Approximate Shortest Paths in Polygons with Violations

2020 ◽  
Vol 30 (01) ◽  
pp. 79-95
Author(s):  
Binayak Dutta ◽  
Sasanka Roy
Keyword(s):  
Approximation Algorithm ◽  
Shortest Path ◽  
Shortest Paths ◽  
Simple Polygon ◽  
Geometric Parameters

We study the shortest [Formula: see text]-violation path problem in a simple polygon. Let [Formula: see text] be a simple polygon in [Formula: see text] with [Formula: see text] vertices and let [Formula: see text] be a pair of points in [Formula: see text]. Let [Formula: see text] represent the interior of [Formula: see text]. Let [Formula: see text] represent the exterior of [Formula: see text]. For an integer [Formula: see text], the shortest [Formula: see text]-violation path problem in [Formula: see text] is the problem of computing the shortest path from [Formula: see text] to [Formula: see text] in [Formula: see text], such that at most [Formula: see text] path segments are allowed to be in [Formula: see text]. The subpaths of a [Formula: see text]-violation path are not allowed to bend in [Formula: see text]. For any [Formula: see text], we present a [Formula: see text] factor approximation algorithm for the problem that runs in [Formula: see text] time. Here [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text] are geometric parameters.

AN OPTIMAL ALGORITHM FOR THE TWO-GUARD PROBLEM

1996 ◽  
Vol 06 (01) ◽  
pp. 15-44 ◽  
Author(s):  
PAUL J. HEFFERNAN
Keyword(s):  
Shortest Path ◽  
Optimal Algorithm ◽  
Shortest Paths ◽  
Simple Polygon ◽  
The Other ◽  
Optimal Solutions ◽  
Decision Algorithm ◽  
Constructive Algorithms ◽  
Shortest Path Trees

In this paper we give optimal solutions for several versions of the two-guard problem. Given a simple polygon P with vertices s and t, the straight walk problem asks whether we can move two points monotonically on P from s to t, one clockwise and one counter-clockwise, such that the points are always co-visible. In the counter walk problem, both points move clockwise, one from s to t and the other from t to s. We provide Θ(n)-time constructive algorithms for both problems. We also give a Θ(n)-time decision algorithm for a version called the general walk problem. We obtain our results by examining the structure of the restrictions placed on the motion of the two points, and by employing properties of shortest paths and shortest path trees.

scholarly journals Computing nearest neighbour interchange distances between ranked phylogenetic trees

2021 ◽  
Vol 82 (1-2) ◽  
Author(s):  
Lena Collienne ◽  
Alex Gavryushkin
Keyword(s):  
Cancer Research ◽  
Computational Complexity ◽  
Phylogenetic Tree ◽  
Shortest Path ◽  
Phylogenetic Trees ◽  
Shortest Paths ◽  
Nearest Neighbour ◽  
Tree Inference ◽  
Subtree Prune And Regraft ◽  
Comparison Algorithms

AbstractMany popular algorithms for searching the space of leaf-labelled (phylogenetic) trees are based on tree rearrangement operations. Under any such operation, the problem is reduced to searching a graph where vertices are trees and (undirected) edges are given by pairs of trees connected by one rearrangement operation (sometimes called a move). Most popular are the classical nearest neighbour interchange, subtree prune and regraft, and tree bisection and reconnection moves. The problem of computing distances, however, is $${\mathbf {N}}{\mathbf {P}}$$ N P -hard in each of these graphs, making tree inference and comparison algorithms challenging to design in practice. Although anked phylogenetic trees are one of the central objects of interest in applications such as cancer research, immunology, and epidemiology, the computational complexity of the shortest path problem for these trees remained unsolved for decades. In this paper, we settle this problem for the ranked nearest neighbour interchange operation by establishing that the complexity depends on the weight difference between the two types of tree rearrangements (rank moves and edge moves), and varies from quadratic, which is the lowest possible complexity for this problem, to $${\mathbf {N}}{\mathbf {P}}$$ N P -hard, which is the highest. In particular, our result provides the first example of a phylogenetic tree rearrangement operation for which shortest paths, and hence the distance, can be computed efficiently. Specifically, our algorithm scales to trees with tens of thousands of leaves (and likely hundreds of thousands if implemented efficiently).

scholarly journals Dynamic Shortest Paths Methods for the Time-Dependent TSP

Algorithms ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 21
Author(s):  
Christoph Hansknecht ◽  
Imke Joormann ◽  
Sebastian Stiller
Keyword(s):  
Column Generation ◽  
Traveling Salesman Problem ◽  
Shortest Path ◽  
Valid Inequalities ◽  
Shortest Paths ◽  
Traveling Salesman ◽  
Time Dependent ◽  
Full Generality ◽  
Branching Rule ◽  
The Traveling Salesman Problem

The time-dependent traveling salesman problem (TDTSP) asks for a shortest Hamiltonian tour in a directed graph where (asymmetric) arc-costs depend on the time the arc is entered. With traffic data abundantly available, methods to optimize routes with respect to time-dependent travel times are widely desired. This holds in particular for the traveling salesman problem, which is a corner stone of logistic planning. In this paper, we devise column-generation-based IP methods to solve the TDTSP in full generality, both for arc- and path-based formulations. The algorithmic key is a time-dependent shortest path problem, which arises from the pricing problem of the column generation and is of independent interest—namely, to find paths in a time-expanded graph that are acyclic in the underlying (non-expanded) graph. As this problem is computationally too costly, we price over the set of paths that contain no cycles of length k. In addition, we devise—tailored for the TDTSP—several families of valid inequalities, primal heuristics, a propagation method, and a branching rule. Combining these with the time-dependent shortest path pricing we provide—to our knowledge—the first elaborate method to solve the TDTSP in general and with fully general time-dependence. We also provide for results on complexity and approximability of the TDTSP. In computational experiments on randomly generated instances, we are able to solve the large majority of small instances (20 nodes) to optimality, while closing about two thirds of the remaining gap of the large instances (40 nodes) after one hour of computation.

FINDING AN OPTIMAL BRIDGE BETWEEN TWO POLYGONS

2002 ◽  
Vol 12 (03) ◽  
pp. 249-261 ◽  
Author(s):  
XUEHOU TAN
Keyword(s):  
Hierarchical Structure ◽  
Shortest Path ◽  
Line Segment ◽  
Euclidean Distance ◽  
Geodesic Distance ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Search Trees ◽  
Bridge Problem ◽  
Path Queries

Let π(a,b) denote the shortest path between two points a, b inside a simple polygon P, which totally lies in P. The geodesic distance between a and b in P is defined as the length of π(a,b), denoted by gd(a,b), in contrast with the Euclidean distance between a and b in the plane, denoted by d(a,b). Given two disjoint polygons P and Q in the plane, the bridge problem asks for a line segment (optimal bridge) that connects a point p on the boundary of P and a point q on the boundary of Q such that the sum of three distances gd(p′,p), d(p,q) and gd(q,q′), with any p′ ∈ P and any q′ ∈ Q, is minimized. We present an O(n log 3 n) time algorithm for finding an optimal bridge between two simple polygons. This significantly improves upon the previous O(n2) time bound. Our result is obtained by making substantial use of a hierarchical structure that consists of segment trees, range trees and persistent search trees, and a structure that supports dynamic ray shooting and shortest path queries as well.

scholarly journals An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d Domains

2013 ◽  
Vol 50 (1) ◽  
pp. 124-184 ◽  
Author(s):  
Lyudmil Aleksandrov ◽  
Hristo Djidjev ◽  
Anil Maheshwari ◽  
Jörg-Rüdiger Sack
Keyword(s):  
Approximation Algorithm ◽  
Shortest Paths

scholarly journals The Effect of Route-choice Strategy on Transit Travel Time Estimates

2019 ◽  
Author(s):  
Nate Wessel ◽  
Steven Farber
Keyword(s):  
Travel Time ◽  
Shortest Path ◽  
Imperfect Information ◽  
Optimal Path ◽  
Shortest Paths ◽  
Route Choice ◽  
Travel Times ◽  
Selection Strategies ◽  
Time Estimates ◽  
Average Travel Time

Estimates of travel time by public transit often rely on the calculation of a shortest-path between two points for a given departure time. Such shortest-paths are time-dependent and not always stable from one moment to the next. Given that actual transit passengers necessarily have imperfect information about the system, their route selection strategies are heuristic and cannot be expected to achieve optimal travel times for all possible departures. Thus an algorithm that returns optimal travel times at all moments will tend to underestimate real travel times all else being equal. While several researchers have noted this issue none have yet measured the extent of the problem. This study observes and measures this effect by contrasting two alternative heuristic routing strategies to a standard shortest-path calculation. The Toronto Transit Commission is used as a case study and we model actual transit operations for the agency over the course of a normal week with archived AVL data transformed into a retrospective GTFS dataset. Travel times are estimated using two alternative route-choice assumptions: 1) habitual selection of the itinerary with the best average travel time and 2) dynamic choice of the next-departing route in a predefined choice set. It is shown that most trips present passengers with a complex choice among competing itineraries and that the choice of itinerary at any given moment of departure may entail substantial travel time risk relative to the optimal outcome. In the context of accessibility modelling, where travel times are typically considered as a distribution, the optimal path method is observed in aggregate to underestimate travel time by about 3-4 minutes at the median and 6-7 minutes at the \nth{90} percentile for a typical trip.

scholarly journals A MODIFIED GENETIC ALGORITHM FOR FINDING FUZZY SHORTEST PATHS IN UNCERTAIN NETWORKS

2016 ◽  
Vol XLI-B2 ◽  
pp. 299-304 ◽  
Author(s):  
A. A. Heidari ◽  
M. R. Delavar
Keyword(s):  
Genetic Algorithm ◽  
Shortest Path ◽  
Shortest Paths ◽  
The Body ◽  
Shortest Path Problems ◽  
Conventional Procedure ◽  
Path Lengths ◽  
The Cost ◽  
Uncertain Networks ◽  
Exploration Exploitation

In realistic network analysis, there are several uncertainties in the measurements and computation of the arcs and vertices. These uncertainties should also be considered in realizing the shortest path problem (SPP) due to the inherent fuzziness in the body of expert's knowledge. In this paper, we investigated the SPP under uncertainty to evaluate our modified genetic strategy. We improved the performance of genetic algorithm (GA) to investigate a class of shortest path problems on networks with vague arc weights. The solutions of the uncertain SPP with considering fuzzy path lengths are examined and compared in detail. As a robust metaheuristic, GA algorithm is modified and evaluated to tackle the fuzzy SPP (FSPP) with uncertain arcs. For this purpose, first, a dynamic operation is implemented to enrich the exploration/exploitation patterns of the conventional procedure and mitigate the premature convergence of GA technique. Then, the modified GA (MGA) strategy is used to resolve the FSPP. The attained results of the proposed strategy are compared to those of GA with regard to the cost, quality of paths and CPU times. Numerical instances are provided to demonstrate the success of the proposed MGA-FSPP strategy in comparison with GA. The simulations affirm that not only the proposed technique can outperform GA, but also the qualities of the paths are effectively improved. The results clarify that the competence of the proposed GA is preferred in view of quality quantities. The results also demonstrate that the proposed method can efficiently be utilized to handle FSPP in uncertain networks.

k-PAIRS NON-CROSSING SHORTEST PATHS IN A SIMPLE POLYGON

1999 ◽  
Vol 09 (06) ◽  
pp. 533-552 ◽  
Author(s):  
EVANTHIA PAPADOPOULOU
Keyword(s):  
Outer Boundary ◽  
Shortest Paths ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Polygonal Domain ◽  
Total Cost

This paper presents a simple O(n+k) time algorithm to compute the set of knon-crossing shortest paths between k source-destination pairs of points on the boundary of a simple polygon of n vertices. Paths are allowed to overlap but are not allowed to cross in the plane. A byproduct of this result is an O(n) time algorithm to compute a balanced geodesic triangulation which is easy to implement. The algorithm extends to a simple polygon with one hole where source-destination pairs may appear on both the inner and outer boundary of the polygon. In the latter case, the goal is to compute a collection of non-crossing paths of minimum total cost. The case of a rectangular polygonal domain where source-destination pairs appear on the outer and one inner boundary12 is briefly discussed.

A Genetic Algorithms Approach for Inverse Shortest Path Length Problems

2014 ◽  
Vol 4 (4) ◽  
pp. 36-54 ◽  
Author(s):  
António Leitão ◽  
Adriano Vinhas ◽  
Penousal Machado ◽  
Francisco Câmara Pereira
Keyword(s):  
Genetic Algorithm ◽  
Shortest Path ◽  
Path Length ◽  
Shortest Paths ◽  
Research Subject ◽  
Relevant Research ◽  
Real Cost ◽  
Shortest Path Problems ◽  
The Past ◽  
Specific Outcome

Inverse Combinatorial Optimization has become a relevant research subject over the past decades. In graph theory, the Inverse Shortest Path Length problem becomes relevant when people don't have access to the real cost of the arcs and want to infer their value so that the system has a specific outcome, such as one or more shortest paths between nodes. Several approaches have been proposed to tackle this problem, relying on different methods, and several applications have been suggested. This study explores an innovative evolutionary approach relying on a genetic algorithm. Two scenarios and corresponding representations are presented and experiments are conducted to test how they react to different graph characteristics and parameters. Their behaviour and differences are thoroughly discussed. The outcome supports that evolutionary algorithms may be a viable venue to tackle Inverse Shortest Path problems.

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