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Euler tour technique
About: Euler tour technique is a research topic. Over the lifetime, 93 publications have been published within this topic receiving 15132 citations.
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TL;DR: The algorithm can be used as a building block for solving other distributed graph problems, and can be slightly modified to run on a strongly-connected diagraph for generating the existent Euler trail or to report that no Euler trails exist.
13,828 citations
TL;DR: It is shown that the ascent procedure can be viewed as a generalization of both the Chu-Liu-Edmonds directed spanning tree algorithm and the Bilde-Krarup-Erlenkotter ascent algorithm for the plant location problem.
Abstract: The Steiner tree problem on a directed graph (STDG) is to find a directed subtree that connects a root node to every node in a designated node setV. We give a dual ascent procedure for obtaining lower bounds to the optimal solution value. The ascent information is also used in a heuristic procedure for obtaining feasible solutions to the STDG. Computational results indicate that the two procedures are very effective in solving a class of STDG's containing up to 60 nodes and 240 directed/120 undirected arcs. The directed spanning tree and uncapacitated plant location problems are special cases of the STDG. Using these relationships, we show that our ascent procedure can be viewed as a generalization ofboth the Chu-Liu-Edmonds directed spanning tree algorithm and the Bilde-Krarup-Erlenkotter ascent algorithm for the plant location problem. The former comparison yields a dual ascent interpretation of the steps of the directed spanning tree algorithm.
369 citations
TL;DR: In this paper, the authors present a simple implementation with O(logn) per tree operation, wheren is the number of tree vertices, where n is the size of the network.
Abstract: Thedynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges (linking) and splitting by deleting edges (cutting), while at the same time allowing reporting of certain combinations of vertex or edge values. For many applications of dynamic trees, values must be combined along paths. For other applications, values must be combined over entire trees. For the latter situation, an idea used originally in parallel graph algorithms, to represent trees by Euler tours, leads to a simple implementation with a time of O(logn) per tree operation, wheren is the number of tree vertices. We apply this representation to the implementation of two versions of the network simplex algorithm, resulting in a time of O(logn) per pivot, wheren is the number of vertices in the problem network.
112 citations
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27 Aug 2011TL;DR: O (log | V |) time algorithms are given using a linear number of processors on a concurrent-read concurrent-write parallel RAM to solve the problem of finding Euler tours in directed and undirected Euler graphs.
Abstract: The problem of finding Euler tours in directed and undirected Euler graphs is considered. O (log | V |) time algorithms are given using a linear number of processors on a concurrent-read concurrent-write parallel RAM.
74 citations
TL;DR: It is shown that encircling the Chinese postman tour in a random direction is an optimal search strategy if and only if the graph is weakly Eulerian (i.e it consists of several Eulerians connected in a tree-like structure).
Abstract: Consider a search game with an immobile hider in a graph. A Chinese postman tour is a closed trajectory which visits all the points of the graph and has minimal length. We show that encircling the Chinese postman tour in a random direction is an optimal search strategy if and only if the graph is weakly Eulerian (i.e it consists of several Eulerian curves connected in a tree-like structure).
63 citations