Dijkstra's Algorithm
Dijkstra's Algorithm is a method for finding the shortest path between nodes in a graph. It starts at a specified node and explores the neighboring nodes, updating the shortest path to each node as it progresses. The algorithm continues until it has visited all nodes and determined the shortest path to each.
3 Key excerpts on "Dijkstra's Algorithm"
eBook - ePub- Stephen Wise(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
...When route planning was something which was done in the offices of haulage companies when planning journeys for their fleet, the speed of the algorithm was not so crucial. However, route planning is now something which people routinely do during journeys using satellite navigation software in their cars or on their phones and so the algorithms need to be fast. FIGURE 11.21 Roads which would be considered by the Dijkstra algorithm in finding the shortest route between Sheffield and Manchester in the North of England. (Crown Copyright Ordnance Survey, all rights reserved.) One of the commonest algorithms is called A* (pronounced A-star). The A* algorithm differs from the Dijkstra algorithm in two ways: 1. At each step, instead of searching all the remaining nodes to see which should be used next, it only searches those with a direct connection to the current node. 2. In deciding which node to add to the current route, the algorithm takes into account how near each one is to the destination as well as how far it is back to the origin. The first change alone will make a large difference to the speed of the algorithm as most nodes in a transport network have direct links to only three or four other nodes. It is rare to find a road junction at which more than four roads meet. This means that at each step of the algorithm, instead of searching what could be hundreds of nodes, the algorithm only considers a few. To understand how the second condition is implemented, it is useful to introduce the notation which is usually used to describe the various route-finding algorithms. When choosing the next node, the Dijkstra algorithm calculates a function, g (n), for each candidate node which is made up of the distance to the current node along the route, and the distance from the current node to node n...
eBook - ePubLinear Optimization and Duality
A Modern Exposition
- Craig A. Tovey(Author)
- 2020(Publication Date)
- Chapman and Hall/CRC(Publisher)
...Find anO(|V||E|)algorithm to find the minimum over all cycles in the graphG=(V,E)of the mean cost. Hint: Letv∈V. Show that if the minimum mean cost equals 0, this value equalsminw∈Vmax0≤k<|V|πwn−πwkn−kwhereπjk≡the cost of a shortest path fromvtojconsisting of exactlykarcs.(*) The Bellman-FordAlgorithm 12.5.1is a dual algorithm from the perspective of the linear program (12.25) to (12.28), because it operates on the dual variable vectorπ. Dijkstra’s Algorithm12.6.1 is also a dual algorithm from this perspective. Find a primal algorithm that is competitive.I ProblemsThink of a real-world shortest path problem in which the paths may not repeat arcs but need not be simple. Think of a real-world shortest path problem in which the paths may contain repeated arcs.Show how any problem involving a finite sequence of binary decisions with deterministic outcomes can be modeled as a shortest path problem. Explain why, in general, such a model is useless.In a weightedmixedgraph, every edge has a weight. However, some edges are directed and the other edges are undirected. Prove that finding a minimum weight simple path in a mixed graph is NP-hard, as is the detection of negative weight cycles [8].The name “Edsger Dijkstra” is particularly appealing to formulators of linear and integer programs because it contains the oft-used string “ijk.” Find another name that is more appealing.The original Rubik’s Cube puzzle is a3×3×3cube with 91×1cubes visible on each facet. The goal of the puzzle is to rotate facets to make each facet monochromatic. Consider the problem of minimizing the number of rotations to reach the goal configuration. Explain why in principle this is a shortest path problem. Hypothesize why it would be computationally impractical. Prove your hypothesis....
eBook - ePubOperations Research Using Excel
A Case Study Approach
- Vikas Singla(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
...Thus: Minimize ∑ i ∑ j C ij X ij where c ij represents the distance between two nodes from i to j and x ij indicated whether a city has been visited or not taking a value of 1 if it is in the shortest path and 0 if not. Objective of minimization is subject to constraints of net flow of each city being visited. From the source there is only outward movement, so net flow would be positive 1; for destination node, there is only inward movement from one city, making a net flow of −1; for intermediary nodes, there would be incoming and outgoing making a net flow of zero. This is shown in the following equations for each of the node following a rule of (Flow out of node) – (Flow into a node) = Net supply at a node. Thus for: Southfield (S) : x sm + x sel + x sgin = 1 Marysville (M) : x mgnc + x mel – x ms = 0 East Liberty (EL) : x elgin + x elgnc + x elsnc – x elm – x els = 0 Greensburg (Gin) : x ginsnc + x gint – x gins – x ginel = 0 Greensboro (Gnc) : x gncl +. x gncsnc – x gncm – x gncel = 0 Swepsonville (Snc) : x sncl + x snct – x sncgnc – x sncel – x sncgin = 0 Timmonsville (T) : x tl – x gint – x snct = 0 Lincoln (L) : – x gncl – x sncl – x tl = –1 Dijkstra’s algorithm provides a quantitative and precise method of achieving the aim of finding the shortest path. The algorithm is important in providing bases for further advanced analysis. An example is illustrated to understand the application of Dijkstra’s algorithm. By using distance between various auto plants of Honda as given in Table 7.14, the following graph (Figure 7.7) shows connectivity between various stations and routes a vehicle can adopt by initiating its journey from Southfield (S), Michigan. FIGURE 7.7 Network diagram According to the algorithm, the node or station from which journey would begin becomes the active node. In this case, Southfield (S) is the first station. From S, the vehicle can move to M by traversing a distance of 36; to EL by travelling a distance of 53 and to Gin distance would be 156...
