On finding optimal and near-optimal lineal spanning trees

The search for good lineal, or depth-first, spanning trees is an important aspect in the implementation of a wide assortment of graph algorithms. We consider the complexity of findingoptimal lineal spanning trees under various notions of optimality. In particular, we show that several natural problems, such as constructing a shortest or a tallest lineal tree, are NP-hard. We also address the issue of polynomial-time, near-optimization strategies for these difficult problems, showing that efficient absolute approximation algorithms cannot exist unlessP = NP.     

On finding spanning trees with few leaves

The problem of finding a spanning tree with few leaves is motivated by the design of communication networks, where the cost of the devices depends on their routing functionality (ending, forwarding, or routing a connection). Besides this application, the problem has its own theoretical importance as a generalization of the Hamiltonian path problem. Lu and Ravi showed that there is no constant factor approximation for minimizing the number of leaves of a spanning tree, unless P=NP. Thus instead of minimizing the number of leaves, we are going to deal with maximizing the number of non-leaves: we give a linear-time 2-approximation for arbitrary graphs, a -approximation for claw-free graphs, and a -approximation for cubic graphs.     

Balancing minimum spanning trees and shortest-path trees

We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and a>0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+2 times the shortest-path distance, and yet the total weight of the tree is at most 1+2/ times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex.Current research supported by NSF Research Initiation Award CCR-9307462. This work was done while this author was supported by NSF Grants CCR-8906949, CCR-9103135, and CCR-9111348.Part of this work was done while this, author was at the University of Maryland Institute for Advanced Computer Studies (UMIACS) and supported by NSF Grants CCR-8906949 and CCR-9111348.     

A robust model for finding optimal evolutionary trees

Constructing evolutionary trees for species sets is a fundamental problem in computational biology. One of the standard models assumes the ability to compute distances between every pair of species, and seeks to find an edge-weighted treeT in which the distanced _ij ^T in the tree between the leaves ofT corresponding to the speciesi andj exactly equals the observed distance,d _ij . When such a tree exists, this is expressed in the biological literature by saying that the distance function or matrix isadditive, and trees can be constructed from additive distance matrices in0(n ²) time. Real distance data is hardly ever additive, and we therefore need ways of modeling the problem of finding the best-fit tree as an optimization problem.In this paper we present several natural and realistic ways of modeling the inaccuracies in the distance data. In one model we assume that we have upper and lower bounds for the distances between pairs of species and try to find an additive distance matrix between these bounds. In a second model we are given a partial matrix and asked to find if we can fill in the unspecified entries in order to make the entire matrix additive. For both of these models we also consider a more restrictive problem of finding a matrix that fits a tree which is not only additive but alsoultrametric. Ultrametric matrices correspond to trees which can be rooted so that the distance from the root to any leaf is the same. Ultrametric matrices are desirable in biology since the edge weights then indicate evolutionary time. We give polynomial-time algorithms for some of the problems while showing others to be NP-complete. We also consider various ways of fitting a given distance matrix (or a pair of upper- and lower-bound matrices) to a tree in order to minimize various criteria of error in the fit. For most criteria this optimization problem turns out to be NP-hard, while we do get polynomial-time algorithms for some.Supported by DIMACS under NSF Contract STC-88-09648.Supported by NSF Grant CCR-9108969.This work was begun while this author was visiting DIMACS in July and August 1992, and was supported in part by the U.S. Department of Energy under Contract DE-AC04-76DP00789.     

Complexities of some interesting problems on spanning trees

This paper deals with the complexity issues of some new interesting spanning tree problems. Here we define some new spanning tree problems by imposing various constraints and restrictions on graph parameters and present relevant results. Also we introduce a new notion of “set version” of some decision problems having integer K<|V| as a parameter in the input instance, where we replace K by a set X⊆|V|. For example, the set version of Maximum Leaf Spanning Tree problem asks whether there exists a spanning tree in G that contains X as a subset of the leaf set. We raise the issue of whether the set versions of NP-complete problems are as hard as the original problems and prove that although in some cases the set versions are easier to solve, this is not necessarily true in general.     

Hierarchical spanning trees and distributing on incomplete hypercubes

Incomplete hypercubes are gaining increasing attention as one of the possible solutions for the limitation on the number of nodes in the hypercubes. Distributing, in which one node sends distinct messages to distinct nodes, and its reverse operation are frequently used operations on data parallel computation. In this paper, we propose two types of spanning trees in incomplete hypercubes (composed of an n-cube and a k-cube): the binomial hierarchical spanning tree-binomial HST, and the balanced hierarchical spanning tree-balanced HST, for distributing (and its reverse operation). Distributing algorithms based on a balanced HST take better advantage of overlap between communication ports, or have speedup up to k/2 or n/2 over that based on the binomial HST when concurrently communication on all ports are possible, from the view point of communication complexity. An algorithm to construct hierarchical spanning trees in general incomplete hypercubes, which consist of an arbitrary number of nodes, is also devised.     

On the number of spanning trees of circulant graphs

In this paper, we derive a simple formula for the number of spanning trees of the circulant graphs. Some special cases of the circulant graphs are also taken into account.     

A parallel algorithm for the enumeration of the spanning trees of a graph

As is well known, the strategy of divide-and-conquer is widely used in problem solving. The method of partitioning is also a fundamental strategy for the design of a parallel algorithm. The problem of enumerating the spanning trees of a graph arises in several contexts such as computer-aided design and computer networks. A parallel algorithm for solving the problem is presented in this paper. It is based on the principle of the inclusion and exclusion of sets, and not directly based on the partitioning of the graph itself. The results of the preliminary experiments on a MIMD system appear promising.     

Parallel construction of optimal independent spanning trees on Cartesian product of complete graphs

A set of k spanning trees rooted at the same vertex r in a graph G is said to be independent if for each vertex x other than r, the k paths from r to x, one path in each spanning tree, are internally disjoint. Using independent spanning trees (ISTs) one can design fault-tolerant broadcasting schemes and increase message security in a network. Thus, the problem of ISTs on graphs has been received much attention. Recently, Yang et al. proposed a parallel algorithm for generating optimal ISTs on the hypercube. In this paper, we propose a similar algorithm for generating optimal ISTs on Cartesian product of complete graphs. The algorithm can be easily implemented in parallel or distributed systems. Moreover, the proof of its correctness is simpler than that of Yang et al.     

Hardness and approximation results for packing steiner trees

We study approximation algorithms and hardness of approximation for several versions of the problem of packing Steiner trees. For packing edge-disjoint Steiner trees of undirected graphs, we show APX-hardness for four terminals. For packing Steiner-node-disjoint Steiner trees of undirected graphs, we show a logarithmic hardness result, and give an approximation guarantee ofO (√n logn), wheren denotes the number of nodes. For the directed setting (packing edge-disjoint Steiner trees of directed graphs), we show a hardness result of Θ(m ^1/3/−ɛ) and give an approximation guarantee ofO(m ^1/2/+ɛ), wherem denotes the number of edges. We have similar results for packing Steiner-node-disjoint priority Steiner trees of undirected graphs. Supported by NSERC Grant No. OGP0138432. Supported by an NSERC postdoctoral fellowship, Department of Combinatorics and Optimization at University of Waterloo, and a University start-up fund at University of Alberta.     

Approximately optimal trees for group key management with batch updates

We investigate the group key management problem for broadcasting applications. Previous work showed that, in handling key updates, batch rekeying can be more cost effective than individual rekeying. One model for batch rekeying is to assume that every user has probability p$p$ of being replaced by a new user during a batch period with the total number of users unchanged. Under this model, it was recently shown that an optimal key tree can be constructed in linear time when p$p$ is a constant and in O(n⁴)$O\left(n^4\right)$ time when p→0$p\to 0$. In this paper, we investigate more efficient algorithms for the case p→0$p\to 0$, i.e., when membership changes are sparse. We design an O(n)$O\left(n\right)$ heuristic algorithm for the sparse case and show that it produces a nearly 2-approximation to the optimal key tree. Simulation results show that its performance is even better in practice. We also design a refined heuristic algorithm and show that it achieves an approximation ratio of 1+?$1+?$ for any fixed ?>0$?>0$ and n$n$, as p→0$p\to 0$. Finally, we give another approximation algorithm for any p∈(0,0.693)$p\in (0,0.693)$ which is shown to be quite good by our simulations.     

字典乘积网络的支撑树计数

支撑树个数是边失效下网络可靠性分析与设计的一个重要性能参考指标,本文利用字典乘积的方法来构建网络,通过这种方法我们很容易由若干特定规模较小网络来构建规模较大的网络,并得到它的一个紧的支撑树计数解析公式,这样的计数公式仅仅依赖于小网络的性能参数,如:结点的度数、小网络的阶数、小网络的支撑树数目.     

Constructing independent spanning trees for locally twisted cubes

The independent spanning trees (ISTs) problem attempts to construct a set of pairwise independent spanning trees and it has numerous applications in networks such as data broadcasting, scattering and reliable communication protocols. The well-known ISTs conjecture, Vertex/Edge Conjecture, states that any n-connected/n-edge-connected graph has n vertex-ISTs/edge-ISTs rooted at an arbitrary vertex r. It has been shown that the Vertex Conjecture implies the Edge Conjecture. In this paper, we consider the independent spanning trees problem on the n-dimensional locally twisted cube LTQ_n. The very recent algorithm proposed by Hsieh and Tu (2009) [12] is designed to construct n edge-ISTs rooted at vertex 0 for LTQ_n. However, we find out that LTQ_n is not vertex-transitive when n≥4; therefore Hsieh and Tu’s result does not solve the Edge Conjecture for LTQ_n. In this paper, we propose an algorithm for constructing n vertex-ISTs for LTQ_n; consequently, we confirm the Vertex Conjecture (and hence also the Edge Conjecture) for LTQ_n.     

Assembling approximately optimal binary search trees efficiently using arithmetics

We introduce a new algorithm for computing an approximately optimal binary search tree with known access probabilities or weights on items. The algorithm is simple to implement and it has two contributions. First, a randomized variant of the algorithm produces a binary search tree with expected performance that improves the previous theoretical guarantees (the performance is dependent on the value of the input random variable). More precisely, if p is the probability of accessing an item, then under expectation the item is found after searching lg1/p+0.087+lg₂(1+pmax) nodes, where pmax is the maximal probability among items. The previous best bound was lg1/p+1, albeit deterministic. For the optimal tree our upper bound implies a non-constructive performance bound of H+0.087+lg₂(1+pmax), where H is the entropy on the item distribution and the previous bound was H+1. The second contribution of the algorithm is a low cost in i/o models of cost such as the cache-oblivious model, while attaining simultaneously the above bound for the produced tree.     

On finding cycle bases and fundamental cycle bases with a shortest maximal cycle

An undirected biconnected graph G with nonnegative integer lengths on the edges is given. The problem we consider is that of finding a cycle basis B of G such that the length of the longest cycle included in B is the smallest among all cycle bases of G. We first observe that Horton's algorithm [SIAM J. Comput. 16 (2) (1987) 358-366] provides a fast solution of the problem that extends the one given by Chickering et al. [Inform. Process. Lett. 54 (1995) 55-58] for uniform graphs. On the other hand we show that, if the basis is required to be fundamental, then the problem is NP-hard and cannot be approximated within 2−?, ∀?>0, even with uniform lengths, unless P=NP. This problem remains NP-hard even restricted to the class of complete graphs; in this case it cannot be approximated within 13/11−?, ∀?>0, unless P=NP; it is instead approximable within 2 in general, and within 3/2 if the triangle inequality holds.