scholarly journals DEPTH FIRST SEARCH (DFS) UNTUK MENENTUKAN DIAMETER GRAF HIRARKI

2019 ◽  
Vol 11 (2) ◽  
pp. 29
Author(s):  
Nugroho Arif Sudibyo ◽  
Ardymulya Iswardani
Keyword(s):  
Shortest Path ◽  
Shortest Paths ◽  
Depth First Search ◽  
Longest Path

Let G = (V, E) be a graph. The distance d (u, v) between two vertices u and v is the length of the shortest path between them. The diameter of the graph is the length of the longest path of the shortest paths between any two graph vertices (u ,v) of a graph, . In this paper we propose algorithms for finding diameter of a hierarchy graph using DFS. Diameter of the hierarchy graph using DFS algoritm is four.

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 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.

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.

FINDING THE SHORTEST PATHS ON SURFACES BY FAST GLOBAL APPROXIMATION AND PRECISE LOCAL REFINEMENT

1996 ◽  
Vol 10 (06) ◽  
pp. 643-656 ◽  
Author(s):  
RON KIMMEL ◽  
NAHUM KIRYATI
Keyword(s):  
Shortest Path ◽  
Optimal Path ◽  
Shortest Paths ◽  
Geodesic Curvature ◽  
Geodesic Curve ◽  
Computationally Efficient ◽  
Global Approximation ◽  
Curve Shortening Flow ◽  
Curve Shortening ◽  
Fixed End

Finding the shortest path between points on a surface is a challenging global optimization problem. It is difficult to devise an algorithm that is computationally efficient, locally accurate and guarantees to converge to the globally shortest path. In this paper a two stage coarse-to-fine approach for finding the shortest paths is suggested. In the first stage the algorithm of Ref. 10 that combines a 3D length estimator with graph search is used to rapidly obtain an approximation to the globally shortest path. In the second stage the approximation is refined to become a shorter geodesic curve, i.e., a locally optimal path. This is achieved by using an algorithm that deforms an arbitrary initial curve ending at two given surface points via geodesic curvature shortening flow. The 3D curve shortening flow is transformed into an equivalent 2D one that is implemented using an efficient numerical algorithm for curve evolution with fixed end points, introduced in Ref. 9.

AN OPTIMAL MORPHING BETWEEN POLYLINES

2002 ◽  
Vol 12 (03) ◽  
pp. 217-228 ◽  
Author(s):  
SERGEI BESPAMYATNIKH
Keyword(s):  
Shortest Path ◽  
Minimum Width ◽  
Longest Path ◽  
Previous Algorithm

We address the problem of continuously transforming or morphing one simple polyline into another so that every point p of the initial polyline moves to a point q of the final polyline using the geodesic shortest path from p to q. We optimize the width of the morphing, that is, the longest path made by a point on the polyline. We present an algorithm for finding the minimum width morphing in O(n2) time using O(n) space, where n is the total number of vertices of polylines. This improves the previous algorithm7 by a factor of log 2 n.

An Efficient K-Shortest Paths Based Routing Algorithm

2012 ◽  
Vol 532-533 ◽  
pp. 1775-1779
Author(s):  
Jian Lian ◽  
Yan Zhang ◽  
Cheng Jiang Li
Keyword(s):  
Computer Networks ◽  
Shortest Path ◽  
Traffic Engineering ◽  
Routing Algorithm ◽  
Shortest Paths ◽  
Routing Algorithms ◽  
Dijkstra Algorithm ◽  
Link State

We present an efficient K-shortest paths routing algorithm for computer networks. This Algorithm is based on enhancements to currently used link-state routing algorithms such as OSPF and IS-IS, which are only focusing on finding the shortest path route by adopting Dijkstra algorithm. Its desire effect to achieve is through the use of K-shortest paths algorighm, which has been implemented successfully in some fileds like traffic engineering. The correctness of this Algorithm is discussed at the same time as long as the comparison with OSPF.

Design of High Speed Reconfigurable Distributed Life Time Efficient Routing Algorithm in Wireless Sensor Network

2020 ◽  
Vol 17 (9) ◽  
pp. 3860-3866
Author(s):  
M. L. Umashankar ◽  
S. Mallikarjunaswamy ◽  
M. V. Ramakrishna
Keyword(s):  
Shortest Path ◽  
High Speed ◽  
Routing Algorithm ◽  
Shortest Paths ◽  
Life Time ◽  
Wireless Sensor ◽  
Signal Interference ◽  
Energy Efficient Routing ◽  
Shortest Path Routing ◽  
Routing Techniques

Designing an energy-efficient routing makes the Wireless Sensor Networks (WSN) more effective and attractive for different applications. The WSN communication system power consumption mainly depends on three aspects such as routing cost computation, signal interference, and routing distance. All three factors are equally important in order to improve the network performance. The system reliability and deployment cost depends on the energy efficiency of the WSN. The energy related cost assignment and shortest paths identification are used in existing routing techniques. In the existing routing techniques maximum achievable lifetime and optimal link cost are low. Hence greatest possible performance can be achieved in distributed routing algorithm by finding shortest path. Maximum lifetime and best cost link can be generally obtained using distributed shortest path routing algorithm. In this paper high speed reconfigurable distributed Lifetime-Efficient Routing algorithm is designed to provide route selection outline with low complexity and obtain better performance compared to existing routing algorithm.

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