A Fire Fighter's Problem

Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed $v>1$. How large must $v$ be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results. First, we analyze the natural curve \mbox{FF}_v that develops when the fighter keeps building, at speed $v$, a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function $(e^{w Z} - s \, Z)^{-1}$, where $w$ and $s$ are real functions of $v$. For $v>v_c=2.6144 \ldots$ all zeroes are complex conjugate pairs. If $\phi$ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs $\Theta( 1/\phi)$ rounds before the fire is contained. As $v$ decreases towards $v_c$ these two zeroes merge into a real one, so that argument $\phi$ goes to~0. Thus, curve \mbox{FF}_v does not contain the fire if the fighter moves at speed $v=v_c$. (That speed $v>v_c$ is sufficient for containing the fire has been proposed before by Bressan et al. [7], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that any curve that visits the four coordinate half-axes in cyclic order, and in inreasing distances from the origin, needs speed $v>1.618\ldots$, the golden ratio, in order to contain the fire. Keywords: Motion Planning, Dynamic Environments, Spiralling strategies, Lower and upper boundsComment: A preliminary version of the paper was presented at SoCG 201

Heuristics for optimum binary search trees and minimum weight triangulation problems

AbstractIn this paper we establish new bounds on the problem of constructing optimum binary search trees with zero-key access probabilities (with applications e.g. to point location problems). We present a linear-time heuristic for constructing such search trees so that their cost is within a factor of 1 + ε from the optimum cost, where ε is an arbitrary small positive constant. Furthermore, by using an interesting amortization argument, we give a simple and practical, linear-time implementation of a known greedy heuristics for such trees.The above results are obtained in a more general setting, namely in the context of minimum length triangulations of so-called semi-circular polygons. They are carried over to binary search trees by proving a duality between optimum (m − 1)-way search trees and minimum weight partitions of infinitely-flat semi-circular polygons into m-gons. With this duality we can also obtain better heuristics for minimum length partitions of polygons by using known algorithms for optimum search trees

Bamboo Garden Trimming Problem (Perpetual Maintenance of Machines with Different Attendance Urgency Factors)

International audienc

Shortcuts for the Circle

Let C be the unit circle in R^2. We can view C as a plane graph whose vertices are all the points on C, and the distance between any two points on C is the length of the smaller arc between them. We consider a graph augmentation problem on C, where we want to place k >= 1 shortcuts on C such that the diameter of the resulting graph is minimized. We analyze for each k with 1 <= k <= 7 what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of k. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is 2 + Theta(1/k^(2/3)) for any k

Efficient Assignment of Identities in Anonymous Populations

We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size $n$ of the population is embedded in the transition function. Our labeling protocols are silent w.h.p., i.e., eventually each agent reaches its final state and remains in it forever w.h.p., as well as safe, i.e., never update the label assigned to any single agent. We first present a fast, silent w.h.p.and safe labeling protocol for which the required number of interactions is asymptotically optimal, i.e., $O(n \log n/\epsilon)$ w.h.p. It uses $(2+\epsilon)n+O(n^c)$ states, for any $c1-\frac 1n$, uses $\ge n+\sqrt {n-1} -1$ states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Furthermore, we show that for any silent and safe labeling protocol utilizing $n+t<2n$ states the expected number of interactions required to achieve a valid labeling is $\ge \frac{n^2}{t+1}$

Pushing the Online Boolean Matrix-vector Multiplication conjecture off-line and identifying its easy cases

Henzinger et al. posed the so-called Online Boolean Matrix-vector Multiplication (OMv) conjecture and showed that it implies tight hardness results for several basic dynamic or partially dynamic problems [STOC'15]. We first show that the OMv conjecture is implied by a simple off-line conjecture that we call the MvP conjecture. We then show that if the definition of the OMv conjecture is generalized to allow individual (i.e., it might be different for different matrices) polynomial-time preprocessing of the input matrix, then we obtain another conjecture (called the OMvP conjecture) that is in fact equivalent to our MvP conjecture. On the other hand, we demonstrate that the OMv conjecture does not hold in restricted cases where the rows of the matrix or the input vectors are clustered, and develop new efficient randomized algorithms for such cases. Finally, we present applications of our algorithms to answering graph queries

Perpetual maintenance of machines with different urgency requirements

A garden $G$ is populated by $n\ge 1$ bamboos $b_1, b_2, ..., b_n$ with the respective daily growth rates $h_1 \ge h_2 \ge \dots \ge h_n$. It is assumed that the initial heights of bamboos are zero. The robotic gardener maintaining the garden regularly attends bamboos and trims them to height zero according to some schedule. The Bamboo Garden Trimming Problem (BGT) is to design a perpetual schedule of cuts to maintain the elevation of the bamboo garden as low as possible. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. The objective is to design a perpetual schedule of servicing which minimizes the maximum (weighted) waiting time for servicing. We consider two variants of BGT. In discrete BGT the robot trims only one bamboo at the end of each day. In continuous BGT the bamboos can be cut at any time, however, the robot needs time to move from one bamboo to the next. For discrete BGT, we show a simple $4$-approximation algorithm and, by exploiting relationship between BGT and the classical Pinwheel scheduling problem, we derive a $2$-approximation algorithm for the general case and a tighter approximation when the growth rates are balanced. A by-product of this last approximation algorithm is that it settles one of the conjectures about the Pinwheel problem. For continuous BGT, we propose approximation algorithms which achieve approximation ratios $O(\log (h_1/h_n))$ and $O(\log n)$

Efficiently Correcting Matrix Products

We study the problem of efficiently correcting an erroneous product of two $n\times n$ matrices over a ring. Among other things, we provide a randomized algorithm for correcting a matrix product with at most $k$ erroneous entries running in $\tilde{O}(n^2+kn)$ time and a deterministic $\tilde{O}(kn^2)$-time algorithm for this problem (where the notation $\tilde{O}$ suppresses polylogarithmic terms in $n$ and $k$).Comment: Fixed invalid reference to figure in v