path trees per node, an extra effort of $O(n^{2})$. This way it can execute efficiently only the aforementioned distance updates, by picking the arcs $(w,t)$ out of these trees. The time complexity order per distance update and path extraction is similar as in other algorithms. An implementation with a data structure of heaps is possible, but a bucket-type data structure may be more appropriate. The implied number of distance updates does not exceed $nm_{0}$ ($m_{0}$ being the total number of shortest path arcs), but is frequently much lower. In extreme cases the new algorithm applies $O(n^{2})$ distance updates, whereas known algorithms require $\Omega( n ^{3})$ updates. The algorithm is especially suited for undirected graphs; here the construction of one tree per node is sufficient and the computation times halve.     

多约束最短链路分离路径精确算法

在通信的源和目的间寻找两条(主用和备用)链路分离的QoS路径是提供可靠QoS路由的重要途径.现有求解多约束链路分离路径对(multi-constrained link-disjoint path pair,简称MCLPP)的算法难以保证求得存在于任意网络中的可行解和最优解.为解决这一问题,分析了MCLPP问题最优解的性质,提出了精确算法的设计原则,在此基础上给出了求解MCLPP问题的精确算法(link-disjoint optimal multi-constrained paths algorithm,简称LIDOMPA算法),可对任意网络求解客观存在的多约束最短链路分离路径对.为了降低算法的复杂性,引入了候选最优解、紧缩的约束向量和结构化的路径支配3种关键方法,在保障算法精确性的同时,有效地降低了LIDOMPA的搜索空间.大量的实验结果表明,LIDOMPA的求解能力优于现有算法,同时可以实现较低的算法执行时间开销.     

Fast-Marching Methods for Curvature Penalized Shortest Paths

We introduce numerical schemes for computing distances and shortest paths with respect to several planar paths models, featuring curvature penalization and data-driven velocity: the Dubins car, the Euler/Mumford elastica, and a two variants of the Reeds–Shepp car. For that purpose, we design monotone and causal discretizations of the associated Hamilton–Jacobi–Bellman PDEs, posed on the three-dimensional domain \({\mathbb R}^2 \times {\mathbb S}^1\). Our discretizations involve sparse, adaptive and anisotropic stencils on a cartesian grid, built using techniques from lattice geometry. A convergence proof is provided, in the setting of discontinuous viscosity solutions. The discretized problems are solvable in a single pass using a variant of the fast-marching algorithm. Numerical experiments illustrate the applications of our schemes in motion planning and image segmentation.