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Yen's algorithm computes single-source K-shortest loopless paths for a graph with non-negative edge cost.[1] The algorithm was published by Jin Y. Yen in 1971 and employs any shortest path algorithm to find the best path, then proceeds to find K − 1 deviations of the best path.[2] Template:Tree search algorithm

Algorithm

Terminology and notation

Notation Description
The size of the graph, i.e., the amount of nodes in the network.
The node of the graph, where ranges from to . This means that is the source node of the graph and is the sink node of the graph.
The cost of the edge between and , assuming that and .
The shortest path from to , where ranges from to . Then , where is the 2nd node of the shortest path and is the 3rd node of the shortest path, and so on.
A deviation path from at node , where ranges from to . Note that the maximum value of is , which is the node just before the sink in the shortest path. This means that the deviation path cannot deviate from the shortest path at the sink. The paths and follow the same path until the node, then edge is different from any path in , where ranges from to .
The root path of that follows that until the node of .
The spur path of that starts at the node of and ends at the sink.

Description

The algorithm can be broken down into two parts, determining the first k-shortest path, , and then determining all other k-shortest paths. It is assumed that the container will hold the k-shortest path, whereas the container , will hold the potential k-shortest paths. To determine , the shortest path from the source to the sink, any efficient shortest path algorithm can be used.

To find the , where ranges from to , the algorithm assumes that all paths from to have previously been found. The iteration can be divided into two processes, finding all the deviations and choosing a minimum length path to become . Note that in this iteration, ranges from to .

The first process can be further subdivided into three operations, choosing the , finding , and then adding to the container . The root path, , is chosen by finding the subpath in that follows the first i nodes of , where ranges from to . Then, if a path is found, the cost of edge of is set to infinity. Next, the spur path, , is found by computing the shortest path from the spur node, node , to the sink. The removal of previous used edges from to ensures that the spur path is different. , the addition of the root path and the spur path, is added to . Next, the edges that were removed, i.e. had their cost set to infinity, are restored to their initial values.

The second process determines a suitable path for by finding the path in container with the lowest cost. This path is removed from container and inserted into container and the algorithm continues to the next iteration. Note that if the amount of paths in container equal or exceed the amount of k-shortest paths that still need to be found, then the necessary paths of container are added to container and the algorithm is finished.

Pseudocode

The algorithm assumes that the Dijkstra algorithm is used to find the shortest path between two nodes, but any shortest path algorithm can be used in its place.

function YenKSP(Graph, source, sink, K):
   // Determine the shortest path from the source to the sink.
   A[0] = Dijkstra(Graph, source, sink);
   // Initialize the heap to store the potential kth shortest path.
   B = [];
   
   for k from 1 to K:
       // The spur node ranges from the first node to the next to last node in the shortest path.
       for i from 0 to size(A[k − 1]) − 1:
           
           // Spur node is retrieved from the previous k-shortest path, k − 1.
           spurNode = A[k-1].node(i);
           // The sequence of nodes from the source to the spur node of the previous k-shortest path.
           rootPath = A[k-1].nodes(0, i);
           
           for each path p in A:
               if rootPath == p.nodes(0, i):
                   // Remove the links that are part of the previous shortest paths which share the same root path.
                   remove p.edge(i, i + 1) from Graph;
           
           // Calculate the spur path from the spur node to the sink.
           spurPath = Dijkstra(Graph, spurNode, sink);
           
           // Entire path is made up of the root path and spur path.
           totalPath = rootPath + spurPath;
           // Add the potential k-shortest path to the heap.
           B.append(totalPath);
           
           // Add back the edges that were removed from the graph.
           restore edges to Graph;
                   
       // Sort the potential k-shortest paths by cost.
       B.sort();
       // Add the lowest cost path becomes the k-shortest path.
       A[k] = B[0];
       B.pop();
   
   return A;

Example

Yen's k-shortest path algorithm, K = 3, A to F
Yen's k-shortest path algorithm, K = 3, A to F

The example uses Yen's K-Shortest Path Algorithm to compute three paths from to . Dijkstra's algorithm is used to calculate the best path from to , which is with cost 5. This path is appended to container and becomes the first k-shortest path, .

Node of becomes the spur node with a root path of itself, . The edge, , is removed because it coincides with the root path and a path in container . Dijkstra's algorithm is used to compute the spur path , which is , with a cost of 8. is added to container as a potential k-shortest path.

Node of becomes the spur node with . The edge, , is removed because it coincides with the root path and a path in container . Dijkstra's algorithm is used to compute the spur path , which is , with a cost of 7. is added to container as a potential k-shortest path.

Node of becomes the spur node with a root path, . The edge, , is removed because it coincides with the root path and a path in container . Dijkstra's algorithm is used to compute the spur path , which is , with a cost of 8. is added to container as a potential k-shortest path.

Of the three paths in container B, is chosen to become because it has the lowest cost of 7. This process is continued to the 3rd k-shortest path. However, within this 3rd iteration, note that some spur paths do not exist.And the path that is chosen to become is .

Features

Space complexity

To store the edges of the graph, the shortest path list , and the potential shortest path list , memory addresses are required.[2] At worse case, the every node in the graph has an edge to every other node in the graph, thus addresses are needed. Only addresses are need for both list and because at most only paths will be stored,[2] where it is possible for each path to have nodes.

Time complexity

The time complexity of Yen's algorithm is dependent on the shortest path algorithm used in the computation of the spur paths, so the Dijkstra algorithm is assumed. Dijkstra's algorithm has a worse case time complexity of , but using a Fibonacci heap it becomes ,[3] where is the amount of edges in the graph. Since Yen's algorithm makes calls to the Dijkstra in computing the spur paths, the time complexity becomes .[4]

Improvements

Yen's algorithm can be improved by using a heap to store the potential k-shortest paths of list . Using a heap as apposed to a list will improve the performance of the algorithm, but not the complexity.[5] One method to slightly decrease complexity is to skip the nodes where there are non-existent spur paths. This case is produced when all the spur paths from a spur node have been used in the previous . Also, if container has paths of minimum length, in reference to those in container , then they can be extract and inserted into container since no shorter paths will be found.

Lawler's modification

Eugene Lawler proposed a modification to Yen's algorithm in which duplicates path are not calculated as opposed to the original algorithm where they are calculated and then discarded when they are found to be duplicates.[6] These duplicates paths result from calculating spur paths of nodes in the root of . For instance, deviates from at some node . Any spur path, where , that is calculated will be a duplicate because they have already been calculated during the iteration. Therefore, only spur paths for nodes that were on the spur path of must be calculated, i.e. only where ranges from to . To perform this operation for , a record is needed to identify the node where branched from .

See also

References

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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang