This paper presents overview of algorithms for the **shortest** **path** **problem**. It explains Dijkstra algorithm and Floyd-Warshalls algorithm that are explained on examples of smaller graphs. Definition of graphs, basic methods of graph presentation and their basic division are explained in fist part of the paper. Dijkstra algorithm is implemented in Java programing language. Program was made in InteliJ development environment. Algorithm is implemented on the example of graph which is made of eight nodes. Graph symbolizes smaller information and communication traffic network. Goal of program is to calculate delays from source node to all other nodes in network. Routing, in Internet traffic, is the process of choosing a **path** over which to send packets from source router to destination. Goal is for packets to be delivered as soon as possible, for decision making we are using **shortest** **path** algorithms as Dijsktra algorithm implemented in this paper.

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The report for this study consists of four chapters. Chapter 1 presents an introduction to the study, **problem** background, objective, scope and significance of this study. Chapter 2 reviews the **Shortest** **Path** **Problem** Types and Applications, classic methods to solving the **Shortest** **Path** **Problem**, heuristic algorithms for the **Shortest** **Path** **Problem**, and the Gravitational Search Algorithm, which are the fundamental parts in the method proposed in this study. Chapter 3 discusses the methodology used in this study. Chapter 4 presents the results and evaluation of the results. And finally Chapter 5 is the Conclusion.

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The standard **shortest** **path** **problem** only considers the cost on the arc, and it doesn’t consider the cost of switch- ing between arcs. But in some applications such as trans- portation networks, the switch cost between arcs does exist and is important. So it is necessary to solve the **problem** by considering switch cost between arcs. This paper will focus on this point. To explain the **problem** better, we deﬁne some notations ﬁrst.

In graph theory, the **Shortest** **Path** **Problem** (SPP) is one of the most famous problems studied and being studied by researchers. The SPP is the **problem** of finding a **path** between two vertices (or nodes) in a digraph such that the sum of the weights of its constituent edges is minimized. Thus the core **problem** is to find the **shortest** **path** from a source vertex S to a single destination vertex D in a directed graph and to compute the corresponding min cost. **Shortest** **Path** problems (SPP) are among the fundamental problems studied in Computational Geometry, Graph Algorithms, Geographical Information Systems (GIS), Network Optimization etc. to list a few only out of many. Sometimes the network of a real life communication

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1226 | P a g e The **shortest** **path** **problem** is the **problem** of finding a **path** between two veracities or nodes in a graph such that sum of weights of its constituent edges are minimized. **Shortest** **path** problems (SPPs) are widespread in practical applications (e.g., logistics, transportation, robot **path** planning[4,5], vehicle routing[6], no-wait flow shop scheduling[7], etc.). SPP consists of finding the **shortest** **path** from the source node to the sink node in a graph. Generally, the distance, time or price of traversing of each arc is called cost. There are a large number of paths in the graph.

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Single source **shortest** **path** **problem** finds application in large domains of scientific and real world. Common applications of these algorithms are in network routing [6], VLSI design, robotics and transportation, they are also used for directions between physical locations like in google maps. Here all the applications mentioned generally involve positive weights but some applications are there where weights can be negative like currency exchange arbitrage and some other areas where, edge represents something other than merely distance between two entities. In such application areas Bellman-Ford algorithm can be used. Bellman-Ford algorithm[12] is applicable on graphs with negative weights and can also detect negative cycles where majority of algorithms fail. Bellman-Ford is also used in wireless sensor networks and other ad hoc networks as distributed Bellman Ford [7] can be used there. Distributed Bellman-Ford is also used as first ARPANET routing algorithm in 1969 [14].

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Abstract. In a network the arc lengths may represent time or cost. In practical situations, it is reasonable to assume that each arc length is a type-2 discrete fuzzy set. We called it the type-2 discrete fuzzy **shortest** **path** **problem**. In this paper we proposed an algorithm for finding **shortest** **path** and **shortest** **path** length from source node to destination node using type reduction method. We have compared our result with other measures like Hamming, Normalized Hamming, Exponential type distance measure also. An illustrative example also included to demonstrate our proposed approach.

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The fuzzy **shortest** **path** **problem** is an extension of fuzzy numbers and it has many real life applications in the field of communication, robotics, scheduling and transportation. Dubois [4] introduced the fuzzy **shortest** **path** **problem** for the first time. Klein [6] introduced a new model to solve the fuzzy **shortest** **path** **problem** for sub-modular functions. Lin and Chern [8] introduced a new design to find the fuzzy **shortest** **path** **problem** on single most vital arc length in a network by using dynamics programming approach. Li et.al. [9] solved the fuzzy **shortest** **path** problems by using neural network approach. Chuang et al. [3] used two steps to find the **shortest** **path** from origin to destination.

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In traditional **Shortest** **Path** **Problem** (SPP) it is always determined that the parameters (Time, Cost and Distance etc.) are fixed between different nodes. But in real life situations where uncertain parameters exist, parameters are considered as fuzzy numbers. In this paper, we explained the application scope of the given fuzzy ranking function. Using this method we can determine both the fuzzy **shortest** **path** and fuzzy **shortest** distance from origin to destination.

At the beginning of this thesis some elementary terms from graph theory are defined as a foundation for the **shortest** **path** **problem**. A few varieties of the aforementioned **problem** are described, as well as the method of representing graphs in computer memory. After that, Floyd-Warshall algorithm for all-pairs **shortest**-**path** **problem** is analized in detail. Dyna- mic programming is also defined as Floyd-Warshall algorithm is a prime example of such a method. Additionally, CUDA platform for hybrid CPU-GPU computing is analized - it’s architecture, it’s advantages and disadvantages, but also, since it is a parallel computing platform, the history of parallel computing is given. Based on all the conclusions, three versions of Floyd-Warshall algorithm have been implemented: a simple C++ implementation of the algorithm which is analogous to the given pseudocode, a simple CUDA implementation which used the advantages of the graphics card architecture and an advanced (blocked) CUDA implementation which further made use of the Floyd-Warshall’s algorithm parallel nature. At the end, the experimental results of all three implementations on different sized graphs and graphs of various densities have been laid out.

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In this paper, I examine Dijkstra’s algorithm in the context of solving the **shortest** **path** **problem**. The mathematical modeling of this **problem** has to do with the theory of graphs. The whole system is programmed as client system using Google Services and Android OS. Google Services are used to determine the optimum **path** on Google map and solve the **shortest** **path** **problem** as alternative of Dijkstra's Algorithm.

All-Pairs **Shortest** **Path** **Problem** (APSP) finds a large number of practical applications in real world. This paper presents a blocked parallel approach for APSP using an open standard framework OpenCL, which provides development environment for utilizing heterogeneous computing elements of computer system and to take advantage of massive parallel capabilities of multi-core processors such as graphics processing unit (GPU) and CPU. This blocked parallel approach exploits the local shared memory of GPU, thereby enhancing the overall performance. The proposed solution is for directed and dense graphs with no negative cycles and is based on blocked Floyd Warshall (FW) and Kleene‟s algorithm. Like Floyd Warshall this approach is also in-place and therefore requires no extra memory.

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Abstract. This paper proposes a method to solve **shortest** **path** problems in route choice processes when each link's travel time is a fuzzy number, called the Perceived Travel Time (PTT). The PTT is a subjective travel time perceived by a driver. The algorithm solves the fuzzy **shortest** **path** **problem** (FSPA) for drivers in the presence of uncertainty regarding route travel time. For congested networks, the method is able to nd the **shortest** **path** in terms of perceived travel time and degree of saturation (congestion) along routes at the same time. The FSPA can be used to support the fuzzication of trac assignment algorithms. The applicability of the resulting FSPA for the trac assignment was tested in conjunction with incremental trac loading and was applied to a large-scale real network. The results of the trac assignment based on the FSPA, User Equilibrium (UE) and a stochastic loading network model (Dial's assignment algorithm) were compared to the observed volume for certain links in the network. We conclude that the proposed method oers better accuracy than the UE or Dial's assignment algorithm for the network under testing. Keywords: Fuzzy sets; Perceived travel time; Route choice; **Shortest** **path**; Urban network.

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Abstract— All-pairs **shortest** **path** **problem**(APSP) finds a large number of practical applications in real world. We owe to present a highly parallel and recursive solution for solving APSP **problem** based on Kleene’s algorithm. The proposed parallel approach for APSP is implemented using an open standard framework OpenCL which provides a development environment for utilizing massive parallel capabilities of Multi core CPU and Many-Core-Processors such as Graphics Processing Unit (GPU). Moreover due to inherent nature of data reuse in the algorithm, shared memory of these processors is exploited to achieve considerable speedup. Our experiments demonstrate a speedup gain up to 521x over NVIDIA GeForce GT 630M GPU and a speedup up to 10x over Intel Core i3-2310M CPU.The proposed OpenCL solution for APSP is for directed and dense graphs with no negative cycles. Like Floyd-Warshall (FW), this approach is also in-place in nature and therefore requires no extra space.

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A graph is a set of vertices and a collection of edges that each connects a pair of vertices. The most intuitive graph- processing **problem** is the **shortest**-**path** **problem** which is concerned with finding the lowest-cost way to get from one vertex to another [1]. The graphs under consideration are edge-weighted and have an associated **path** weight, the value of which is the sum of the weights of the path's edges. A prominent application of the **shortest**-**path** **problem** is routing in multi-hop wireless networks which employs **shortest**-**path** and graph theory-based algorithms such as dynamic programming (DP) algorithms. Every wireless channel between two nodes (vertices) is viewed as a link (edge). In dynamic (adaptive) routing, the nodes need to be notified to recalculate their routes in prompt response to topology changes. This calls for intensive parallel processing in the nodes [2] to prevent bottlenecks which counteract the high speed merits of wireless networks. The ever-increasing capacity of wireless transmission [3] necessitates node switching systems with capacity and rapid response to meet this speed demand. In this paper, we attempt to calculate the best **path** in each vertex by employing systolic parallel dynamic programming assuming general directed acyclic graphs (DAGs). Since a DAG is a directed graph with no

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Abstract: Membrane Computing is inspired from biological cell activities as a new distributed parallel computational framework, which can be used for decreasing the time complexity of problems with high execution time. Since the usual way to reduce the time complexity of Artificial Intelligence (AI) problems is using parallel algorithms, Membrane Computing can be extensively applied. On the other hand, **shortest** **path** **problem** (SPP) is the most broadly method to solve the problems in AI. There have been a variety of algorithms presented, that could find a solution in a desirable time but usually are not accurate, or they are accurate but they are too slow. In Membrane Computing technique, the normal way for reducing the time complexity is using P system with active membranes that the number of membranes increases during the computation; thus, the time is traded against the space. This paper presents the first Membrane Computing technique for solving the SPP using P system with membranes division by a breadth first exploring on a grid. The theorems show the run times for breadth-first search and SPP are significantly reduced.

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Abstract. The **shortest** **path** **problem** is an important classical network optimization **problem** arising from many applications including robotics, networking, and transportation. In most situations, however, some issues of a network-theoretic **problem** may be uncertain. In conventional **shortest** **path** problems, there always an assumption that one who takes the decision is certain about the parameters (distance, time etc.) between different possible vertices in the network G={V,E}. But while considering the real time cases, the possibility of existence of uncertainty about the parameters between different nodes is always high. In those situations, the representation of parameters are given by fuzzy numbers and here we consider the generalized trapezoidal fuzzy numbers, can be dealt with the uncertainty using fuzzy set theory. In order to provide solution for the uncertain **shortest** **path** **problem**, we proposed Hybrid Ant based optimization with ranking of generalized trapezoidal fuzzy numbers, which is proposed recently, as a fitness function. The proposed model is implemented using MATLAB with the test network of 30 nodes and the results reports that the algorithm converges in a more reasonable time in comparison with conventional approaches.

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The on-line **shortest** **path** **problem** is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a **path** between two distinguished vertices such that the loss of the chosen **path** (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit **problem**, after choosing a **path**, the decision maker learns only the weights of those edges that belong to the chosen **path**. For this **problem**, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best **path**, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1/ √ n and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with complexity that is linear in the number of rounds n (i.e., the average complexity per round is constant) and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen **path** at a total of m n time instances. Another extension is shown where the decision maker competes against a time-varying **path**, a generalization of the **problem** of tracking the best expert. A version of the multi-armed bandit setting for **shortest** **path** is also discussed where the decision maker learns only the total weight of the chosen **path** but not the weights of the individual edges on the **path**. Applications to routing in packet switched networks along with simulation results are also presented.

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The order inducing variable, u c (⊆ 0,1) determines the k lm largest edge 〈u c , φ c 〉 which is included in a **path** from source to sink of the network such that for any two edges included in a **path**, u (h%) ≥ u (h) ; whereas for each TrIFN φ h , its membership grade represents the acceptance degree to which an edge i − j will be included in the **shortest** **path** between source to sink and it’s non-membership grade represents the rejection degree to which an edge i − j will be accepted from the **shortest** **path** between source to sink. The new algorithm is detailed below:

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The Minimal spanning tree algorithm in [10] deals with linking the nodes of a network, directly or indirectly, using the **shortest** length of connecting branches. A typical application occurs in the construction of paved roads that link several rural towns. The road between two towns may pass through one or more towns. The most economical design of the road system calls for minimizing the total miles of paved roads, a result that is achieved by implementing the minimal spanning tree algorithm. When it is not possible to identify the distances between two places accurately, neu- trosophic theory is used in that situation. Here, we introduce the concept of Bipolar Neutrosophic numbers as distances(lengths) in Minimal Spanning Tree algorithm to find the **shortest** route in a network.

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