Cache Oblivious Search Trees via Binary Trees of Small Height

Abstract

We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and Farach-Colton and has the same complexity bounds. In particular, our data structure avoids the use of weight balanced B-trees, and can be implemented as just a single array of data elements, without the use of pointers. The structure also improves space utilization.<br /> <br />For storing n elements, our proposal uses (1+epsilon)n times the element size of memory, and performs searches in worst case O(log_B n) memory transfers, updates in amortized O((log^2 n)/(epsilon B)) memory transfers, and range queries in worst case O(log_B n + k/B) memory transfers, where k is the size of the output.<br /> <br />The basic idea of our data structure is to maintain a dynamic binary tree of height log n + O(1) using existing methods, embed this tree in a static binary tree, which in turn is embedded in an array in a cache oblivious fashion, using the van Emde Boas layout of Prokop.<br /> <br />We also investigate the practicality of cache obliviousness in the area of search trees, by providing an empirical comparison of different methods for laying out a search tree in memory.<br /> <br />The source code of the programs, our scripts and tools, and the data we present here are available online under ftp.brics.dk/RS/01/36/Experiments/.

Insertion Sort is O(n log n)

“…This idea has found recent application in external memory and cache-oblivious algorithms in the packed memory structure of Bender, Demaine and Farach-Colton [1] and later used in [2][3][4].…”

Cache Oblivious Distribution Sweeping

“…Bender et al [6], gave a proposal for cache oblivious search trees with search cost matching that of standard (cache aware) B-trees [4]. Simpler cache oblivious search trees with complexities matching that of [6] were presented in [7,10]. Cache-oblivious data structures based on on exponential structures are presented in [5].…”

Cache-Oblivious Dictionaries and Multimaps with Negligible Failure Probability

“…Since the introduction of the cache-oblivious framework by Frigo et al [14], several cache-oblivious algorithms have subsequently been presented, including cache-oblivious B-trees [5], cache-oblivious binary search trees [7], and cacheoblivious sorting [8]. Pagh et al [26] describe a scheme for cache-oblivious hashing, which is based on linear probing and achieves O(1) expected-time performance for lookups and updates, but it does not achieve constant time bounds for any of these operations in the worst case.…”