Rolf Fagerberg

On the limits of cache-obliviousness

In this paper, we present lower bounds for permuting and sorting in the cache-oblivious model. We prove that (1) I/O optimal cache-oblivious comparison based sorting is not possible without a tall cache assumption, and (2) there does not exist an I/O optimal cache-oblivious algorithm for permuting, not even in the presence of a tall cache assumption.Our results for sorting show the existence of an inherent trade-off in the cache-oblivious model between the strength of the tall cache assumption and the overhead for the case M » B, and show that Funnelsort and recursive binary mergesort are optimal algorithms in the sense that they attain this trade-off.

Funnel Heap - A Cache Oblivious Priority Queue

The cache oblivious model of computation is a two-level memory model with the assumption that the parameters of the model are unknown to the algorithms. A consequence of this assumption is that an algorithm efficient in the cache oblivious model is automatically efficient in a multi-level memory model. Arge et al. recently presented the first optimal cache oblivious priority queue, and demonstrated the importance of this result by providingthe first cache oblivious algorithms for graph problems. Their structure uses cache oblivious sorting and selection as subroutines. In this paper, we devise an alternative optimal cache oblivious priority queue based only on binary merging. We also show that our structure can be made adaptive to different usage profiles.

Optimal sparse matrix dense vector multiplication in the I/O-model

We analyze the problem of sparse-matrix dense-vector multiplication (SpMV) in the I/O-model. The task of SpMV is to compute y := Ax, where A is a sparse N x N matrix and x and y are vectors. Here, sparsity is expressed by the parameter k that states that A has a total of at most kN nonzeros, i.e., an average number of k nonzeros per column. The extreme choices for parameter k are well studied special cases, namely for k=1 permuting and for k=N dense matrix-vector multiplication. We study the worst-case complexity of this computational task, i.e., what is the best possible upper bound on the number of I/Os depending on k and N only. We determine this complexity up to a constant factor for large ranges of the parameters. By our arguments, we find that most matrices with kN nonzeros require this number of I/Os, even if the program may depend on the structure of the matrix. The model of computation for the lower bound is a combination of the I/O-models of Aggarwal and Vitter, and of Hong and Kung. We study two variants of the problem, depending on the memory layout of A. If A is stored in column major layout, SpMV has I/O complexity Θ(min{kN B(1+logM/BN max{M,k}), kN}) for k ≤ N1-e and any constant 1> e > 0. If the algorithm can choose the memory layout, the I/O complexity of SpMV is Θ(min{kN B(1+logM/BN kM), kN]) for k ≤ 3√N. In the cache oblivious setting with tall cache assumption M ≥ B1+e, the I/O complexity is Ο(kN B(1+logM/B N k)) for A in column major layout.

Engineering a cache-oblivious sorting algorithm

This paper is an algorithmic engineering study of cache-oblivious sorting. We investigate by empirical methods a number of implementation issues and parameter choices for the cache-oblivious sorting algorithm Lazy Funnelsort and compare the final algorithm with Quicksort, the established standard for comparison-based sorting, as well as with recent cache-aware proposals. The main result is a carefully implemented cache-oblivious sorting algorithm, which, our experiments show, can be faster than the best Quicksort implementation we are able to find for input sizes well within the limits of RAM. It is also at least as fast as the recent cache-aware implementations included in the test. On disk, the difference is even more pronounced regarding Quicksort and the cache-aware algorithms, whereas the algorithm is slower than a careful implementation of multiway Mergesort, such as TPIE.

Computing the Quartet Distance between Evolutionary Trees in Time O(n log n)

Abstract Evolutionary trees describing the relationship for a set of species are central in evolutionary biology, and quantifying differences between evolutionary trees is therefore an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook, McMorris, and Meacham. The quartet distance between two unrooted evolutionary trees is the number of quartet topology differences between the two trees, where a quartet topology is the topological subtree induced by four species. In this paper we present an algorithm for computing the quartet distance between two unrooted evolutionary trees of n species, where all internal nodes have degree three, in time O(n log n. The previous best algorithm for the problem uses time O(n2).

Cache-oblivious string dictionaries

We present static cache-oblivious dictionary structures for strings which provide analogues of tries and suffix trees in the cache-oblivious model. Our construction takes as input either a set of strings to store, a single string for which all suffixes are to be stored, a trie, a compressed trie, or a suffix tree, and creates a cache-oblivious data structure which performs prefix queries in <i>O</i>(log<inf><i>B</i></inf><i>n</i> + |<i>P</i>|/<i>B</i>) I/Os, where <i>n</i> is the number of leaves in the trie, <i>P</i> is the query string, and <i>B</i> is the block size. This query cost is optimal for unbounded alphabets. The data structure uses linear space.

Cache-Oblivious Data Structures and Algorithms for Undirected Breadth-First Search and Shortest Paths

We present improved cache-oblivious data structures and algorithms for breadth-first search and the single-source shortest path problem on undirected graphs with non-negative edge weights. Our results removes the performance gap between the currently best cache-aware algorithms for these problems and their cache-oblivious counterparts. Our shortest-path algorithm relies on a new data structure, called bucket heap, which is the first cache-oblivious priority queue to efficiently support a weak DecreaseKey operation.

The cost of cache-oblivious searching

Tight bounds on the cost of cache-oblivious searching are proved. It is shown that no cache-oblivious search structure can guarantee that a search performs fewer than lg e log/sub B/N block transfers between any two levels of the memory hierarchy. This lower bound holds even if all of the block sizes are limited to be powers of 2. A modified version of the van Emde Boas layout is proposed, whose expected block transfers between any two levels of the memory hierarchy arbitrarily close to [lg e + O(lg lg B/ lgB)] logB N + O(1). This factor approaches lg e /spl ap/ 1.443 as B increases. The expectation is taken over the random placement of the first element of the structure in memory. As searching in the disk access model (DAM) can be performed in log/sub B/N + 1 block transfers, this result shows a separation between the 2-level DAM and cache-oblivious memory-hierarchy models. By extending the DAM model to k levels, multilevel memory hierarchies can be modeled. It is shown that as k grows, the search costs of the optimal k-level DAM search structure and of the optimal cache-oblivious search structure rapidly converge. This demonstrates that for a multilevel memory hierarchy, a simple cache-oblivious structure almost replicates the performance of an optimal parameterized k-level DAM structure.

Online Sorted Range Reporting

We study the following one-dimensional range reporting problem: On an array A of n elements, support queries that given two indices i ≤ j and an integer k report the k smallest elements in the subarray A[i..j] in sorted order. We present a data structure in the RAM model supporting such queries in optimal O(k) time. The structure uses O(n) words of space and can be constructed in O(n logn) time. The data structure can be extended to solve the online version of the problem, where the elements in A[i..j] are reported one-by-one in sorted order, in O(1) worst-case time per element. The problem is motivated by (and is a generalization of) a problem with applications in search engines: On a tree where leaves have associated rank values, report the highest ranked leaves in a given subtree. Finally, the problem studied generalizes the classic range minimum query (RMQ) problem on arrays.

Dynamic Representations of Sparse Graphs

We present a linear space data structure for maintaining graphs with bounded arboricity--a large class of sparse graphs containing e.g. planar graphs and graphs of bounded treewidth--under edge insertions, edge deletions, and adjacency queries. The data structure supports adjacency queries in worst case O(c) time, and edge insertions and edge deletions in amortized O(1) and O(c+log n) time, respectively, where n is the number of nodes in the graph, and c is the bound on the arboricity.