We present a simple $O(n^4)$-time algorithm for computing optimal search trees with two-way comparisons. The only previous solution to this problem, by Anderson et al., has the same running time, but is significantly more complicated and is restricted to the variant where only successful queries are allowed. Our algorithm extends directly to solve the standard full variant of the problem, which also allows unsuccessful queries and for which no polynomial-time algorithm was previously known. The correctness proof of our algorithm relies on a new structural theorem for two-way-comparison search trees.
Search trees are commonly used to implement access operations to a set of stored keys. If this set is static and the probabilities of membership queries are known in advance, then one can precompute an optimal search tree, namely one that minimizes the expected access cost. For a non-key query, a search tree can determine its approximate location by returning the inter-key interval containing the query. This is in contrast to other dictionary data structures, like hash tables, that only report a failed search. We address the question "what is the additional cost of determining approximate loca…
在本文中，我们研究的问题是否静态搜索树应该是不平衡的。我们提出了几种方法来重建一个不平衡的k元搜索树$T$到一个新的树$R$，保留了$T$的许多属性，同时具有高度的$\log_k n +1$，这是一个单位的最佳高度。更具体地说，我们表明，它是可能的，以确保$R$中的元素的深度不超过其深度在$T$加上最多$\log_k \log_k n +2$。同时，可以保证树$R$中的平均访问时间$P(R)$不大于树$T$中的平均访问时间$P(T)$加上$O(\log_k P(T))$。这表明，对于大多数应用程序，平衡树总是比不平衡树更好的选择，因为平衡树具有相似的平均访问时间和更好的最坏情况访问时间。
The next-generation non-volatile memory (NVM) is striding into computer systems as a new tier as it incorporates both DRAM's byte-addressability and disk's persistency. Researchers and practitioners have considered building persistent memory by placing NVM on the memory bus for CPU to directly load and store data. As a result, cache-friendly data structures have been developed for NVM. One of them is the prevalent B+-tree. State-of-the-art in-NVM B+-trees mainly focus on the optimization of write operations (insertion and deletion). However, search is of vital importance for B+-tree. Not only …
In 1971, Knuth gave an $O(n^2)$-time algorithm for the classic problem of finding an optimal binary search tree. Knuth's algorithm works only for search trees based on 3-way comparisons, while most modern computers support only 2-way comparisons (e.g., $<, \le, =, \ge$, and $>$). Until this paper, the problem of finding an optimal search tree using 2-way comparisons remained open -- poly-time algorithms were known only for restricted variants. We solve the general case, giving (i) an $O(n^4)$-time algorithm and (ii) an $O(n \log n)$-time additive-3 approximation algorithm. Also, for finding op…
We present a technique that allows for improving on some relative greedy procedures by well-chosen (non-oblivious) local search algorithms. Relative greedy procedures are a particular type of greedy algorithm that start with a simple, though weak, solution, and iteratively replace parts of this starting solution by stronger components. Some well-known applications of relative greedy algorithms include approximation algorithms for Steiner Tree and, more recently, for connectivity augmentation problems. The main application of our technique leads to a $(1.5+ε)$-approximation for Weighted Tree Au…
This paper describes the shortest path problem in weighted graphs and examines the differences in efficiency that occur when using Dijkstra's algorithm with a Fibonacci heap, binary heap, and self-balancing binary tree. Using C++ implementations of these algorithm variants, we find that the fastest method is not always the one that has the lowest asymptotic complexity. Reasons for this are discussed and backed with empirical evidence.
Search trees on trees (STTs) generalize the fundamental binary search tree (BST) data structure: in STTs the underlying search space is an arbitrary tree, whereas in BSTs it is a path. An optimal BST of size $n$ can be computed for a given distribution of queries in $O(n^2)$ time [Knuth 1971] and centroid BSTs provide a nearly-optimal alternative, computable in $O(n)$ time [Mehlhorn 1977]. By contrast, optimal STTs are not known to be computable in polynomial time, and the fastest constant-approximation algorithm runs in $O(n^3)$ time [Berendsohn, Kozma 2022]. Centroid trees can be defined for…
Ordered set (and map) is one of the most used data type. In addition to standard set operations, like insert, delete and contains, it can provide set-set operations such as union, intersection, and difference. Each of these set-set operations is equivalent to batched operations: the data structure should process a set of operations insert, delete, and contains. It is obvious that we want these "large" operations to be parallelized. Typically, these sets are implemented with the trees of logarithmic height, such as 2-3 tree, Treap, AVL tree, Red-Black tree, etc. Until now, little attention was …
In this paper we implement an n-ary Huffman Encoding and Decoding application using different degrees of tree structures. Our goal is to compare the performance of the algorithm in terms of compression ratio, decompression speed and weighted path length when using higher degree trees, compared to the 2-ary Huffman Code. The Huffman tree degrees that we compare are 2-ary, 3-ary, 4-ary, 5-ary, 6-ary, 7-ary, 8-ary and 16-mal. We also present the impact that branch prediction has on the performance of the n-ary Huffman Decoding.
Jet Propulsion Laboratory, California Institute of Technology（喷气推进实验室，加州理工学院）
K-Nearest Neighbors (KNN) search is a fundamental algorithm in artificial intelligence software with applications in robotics, and autonomous vehicles. These wide-ranging applications utilize KNN either directly for simple classification or combine KNN results as input to other algorithms such as Locally Weighted Learning (LWL). Similar to binary trees, kd-trees become unbalanced as new data is added in online applications which can lead to rapid degradation in search performance unless the tree is rebuilt. Although approximate methods are suitable for graphics applications, which prioritize q…
University of Sydney（悉尼大学）；King Fahd University of Petroleum and Minerals（法赫德国王石油矿产大学）
Partitioning trees are efficient data structures for $k$-nearest neighbor search. Machine learning libraries commonly use a special type of partitioning trees called $k$d-trees to perform $k$-nn search. Unfortunately, $k$d-trees can be ineffective in high dimensions because they need more tree levels to decrease the vector quantization (VQ) error. Random projection trees rpTrees solve this scalability problem by using random directions to split the data. A collection of rpTrees is called rpForest. $k$-nn search in an rpForest is influenced by two factors: 1) the dispersion of points along the …
