scholarly journals Better Distance Preservers and Additive Spanners

2021 ◽  
Vol 17 (4) ◽  
pp. 1-24
Author(s):  
Greg Bodwin ◽  
Virginia Vassilevska Williams
Keyword(s):  
Shortest Path ◽  
Upper Bound ◽  
Error Function ◽  
Shortest Paths ◽  
Prior Work ◽  
Input Graph ◽  
Original Graph ◽  
New Construction ◽  
The Right ◽  
Unweighted Graph

We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers , which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work on distance preservers has exploited only a simple structural property of shortest paths, called consistency , stating that one can break shortest path ties such that no two paths intersect, split apart, and then intersect again later. We prove that consistency alone is not enough to understand distance preservers, by showing both a lower bound on the power of consistency and a new general upper bound that polynomially surpasses it. Specifically, our new upper bound is that any p demand pairs in an n -node undirected unweighted graph have a distance preserver on O( n 2/3 p 2/3 + np 1/3 edges. We leave a conjecture that the right bound is O ( n 2/3 p 2/3 + n ) or better. The second part of this paper leverages these distance preservers in a new construction of additive spanners , which are subgraphs that preserve all pairwise distances up to an additive error function. We give improved error bounds for spanners with relatively few edges; for example, we prove that all graphs have spanners on O(n) edges with + O ( n 3/7 + ε ) error. Our construction can be viewed as an extension of the popular path-buying framework to clusters of larger radii.

scholarly journals The Provable Virtue of Laziness in Motion Planning

2019 ◽  
Author(s):  
Nika Haghtalab ◽  
Simon Mackenzie ◽  
Ariel D. Procaccia ◽  
Oren Salzman ◽  
Siddhartha Srinivasa
Keyword(s):  
Lower Bound ◽  
Motion Planning ◽  
Shortest Path ◽  
Upper Bound ◽  
Probabilistic Model ◽  
Shortest Paths ◽  
Worst Case ◽  
Asymptotically Optimal ◽  
Cluttered Environments ◽  
Running Time

The Lazy Shortest Path (LazySP) class consists of motion-planning algorithms that only evaluate edges along candidate shortest paths between the source and target. These algorithms were designed to minimize the number of edge evaluations in settings where edge evaluation dominates the running time of the algorithm such as manipulation in cluttered environments and planning for robots in surgical settings; but how close to optimal are LazySP algorithms in terms of this objective? Our main result is an analytical upper bound, in a probabilistic model, on the number of edge evaluations required by LazySP algorithms; a matching lower bound shows that these algorithms are asymptotically optimal in the worst case.

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.

scholarly journals Coefficient inequalities for a class of analytic functions associated with the lemniscate of Bernoulli

2018 ◽  
Vol 37 (4) ◽  
pp. 83-95
Author(s):  
Trailokya Panigrahi ◽  
Janusz Sokól
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Lemniscate Of Bernoulli ◽  
Toeplitz Determinant ◽  
Second Hankel Determinant ◽  
Coefficient Inequalities ◽  
The Right

In this paper, a new subclass of analytic functions ML_{\lambda}^{*}  associated with the right half of the lemniscate of Bernoulli is introduced. The sharp upper bound for the Fekete-Szego functional |a_{3}-\mu a_{2}^{2}|  for both real and complex \mu are considered. Further, the sharp upper bound to the second Hankel determinant |H_{2}(1)| for the function f in the class ML_{\lambda}^{*} using Toeplitz determinant is studied. Relevances of the main results are also briefly indicated.

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.

The Shortest Path Algorithm Dynamic Visualization Realization

2013 ◽  
Vol 694-697 ◽  
pp. 2291-2294
Author(s):  
Xiao Bo Yang ◽  
Bang Ze Chen
Keyword(s):  
Shortest Path ◽  
Implementation Process ◽  
Object Oriented ◽  
Visual Image ◽  
The Other ◽  
Dynamic Visualization ◽  
Object Oriented Method ◽  
Shortest Path Algorithm ◽  
Source Point ◽  
The Right

By using object oriented method design graph vertices into class, and in this foundation to increase visual member, realize from the source point to the other vertex of the shortest path algorithm of dynamic visualization. Around the two sync window animation, the left window with thick lines drawn through the vertices and edges, " revealed the source point to the other vertex of the shortest path " list box display shortest path sequence and path length, the right window demonstration algorithm dynamic implementation process, and in the " S " text box to display the current most find shortest path vertices. The system has friendly interface, visual image.

scholarly journals Intelligent Policy Framework for Georouting Using Spatial Database

2018 ◽  
Vol 7 (3.6) ◽  
pp. 5
Author(s):  
Swasti Saxena ◽  
B Amutha ◽  
Abhinaba Das
Keyword(s):  
Climate Change ◽  
Carbon Footprint ◽  
Shortest Path ◽  
Policy Framework ◽  
Energy Sources ◽  
Tourist Attraction ◽  
Android Application ◽  
Shortest Route ◽  
The Right ◽  
Day By Day

In today’s world of time effectiveness and climate change it is a necessity to take the right decision at the right time and act responsibly! Our carbon footprint is increasing day by day and our injudicious use of non-renewable sources of energy are posing a threat to get these energy sources exhausted soon! It is our responsibility to take an initiative towards it from our end.In this paper I am going to discuss about an Android Application which I have developed that will solve the problem of finding the shortest route from source to destination. This will lead to less wastage of car fuel and time to find the shortest path!This application has a two fold benefit  i.e. one of finding the shortest path from source and destination and second is acting as a world tour guide. It will not only show you the places of tourist attraction in that place but will also facilitate in providing suggestions for the cheapest means of transport that should be taken in order reach destination in minimum possible time and by spending minimum amount of money on travel!

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.

A Generic Approach on How to Formally Specify and Model Check Path Finding Algorithms: Dijkstra, A* and LPA*

2020 ◽  
Vol 30 (10) ◽  
pp. 1481-1523
Author(s):  
Kazuhiro Ogata
Keyword(s):  
Model Checking ◽  
Shortest Path ◽  
Formal Specification ◽  
Model Check ◽  
Shortest Paths ◽  
Path Finding ◽  
Property A ◽  
Goal Node ◽  
The Cost ◽  
Start Node

The paper describes how to formally specify three path finding algorithms in Maude, a rewriting logic-based programming/specification language, and how to model check if they enjoy desired properties with the Maude LTL model checker. The three algorithms are Dijkstra Shortest Path Finding Algorithm (DA), A* Algorithm and LPA* Algorithm. One desired property is that the algorithms always find the shortest path. To this end, we use a path finding algorithm (BFS) based on breadth-first search. BFS finds all paths from a start node to a goal node and the set of all shortest paths is extracted. We check if the path found by each algorithm is included in the set of all shortest paths for the property. A* is an extension of DA in that for each node [Formula: see text] an estimation [Formula: see text] of the distance to the goal node from [Formula: see text] is used and LPA* is an incremental version of A*. It is known that if [Formula: see text] is admissible, A* always finds the shortest path. We have found a possible relaxed sufficient condition. The relaxed condition is that there exists the shortest path such that for each node [Formula: see text] except for the start node on the path [Formula: see text] plus the cost to [Formula: see text] from the start node is less than the cost of any non-shortest path to the goal from the start. We informally justify the relaxed condition. For LPA*, if the relaxed condition holds in each updated version of a graph concerned including the initial graph, the shortest path is constructed. Based on the three case studies for DA, A* and LPA*, we summarize the formal specification and model checking techniques used as a generic approach to formal specification and model checking of path finding algorithms.

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