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32-434: SSSP may refer to: Technology [ edit ] Samsung Smart Signage Platform , a smart TV Single-source shortest paths ; finding a path between two vertices in a graph, such that the sum of the weights is minimized Entertainment [ edit ] Sawaare Sabke Sapne... Preeto , an Indian television series Learned societies [ edit ] Society for

64-792: A real-valued weight function f : E → R {\displaystyle f:E\rightarrow \mathbb {R} } , and an undirected (simple) graph G {\displaystyle G} , the shortest path from v {\displaystyle v} to v ′ {\displaystyle v'} is the path P = ( v 1 , v 2 , … , v n ) {\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})} (where v 1 = v {\displaystyle v_{1}=v} and v n = v ′ {\displaystyle v_{n}=v'} ) that over all possible n {\displaystyle n} minimizes

96-417: A fundamental concept in graph theory and operations research, often used to model problems involving the transportation of goods, liquids, or information through a network. A network flow problem typically involves a directed graph where each edge represents a pipe, wire, or road, and each edge has a capacity, which is the maximum amount that can flow through it. The goal is to find a feasible flow that maximizes

128-426: A negative cycle. Some shortest-paths algorithms can be used for this purpose: Many problems can be framed as a form of the shortest path for some suitably substituted notions of addition along a path and taking the minimum. The general approach to these is to consider the two operations to be those of a semiring . Semiring multiplication is done along the path, and the addition is between paths. This general framework

160-417: A nondeterministic abstract machine as a graph where vertices describe states and edges describe possible transitions, shortest path algorithms can be used to find an optimal sequence of choices to reach a certain goal state, or to establish lower bounds on the time needed to reach a given state. For example, if vertices represent the states of a puzzle like a Rubik's Cube and each directed edge corresponds to

192-421: A road segment depends on many factors such as the amount of traffic (origin-destination matrix), road work, weather, accidents and vehicle breakdowns. A more realistic model of such a road network is a stochastic time-dependent (STD) network. There is no accepted definition of optimal path under uncertainty (that is, in stochastic road networks). It is a controversial subject, despite considerable progress during

224-445: A single move or turn, shortest path algorithms can be used to find a solution that uses the minimum possible number of moves. In a networking or telecommunications mindset, this shortest path problem is sometimes called the min-delay path problem and usually tied with a widest path problem . For example, the algorithm may seek the shortest (min-delay) widest path, or widest shortest (min-delay) path. A more lighthearted application

256-516: A specific set of vertices to be included in the path, which makes the problem similar to the Traveling Salesman Problem (TSP). The TSP is the problem of finding the shortest path that goes through every vertex exactly once, and returns to the start. The problem of finding the longest path in a graph is also NP-complete. The Canadian traveller problem and the stochastic shortest path problem are generalizations where either

288-442: Is a representation of the primitive path network within the framework of Reptation theory . The widest path problem seeks a path so that the minimum label of any edge is as large as possible. Other related problems may be classified into the following categories. Unlike the shortest path problem, which can be solved in polynomial time in graphs without negative cycles, shortest path problems which include additional constraints on

320-601: Is adjacent to v i + 1 {\displaystyle v_{i+1}} for 1 ≤ i < n {\displaystyle 1\leq i<n} . Such a path P {\displaystyle P} is called a path of length n − 1 {\displaystyle n-1} from v 1 {\displaystyle v_{1}} to v n {\displaystyle v_{n}} . (The v i {\displaystyle v_{i}} are variables; their numbering here relates to their position in

352-599: Is also sometimes called the single-pair shortest path problem , to distinguish it from the following variations: These generalizations have significantly more efficient algorithms than the simplistic approach of running a single-pair shortest path algorithm on all relevant pairs of vertices. Several well known algorithms exist for solving this problem and its variants. Additional algorithms and associated evaluations may be found in Cherkassky, Goldberg & Radzik (1996) . An algorithm using topological sorting can solve

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384-460: Is known as the algebraic path problem . Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures. More recently, an even more general framework for solving these (and much less obviously related problems) has been developed under the banner of valuation algebras . In real-life, a transportation network is usually stochastic and time-dependent. The travel duration on

416-549: Is that the road network is static, so the preprocessing phase can be done once and used for a large number of queries on the same road network. The algorithm with the fastest known query time is called hub labeling and is able to compute shortest path on the road networks of Europe or the US in a fraction of a microsecond. Other techniques that have been used are: For shortest path problems in computational geometry , see Euclidean shortest path . The shortest multiple disconnected path

448-403: Is the games of " six degrees of separation " that try to find the shortest path in graphs like movie stars appearing in the same film. Other applications, often studied in operations research , include plant and facility layout, robotics , transportation , and VLSI design. A road network can be considered as a graph with positive weights. The nodes represent road junctions and each edge of

480-522: The Study of Social Problems , a learned society dedicated to research in the social sciences Society for the Scientific Study of Psychopathy , a learned society dedicated to studying psychopathy Soil Science Society of Poland , a learned society dedicated to soil science research Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

512-449: The computers may be selfish: a computer might tell us that its transmission time is very long, so that we will not bother it with our messages. A possible solution to this problem is to use a variant of the VCG mechanism , which gives the computers an incentive to reveal their true weights. In some cases, the main goal is not to find the shortest path, but only to detect if the graph contains

544-654: The definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v 1 , v 2 , … , v n ) ∈ V × V × ⋯ × V {\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})\in V\times V\times \cdots \times V} such that v i {\displaystyle v_{i}}

576-462: The desired solution path are called Constrained Shortest Path First , and are harder to solve. One example is the constrained shortest path problem, which attempts to minimize the total cost of the path while at the same time maintaining another metric below a given threshold. This makes the problem NP-complete (such problems are not believed to be efficiently solvable for large sets of data, see P = NP problem ). Another NP-complete example requires

608-404: The flow from a source node to a sink node. Shortest Path Problems can be used to solve certain network flow problems, particularly when dealing with single-source, single-sink networks. In these scenarios, we can transform the network flow problem into a series of shortest path problems. The all-pairs shortest path problem finds the shortest paths between every pair of vertices v , v' in

640-462: The graph is associated with a road segment between two junctions. The weight of an edge may correspond to the length of the associated road segment, the time needed to traverse the segment, or the cost of traversing the segment. Using directed edges it is also possible to model one-way streets. Such graphs are special in the sense that some edges are more important than others for long-distance travel (e.g. highways). This property has been formalized using

672-401: The graph is not completely known to the mover, changes over time, or where actions (traversals) are probabilistic. Sometimes, the edges in a graph have personalities: each edge has its own selfish interest. An example is a communication network, in which each edge is a computer that possibly belongs to a different person. Different computers have different transmission speeds, so every edge in

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704-509: The graph. The all-pairs shortest paths problem for unweighted directed graphs was introduced by Shimbel (1953) , who observed that it could be solved by a linear number of matrix multiplications that takes a total time of O ( V ) . Shortest path algorithms are applied to automatically find directions between physical locations, such as driving directions on web mapping websites like MapQuest or Google Maps . For this application fast specialized algorithms are available. If one represents

736-440: The network has a numeric weight equal to the number of milliseconds it takes to transmit a message. Our goal is to send a message between two points in the network in the shortest time possible. If we know the transmission-time of each computer (the weight of each edge), then we can use a standard shortest-paths algorithm. If we do not know the transmission times, then we have to ask each computer to tell us its transmission-time. But,

768-437: The notion of highway dimension. There are a great number of algorithms that exploit this property and are therefore able to compute the shortest path a lot quicker than would be possible on general graphs. All of these algorithms work in two phases. In the first phase, the graph is preprocessed without knowing the source or target node. The second phase is the query phase. In this phase, source and target node are known. The idea

800-453: The past decade. One common definition is a path with the minimum expected travel time. The main advantage of this approach is that it can make use of efficient shortest path algorithms for deterministic networks. However, the resulting optimal path may not be reliable, because this approach fails to address travel time variability. To tackle this issue, some researchers use travel duration distribution instead of its expected value. So, they find

832-400: The probability distribution of total travel duration using different optimization methods such as dynamic programming and Dijkstra's algorithm . These methods use stochastic optimization , specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length. The terms travel time reliability and travel time variability are used as opposites in

864-434: The sequence and needs not to relate to any canonical labeling of the vertices.) Let E = { e i , j } {\displaystyle E=\{e_{i,j}\}} where e i , j {\displaystyle e_{i,j}} is the edge incident to both v i {\displaystyle v_{i}} and v j {\displaystyle v_{j}} . Given

896-441: The shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length or distance of each segment. The shortest path problem can be defined for graphs whether undirected , directed , or mixed . Here it is defined for undirected graphs; for directed graphs

928-452: The single-source shortest path problem in time Θ( E + V ) in arbitrarily-weighted directed acyclic graphs. The following table is taken from Schrijver (2004) , with some corrections and additions. A green background indicates an asymptotically best bound in the table; L is the maximum length (or weight) among all edges, assuming integer edge weights. Finds a negative cycle or calculates distances to all vertices. Network flows are

960-421: The sum ∑ i = 1 n − 1 f ( e i , i + 1 ) . {\displaystyle \sum _{i=1}^{n-1}f(e_{i,i+1}).} When each edge in the graph has unit weight or f : E → { 1 } {\displaystyle f:E\rightarrow \{1\}} , this is equivalent to finding the path with fewest edges. The problem

992-500: The title SSSP . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=SSSP&oldid=1049875318 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Shortest path problem#Single-source shortest paths The problem of finding

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1024-442: The transportation research literature: the higher the variability, the lower the reliability of predictions. To account for variability, researchers have suggested two alternative definitions for an optimal path under uncertainty. The most reliable path is one that maximizes the probability of arriving on time given a travel time budget. An α-reliable path is one that minimizes the travel time budget required to arrive on time with

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