Cutting Bamboo down to Size

This paper studies the problem of programming a robotic panda gardener to keep a bamboo garden from obstructing the view of the lake by your house. The garden consists of n bamboo stalks with known daily growth rates and the gardener can cut at most one bamboo per day. As a computer scientist, you found out that this problem has already been formalized in [G?sieniec et al., SOFSEM\u2717] as the Bamboo Garden Trimming (BGT) problem, where the goal is that of computing a perpetual schedule (i.e., the sequence of bamboos to cut) for the robotic gardener to follow in order to minimize the makespan, i.e., the maximum height ever reached by a bamboo. Two natural strategies are Reduce-Max and Reduce-Fastest(x). Reduce-Max trims the tallest bamboo of the day, while Reduce-Fastest(x) trims the fastest growing bamboo among the ones that are taller than x. It is known that Reduce-Max and Reduce-Fastest(x) achieve a makespan of O(log n) and 4 for the best choice of x = 2, respectively. We prove the first constant upper bound of 9 for Reduce-Max and improve the one for Reduce-Fastest(x) to (3+?5)/2 < 2.62 for x = 1+1/?5. Another critical aspect stems from the fact that your robotic gardener has a limited amount of processing power and memory. It is then important for the algorithm to be able to quickly determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a Trimming Oracle data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by Reduce-Max and Reduce-Fastest(x)

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 $O(\log n)$

The Variable-Processor Cup Game

The problem of scheduling tasks on $p$ processors so that no task ever gets too far behind is often described as a game with cups and water. In the $p$-processor cup game on $n$ cups, there are two players, a filler and an emptier, that take turns adding and removing water from a set of $n$ cups. In each turn, the filler adds $p$ units of water to the cups, placing at most $1$ unit of water in each cup, and then the emptier selects $p$ cups to remove up to $1$ unit of water from. The emptier's goal is to minimize the backlog, which is the height of the fullest cup. The $p$-processor cup game has been studied in many different settings, dating back to the late 1960's. All of the past work shares one common assumption: that $p$ is fixed. This paper initiates the study of what happens when the number of available processors $p$ varies over time, resulting in what we call the \emph{variable-processor cup game}. Remarkably, the optimal bounds for the variable-processor cup game differ dramatically from its classical counterpart. Whereas the $p$-processor cup has optimal backlog $\Theta(\log n)$, the variable-processor game has optimal backlog $\Theta(n)$. Moreover, there is an efficient filling strategy that yields backlog $\Omega(n^{1 - \epsilon})$ in quasi-polynomial time against any deterministic emptying strategy. We additionally show that straightforward uses of randomization cannot be used to help the emptier. In particular, for any positive constant $\Delta$, and any $\Delta$-greedy-like randomized emptying algorithm $\mathcal{A}$, there is a filling strategy that achieves backlog $\Omega(n^{1 - \epsilon})$ against $\mathcal{A}$ in quasi-polynomial time

An aesthetic for sustainable interactions in product-service systems?

Copyright @ 2012 Greenleaf PublishingEco-efficient Product-Service System (PSS) innovations represent a promising approach to sustainability. However the application of this concept is still very limited because its implementation and diffusion is hindered by several barriers (cultural, corporate and regulative ones). The paper investigates the barriers that affect the attractiveness and acceptation of eco-efficient PSS alternatives, and opens the debate on the aesthetic of eco-efficient PSS, and the way in which aesthetic could enhance some specific inner qualities of this kinds of innovations. Integrating insights from semiotics, the paper outlines some first research hypothesis on how the aesthetic elements of an eco-efficient PSS could facilitate user attraction, acceptation and satisfaction

Water Integration for Squamscott Exeter (WISE): Preliminary Integrated Plan, Final Technical Report

This document introduces the goals, background and primary elements of an Integrated Plan for the Lower Exeter and Squamscott River in the Great Bay estuary in southern New Hampshire. This Plan will support management of point (wastewater treatment plant) and nonpoint sources in the communities of Exeter, Stratham and Newfields. The Plan also identifies and quantifies the advantages of the use of green infrastructure as a critical tool for nitrogen management and describes how collaboration between those communities could form the basis for an integrated plan. The Plan will help communities meet new wastewater and proposed stormwater permit requirements. Critical next steps are need before this Plan will fulfill the 2018 Nitrogen Control Plan requirements for Exeter and proposed draft MS4 requirements for both Stratham and Exeter. These next steps include conducting a financial capability assessment, development of an implementation schedule and development of a detailed implementation plan. The collaborative process used to develop this Plan was designed to provide decision makers at the local, state and federal levels with the knowledge they need to trust the Plan’s findings and recommendations, and to enable discussions between stakeholders to continue the collaborative process. This Plan includes the following information to guide local response to new federal permit requirements for treating and discharging stormwater and wastewater: Sources of annual pollutant load quantified by type and community; Assessment and evaluation of different treatment control strategies for each type of pollutant load; Assessment and evaluation of nutrient control strategies designed to reduce specific types of pollutants; Evaluation of a range of point source controls at the wastewater treatment facility based on regulatory requirements; Costs associated with a range of potential control strategies to achieve reduction of nitrogen and other pollutants of concern; and A preliminary implementation schedule with milestones for target load reductions using specific practices for specific land uses at points in time; Recommendations on how to implement a tracking and accounting program to document implementation; Design tools such as BMP performance curves for crediting the use of structural practices to support nitrogen accounting requirements; and Next Steps for how to complete this Plan

Vegetative regeneration and distribution of Fallopia japonica and Fallopia x bohemica: implications for control and management

Fallopia japonica (Houtt.) Ronse Decraene (Japanese knotweed), an introduced, invasive, rhizomatous perennial plant, has become an increasing problem for nature conservation and land management in both rural and urban areas in the British Isles. In the native range of the plant, Japan, Taiwan and northern China, a number of varieties are recorded. Three congeners of F. japonica are present in the British Isles, F. sachalinensis, F. japonica var. conipacta and F. baldschuanica in addition to a hybrid F. x bohemica. An investigation by postal survey of the distribution of the hybrid F. x bohemica has identified 131 records for the British Isles. Both male and female plants of F. x bohemica have been recorded. Current understanding suggests that only female plants of F. japonica are present in the British Isles, inferring that the only means of reproduction is through vegetative regeneration. High rates of regeneration were recorded in this study for stem and rhizome material for both F. japonica and F. x bohemica in an aquatic and terrestrial environment. Implications of vegetative regeneration are discussed in terms of current management practices and future methods of control. A combination of digging with a mechanical excavator followed by spraying with the herbicide glyphosate decreased the time required to achieve an effective level of control of F. japonica compared to spraying alone. Fragmentation of the rhizome system through digging resulted in an increase in stem density allowing a more effective delivery of herbicide. Implications in terms of costs for F. japonica treatment on sites awaiting re-development are discussed. Analysis of data collated from surveys of F. japonica in Swansea using a Geographical Information System suggest that the primary habitats infested are waste ground and stream and river banks. Results suggest that disturbance, both by natural means and by human intervention has been the primary cause of spread of F. japonica in the British Isles. Management strategies are proposed which take account of these results and measures are put forward to help prevent future infestations

Optimal Time-Backlog Tradeoffs for the Variable-Processor Cup Game

The \emph{$p$-processor cup game} is a classic and widely studied scheduling problem that captures the setting in which a $p$-processor machine must assign tasks to processors over time in order to ensure that no individual task ever falls too far behind. The problem is formalized as a multi-round game in which two players, a filler (who assigns work to tasks) and an emptier (who schedules tasks) compete. The emptier's goal is to minimize backlog, which is the maximum amount of outstanding work for any task. Recently, Kuszmaul and Westover (ITCS, 2021) proposed the \emph{variable-processor cup game}, which considers the same problem, except that the amount of resources available to the players (i.e., the number $p$ of processors) fluctuates between rounds of the game. They showed that this seemingly small modification fundamentally changes the dynamics of the game: whereas the optimal backlog in the fixed $p$-processor game is $\Theta(\log n)$, independent of $p$, the optimal backlog in the variable-processor game is $\Theta(n)$. The latter result was only known to apply to games with \emph{exponentially many} rounds, however, and it has remained an open question what the optimal tradeoff between time and backlog is for shorter games. This paper establishes a tight trade-off curve between time and backlog in the variable-processor cup game. Importantly, we prove that for a game consisting of $t$ rounds, the optimal backlog is $\Theta(n)$ if and only if $t \ge \Omega(n^3)$. Our techniques also allow for us to resolve several other open questions concerning how the variable-processor cup game behaves in beyond-worst-case-analysis settings.Comment: 40 pages, published in International Conference on Automata, Languages, and Programming (ICALP), 2022. Abstract abridged for arXiv submission: see paper for full abstract. Updated to acknowledge additional fundin