Torben Hagerup

Sorting in Linear Time

We show that a unit-cost RAM with a word length ofwbits can sortnintegers in the range 0?2w?1 inO(nloglogn) time for arbitraryw?logn, a significant improvement over the bound ofO(nlogn) achieved by the fusion trees of Fredman and Willard. Provided thatw?(logn)2+?for some fixed?>0, the sorting can even be accomplished in linear expected time with a randomized algorithm. Both of our algorithms parallelize without loss on a unit-cost PRAM with a word length ofwbits. The first one yields an algorithm that usesO(logn) time andO(nloglogn) operations on a deterministic CRCW PRAM. The second one yields an algorithm that usesO(logn) expected time andO(n) expected operations on a randomized EREW PRAM, provided thatw?(logn)2+?for some fixed?>0. Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting of multiple-precision integers represented in several words.

Algorithm Theory - SWAT 2004

Frigo, Leiserson, Prokop and Ramachandran in 1999 introduced the ideal-cache model as a formal model of computation for developing algorithms in environments with multiple levels of caching, and coined the terminology of cache-oblivious algorithms. Cache-oblivious algorithms are described as standard RAM algorithms with only one memory level, i.e. without any knowledge about memory hierarchies, but are analyzed in the two-level I/O model of Aggarwal and Vitter for an arbitrary memory and block size and an optimal off-line cache replacement strategy. The result are algorithms that automatically apply to multi-level memory hierarchies. This paper gives an overview of the results achieved on cache-oblivious algorithms and data structures since the seminal paper by Frigo et al.

Improved Shortest Paths on the Word RAM

Thorup recently showed that single-source shortest-paths problems in undirected networks with n vertices, m edges, and edge weights drawn from {0, . . . , 2w -1} can be solved in O(n+m) time and space on a unit-cost random-access machine with a word length of w bits. His algorithm works by traversing a so-called component tree. Two new related results are provided here. First, and most importantly, Thorups approach is generalized from undirected to directed networks. The resulting time bound, O(n + m log w), is the best deterministic linear-space bound known for sparse networks unless w is superpolynomial in log n. As an application, all-pairs shortest-paths problems in directed networks with n vertices, m edges, and edge weights in {-2w, . . . , 2w} can be solved in O(nm + n2 log log n) time and O(n + m) space (not counting the output space). Second, it is shown that the component tree for an undirected network can be constructed in deterministic linear time and space with a simple algorithm, to be contrasted with a complicated and impractical solution suggested by Thorup. Another contribution of the present paper is a greatly simplified view of the principles underlying algorithms based on component trees.

Characterizing Multiterminal Flow Networks and Computing Flows in Networks of Small Treewidth

We show that if a flow network haskinput/output terminals (for the traditional maximum-flow problem,k=2), its external flow pattern (the possible values of flow into and out of the terminals) has two characterizations of size independent of the total number of vertices: a set of 2k+1 inequalities inkvariables representing flow values at the terminals, and a mimicking network with at most 22kvertices and the same external flow pattern as the original network. For the case in which the underlying graph has bounded treewidth, we present sequential and parallel algorithms that can compute these characterizations as well as a flow consistent with any desired feasible external flow (including a maximum flow between two given terminals). For constantk, the sequential algorithm runs inO(n) time onn-vertex networks, and the parallel algorithm runs inO(logn) time on an EREW PRAM withO(n/logn) processors if an explicit tree decomposition of the network of sizeO(n) is given; if not, known algorithms can compute such a tree decomposition inO((logn)2) time usingO(n/(logn)2) processors.

Simpler linear-time kernelization for planar dominating set

We describe a linear-time algorithm that inputs a planar graph G and outputs a planar graph of size O(k) and with domination number k, where k is the domination number of G, i.e., the size of a smallest dominating set in G. In the language of parameterized computation, the new algorithm is a linear-time kernelization for the NP-complete Planar Dominating Set problem that produces a kernel of linear size. Such an algorithm was previously known (van Bevern et al., these proceedings), but the new algorithm and its analysis are considerably simpler.

Simple Minimal Perfect Hashing in Less Space

A minimal perfect hash function for a set S is an injective mapping from S to {0, . . . |S|-1}. Taking as our model of computation a unit-cost RAM with a word length of w bits, we consider the problem of constructing minimal perfect hash functions with constant evaluation time for arbitrary subsets of U = {0, . . . 2w - 1}. Pagh recently described a simple randomized algorithm that, given a set S ⊆ U of size n, works in O(n) expected time and computes a minimal perfect hash function for S whose representation, besides a constant number of words, is a table of at most (2+Ɛ)n integers in the range {0,. . ., n-1}, for arbitrary fixed Ɛ > 0. Extending his method, we show how to replace the factor of 2 + Ɛ by 1 + Ɛ.

An Efficient Quasidictionary

We define a quasidictionary to be a data structure that supports the following operations: check-in(v) inserts a data item v and returns a positive integer tag to be used in future references to v; check-out(x) deletes the data item with tag x; access(x) inspects and/or modifies the data item with tag x. A quasidictionary is similar to a dictionary, the difference being that the names identifying data items are chosen by the data structure rather than by its user. We describe a deterministic quasidictionary that executes the operations check-in and access in constant time and check-out in constant amortized time, works in linear space, and uses only tags bounded by the maximum number of data items stored simultaneously in the quasidictionary since it was last empty.

Online and offline access to short lists

We consider the list-update problem introduced by Sleator and Tarjan, specializing it to the case of accesses only and focusing on short lists. We describe a new optimal offline algorithm, faster than the best previous algorithm when the number of accesses is sufficiently large relative to the number l of items. We also give a simple optimal offline algorithm for l = 3. Taking cl to denote the best competitive ratio of a randomized online algorithm for the list-access problem with l items, we demonstrate that c3 = 6/5 and give new upper and lower bounds on c4. Finally we prove a strengthened lower bound for general l.

Parallel preprocessing for path queries without concurrent reading

We consider the problem of preprocessing a tree T with edge labels drawn from a semigroup such that subsequent queries for the semigroup product of the edge labels on a path in T can be answered efficiently. A sequential algorithm exhibiting an optimal trade-off between preprocessing time and query time was described by Chazelle. A parallelization of the preprocessing part of Chazelles algorithm for the exclusive-read exclusive-write parallel RAM (EREW PRAM) was announced by Alon and Schieber, but few details were provided. Later a different solution, complete with all details, was described by Thorup, but it requires the stronger concurrent-read exclusive-write PRAM. We describe a simple algorithm for the EREW PRAM.

Tree Decompositions of Small Diameter

Motivated by applications in parallel and dynamic graph algorithms, we investigate the tradeoff between width and diameter of tree decompositions. For all integers n, k and K with 1 ≤ k ≤ K ≤ n− 1, denote by D(n, k, K) the maximum, over all n-vertex graphs G of treewidth k, of the smallest diameter of a tree decomposition of G of width K. We determine D(n, k, K), up to a constant factor, for all values of n, k and K. When K is bounded by a constant (the case of greatest practical relevance), D(n, k, K) is θ(n) for K ≤ 2k-1, θ(√n) for 2k ≤ K ≤ 3k−2, and θ(log n) for K ≥ 3k−1. We provide much more accurate bounds for the case K ≤ 2k−1.