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Dynamic Trees as Search Trees via Euler Tours, Applied to the Network Simplex Algorithm
 Mathematical Programming
, 1997
"... The dynamic 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 ..."
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The dynamic 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, we show that an idea used originally in parallel graph algorithms, to represent trees by Euler tours, leads to a simple implementation with a time of O(log n) per tree operation, where n 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(log n) per pivot, where n is the number of vertices in the problem network.
Modifying Networks to Obtain Low Cost Trees
 In WG: GraphTheoretic Concepts in Computer Science, International Workshop WG
, 1996
"... We consider the problem of reducing the edge lengths of a given network so that the modified network has a spanning tree of small total length. It is assumed that each edge e of the given network has an associated function Ce that specifies the cost of shortening the edge by a given amount and that ..."
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Cited by 5 (4 self)
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We consider the problem of reducing the edge lengths of a given network so that the modified network has a spanning tree of small total length. It is assumed that each edge e of the given network has an associated function Ce that specifies the cost of shortening the edge by a given amount and that there is a budget B on the total reduction cost. The goal is to develop a reduction strategy satisfying the budget constraint so that the total length of a minimum spanning tree in the modified network is the smallest possible over all reduction strategies that obey the budget constraint. We show that in general the problem of computing optimal reduction strategy for modifying the network as above is NPhard and present the first polynomial time approximation algorithms for the problem, where the cost functions Ce are allowed to be taken from a broad class of functions. We also present improved approximation algorithms for the class of treewidthbounded graphs when the cost functions are li...
Improving Steiner Trees of a Network Under Multiple Constraints (Extended Abstract)
, 1997
"... We consider the problem of decreasing the edge weights of a given network so that the modified network has a Steiner tree in which two performance measures are simultaneously optimized. We formulate these problems, referred to as bicriteria network improvement problems, by specifying a budget on t ..."
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Cited by 2 (1 self)
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We consider the problem of decreasing the edge weights of a given network so that the modified network has a Steiner tree in which two performance measures are simultaneously optimized. We formulate these problems, referred to as bicriteria network improvement problems, by specifying a budget on the total modification cost, a constraint on one of the performance measures and using the other performance measure as a minimization objective. Network improvement problems are known to be NPhard even when only one performance measure is considered. We present the first polynomial time approximation algorithms for bicriteria network improvement problems. Our approximation algorithms are for two pairs of performance measures, namely (diameter, total cost) and (degree, total cost). These algorithms produce solutions which are within a logarithmic factor of the optimum value of the minimizat...