Parallel construction of multidimensional binary search trees

Multidimensional binary search tree (abbreviated k-d tree) is a popular data structure for the organization and manipulation of spatial data. The data structure is useful in several applications including graph partitioning, hierarchical applications such as molecular dynamics and n-body simulations, and databases. In this paper, we study efficient parallel construction of k-d trees on coarse-grained distributed memory parallel computers. We consider several algorithms for parallel k-d tree construction and analyze them theoretically and experimentally, with a view towards identifying the algorithms that are practically efficient. We have carried out detailed implementations of all the algorithms discussed on the CM-5 and report on experimental results     

Smoothed analysis of binary search trees

Binary search trees are one of the most fundamental data structures. While the height of such a tree may be linear in the worst case, the average height with respect to the uniform distribution is only logarithmic. The exact value is one of the best studied problems in average-case complexity.     

Efficient algorithms to create and maintain balanced and threaded binary search trees

The algorithm proposed by Chang and lyengar to perfectly balance binary search trees has been modified to not only balance but also thread binary search trees. Threads are constructed in the same sequence as normal pointers during the balancing process. No extra workspace is necessary, and the running time is also linear for the modified algorithm. Such produced tree structure has minimal average path length for fast information retrieval, and threads to facilitate more flexible and efficient traversing schemes. Maintenance and manipulation of the data structure are discussed and relevant algorithms given.     

A function for generating nearly balanced binary search trees from sets of non-random keys

A function for generating nearly balanced binary search trees from sets of non-random keys is described. This function can be used in hashing organization where collisions are resolved by construction of binary trees. In particular it can be used as the secondary function in the relatively new technique of dynamic hashing.     

Reductions in binary search trees

We analyze two bottom-up reduction algorithms over binary trees that represent replaceable data within a certain system, assuming the binary search tree (BST) probabilistic model. These reductions are based on idempotent and nilpotent operators, respectively. In both cases, the average size of the reduced tree, as well as the cost to obtain it, is asymptotically linear with respect to the size of the original tree. Additionally, the limiting distributions of the size of the trees obtained by means of these reductions satisfy a central limit law of Gaussian type.     

Partial match queries in relaxed multidimensional search trees

Partial match queries arise frequently in the context of large databases, where each record contains a distinct multidimensional key, that is, the key of each record is aK-tuple of values. The components of a key are called thecoordinates orattributes of the key. In a partial match query we specify the value ofs attributes, 0<s<K, and leave the remainingK —s attributes unspecified. The goal is to retrieve all the records in the database that match the specified attributes. In this paper we present several results about the average performance and variance of partial matches in relaxedK-dimensional trees (search trees and digital tries). These data structures are variants of the well knownK d-trees andK d-tries. In relaxed trees the sequence of attributes used to guide a query is explicitly stored at the nodes of the tree and randomly generated and, in general, will be different for different search paths. In the standard variants, the sequence of attributes that guides a query examines the attributes in a cyclic fashion, fixed and identical for all search paths. We show that the probabilistic analysis of the relaxed multidimensional trees is very similar to that of standardK d-trees andK d-tries, and also to the analysis of quadtrees. In fact, besides the average cost and variance of partial match in relaxedK d-trees andK d-tries, we also obtain the variance of partial matches in two-dimensional quadtrees. We also compute the average cost of partial matches in other relaxed multidimensional digital tries, namely, relaxedK d-Patricia and relaxedK d-digital search trees. This research was supported by Acciones Integradas Hispano-Austríacas HU1997-0016 (Austrian-Spanish Scientific Exchange Program). The first author was also supported by ESPRIT LTR 20244 (ALCOM IT), CICYT TIC97-1475-CE, DGES PB95-0787 (KOALA), and CIRIT 1997SGR-00366 (SGR). The second author was also supported by the Austrian Research Society (FWF) under Project P12599-MAT. Online publication October 13, 2000.     

改进二进制布谷鸟搜索算法求解多维背包问题

针对多约束组合优化问题--多维背包问题(MKP),提出了一种改进二进制布谷鸟搜索(MBCS)算法.首先,采用经典的二进制代码变换公式构建了二进制布谷鸟搜索(BCS)算法.其次,引入病毒生物进化机制和病毒感染操作,一方面赋予布谷鸟鸟巢位置自变异机制增加种群多样性;一方面将布谷鸟鸟巢位置所组成的主群体的纵向全局搜索和病毒群体的横向局部搜索进行动态结合,进一步提高了算法的收敛速度,降低了陷入局部极值的概率.再次,针对MKP特点设计了不可行解的混合修复策略.最后将MBCS算法同量子遗传算法(QGA)、二进制粒子群优化(BPSO)算法、BCS算法就来源于ELIB数据库和OR_LIB数据库的15个算例进行了仿真对比.实验结果表明,所提算法计算误差均小于1%,标准差小于170,相比这3种算法具有相对更好的寻优精度和求解稳定性,是一种求解多维背包等NP难问题有效的算法.     

Improving time and space efficiency in generalized binary search trees

Summary This paper deals with main memory data structures for which time and space performance are simultaneously considered. The main contribution is a new data structure called Generalised Binary Search Tree (GBS-tree) together with searching and updating algorithms on this structure. GBS-trees generalise different data structures based on binary trees that have appeared in the literature. A k-t GBS-tree allows up to t keys per node and subtrees in the tree's fringe of exactly 2k-1 full nodes are kept balanced. Their time and space performances are analysed in depth. The time performance is expressed in terms of the average and the variance of the number of binary comparisons between a given key and keys already in the structure. The space performance measures both the space used to space generated ratio (space utilization factor) and the pointers to keys ratio of these trees. The analysis shows that the time performance always improves when GBS-trees of higher order are considered. In the absence of balancing techniques, larger values of t, which produces smaller pointers to key ratios, induce unacceptably poor space utilizations factors. We show that both pointers to keys ratio and space utilization factor improve when larger values of k are used. Thus, local balancing techniques are adequate, not only for time performance improvement, but also, for space performance improvement.     

Load balanced and optimal disk allocation strategy for partial match queries on multidimensional files

A multidimensional file is one whose data are characterized by several attributes, each specified in a given domain. A partial match query on a multidimensional file extracts all data whose attributes match the values of one or more attributes specified in the query. The disk allocation problem of a multidimensional file F on a database system with multiple disks accessible in parallel is the problem of distributing F among the disks such that the data qualifying for each partial match query are distributed as evenly as possible among the disks of the system. We propose an optimal solution to this problem for multidimensional files with pairwise prime domains based on a large and flexible class of maximum distance separable codes, namely, the redundant residue codes. We also introduce a new family of residue codes, called the redundant nonpairwise prime residue codes, to deal with files whose attribute domains are nonpairwise prime.     

Distributed hierarchical search for balanced energy consumption routing spanning trees in wireless sensor networks

We provide a new heuristic method approach to search for degree-balanced and small weight routing spanning trees in a network. The method is a modification of Kruskal’s minimum spanning tree search algorithm and is based on a distributed search by hierarchical clusters. It provides spanning trees with a lower maximum weighted degree, a bigger diameter, and can be used for balanced energy consumption routing in wireless sensor networks (WSN’s). The method can be naturally implemented in parallel or as a simple locally distributed algorithm. Simulations for a realistic case scenario WSN are done based on the transmission energy matrix. The simulation results show that the proposed approach can extend the functional lifetime of a WSN in terms of sensor transmission energy by 3–4 times. We also show that the results can be further improved by using a preliminary clustering of the input network.     

On the costs of optimal and near-optimal binary search trees

Summary We show that the cost of an optimal binary search tree can vary substantially, depending only on the left-to-right order imposed on the probabilities. We also prove that the costs of some common classes of near-optimal trees cannot be bounded above by the cost of an optimal tree plus a constant. This work was supported by the National Research Council of Canada, while the author was at the University of Waterloo     

The analysis of heuristics for search trees

We analyze the performance of search trees built under a variety of insertion heuristics. The main results are a method to obtain asymptotic expressions for the moments of the distribution of the search time, and a proof that this distribution is asymptotically normal.     

A method for constructing binary search trees by making insertions at the root

It is possible to construct a binary search tree by inserting items at the root instead of adding them as leaves. When used for sorting, the method has several desirable properties, including (a) fewer comparisons in the best case, (b) fewer comparisons in the worst case, (c) a reduced variance, and (d) good performance when the items are already nearly sorted or nearly reverse sorted. For applications in which the tree is searched for existing items as well as having new items added to it (e.g., in the construction of a symbol table), the tree can be made to exhibit stacklike behavior, so that the fewest comparisons are required to locate the most recently used items.     

Performance analysis of partial-match search algorithms for BD trees

Database systems are becoming increasingly popular for answering queries. Partial-match search queries are an important class of queries in such a system. Several storage structures have been proposed to answer these queries efficiently. The BD tree is an example of such a storage structure. A previous study indicated that the k-d tree performance is better than that of the BD tree for partial-match search queries. A recent paper reported some improved algorithms. However, it is unclear whether the improved algorithms show the BD tree in a favourable light for partial-match search queries. This paper explores the performance of these algorithms and compares their performance to that of the k-d tree. Since the BD tree construction process uses some heuristics to make it a better balanced tree, this paper also evaluates the effect of these heuristics on the partial-match search algorithms. The major conclusions of this study are that the BD tree performance for partial-match search is better than that of the k-d tree when an improved algorithm is used for partial-match search, and only the DZ expression rearrangement heuristic has substantial effect on partial-match search performance.     

Analysis of the standard deletion algorithms in exact fit domain binary search trees

It is well known that the expected search time in anN node binary search tree generated by a random sequence of insertions isO(logN). Little has been published about the asymptotic cost when insertions and deletions are made following the usual algorithms with no attempt to retain balance. We show that after a sufficient number of updates, each consisting of choosing an element at random, removing it, and reinserting the same value, that the average search cost is Θ(N ^1/2).