Unique Shortest Path Research Articles

An efficient oracle for counting shortest paths in planar graphs

We study the counting shortest paths problem: find the number ns(u,v) of shortest paths from a vertex u to a vertex v in a graph of n vertices. Bezáková and Searns [ISAAC 2018] give an oracle which, given u and v in a planar graph, answers ns(u,v) in O(n) time and O(n1.5) memory space. Applying this oracle directly, it takes O(n) time to answer whether there is a unique shortest path from u to v. We propose an oracle which answers ns(u,v) in O(n) time and O(n1.5) space. Our oracle answers whether there is a unique shortest path from u to v in O(log⁡n) time. Computational studies show that our oracle is faster than the oracle of Bezáková and Searns for large graphs. A key technique to speed up the query time of our oracle is applying Voronoi diagrams on planar graphs, an advanced data structure widely used in shortest distance oracles. Based on Voronoi diagrams on planar graphs, significant theoretical improvements have been made for distance oracles. Our studies confirm that Voronoi diagrams are efficient data structures for distance oracles in practice.

Shortest paths and convex hulls in 2D complexes with non-positive curvature

Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many algorithms and properties of shortest paths and convex hulls in Euclidean space fail to transfer over. We give an algorithm, using linear programming, to compute the convex hull of a set of points in a 2-dimensional CAT(0) polyhedral complex with a single vertex. We explore the use of shortest path maps to answer single-source shortest path queries in 2-dimensional CAT(0) polyhedral complexes, and we unify efficient solutions for 2-manifold and rectangular cases.

Embedding IP Unique Shortest Path Topology on a Wavelength-Routed Network: Normal and Survivable Design

In this paper, we address the network virtualization problem of embedding a unique shortest path-based IP topology using lightpaths in a wavelength-routed network. We present an integer linear programming formulation and propose a 2-phase heuristic approach to solve this problem. We extend the model and the heuristic by addressing survivability in an integrated cross-layer framework, where the objective is to allocate a light-path topology that remains connected in the event of any single physical link failure while providing the IP network with unique shortest paths for all node-pairs. We consider a number of measures to show effectiveness of our approach and to discuss the impact on normal and survivable topology design, in terms of the number of transreceivers deployed.

Shortest path problem in rectangular complexes of global nonpositive curvature

CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x,y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees ⩾4). Namely, we show that for a CAT(0) rectangular complex K with n vertices, one can construct a data structure D of size O(n2) so that, given any two points x,y∈K, the shortest path γ(x,y) between x and y can be computed in O(d(p,q)) time, where p and q are vertices of two faces of K containing the points x and y, respectively, such that γ(x,y)⊂K(I(p,q)) and d(p,q) is the distance between p and q in the underlying graph of K. If K is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer two-point shortest path queries in O(d(p,q)logΔ) time, where Δ is the maximal degree of a vertex of G(K). Finally, if K is a squaregraph, then one can construct a data structure D of size O(nlogn) and answer two-point shortest path queries in O(d(p,q)) time.

Predicting Cell Cycle Regulated Genes by Causal Interactions

The fundamental difference between classic and modern biology is that technological innovations allow to generate high-throughput data to get insights into molecular interactions on a genomic scale. These high-throughput data can be used to infer gene networks, e.g., the transcriptional regulatory or signaling network, representing a blue print of the current dynamical state of the cellular system. However, gene networks do not provide direct answers to biological questions, instead, they need to be analyzed to reveal functional information of molecular working mechanisms. In this paper we propose a new approach to analyze the transcriptional regulatory network of yeast to predict cell cycle regulated genes. The novelty of our approach is that, in contrast to all other approaches aiming to predict cell cycle regulated genes, we do not use time series data but base our analysis on the prior information of causal interactions among genes. The major purpose of the present paper is to predict cell cycle regulated genes in S. cerevisiae. Our analysis is based on the transcriptional regulatory network, representing causal interactions between genes, and a list of known periodic genes. No further data are used. Our approach utilizes the causal membership of genes and the hierarchical organization of the transcriptional regulatory network leading to two groups of periodic genes with a well defined direction of information flow. We predict genes as periodic if they appear on unique shortest paths connecting two periodic genes from different hierarchy levels. Our results demonstrate that a classical problem as the prediction of cell cycle regulated genes can be seen in a new light if the concept of a causal membership of a gene is applied consequently. This also shows that there is a wealth of information buried in the transcriptional regulatory network whose unraveling may require more elaborate concepts than it might seem at first.

Inapproximability results for the inverse shortest paths problem with integer lengths and unique shortest paths

AbstractWe study the complexity of two inverse shortest paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph D = (V, A), the task is to find positive integer arc lengths such that the given paths are uniquely determined shortest paths between their respective terminals. In the first problem we seek for arc lengths that minimize the length of the longest of the prescribed paths. In the second problem, the length of the longest arc is to be minimized. We show that it is 𝒩𝒫‐hard to approximate the minimal longest path length within a factor less than 8/7 or the minimal longest arc length within a factor less than 9/8. This answers the (previously) open question whether these problems are 𝒩𝒫‐hard or not. We also present a simple algorithm that achieves an 𝒪(∣V∣)‐approximation guarantee for both variants. Both ISP problems arise in the planning of telecommunication networks with shortest path routing protocols. Our results imply that it is 𝒩𝒫‐hard to decide whether a given path set can be realized with a real shortest path routing protocol such as OSPF, IS‐IS, or RIP. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 29–36 2007

Internet Routing and Related Topology Issues

In most domains of the Internet network, the traffic demands are routed on a single-path defined as the shortest one according to a set of administrative weights. Most of the time, the values set by the administrator (or the default ones) are such that there are many paths of the same length between the extremities of some demands. However, if the shortest paths are not unique, it might become difficult for an Internet domain administrator to predict and control the traffic flows in the network. Moreover, the sequence order of packets can be changed when many paths are used leading to some end-to-end delays. It is hence an important issue to ensure that each shortest path is unique according to a given set of administrative weights. We show that it is possible to determine a set of small integer weights (smaller than 6 times the radius of the network) such that all links are used and every demand is routed on a unique shortest path. Above and beyond this uniqueness requirement, network administrators wishing to exploit the available resources would like to control the whole routing pattern. The problem they face consists of determining a set of weights enforcing a given routing policy. We formulate this problem using linear programs, and we show how integer weights can be computed by heuristics with guaranteed worst-case performances. Some conditions on the given routing, necessary for the existence of a solution, are derived. Both necessary and sufficient conditions are also provided, together with some other useful properties, in the case of particular graphs such as cycles and cacti.

Duality and Distance Constraints for the Nonlinear p-Center Problem and Covering Problem on a Tree Network

The problem of locating a fixed number, p, of facilities (centers) on a network, where there are constraints on the center locations and where the centers provide a service to customers (demand points) located at vertices of the network is addressed. The cost or “loss” of servicing a given demand point is a nonlinear function of the distance between the demand point and the closest center. We consider the case where the network has special structure (a tree network), i.e., there is a unique shortest path between any two points on the network. We also provide and interpret a dual to this problem and give polynomially bounded procedures for solving both problems. The primal location problem is solved with the aid of a related problem for which we also give a dual.

Geodetic orientations of complete k-partite graphs

Ore defined a graph to be geodetic if and only if there is a unique shortest path between two points, and posed the problem of characterizing such graphs. Here this problem is studied in the context of oriented graphs and such geodetic orientations are characterized first for complete graphs (geodetic tournaments), then for complete bipartite and complete tripartite graphs, and finally for complete k-partite graphs.