SODA

We present several new results regarding λ_s(n), the maximum length of a Davenport--Schinzel sequence of order s on n distinct symbols.

First, we prove that

[EQUATION]

where t = [(s - 2)/2], and α(n) denotes the inverse Ackermann function. The previous ...

In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time O(mn), dates back to König's work in 1916 (here m = nd is the number of edges in ...

In the budgeted learning problem, we are allowed to experiment on a set of alternatives (given a fixed experimentation budget) with the goal of picking a single alternative with the largest possible expected payoff. Constant factor approximation ...

In this paper, we consider the restless bandit problem, which is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit problem in decision theory. In its ultimate generality, the restless bandit problem is known to ...

The online multi-armed bandit problem and its generalizations are repeated decision making problems, where the goal is to select one of several possible decisions in every round, and incur a cost associated with the decision, in such a way that the ...

We study the cover time of random geometric graphs. Let I ( d ) = [0, 1] ^d denote the unit torus in d dimensions. Let D ( x, r ) denote the ball (disc) of radius r . Let ý _d be the volume of the unit ball D (0, 1) in d dimensions. A random ...

We analyze the complexity of the Walk on Spheres algorithm for simulating Brownian Motion in a domain Ω ⊂R^d. The algorithm, which was first proposed in the 1950s, produces samples from the hitting probability distribution of the Brownian Motion process ...

In sorting situations where the final destination of each item is known, it is natural to repeatedly choose items and place them where they belong, allowing the intervening items to shift by one to make room. (In fact, a special case of this algorithm ...

Monotonic surfaces spanning finite regions of Z^d arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show ...

The hitting time of a classical random walk (Markov chain) is the time required to detect the presence of -- or equivalently, to find -- a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the ...

Pebbles are placed on some vertices of a directed graph. Is it possible to move each pebble along at most one edge of the graph so that in the final configuration no pebble is left on its own? We give an O(mn)-time algorithm for solving this problem, ...

We prove that every graph with maximum degree Δ can be properly (Δ + 1)-coloured so that no colour appears more than O(log Δ / log log Δ) times in the neighbourhood of any vertex. This is best possible up to the constant factor in the O(−) term. We also ...

We prove that given an undirected graph G on n vertices and an integer k, one can compute in polynomial time in n a graph G' with at most 5k² + k vertices and an integer k' such that G has a feedback vertex set of size at most k iff G' has a feedback ...

Gimbel and Thomassen asked whether 3-colorability of a triangle-free graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a linear-time algorithm for every surface which combined with previous results gives a ...

A digital search tree (DST) -- one of the most fundamental data structures on words -- is a digital tree in which keys (strings, words) are stored directly in (internal) nodes. Such trees find myriad of applications from the popular Lempel-Ziv'78 data ...

Computer Science has historically been strong on data structures and weak on inference from data, whereas Statistics has historically been weak on data structures and strong on inference from data. One way to draw on the strengths of both disciplines is ...

We establish the first nontrivial lower bounds on time-space tradeoffs for the selection problem. We prove that any comparison-based randomized algorithm for finding the median requires Ω(n log log_sn) expected time in the RAM model (or more generally in ...

We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated ...

Let S be a surface embedded in space in such a way that each point has a neighborhood within which the surface is a terrain. Then S projects to an immersed surface in the plane, the boundary of which is a (possibly self-intersecting) curve. Under what ...

We establish a bound of O(n²k^1+ε), for any ε > 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R³ with a total of n facets, and present a randomized algorithm which computes the boundary of ...

We give the first optimal solution to a standard problem in computational geometry: three-dimensional halfspace range reporting. We show that n points in 3-d can be stored in a linear-space data structure so that all k points inside a query halfspace ...

The assignment problem concerns finding the minimum-cost perfect matching in a complete weighted n x n bipartite graph. Any algorithm for this classical question clearly requires Ω(n²) time, and the best known one (Edmonds and Karp, 1972) finds solution ...

We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay ...

Let A be a 0/1 matrix of size m x n, and let p be the density of A (i.e., the number of ones divided by m · n). We show that A can be approximated in the cut norm within ε · mnp by a sum of cut matrices (of rank 1), where the number of summands is ...

The hypergraph of the Fano plane is the unique 3-uniform hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. This hypergraph is not 2-colorable, but becomes so on deleting any hyperedge from it. We ...

Thomason and Chung, Graham, and Wilson were the first to systematically study quasi-random graphs and hypergraphs, and proved that several properties of random graphs imply each other in a deterministic sense. Their concepts of quasi-randomness match ...

We give an O(n log² n)-time, linear-space algorithm that, given a directed planar graph with positive and negative arc-lengths, and given a node s, finds the distances from s to all nodes. The best previously known algorithm requires O(n log³ n) time ...

We present an Õ(m) (near-linear) time Monte Carlo algorithm for constructing the cactus data structure, a useful representation of all the global minimum edge cuts of an undirected graph. Our algorithm represents a fundamental improvement over the best ...

This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w · x - θ). We consider halfspaces over the continuous domain Rⁿ (endowed with the standard multivariate Gaussian ...

We consider an unweighted undirected graph with n vertices, m edges, and edge-connectivity 2k. The weak edge orientation problem requires that the edges of this graph be oriented so the resulting directed graph is at least k edge-connected. Nash-...