Prograph Based Analysis of Single Source Shortest Path Problem with Few Distinct Positive Lengths

This article presents a novel shortest path algorithm for connected networks with nonnegative edge weights. The worst case running time of the single source shortest path version of the algorithm is O(max(m, )) where m is the number of edges of the input network and is the normalized eccentricity of the source vertex vs. The pseudo‐polynomial nature of the time complexity is overcome with a simple speed‐up technique. The proposed approach can be implemented on a wide class of shortest path problems. Approximate solutions can be easily maintained in the preprocessing phase. An experimental efficiency analysis shows that the proposed approach outperforms existing algorithms in total computing time. The proposed algorithm is efficient for all classes of networks and particularly for networks with small diameter.

We present an experimental study of the single source shortest path problem with non-negative edge weights (NSSP) on large-scale graphs using the Δ-stepping parallel algorithm. We report performance results on the Cray MTA-2, a multithreaded parallel computer. The MTA-2 is a high-end shared memory system offering two unique features that aid the efficient parallel implementation of irregular algorithms: the ability to exploit fine-grained parallelism, and low-overhead synchronization primitives. Our implementation exhibits remarkable parallel speedup when compared with competitive sequential algorithms, for low-diameter sparse graphs. For instance, Δ-stepping on a directed scale-free graph of 100 million vertices and 1 billion edges takes less than ten seconds on 40 processors of the MTA-2, with a relative speedup of close to 30. To our knowledge, these are the first performance results of a shortest path problem on realistic graph instances in the order of billions of vertices and edges.

Labeling methods for partially ordered paths

The shortest path problem (SPP) constitutes one of the most frequently encountered classes of problems in graph theory. It is certainly the most fundamental of components in the fields of transportation and communication networks. Shortest path problems may be encountered directly, possibly as a result of a clever formulation of a problem not at first sight involving shortest paths, or indirectly as a subproblem in the solution of a more complicated optimization problem. This use of shortest path problems as subroutines motivates the search for algorithms with good theoretical bounds on running time. We also seek computer implementations whose empirical performances are rapid in spite of perhaps weak theoretical bounds for the algorithms they implement. As testimony to the importance of shortest path and related problems, a large number of surveys, annotated bibliographies, and reviews have appeared over the past thirty years. Among them are those by Dreyfus (1969), Pierce (1975), Golden and Magnanti (1977), and Gallo and Pallottino (1988).

We present a method for solving the transshipment problem - also known as uncapacitated minimum cost flow - up to a multiplicative error of $1 + \varepsilon$ in undirected graphs with non-negative edge weights using a tailored gradient descent algorithm. Using $\tilde{O}(\cdot)$ to hide polylogarithmic factors in $n$ (the number of nodes in the graph), our gradient descent algorithm takes $\tilde O(\varepsilon^{-2})$ iterations, and in each iteration it solves an instance of the transshipment problem up to a multiplicative error of $\operatorname{polylog} n$. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a randomized rounding scheme, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining the following results: (1) Broadcast CONGEST model: $(1 + \varepsilon)$-approximate SSSP using $\tilde{O}((\sqrt{n} + D)\varepsilon^{-3})$ rounds, where $ D $ is the (hop) diameter of the network. (2) Broadcast congested clique model: $(1 + \varepsilon)$-approximate transshipment and SSSP using $\tilde{O}(\varepsilon^{-2})$ rounds. (3) Multipass streaming model: $(1 + \varepsilon)$-approximate transshipment and SSSP using $\tilde{O}(n)$ space and $\tilde{O}(\varepsilon^{-2})$ passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative edge weights that are polynomially bounded in $n$; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights.

Shortest path problem with uncertain arc lengths

This thesis proposes an algorithm that involves Dijkstra's algorithm to solve the Single Source Shortest Path Problem in a graph where the length of each edge is positive, and the specific case where there is a distinct number of possible lengths. Given a graph where there are n vertices, m edges, and K distinct edge lengths, the proposed algorithm runs in O(m ) time if nK ≤ 2m and O(m log nKm ) time if nK > 2m. The algorithm is tested and compared with some of the fastest algorithms for the Single Source Shortest Path Problem for graphs with arbitrary lengths. The results verify the theoretical conclusions that the proposed algorithm is better, in the case where K is small, than the compared algorithms that did not take advantage of having few distinct edge lengths.

The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n) time, where n is the number of vertices in the input graph G. Peleg and Rubinovich, FOCS'99, showed a lower bound of Ω(D + √n) for this problem, where D is the hop-diameter of G.

In the field of research on Deoxyribonucleic acid (DNA), graph theory can be applied to model the structure of a DNA molecule. In particular, the shortest path problem in graph theory can be used to identify the shortest path between vertices of a graph. Hence, it is possible to apply the shortest path problem for minimization of a DNA string so that the time taken for computation of genome assembly can be reduced. This paper presents a method to represent a DNA string graphically where the shortest path is calculated for the graph generated from the DNA string, and the shortest path is then used to minimize the DNA string. In this research, a DNA string is presented in graphical form by using base pairs of length two as the vertices where the initial bases used are Adenine (A) and Guanine (G) which are the main bases in purine. The number of base pairs between adjacent vertices in the DNA string is represented by the edges. The graph is then reduced by following a given set of rules where the shortest path is calculated for all start and end vertices of the reduced graph. Next, the simplification of the graph is done based on the shortest paths obtained by removing all the untraversed paths where the Euler path for the simplified graph is used to form a minimized DNA string. The result shows that a minimized DNA string can be obtained by simplifying the graph of the DNA string using the shortest path problem

We study the 2-Disjoint Shortest Paths (2-DSP) problem: given a directed weighted graph and two terminal pairs (s₁,t₁) and (s₂,t₂), decide whether there exist vertex-disjoint shortest paths between each pair. Building on recent advances in disjoint shortest paths for DAGs and undirected graphs (Akmal et al. 2024), we present an O(mn log n)-time algorithm for this problem in weighted directed graphs that do not contain negative or zero weight cycles. This algorithm presents a significant improvement over the previously known O(m⁵n)-time bound (Berczi et al. 2017). Our approach exploits the algebraic structure of polynomials that enumerate shortest paths between terminal pairs. A key insight is that these polynomials admit a recursive decomposition, enabling efficient evaluation via dynamic programming over fields of characteristic two. Furthermore, we demonstrate how to report the corresponding paths in O(mn² log n)-time. In addition, we extend our techniques to a more general setting: given two terminal pairs (s₁, t₁) and (s₂, t₂) in a directed graph, find the minimum possible number of vertex intersections between any shortest path from s₁ to t₁ and s₂ to t₂. We call this the Minimum 2-Disjoint Shortest Paths (Min-2-DSP) problem. We provide in this paper the first efficient algorithm for this problem, including an O(m² n³)-time algorithm for directed graphs with positive edge weights, and an O(m+n)-time algorithm for DAGs and undirected graphs. Moreover, if the number of intersecting vertices is at least one, we show that it is possible to report the paths in the same O(m+n)-time. This is somewhat surprising, as there is no known o(mn) time algorithm for explicitly reporting the paths if they are vertex-disjoint, and is left as an open problem in (Akmal et al. 2024).

The problem of processing shortest pa th queries in graphs arises in application areas such as intelligent t ranspor ta t ion system (ITS), geographic information system (GIS), and robotics. In this paper, we present an efficient algorithmic solution for processing shortest pa th queries in undirected planar graphs with non-negative edge weights. Previous algorithms for this problem on planar graphs all have a tradeoff between the query t ime and the da ta structure space used by the solutions. The previously best known trade-off on an n-vertex planar graph G in the worst case is O(v/-r) t ime for a pa th length query and O(v/-r + L) t ime for report ing an actual shortest path, with a da ta structure of O(n2/x/~) space, where r is an integer parameter with 1 < r _< n and L is the number of edges on the output shortest path. We present a new scheme, called frame search, that exploits the graph planari ty in a novel fashion. Wi th this scheme, we build an improved da ta structure of O(n + pv/-r + p2/r) space (in O(n + p2/v/~ + pr 3/4) time), where p is the minimum cardinality of a face-covering of the planar embedding of G with 1 < p < O(n), and r is an integer parameter with 1 < r < p. This da ta structure enables us to process each length query in O(v/-rlog r + a(n)) t ime and report an actual shortest path in an addit ional O(L) time, where a(n) is the inverse of Ackermann's function. In the worst case, our approach reduces the previously best query t ime by a factor of up to O(n 1/4) if the same amount of space is used. Our technique can also be applied to improve the previous query algorithms for some geometric shortest pa th problems on the plane.

Let G = (V, E, w) be a directed graph, where w : V ← R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc(u, v) denote the maximum bottleneck weight of a path from u to v having length d(u, v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n2.86).

Finding shortest distances and paths have always been a crucial area of study. In this article, we have proposed a linear time pre-processing method for optimization of shortest path and distance algorithms, for an undirected weighted graph with non-negative edges. In this approach, certain parts of a graph are grouped together as subgraphs termed as optimized deterministic routing areas (ODRAs). Each ODRA has a unique representative element called optimized proxy. The input graph is reduced by replacing ODRAs with optimized proxies and the new reduced graph is used for the shortest path and distance queries. This is a linear time pre-processing stage which helps in significant decrease in computation time for finding shortest distances and paths in graphs. The lower bound and upper bound of time complexity of the proposed algorithm is O(n/2) and O(n) respectively.

This article analyzes problems of determining the shortest path and optimal route amongst the given stoppages. The model of the problem is presented as a directional graph, where nodes are pickup points (termed as stoppage point in database) and crossings outside stoppage points and edges are roads among stoppage points and crossings. Each node has some information attached to it: stoppageId, stoppageName, latitude, longitude and numberOfPassengers of the stop, maintenance organizations, and mark(s) of the crossing(s). All pickup points are connected by roads. These roads are considered as the edges of the graph. Edges also have information attached to it: roadId, source, destination, distance, time etc.We have selected Floyd Warshall algorithm to nd the shortest path between two stoppages. This algorithm works in two stages: in rst stage, it nds the shortest path between all stoppages, and in second stage it nds optimized route to visit some of these stoppages. The solution is displayed in the form of shortest distance and time between two locations. The program is written in java language. It uses 3 tables as input from database : nodes, vehicle details and road. This paper gives implementation outcome of Floyd Warshall algorithm to solve the all pairs shortestpath problem for directed road graph system. We have considered an example of a map of Pune. Keyword: Adjacency matrix,intermediate path,optimal route, shortest path,transitive closure I. Introduction Now days in the current scenario companies and schools provide transportation facilities to their employees and students from their organizations to their respective houses and vice versa. The purpose of this article is to present an interface between a traveling agency and its traveller so that, the subscribers can keep a track of the service they have subscribed.A key problem in public conveyance system is the computation of shortest paths between di erent locations for a given region. Sometimes this computation has to be done in real time. During the literature survey we found how di cult it is to nd the shortest path covering all pickup points and how to allocate available transport vehicles to these speci c routes manually. There is always a need to nd optimal path for their operational viability. So, just the idea that our system will help in e cient allocation of vehicles and shortest route creation to save time and fuel consumption. The system scope is to design such system for a small area of the city.

A class of cooperative games arising from shortest path problems is defined. These shortest path games are totally balanced and allow a population-monotonic allocation scheme. Possible methods for obtaining core elements are indicated; first, by relating to the allocation rules in taxation and bankruptcy problems, second, by constructing an explicit rule that takes opportunity costs into account by considering the costs of the second best alternative and that rewards players who are crucial to the construction of the shortest path. The core and the bargaining sets of Davis-Maschler and Mas-Colell are shown to coincide. Finally, noncooperative games arising from shortest path problems are introduced, in which players make bids or claims on paths. The core allocations of the cooperative shortest path game coincide with the payoff vectors in the strong Nash equilibria of the associated noncooperative shortest path game.