arXiv subjects

Mikkel Thorup

Publications and source records attributed to Mikkel Thorup.

82 records · Page 5Linked to original sources

arXiv2011

Minimum k-way cut of bounded size is fixed-parameter tractable

We consider a the minimum k-way cut problem for unweighted graphs with a size bound s on the number of cut edges allowed. Thus we seek to remove as few edges as possible so as to split a graph into k components, or report that this requires cutting more than s edges. We show that this problem is fixed-parameter tractable (FPT) in s. More precisely, for s=O(1), our algorithm runs in quadratic time while we have a different linear time algorithm for planar graphs and bounded genus graphs. Our tractability result stands in contrast to known W[1] hardness of related problems. Without the size bound, Downey et al.[2003] proved that the minimum k-way cut problem is W[1] hard in k even for simple unweighted graphs. Downey et al. asked about the status for planar graphs. Our result implies tractability in k for the planar graphs since the minimum k-way cut of a planar graph is of size at most 6k (more generally, we get tractability in k for any graph class with k-way cuts of size limited by is a function of k, e.g., bounded degree graphs, or simple graphs with an excluded minor). A simple reduction shows that vertex cuts are at least as hard as edge cuts, so the minimum k-way vertex cut is also W[1] hard in terms of k. Marx [2004] proved that finding a minimum k-way vertex cut of size s is also W[1] hard in s. Marx asked about the FPT status with edge cuts, which we prove tractable here. We are not aware of any other cut problem where the vertex version is W[1] hard but the edge version is FPT.

cs.DM↗

arXiv2010

Stream sampling for variance-optimal estimation of subset sums

From a high volume stream of weighted items, we want to maintain a generic sample of a certain limited size $k$ that we can later use to estimate the total weight of arbitrary subsets. This is the classic context of on-line reservoir sampling, thinking of the generic sample as a reservoir. We present an efficient reservoir sampling scheme, $\varoptk$, that dominates all previous schemes in terms of estimation quality. $\varoptk$ provides {\em variance optimal unbiased estimation of subset sums}. More precisely, if we have seen $n$ items of the stream, then for {\em any} subset size $m$, our scheme based on $k$ samples minimizes the average variance over all subsets of size $m$. In fact, the optimality is against any off-line scheme with $k$ samples tailored for the concrete set of items seen. In addition to optimal average variance, our scheme provides tighter worst-case bounds on the variance of {\em particular} subsets than previously possible. It is efficient, handling each new item of the stream in $O(\log k)$ time. Finally, it is particularly well suited for combination of samples from different streams in a distributed setting.

cs.DS↗

arXiv2007

Maximum overhang

How far can a stack of $n$ identical blocks be made to hang over the edge of a table? The question dates back to at least the middle of the 19th century and the answer to it was widely believed to be of order $\log n$. Recently, Paterson and Zwick constructed $n$-block stacks with overhangs of order $n^{1/3}$, exponentially better than previously thought possible. We show here that order $n^{1/3}$ is indeed best possible, resolving the long-standing overhang problem up to a constant factor.

math.HO↗

arXiv2007

On the variance of subset sum estimation

For high volume data streams and large data warehouses, sampling is used for efficient approximate answers to aggregate queries over selected subsets. Mathematically, we are dealing with a set of weighted items and want to support queries to arbitrary subset sums. With unit weights, we can compute subset sizes which together with the previous sums provide the subset averages. The question addressed here is which sampling scheme we should use to get the most accurate subset sum estimates. We present a simple theorem on the variance of subset sum estimation and use it to prove variance optimality and near-optimality of subset sum estimation with different known sampling schemes. This variance is measured as the average over all subsets of any given size. By optimal we mean there is no set of input weights for which any sampling scheme can have a better average variance. Such powerful results can never be established experimentally. The results of this paper are derived mathematically. For example, we show that appropriately weighted systematic sampling is simultaneously optimal for all subset sizes. More standard schemes such as uniform sampling and probability-proportional-to-size sampling with replacement can be arbitrarily bad. Knowing the variance optimality of different sampling schemes can help deciding which sampling scheme to apply in a given context.

cs.DS↗

arXiv2006

Time-Space Trade-Offs for Predecessor Search

We develop a new technique for proving cell-probe lower bounds for static data structures. Previous lower bounds used a reduction to communication games, which was known not to be tight by counting arguments. We give the first lower bound for an explicit problem which breaks this communication complexity barrier. In addition, our bounds give the first separation between polynomial and near linear space. Such a separation is inherently impossible by communication complexity. Using our lower bound technique and new upper bound constructions, we obtain tight bounds for searching predecessors among a static set of integers. Given a set Y of n integers of l bits each, the goal is to efficiently find predecessor(x) = max{y in Y | y <= x}, by representing Y on a RAM using space S. In external memory, it follows that the optimal strategy is to use either standard B-trees, or a RAM algorithm ignoring the larger block size. In the important case of l = c*lg n, for c>1 (i.e. polynomial universes), and near linear space (such as S = n*poly(lg n)), the optimal search time is Theta(lg l). Thus, our lower bound implies the surprising conclusion that van Emde Boas' classic data structure from [FOCS'75] is optimal in this case. Note that for space n^{1+eps}, a running time of O(lg l / lglg l) was given by Beame and Fich [STOC'99].

cs.CC↗

arXiv2005

Sampling to estimate arbitrary subset sums

Starting with a set of weighted items, we want to create a generic sample of a certain size that we can later use to estimate the total weight of arbitrary subsets. For this purpose, we propose priority sampling which tested on Internet data performed better than previous methods by orders of magnitude. Priority sampling is simple to define and implement: we consider a steam of items i=0,...,n-1 with weights w_i. For each item i, we generate a random number r_i in (0,1) and create a priority q_i=w_i/r_i. The sample S consists of the k highest priority items. Let t be the (k+1)th highest priority. Each sampled item i in S gets a weight estimate W_i=max{w_i,t}, while non-sampled items get weight estimate W_i=0. Magically, it turns out that the weight estimates are unbiased, that is, E[W_i]=w_i, and by linearity of expectation, we get unbiased estimators over any subset sum simply by adding the sampled weight estimates from the subset. Also, we can estimate the variance of the estimates, and surpricingly, there is no co-variance between different weight estimates W_i and W_j. We conjecture an extremely strong near-optimality; namely that for any weight sequence, there exists no specialized scheme for sampling k items with unbiased estimators that gets smaller total variance than priority sampling with k+1 items. Very recently Mario Szegedy has settled this conjecture.

cs.DS↗

arXiv2004

Efficient Tree Layout in a Multilevel Memory Hierarchy

We consider the problem of laying out a tree with fixed parent/child structure in hierarchical memory. The goal is to minimize the expected number of block transfers performed during a search along a root-to-leaf path, subject to a given probability distribution on the leaves. This problem was previously considered by Gil and Itai, who developed optimal but slow algorithms when the block-transfer size B is known. We present faster but approximate algorithms for the same problem; the fastest such algorithm runs in linear time and produces a solution that is within an additive constant of optimal. In addition, we show how to extend any approximately optimal algorithm to the cache-oblivious setting in which the block-transfer size is unknown to the algorithm. The query performance of the cache-oblivious layout is within a constant factor of the query performance of the optimal known-block-size layout. Computing the cache-oblivious layout requires only logarithmically many calls to the layout algorithm for known block size; in particular, the cache-oblivious layout can be computed in O(N lg N) time, where N is the number of nodes. Finally, we analyze two greedy strategies, and show that they have a performance ratio between Omega(lg B / lg lg B) and O(lg B) when compared to the optimal layout.

cs.DS↗

arXiv2003

Maintaining Information in Fully-Dynamic Trees with Top Trees

We introduce top trees as a design of a new simpler interface for data structures maintaining information in a fully-dynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we show how to maintain the diameter, center, and median of each tree in the forest. The forest can be updated by insertion and deletion of edges and by changes to vertex and edge weights. Each update is supported in O(log n) time, where n is the size of the tree(s) involved in the update. Also, we show how to support nearest common ancestor queries and level ancestor queries with respect to arbitrary roots in O(log n) time. Finally, with marked and unmarked vertices, we show how to compute distances to a nearest marked vertex. The later has applications to approximate nearest marked vertex in general graphs, and thereby to static optimization problems over shortest path metrics. Technically speaking, top trees are easily implemented either with Frederickson's topology trees [Ambivalent Data Structures for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees, SIAM J. Comput. 26 (2) pp. 484-538, 1997] or with Sleator and Tarjan's dynamic trees [A Data Structure for Dynamic Trees. J. Comput. Syst. Sc. 26 (3) pp. 362-391, 1983]. However, we claim that the interface is simpler for many applications, and indeed our new bounds are quadratic improvements over previous bounds where they exist.

cs.DS↗

arXiv2003

Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some small delta > 0. Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this paper, we give improved randomized rounding schemes for their relaxation, yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation algorithm in general. Our approach hinges on the observation that the problem of designing a randomized rounding scheme for a geometric relaxation is itself a linear programming problem. The paper explores computational solutions to this problem, and gives a proof that for a general class of geometric relaxations, there are always randomized rounding schemes that match the integrality gap.

cs.DS↗

arXiv2003

Dynamic Ordered Sets with Exponential Search Trees

We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully-dynamic linear space data structures. This leads to an optimal bound of O(sqrt(log n/loglog n)) for searching and updating a dynamic set of n integer keys in linear space. Here searching an integer y means finding the maximum key in the set which is smaller than or equal to y. This problem is equivalent to the standard text book problem of maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein: Introduction to Algorithms, 2nd ed., MIT Press, 2001). The best previous deterministic linear space bound was O(log n/loglog n) due Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space. We also get the following worst-case linear space trade-offs between the number n, the word length w, and the maximal key U < 2^w: O(min{loglog n+log n/log w, (loglog n)(loglog U)/(logloglog U)}). These trade-offs are, however, not likely to be optimal. Our results are generalized to finger searching and string searching, providing optimal results for both in terms of n.

cs.DS↗