Computing Shortest Paths Research Articles

Implementation of graph-based interactive 3D vessel segmentation filter

This paper describes an ITK implementation of a 3D vascular segmentation filter using a graph-based energy minimization algorithm. The method first computes the shortest path between two user provided points in the vessel and then performs a graph min-cut based segmentation of the vessel based on the intensity information coupled with the computed path as a spatial constraint. The shortest path computation is adapted from the algorithm in [1] to use a vesselness based weighting function. The min-cut algorithm uses the Boost graph library to calculate the minimal cut. Several examples of applications are provided, along with images of the resulting segmentations.

Applying the improved Chen and Han’s algorithm to different versions of shortest path problems on a polyhedral surface

The computation of shortest paths on a polyhedral surface is a common operation in many computer graphics applications. There are two best known exact algorithms for the “single source, any destination” shortest path problem. One is proposed by Mitchell et al. (1987) [1]. The other is by Chen and Han (1990) [11]. Recently, Xin and Wang (2009) [9] improved the CH algorithm by exploiting a filtering theorem and achieved a practical method that outperforms both the CH algorithm and the MMP algorithm whether in time or in space. In this paper, we apply the improved CH algorithm to different versions of shortest path problems. The contributions of this paper include: (1) For a surface point p ∈ △ v 1 v 2 v 3 , we present an unfolding technique for estimating the distance value at p using the distances at v 1 , v 2 and v 3 . (2) We show that the improved CH algorithm can be naturally extended to the “multiple sources, any destination” version. Also, introducing a well-chosen heuristic factor into the improved CH algorithm will induce an exact solution to the “single source, single destination” version. (3) At the conclusion of multi-source shortest path algorithms, we can use the distance values at vertices to approximately compute the geodesic-distance-based offsets, the Voronoi diagram and the Delaunay triangulation in O ( n ) time. (4) By importing a precision parameter λ , we obtain a precision controlled approximant which varies from the improved CH algorithm to Dijkstra’s algorithm as λ increases from 0 to 1 . Thus, an interesting relationship between them can be naturally established.

Manipulating substances with Physarum polycephalum

The plasmodium of Physarum polycephalum is a large single cell capable for optimal spanning of food sources and avoidance of harmful stimuli. The sophisticated foraging behavior of the plasmodium can be interpreted in terms of computation. When propagating on a substrate with distributed sources of food the plasmodium simulates a general-purpose storage modification machine, approximates varieties of proximity graphs and imitates calculation of shortest path and plane tessellation. The plasmodium's behaviour is determined by the space–time distribution of attracting and repelling sources, and immediately guided by the waves of excitation traveling inside the plasmodium. Due to cytoplasmic streaming a harmless colored substance can be naturally ingested by the plasmodium and distributed inside the protoplasmic network. We show that by controlling the plasmodium's propagation over an uncolored substrate we can ‘fill’ specified areas of the substrate with the color transported by the plasmodium. We experimentally demonstrate that the plasmodium of P. polycephalum excels in adaptive transportation, mixing and transformation of colored food particles. We uncover a range of operations implementable by the plasmodium over color set, and design methods to control mixing and transportation of colors.

Shortest path refinement for motion estimation from tagged MR images.

Magnetic resonance tagging makes it possible to measure the motion of tissues such as muscles in the heart and tongue. The harmonic phase (HARP) method largely automates the process of tracking points within tagged MR images, permitting many motion properties to be computed. However, HARP tracking can yield erroneous motion estimates due to 1) large deformations between image frames, 2) through-plane motion, and 3) tissue boundaries. Methods that incorporate the spatial continuity of motion--so-called refinement or flood-filling methods--have previously been reported to reduce tracking errors. This paper presents a new refinement method based on shortest path computations. The method uses a graph representation of the image and seeks an optimal tracking order from a specified seed to each point in the image by solving a single source shortest path problem. This minimizes the potential errors for those path dependent solutions that are found in other refinement methods. In addition to this, tracking in the presence of through-plane motion is improved by introducing synthetic tags at the reference time (when the tissue is not deformed). Experimental results on both tongue and cardiac images show that the proposed method can track the whole tissue more robustly and is also computationally efficient.

A Border Line-Based Pruning Scheme for Shortest Path Computations

With the progress of IT and mobile positioning technologies, various types of location-based services (LBS) have been proposed and implemented. Finding a shortest path between two nodes is one of the most fundamental tasks in many LBS related applications. So far, there have been many research efforts on the shortest path finding problem. For instance, A * algorithm estimates neighboring nodes using a heuristic function and selects minimum cost node as the closest one to the destination. Pruning method, which is known to outperform the A * algorithm, improves its routing performance by avoiding unnecessary exploration in the search space. For pruning, shortest paths for all node pairs in a map need to be pre-computed, from which a shortest path container is generated for each edge. The container for an edge consists of all the destination nodes whose shortest path passes through the edge and possibly some unnecessary nodes. These containers are used during routing to prune unnecessary node visits. However, this method shows poor performance as the number of unnecessary nodes included in the container increases. In this paper, we focus on this problem and propose a new border line-based pruning scheme for path routing which can reduce the number of unnecessary node visits significantly. Through extensive experiments on randomly-generated, various complexity of maps, we empirically find out optimal number of border lines for clipping containers and compare its performance with other methods.

Solving path problems on the GPU

We consider the computation of shortest paths on Graphic Processing Units (GPUs). The blocked recursive elimination strategy we use is applicable to a class of algorithms (such as all-pairs shortest-paths, transitive closure, and LU decomposition without pivoting) having similar data access patterns. Using the all-pairs shortest-paths problem as an example, we uncover potential gains over this class of algorithms. The impressive computational power and memory bandwidth of the GPU make it an attractive platform to run such computationally intensive algorithms. Although improvements over CPU implementations have previously been achieved for those algorithms in terms of raw speed, the utilization of the underlying computational resources was quite low. We implemented a recursively partitioned all-pairs shortest-paths algorithm that harnesses the power of GPUs better than existing implementations. The alternate schedule of path computations allowed us to cast almost all operations into matrix–matrix multiplications on a semiring. Since matrix–matrix multiplication is highly optimized and has a high ratio of computation to communication, our implementation does not suffer from the premature saturation of bandwidth resources as iterative algorithms do. By increasing temporal locality, our implementation runs more than two orders of magnitude faster on an NVIDIA 8800 GPU than on an Opteron. Our work provides evidence that programmers should rethink algorithms instead of directly porting them to GPU.

A novel single source shortest path algorithm

This paper presents a new algorithm for computing shortest paths in a nearly directed graph G = (V, E). The existing Dijkstra’s algorithm finds the shortest distance between given two vertices. This paper proposes a new algorithm which proves to be better than Dijkstra’s and Floyd’s algorithm in certain cases. The proposed algorithm is a single source shortest path algorithm which finds the shortest path and shortest distance from a single vertex in a graph to all other vertices in the graph.

Network traffic routing using effective bandwidth theory

AbstractOptimal routing in a packet network is a difficult task—the amount of connections makes per‐connection routing impractical and even using aggregated connections global optimisation in the network is infeasible. The currently used algorithms are therefore suboptimal heuristics, using shortest path computation instead of e.g. throughput optimisation. The quality of obtained results depends on the selected link weights, but optimisation of those is again a difficult mixed integer‐programming problem. In this paper we discuss some proposed heuristics using the results of large deviation analysis (effective bandwidths) for throughput optimisation. The more advanced approach to traffic modelling uses more of the available information and potentially allows for finding better solutions. The presented algorithms are tested through simulation and discussion of the results is included. Copyright © 2009 John Wiley & Sons, Ltd.

Continuous monitoring of nearest neighbors on land surface

As geo-realistic rendering of land surfaces is becoming commonplace in geographical information systems (GIS), games and online Earth visualization platforms, a new type of k Nearest Neighbor (kNN) queries, "surface" k Nearest Neighbor (skNN) queries, has emerged and been investigated recently, which extends the traditional kNN queries to a constrained third dimension (i.e., land surface). All existing techniques, however, assume a static environment, limiting their utility in emerging applications (e.g., Location-based Services) where objects move. In this paper, for the first time, we propose two exact methods that can continuously answer skNN queries in a highly dynamic environment which allows for arbitrary movements of data objects. The first method, inspired by the existing techniques in monitoring kNN in road networks [7] maintains an analogous counterpart of the Dijkstra Expansion Tree on land surface, called Surface Expansion Tree (SE-Tree). However, we show the concept of expansion tree for land surface does not work as SE-tree suffers from intrinsic defects: it is fat and short, and hence does not improve the query efficiency. Therefore, we propose a superior approach that partitions SE-Tree into hierarchical chunks of pre-computed surface distances, called Angular Surface Index Tree (ASI-Tree). Unlike SE-tree, ASI-Tree is a well balanced thin and tall tree. With ASI-Tree, we can continuously monitor skNN queries efficiently with low CPU and I/O overheads by both speeding up the surface shortest path computations and localizing the searches. We experimentally verify the applicability and evaluate the efficiency of the proposed methods with both real world and synthetic data sets. ASI-Tree consistently and significantly outperforms SE-Tree in all cases.

Hybrid co-evolutionary particle swarm optimization and noising metaheuristics for the delay constrained least cost path problem

This paper presents a co-evolutionary particle swarm optimization (PSO) algorithm, hybridized with noising metaheuristics, for solving the delay constrained least cost (DCLC) path problem, i.e., shortest-path problem with a delay constraint on the total "cost" of the optimal path. The proposed algorithm uses the principle of Lagrange relaxation based aggregated cost. It essentially consists of two concurrent PSOs for solving the resulting minimization-maximization problem. The main PSO is designed as a hybrid PSO-noising metaheuristics algorithm for efficient global search to solve the minimization part of the DCLC-Lagrangian relaxation by finding multiple shortest paths between a source-destination pair. The auxiliary/second PSO is a co-evolutionary PSO to obtain the optimal Lagrangian multiplier for solving the maximization part of the Lagrangian relaxation problem. For the main PSO, a novel heuristics-based path encoding/decoding scheme has been devised for representation of network paths as particles. The simulation results on several networks with random topologies illustrate the efficiency of the proposed hybrid algorithm for the constrained shortest path computation problems.

상수 삽입 전이 시간을 가지는 양단 우선순위 큐

우선순위 큐는 스케줄링, 정렬, 유전자 검색과 같은 우선순위에 따른 검색, 최단거리 계산과 같은 응용에 사용될 수 있다. 본 논문에서 제안하는 배열을 이용한 양단 우선순위 큐 자료구조는 삽입과 삭제 연산에 각각 O(1) 전이시간과 O(logn) 시간이 걸린다. 본 저자가 알고 있는 한, 지금까지의 배열을 이용한 양단우선순위 큐 알고리즘은 삽입과 삭제에 모두 O(logn) 시간이 걸린다. Priority queues can be used in applications such as scheduling, sorting, retrival based on a priority like gene searching, shortest paths computation. This paper proposes a data structure using array representation for double-ended priority queue in which insertion and deletion takes O(1) amortized time and O(logn) time, respectively. To the author's knowledge, all the published array-based data structures for double ended priority queue support O(logn) time insertion and deletion operations.

Planning wireless networks by shortest path

Transmitters and receivers are the basic elements of wireless networks and are characterized by a number of radio-electrical parameters. The generic planning problem consists of establishing suitable values for these parameters so as to optimize some network performance indicator. The version here addressed, namely the Power Assignment Problem (pap), is the problem of assigning transmission powers to the transmitters of a wireless network so as to maximize the satisfied demand. This problem has relevant practical applications both in radio-broadcasting and in mobile telephony. Typical solution approaches make use of mixed integer linear programs with huge coefficients in the constraint matrix yielding numerical inaccuracy and poor bounds, and so cannot be exploited to solve large instances of practical interest. In order to overcome these inconveniences, we developed a two-phase heuristic to solve large instances of pap, namely a constructive heuristic followed by an improving local search. Both phases are based on successive shortest path computations on suitable directed graphs. Computational tests on a number of instances arising in the design of the national Italian Digital Video Broadcasting (DVB) network are presented.

Energy-Efficient Decentralized Cooperative Routing in Wireless Networks

Wireless adhoc networks transmit information from a source to a destination via multiple hops in order to save energy and, thus, increase the lifetime of battery-operated nodes. The energy savings can be especially significant in cooperative transmission schemes, where several nodes cooperate during one hop to forward the information to the next node along a route to the destination. Finding the best multi-hop transmission policy in such a network which determines nodes that are involved in each hop, is a very important problem, but also a very difficult one especially when the physical wireless channel behavior is to be accounted for and exploited. We model the above optimization problem for randomly fading channels as a decentralized control problem - the channel observations available at each node define the information structure, while the control policy is defined by the power and phase of the signal transmitted by each node. In particular, we consider the problem of computing an energy-optimal cooperative transmission scheme in a wireless network for two different channel fading models: (i) slow fading channels, where the channel gains of the links remain the same for a large number of transmissions, and (ii) fast fading channels, where the channel gains of the links change quickly from one transmission to another. For slow fading, we consider a factored class of policies (corresponding to local cooperation between nodes), and show that the computation of an optimal policy in this class is equivalent to a shortest path computation on an induced graph, whose edge costs can be computed in a decentralized manner using only locally available channel state information (CSI). For fast fading, both CSI acquisition and data transmission consume energy. Hence, we need to jointly optimize over both these; we cast this optimization problem as a large stochastic optimization problem. We then jointly optimize over a set of CSI functions of the local channel states, and a corresponding factored class of control policies corresponding to local cooperation between nodes with a local outage constraint. The resulting optimal scheme in this class can again be computed efficiently in a decentralized manner. We demonstrate significant energy savings for both slow and fast fading channels through numerical simulations of randomly distributed networks.

Engineering multilevel overlay graphs for shortest-path queries

An overlay graph of a given graph G = ( V , E ) on a subset S ⊆ V is a graph with vertex set S and edges corresponding to shortest paths in G . In particular, we consider variations of the multilevel overlay graph used in Schulz et al. [2002] to speed up shortest-path computation. In this work, we follow up and present several vertex selection criteria, along with two general strategies of applying these criteria, to determine a subset S of a graph's vertices. The main contribution is a systematic experimental study where we investigate the impact of selection criteria and strategies on multilevel overlay graphs and the resulting speed-up achieved for shortest-path computation: Depending on selection strategy and graph type, a centrality index criterion, selection based on planar separators, and vertex degree turned out to perform best.

DT-MRI Fiber Tracking: A Shortest Paths Approach

We derive a new fiber tracking algorithm for DT-MRI that parts with the locally "greedy" paradigm intrinsic to conventional tracking algorithms. We demonstrate the ability to precisely reconstruct a diverse range of fiber trajectories in authentic and computer-generated DT-MRI data, for which well-known conventional tracking algorithms are shown to fail. Our approach is to pose fiber tracking as a problem in computing shortest paths in a weighted digraph. Voxels serve as vertices, and edges are included between neighboring voxels. We assign probabilities (weights) to edges using a Bayesian framework. Higher probabilities are assigned to edges that are aligned with fiber trajectories in their close proximity. We compute optimal paths of maximum probability using computationally scalable shortest path algorithms. The salient features of our approach are: global optimality--unlike conventional tracking algorithms, local errors do not accumulate and one "wrong-turn" does not spell disaster; a target point is specified a priori; precise reconstruction is demonstrated for extremely low signal-to-noise ratio; impartiality to which of two endpoints is used as a seed; and, faster computation times than conventional all-paths tracking. We can use our new tracking algorithm in either a single-path tracking mode (deterministic tracking) or an all-paths tracking mode (probabilistic tracking).

Diagnosis of behaviors of interest in partially-observed discrete-event systems

In this paper, we address the problem of diagnosing the behaviors of interest in discrete-event systems. To this end, we introduce the notion of language-diagnosability, based on language specifications that generalizes diagnosability based on event specifications. A polynomial-time algorithm for verifying language-diagnosability is developed. Building upon the verification algorithm, we develop a polynomial-time algorithm for computing the worst case detection delay of a given system. The computation of the worst case detection delay involves the shortest path computation of a weighted, directed graph. We exploit a special weighting structure of the graph resulting from the verification algorithm, which enables an algorithm with a lower complexity than the commonly used Bellman–Ford shortest path algorithm.

SU‐GG‐T‐542: Arc‐Modulated Radiation Therapy (AMRT): A Novel Method for Rotational Radiation Therapy

Purpose: To develop a prototype treatment planning system for Arc‐Modulated Radiation Therapy (AMRT), which delivers the optimal radiation distribution with only one beam rotation. Method and Materials: The AMRT planning starts with inverse planning, typically using 36 equally‐spaced (fixed) beams and producing 36 continuous intensity maps. After inverse planning, an AMRT leaf sequencing algorithm converts the optimized intensity patterns to a sequence of MLC apertures along a single arc. Finally, a homegrown Monte‐Carlo superposition‐based dose calculation engine is used to quickly and accurately evaluate the plan qualities. A big challenge to AMRT is leaf sequencing. Our algorithm is based on a key observation that if each optimized intensity pattern is delivered through its neighboring beam angles within a ±5° range, there is only negligible error on the resulting dose distribution. Thus, the sequencing problem becomes how to guarantee the interconnectedness of the MLC apertures along the single arc, while reproducing or closely approximating the optimized intensity patterns. We model this problem as computing shortest paths in directed acyclic graphs, and solve it efficiently using various algorithmic techniques. Results: We applied our AMRT approach to 18 clinical cases with a wide range of treatment sites. In terms of dose conformity, we observed that AMRT plans outperform gantry‐fixed IMRT plans and rival intensity‐modulated arc therapy (IMAT) plans. In terms of delivery efficiency, AMRT plans are much more efficient and can be delivered in less than 5 minutes for complex head‐and‐neck cases. The AMRT planning time is currently about 1–2 hours. Conclusion: We developed a prototype treatment planning system for AMRT, which delivers the optimal radiation distribution with only one beam rotation. Experimental studies demonstrated the potentials of our new approach to dynamic IMRT which warrants further investigations.

Optimal routing for end-to-end guarantees using Network Calculus

In this paper we show how Network Calculus can be used to compute the optimal route for a flow (w.r.t. end-to-end guarantees on the delay or the backlog) in a network in the presence of cross-traffic. When cross-traffic is independent, the computation is shown to boil down to a functional shortest path problem. Under usual assumptions (concave arrival curves and convex service curves), a simple min-max lemma enables us to solve the problem in polynomial time by a sequence of classical shortest path computations. When cross-traffic perturbs the main flow over more than one node and under blind multiplexing, one can take into account the “Pay Multiplexing Only Once” (PMOO) phenomenon identified in [J.B. Schmitt, F.A. Zdarsky, The disco network calculator: A toolbox for worst case analysis. in: Proceedings of Valuetools’06, 2006; J.B. Schmitt, F.A. Zdarsky, I. Martinovic, Performance bounds in feed-forward networks under blind multiplexing. Technical Report 349/06, University of Kaiserslautern, Germany, 2006.]. It enables us to improve bounds on delays and backlogs, but it makes the computation more involved. We provide a formula which gives a service curve for a path with PMOO conditions. It introduces a new multi-dimensional Network Calculus operator and it generalizes a formula in the above reference. Moreover, when arrival (resp. service) curves are affine (resp. convex), we describe an efficient algorithm to compute this formula. We finally show how to adapt our routing algorithms to have optimal end-to-end guarantees for these new bounds on delays and backlogs which take into account PMOO.

Technical Note—A Note on Parametric Analysis in Linear Assignment

A classic application of the linear assignment problem is the assignment of people to jobs (or jobs to people). In this context, it is interesting to measure competition for jobs and to generate a suitable list of jobs from which a person can choose; the length of the list is a parameter. A known list-generation procedure is based on an interior-point method followed by a parametric analysis. We describe a more efficient procedure, exploiting linear assignment theory and shortest-path computations. Further, we propose an alternative list-generation procedure, based on a special type of dual values for the linear assignment problem.

Dynamic Multi-level Overlay Graphs for Shortest Paths

Multi-level overlay graphs represent a speed-up technique for shortest paths computation which is based on a hierarchical decomposition of a weighted directed graph G. They have been shown to be experimentally efficient, especially when applied to timetable information. However, no theoretical result on the cost of constructing, maintaining and querying multi-level overlay graphs in a dynamic environment is known. In this paper, we show theoretical properties of multi-level overlay graphs that lead us to the definition of a new data structure for the computation and the maintenance of an overlay graph of G while weight decrease or weight increase operations are performed on G. Our solution is theoretically faster than the recomputation from scratch and allows queries that can be performed more efficiently than running Dijkstra’s shortest paths algorithm on G.