Generation of colour-constrained spanning trees with application in symbolic circuit analysis

Yu, Qicheng; Sechen, Carl

doi:10.1002/(sici)1097-007x(199609/10)24:5<597::aid-cta938>3.0.co;2-q

Cited by 2 publications

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“…The problem of finding the k best spanning trees has been applied to NP-hard multicriterion optimization problems for spanning trees [50,96,176], to point process intensity estimation [97], to the analysis of metabolic pathways [7], to image segmentation [177] and classification [63], to the reconstruction of pedigrees from genetic data [52], to the parsing of natural-language text [1], and to the analysis of electronic circuits [189]. This problem is a special case of finding the k best bases of a matroid, which has also been studied [25,34,95,126].…”

Section: Spanning Treesmentioning

confidence: 99%

k-Best Enumeration

Eppstein¹

2016

Encyclopedia of Algorithms

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We survey k-best enumeration problems and the algorithms for solving them, including in particular the problems of finding the k shortest paths, k smallest spanning trees, and k best matchings in weighted graphs.in both the input size and k. There are two general techniques that can be used to achieve this, based on optimal substructures and solution space partitioning. In turn, these methods rely on fast algorithms for the selection problem in certain sets of structured values. SelectionUnderlying many k-best algorithms is the problem of selection, finding the k smallest values from a larger set of values [21]. If there are n values in the set, then selection may be solved in time O(n + k), for instance by quickselect, an algorithm that chooses a random pivot value from the set, partitions the rest of the set into subsets that are less than, equal to, or greater than the pivot, and then recurses into one of these subsets [131].However, in k-best problems, we do not wish to generate all solutions before finding the best of them: we wish to avoid the O(n) part of this O(n + k) time bound. Often, the solution values among which we are selecting the k best have some additional structure that allows finding the k best without examining all solutions. For instance, suppose that the space of solutions can be represented as the vertex set of an (implicitly defined) edge-weighted graph, that the degrees of the vertices in this graph are bounded, and that the quality of any particular solution equals the length of the shortest path in this graph from some designated starting vertex to the solution vertex. In such a case, Dijkstra's algorithm can be used to recover the k smallest of these values in time O(k log k), independent of the size of the graph, without requiring that the whole graph be explored. Other structures that allow the k best solutions to be found more efficiently than enumeration of the whole solution set include representations of the solution space as the set of sums of pairs of values drawn from two sorted sets [76,111], or as the elements of an (implicitly defined) matrix whose rows and columns are sorted [77]. Again, for such structures, the k smallest values can be found efficiently without examining the whole matrix or the whole set of pair sums.A general framework that can be used to describe many of these structured selection problems is the problem of selection in a d-ary heap. A min-heap is a rooted tree, with values associated with its nodes, such that each non-root node has a value that is at least as large as its parent. In order to avoid having to construct all solutions before performing a selection algorithm, and in order to handle situations in which this tree has infinitely many nodes, we will assume that our selection algorithm is given as input an implicit representation of such a heap. That is, its input consists of a binary representation of the root node of the heap together with pointers to two subroutines that take a single node as input: one subroutine to list the children of any...

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Section: Spanning Treesmentioning

confidence: 99%

k-Best Enumeration

Eppstein¹

2016

Encyclopedia of Algorithms

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Efficient approximation of symbolic network functions using matroid intersection algorithms

Sechen

Proceedings of ISCAS'95 - International Symposium on Circuits and Systems

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where and D, are sums of product terms and each product term is the product of circuit element parameters [h]. Due to the rapid growth of the number of product terms in the symbolic network function when the circuit size grows. symbolic network functions have to be approximated during or after the computation. Much attention has been paid to finding an efficient and effective approximation strategy [4]. Amon? the existinp strategies, approximation-during-computation (ADC) strategies art superior because they avoid the burden of generating a huye :[mount of unnecessary data only to be discarded in the aubsequenr approximation stages. 11; this paper we propose a ne% approximation straiep? that directly generales common spanning trees of a tuo-graph i n decreasing order of trce admittance product using inatroid intersection algorithms. I t reduces the total time for computing an approximate symbolic expression i n expanded I'ormat to polynomial with respect to the circuit s i x and the number of sdmple frequenciea i n [he ranye (IC Inter-est Comp'ired to the successful stvaregy of generating %ul!age giaph spanning rrees i n decreasing[tic ,ivcragc time Dei-product i c m grobv.; slower with the size of the two-graph for our new siraieg).b.xpcrimcntal rcsiilts shows clearly betrei-pci-foi-maiicc 0-7803-2570-2195 $4.00 01995 IEEE Problem ForniulationA unified approach of approximate symbolic analysis has bsen recently proposed and implemented [ 111 which enables the dnalysis of much larger analog integrated circuits than before.The mproach is based on the classical two-graph tree enumera-;ion meihod and is outlined below: First, a two-graph consisting of a voltage graph and a currenr graph is constructed for both the numerator and the denominator of the network function such that a c h product term in the cancellation-free symbolic expression corresponds to a common spanning tree of the two-graph and is called a tree admittance product. Second, a sensitivity-based simplification is carried out in which the two-graphs are simplified by edge contraction and deletion based on the numerical contribution of circuit elements for these edges to the network function. Finally. only the product terms with the largest magnitude in the expression. called significant product terms, are computed by generating the common spanning trces of :he simplified two-_graphs i n decreasing order of their tree admittance product unti! certain error control criteria are satisfied. FOP the symbolic cupression to be vaiid over the whole frequency range of interest, ,panning trees containing different numbers of capacitor edges are generated separately. The problem involved in tRe last step abo\e can be abstracted as the following where greater tree admittance product translates to lower weight: Problem 1. Two undirected graphs G , and G, with n nodes each are given such that: I j Both graphs consist of the same set E of m weighted e d g e s . Each edge in E has one "copy" in G , and another "CO/,>" i l l G , 2 ) The topologies of G , and G, are ...

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Generation of colour-constrained spanning trees with application in symbolic circuit analysis

Cited by 2 publications

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k-Best Enumeration

k-Best Enumeration

Efficient approximation of symbolic network functions using matroid intersection algorithms

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