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In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The problem of finding 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 of the 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 the definition of path requires that consecutive vertices be connected by an appropriate directed edge. [1]
Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices such that is adjacent to for . Such a path is called a path of length from to . (The are variables; their numbering here relates to their position in the sequence and needs not to relate to any canonical labeling of the vertices.)
Let where is the edge incident to both and . Given a real-valued weight function , and an undirected (simple) graph , the shortest path from to is the path (where and ) that over all possible minimizes the sum When each edge in the graph has unit weight or , this is equivalent to finding the path with fewest edges.
The problem 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).
Weights | Time complexity | Author |
---|---|---|
+ | O(V2) | Dijkstra 1959 |
+ | O((E + V) log V) | Johnson 1977 (binary heap) |
+ | O(E + V log V) | Fredman & Tarjan 1984 (Fibonacci heap) |
O(E) | Thorup 1999 (requires constant-time multiplication) |
Algorithm | Time complexity | Author |
---|---|---|
Breadth-first search | O(E + V) |
An algorithm using topological sorting can solve the single-source shortest path problem in time Θ(E + V) in arbitrarily-weighted DAGs. [2]
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.
Weights | Algorithm | Time complexity | Author |
---|---|---|---|
Ford 1956 | |||
Bellman–Ford algorithm | Shimbel 1955, Bellman 1958, Moore 1959 | ||
Dantzig 1960 | |||
Dijkstra's algorithm with list | Leyzorek et al. 1957, Dijkstra 1959, Minty (see Pollack & Wiebenson 1960), Whiting & Hillier 1960 | ||
Dijkstra's algorithm with binary heap | Johnson 1977 | ||
Dijkstra's algorithm with Fibonacci heap | Fredman & Tarjan 1984, Fredman & Tarjan 1987 | ||
Quantum Dijkstra algorithm with adjacency list | Dürr et al. 2006 [3] | ||
Dial's algorithm [4] (Dijkstra's algorithm using a bucket queue with L buckets) | Dial 1969 | ||
Johnson 1981, Karlsson & Poblete 1983 | |||
Gabow's algorithm | Gabow 1983, Gabow 1985 | ||
Ahuja et al. 1990 | |||
Thorup | Thorup 2004 |
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Weights | Algorithm | Time complexity | Author |
---|---|---|---|
Ford 1956 | |||
Bellman–Ford algorithm | Shimbel 1955, Bellman 1958, Moore 1959 | ||
Johnson-Dijkstra with binary heap | Johnson 1977 | ||
Johnson-Dijkstra with Fibonacci heap | Fredman & Tarjan 1984, Fredman & Tarjan 1987, adapted after Johnson 1977 | ||
Johnson's technique applied to Dial's algorithm [4] | Dial 1969, adapted after Johnson 1977 | ||
Interior-point method with Laplacian solver | Cohen et al. 2017 | ||
Interior-point method with flow solver | Axiotis, Mądry & Vladu 2020 | ||
Robust interior-point method with sketching | van den Brand et al. 2020 | ||
interior-point method with dynamic min-ratio cycle data structure | Chen et al. 2022 | ||
Based on low-diameter decomposition | Bernstein, Nanongkai & Wulff-Nilsen 2022 | ||
Hop-limited shortest paths | Fineman 2023 |
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Finds a negative cycle or calculates distances to all vertices.
Weights | Algorithm | Time complexity | Author |
---|---|---|---|
Andrew V. Goldberg |
Weights | Algorithm | Time complexity | Author |
---|---|---|---|
Henzinger et al. 1997 |
The all-pairs shortest path problem finds the shortest paths between every pair of vertices v, v' in 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(V4).
Weights | Time complexity | Algorithm |
---|---|---|
+ | O(V3) | Floyd–Warshall algorithm |
Seidel's algorithm (expected running time) | ||
Williams 2014 | ||
+ | O(EV log α(E,V)) | Pettie & Ramachandran 2002 |
O(EV) | Thorup 1999 applied to every vertex (requires constant-time multiplication). |
Weights | Time complexity | Algorithm |
---|---|---|
(no negative cycles) | Floyd–Warshall algorithm | |
Williams 2014 | ||
(no negative cycles) | Quantum search [5] [6] | |
(no negative cycles) | O(EV + V2 log V) | Johnson–Dijkstra |
(no negative cycles) | O(EV + V2 log log V) | Pettie 2004 |
O(EV + V2 log log V) | Hagerup 2000 |
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. [7]
If one represents 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 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 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. [8]
A road network can be considered as a graph with positive weights. The nodes represent road junctions and each edge of 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 the notion of highway dimension. [9] 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 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. [10] Other techniques that have been used are:
For shortest path problems in computational geometry, see Euclidean shortest path.
The shortest multiple disconnected path [11] 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 the desired solution path are called Constrained Shortest Path First, and are harder to solve. One example is the constrained shortest path problem, [12] 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 a specific set of vertices to be included in the path, [13] 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 the graph is not completely known to the mover, changes over time, or where actions (traversals) are probabilistic. [14] [15]
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 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, 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 a negative cycle. Some shortest-paths algorithms can be used for this purpose:
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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 is known as the algebraic path problem. [17] [18] [19]
Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures. [20]
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. [21]
In real-life situations, the transportation network is usually stochastic and time-dependent. In fact, a traveler traversing a link daily may experiences different travel times on that link due not only to the fluctuations in travel demand (origin-destination matrix) but also due to such incidents as work zones, bad weather conditions, accidents and vehicle breakdowns. As a result, a stochastic time-dependent (STD) network is a more realistic representation of an actual road network compared with the deterministic one. [22] [23]
Despite considerable progress during the course of the past decade, it remains a controversial question how an optimal path should be defined and identified in stochastic road networks. In other words, there is no unique definition of an optimal path under uncertainty. One possible and common answer to this question is to find a path with the minimum expected travel time. The main advantage of using this approach is that efficient shortest path algorithms introduced for the deterministic networks can be readily employed to identify the path with the minimum expected travel time in a stochastic network. However, the resulting optimal path identified by this approach may not be reliable, because this approach fails to address travel time variability. To tackle this issue some researchers use distribution of travel time instead of expected value of it so they find the probability distribution of total travelling time using different optimization methods such as dynamic programming and Dijkstra's algorithm . [24] These methods use stochastic optimization, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length. [25] The concept of travel time reliability is used interchangeably with travel time variability in the transportation research literature, so that, in general, one can say that the higher the variability in travel time, the lower the reliability would be, and vice versa.
In order to account for travel time reliability more accurately, two common alternative definitions for an optimal path under uncertainty have been suggested. Some have introduced the concept of the most reliable path, aiming to maximize the probability of arriving on time or earlier than a given travel time budget. Others, alternatively, have put forward the concept of an α-reliable path based on which they intended to minimize the travel time budget required to ensure a pre-specified on-time arrival probability.
The travelling salesman problem, also known as the travelling salesperson problem (TSP), asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research.
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components.
Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later.
In computer science, the Floyd–Warshall algorithm is an algorithm for finding shortest paths in a directed weighted graph with positive or negative edge weights. A single execution of the algorithm will find the lengths of shortest paths between all pairs of vertices. Although it does not return details of the paths themselves, it is possible to reconstruct the paths with simple modifications to the algorithm. Versions of the algorithm can also be used for finding the transitive closure of a relation , or widest paths between all pairs of vertices in a weighted graph.
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined objective function. One well-known variant, which is often used synonymously with the term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals and minimizes the total weight of its edges. Further well-known variants are the Euclidean Steiner tree problem and the rectilinear minimum Steiner tree problem.
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem.
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge (u,v) from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the graph may represent tasks to be performed, and the edges may represent constraints that one task must be performed before another; in this application, a topological ordering is just a valid sequence for the tasks. Precisely, a topological sort is a graph traversal in which each node v is visited only after all its dependencies are visited. A topological ordering is possible if and only if the graph has no directed cycles, that is, if it is a directed acyclic graph (DAG). Any DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time. Topological sorting has many applications, especially in ranking problems such as feedback arc set. Topological sorting is possible even when the DAG has disconnected components.
A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments. In it, any two points can reach each other along a path through the line segments. It can be found as the minimum spanning tree of a complete graph with the points as vertices and the Euclidean distances between points as edge weights.
Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist. It works by using the Bellman–Ford algorithm to compute a transformation of the input graph that removes all negative weights, allowing Dijkstra's algorithm to be used on the transformed graph. It is named after Donald B. Johnson, who first published the technique in 1977.
The Christofides algorithm or Christofides–Serdyukov algorithm is an algorithm for finding approximate solutions to the travelling salesman problem, on instances where the distances form a metric space . It is an approximation algorithm that guarantees that its solutions will be within a factor of 3/2 of the optimal solution length, and is named after Nicos Christofides and Anatoliy I. Serdyukov ; the latter discovered it independently in 1976.
In computer science, the Hopcroft–Karp algorithm is an algorithm that takes a bipartite graph as input and produces a maximum-cardinality matching as output — a set of as many edges as possible with the property that no two edges share an endpoint. It runs in time in the worst case, where is set of edges in the graph, is set of vertices of the graph, and it is assumed that . In the case of dense graphs the time bound becomes , and for sparse random graphs it runs in time with high probability.
Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible.
A geometric spanner or a t-spanner graph or a t-spanner was initially introduced as a weighted graph over a set of points as its vertices for which there is a t-path between any pair of vertices for a fixed parameter t. A t-path is defined as a path through the graph with weight at most t times the spatial distance between its endpoints. The parameter t is called the stretch factor or dilation factor of the spanner.
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems.
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of vertices from an n-vertex graph can partition the graph into disjoint subgraphs each of which has at most vertices.
In computer science, the method of contraction hierarchies is a speed-up technique for finding the shortest-path in a graph. The most intuitive applications are car-navigation systems: a user wants to drive from to using the quickest possible route. The metric optimized here is the travel time. Intersections are represented by vertices, the road sections connecting them by edges. The edge weights represent the time it takes to drive along this segment of the road. A path from to is a sequence of edges ; the shortest path is the one with the minimal sum of edge weights among all possible paths. The shortest path in a graph can be computed using Dijkstra's algorithm but, given that road networks consist of tens of millions of vertices, this is impractical. Contraction hierarchies is a speed-up method optimized to exploit properties of graphs representing road networks. The speed-up is achieved by creating shortcuts in a preprocessing phase which are then used during a shortest-path query to skip over "unimportant" vertices. This is based on the observation that road networks are highly hierarchical. Some intersections, for example highway junctions, are "more important" and higher up in the hierarchy than for example a junction leading into a dead end. Shortcuts can be used to save the precomputed distance between two important junctions such that the algorithm doesn't have to consider the full path between these junctions at query time. Contraction hierarchies do not know about which roads humans consider "important", but they are provided with the graph as input and are able to assign importance to vertices using heuristics.
In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The widest path problem is also known as the maximum capacity path problem. It is possible to adapt most shortest path algorithms to compute widest paths, by modifying them to use the bottleneck distance instead of path length. However, in many cases even faster algorithms are possible.
In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices such that either the number of edges that the path passes through or the sum of the weights of the edges is minimized. The betweenness centrality for each vertex is the number of these shortest paths that pass through the vertex.
The k shortest path routing problem is a generalization of the shortest path routing problem in a given network. It asks not only about a shortest path but also about next k−1 shortest paths. A variation of the problem is the loopless k shortest paths.