Multiway Search Trees Research Articles

Deletion without rebalancing in multiway search trees

Some database systems that use a form of B-tree for the underlying data structure do not do rebalancing on deletion. This means that a bad sequence of deletions can create a very unbalanced tree. Yet such databases perform well in practice. Avoidance of rebalancing on deletion has been justified empirically and by average-case analysis, but to our knowledge, no worst-case analysis has been done. We do such an analysis. We show that the tree height remains logarithmic in the number of insertions, independent of the number of deletions. Furthermore, the amortized time for an insertion or deletion, excluding the search time, is O (1), and nodes are modified by insertions and deletions with a frequency that is exponentially small in their height. The latter results do not hold for standard B-trees. By adding periodic rebuilding of the tree, we obtain a data structure that is theoretically superior to standard B-trees in many ways. Our results suggest that rebalancing on deletion not only is unnecessary but may be harmful.

High-performance IP forwarding with efficient routing-table update

There has been an extensive study in constructing the routing tables during the past few years. Although the existing works have certain advantages, those approaches either use complicated data structures which result in large storage requirement and high complexity for updating/building the forwarding table or they are not scalable to fit in Internet protocol version 6 (IPv6). In this work, we propose a fast forwarding-table construction algorithm. With the modified multiway search tree, we can further reduce the depth of the tree and eliminate the storage for pointers. It leads to reduce the FT size and shorten the routing-table lookup time. While considering the route flaps, the forwarding performance will degrade by only 3.1% with 4000 BGP updates per 30 s in the worst case. Moreover, it is simple enough to fulfill the need of the fast packet forwarding. An extension approach to solve the IPv6 routing lookup is also presented for the future deployment.

The Joint Distribution of Elastic Buckets in Multiway Search Trees

Random search trees are studied when they grow under a general computer memory management scheme. In a general scheme, the space is released in buckets of certain predesignated sizes. For a search tree with branch factor m, the nodes may hold up to $m - 1$ keys. Suppose the buckets of the memory management scheme that can hold less than m keys have key capacities $c_1 , \ldots ,c_r $. The search tree must then be implemented with multitype nodes of these capacities. After n insertions, let $X_n^{(i)} $ be the number of buckets of type i (i.e., of capacity $c_i $,$1 \leqslant i \leqslant p$. The multivariate structure of the tree is investigated. For the vector ${\bf X}_n = ( \leqslant X_n^{(1)} , \ldots ,X_n^{(p)} )^T $, the asymptotic mean and covariance matrix are determined. Under practical memory management schemes, all variances and covariances experience a phase transition: For $3 \leqslant m \leqslant 26$, all variances and covariances are asymptotically linear in n; for higher branch factors the variances and covariances become a superlinear (but ubquadratic) function of n. The joint distribution of ${\bf X}_n $ is shown to be multivariate normal in a range of m. While the tree is growing, conversions between types are necessary. A multivariate problem concerning these conversions with an asymptotic multivariate normal distribution is also studied. The fixed bucket, exact fit, and buddy system allocation schemes will serve as illustrating examples.

Unsuccessful Search in Self-Adjusting Data Structures

This paper introduces a general technique for speeding up unsuccessful search using very little extra space (two bits per key). This technique is applicable to many data structures including linear lists and search trees. For linear lists we obtain on-line algorithms for processing a sequence of successful and unsuccessful searches which are competitive with strong off-line algorithms. In a virtual memory environment our self-adjusting algorithm for multiway search trees is competitive with an optimal static multiway tree and will often outperform the static tree. We are also able to extend several other results for successful search to include both successful and unsuccessful searches.

Self-adjusting multi-way search trees

We introduce a new data structure, the k-forest, which is a self-adjusting multi-way search tree. A k-forest provides an efficient implementation of a weighted Dictionary in a virtual memory environment where the time to access a tree node is much greater than the time to examine data itmes in a node. We achieve results that are natural generalizations of the Working Set and Static Optimality theorems of Sleator and Tarjan for splay trees. Thus we are able to show that even when the lookup frequencies of the elements of the dictionary are not known in advance, a k-forest can equal or exceed the performance of an optimal static multi-way tree which is built knowing the lookup frequencies in advance.

Optimal multiway search trees for variable size keys

This paper considers the construction of optimal search trees for a sequence of n keys of varying sizes, under various cost measures. Constructing optimal search cost multiway trees is NP-hard, although it can be done in pseudo-polynomial time O3 and space O2, where L is the page size limit. An optimal space multiway search tree is obtained in O3 time and O2 space, while an optimal height tree in O(n2 log2n) time and O(n) space both having additionally minimal root sizes. The monotonicity principle does not hold for the above cases. Finding optimal search cost weak B-trees is NP-hard, but a weak B-tree of height 2 and minimal root size can be constructed in O(n log n) time. In addition, if its root is restricted to contain M keys then a different algorithm is applied, having time complexity O(nM log n). The latter solves a problem posed by McCreight.

The construction of optimal multiway search trees and the monotonicity principle

Given Nweighted keysN+1 missing-key weights and a branching factor t the application of dynamic programming yields algorithms for constructing optimal binary search trees (t = 2), optimal multi-way search trees (t>2), and optimal leaf search trees (or alphabetic code trees) with leaf weights only. The basic running time in all cases is O(N 3)(in terms of the number of keys), but it can be reduced to O(N 2) for binary search trees by a “monotonicity” principle which restricts the number of candidates for the root at each step in the construction. This principle can also be applied for multiway search trees when the missing-key weights are zero. However it does not extend to optimal multiway search trees in general, as we demonstrate; in particular, there is no monotonicity principle for alphabetic code trees, contrary to what is claimed in [5].

An experimental study of insertion schemes for classes of multiway search trees

Insertion schemes for various classes of multiway search trees have been implemented in PASCAL and experimentally studied. While the original B-tree insertion scheme does not consider brothers the dense m-ary-tree insertion scheme considers all brothers before splitting an overflowing node. There are many possible schemes in between these two extremes. We study the influence of the number of considered brothers, the splitting of an overflowing node with respect to the storage utilization and the number of input/output operations per insertion.

Optimum multiway search trees

The construction of optimum multiway search trees for n keys, n key weights and n+1 gap weights, is investigated. A new general optimality principle is presented, which can be "tuned" for specific applications. Moreover we consider the affects of three additional constraints, namely height, structural and node search restrictions, which lead to a number of new construction algorithms. In particular we concentrate on the construction of optimum t-ary search trees with linear and binary search within their nodes for which we obtain O(n 3 t) and O(n 3log2 t) time algorithms, respectively. Whether these algorithms are or are not optimal remains an important open problem, even in the binary case.

On the encipherment of search trees and random access files

The securing of information in indexed, random access files by means of privacy transformations must be considered as a problem distinct from that for sequential files. Not only must processing overhead due to encrypting be considered, but also threats to encipherment arising from updating and the file structure itself must be countered. A general encipherment scheme is proposed for files maintained in a paged structure in secondary storage. This is applied to the encipherment of indexes organized as B -trees; a B -tree is a particular type of multiway search tree. Threats to the encipherment of B -trees, especially relating to updating, are examined, and countermeasures are proposed for each. In addition, the effect of encipherment on file access and update, on paging mechanisms, and on files related to the enciphered index are discussed. Many of the concepts presented may be readily transferred to other forms of multiway index trees and to binary search trees.

On the encipherment of search trees and random access files

The securing of information in indexed, random access files by means of privacy transformations must be considered as a problem distinct from that for sequential files. Not only must processing overhead due to encrypting be considered, but also must threats to encipherment arising from updating and the file structure itself be countered. A general encipherment scheme is proposed for files maintained in a paged structure on secondary storage. This is then applied to the encipherment of indexes organized as B-trees, a particular type of multiway search tree. Threats to the encipherment of B-trees, especially relating to updating, are examined, and counter-measures proposed for each. In addition, the effect of encipherment on file access and update, on paging mechanisms, and on files related to the enciphered index are discussed. Many of the concepts presented here may be readily transferred to other forms of multiway index trees and to binary search trees.