Trimming and Gluing Gray Codes

We consider the algorithmic problem of generating each subset of [n]:={1,2,...,n} whose size is in some interval [k,l], 0 <= k <= l <= n, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k=0 and l=n this is the classical problem of generating all 2^n subsets of [n] by element additions/removals, and for k=l this is the classical problem of generating all n over k subsets of [n] by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values n, k, and l, improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant time O(1) per generated set and space O(n). Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the n-dimensional cube Q_n induced by all levels [k,l]. We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of Q_(2k+1) induced by all levels [k-c,k+1+c], c in {0, 1, ...k}, has a Hamilton cycle. We also prove an approximate version of this conjecture, showing that this graph has a cycle that visits a (1-o(1))-fraction of all vertices

A short proof of the middle levels theorem

Consider the graph that has as vertices all bitstrings of length $2n+1$ with exactly $n$ or $n+1$ entries equal to 1, and an edge between any two bitstrings that differ in exactly one bit. The well-known middle levels conjecture asserts that this graph has a Hamilton cycle for any $n\geq 1$. In this paper we present a new proof of this conjecture, which is much shorter and more accessible than the original proof

A constant-time algorithm for middle levels Gray codes

For any integer $n\geq 1$ a middle levels Gray code is a cyclic listing of all $n$-element and $(n+1)$-element subsets of $\{1,2,\ldots,2n+1\}$ such that any two consecutive subsets differ in adding or removing a single element. The question whether such a Gray code exists for any $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T. M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence proof was turned into an algorithm that computes each new set in the Gray code in time $\mathcal{O}(n)$ on average. In this work we present an algorithm for computing a middle levels Gray code in optimal time and space: each new set is generated in time $\mathcal{O}(1)$ on average, and the required space is $\mathcal{O}(n)$

Improving lumber recovery of low-quality hardwoods via finger-jointing technologies

The purpose of this project was to improve hardwood lumber recovery from low quality logs and lumber of Appalachian species by using finger-jointing technologies to create value-added products. Currently, there is an abundance of low quality lumber created by sawmill operations that cannot be efficiently utilized. The high presence of defects in the lumber makes processing this material costly and therefore little market exists to utilize this resource. Creating value added products from this material can help to improve forest health and alleviate the demand of quality wood products.;This project processed a total of 4,800 board feet of low-grade lumber to determine the volume of usable wood contained within low-grade lumber. Four common Appalachian species were salvaged; black cherry, soft maple, red oak, and yellow-poplar; and subsequently finger-jointed, end-to-end to create long usable stock. Lumber was then edge-glued to create solid panels which could be used in furniture manufacturing.;The recovery ratios, size distribution, mechanical and physical properties of different species were investigated and compared. Yellow-poplar produced the highest recovery ratios followed by red oak, cherry, and maple. Finger-jointed, edge-glued panels were created and their mechanical and physical properties were evaluated. Results indicated that the panels could perform suitably for their intended end-use. The recovery ratio of converting rough, low-grade lumber, into solid panels was approximately 33%. Cost/benefit analyses were performed to estimate the profitability of the process. Based on current value of solid edge-glued panels, cherry and red oak were the most profitable species to process

An Exploratory Study of: Designers\u27 Finishing Knowledge and the Communication Flow between Designers and Finishing Professionals

Several fundamental questions about graphic designers\u27 print finishing knowledge motivated this thesis. Is there a gap in knowledge between the fields of finishing and design? What do designers actually know about finishing? Is this knowledge enough to enable them to communicate well with printers/finishers and design printed pieces efficiently so finishing runs smoothly? What happens when mistakes are made due to a lack of education and communication? If designers do need to know more information about finishing, what are the most important concepts? Are there any resources now that designers can use to learn more about finishing? Do printers/finishers have advice for designers about finishing? In order to collect information on this subject, a questionnaire was developed, twenty-two professionals in the Rochester (NY) area graphic arts industry were interviewed and their answers were recorded. The interview results were analyzed and common themes were discovered. Designers do not have adequate finishing knowledge and there is a communication gap between designers and finishing professionals. Those just starting out in the industry with a degree in Graphic Design have not been previously taught about finishing and do not have the necessary knowledge of finishing operations and procedures to make correct file creation decisions. Design professionals rarely communicate with printers/finishers except to acquire a print quote or sign a press approval. Designers also do not know what questions to ask. They see their projects on one-dimensional computer monitors and are not aware of the three-dimensional problems that occur in the finishing world. Designers often do not learn from mistakes made during project workflows because they are not aware of the reasons for those mistakes. Printers/finishers do not share this information with designers, increasing the chances that the mistakes will reoccur. In the current workflow, designers and printers/finishers often do not interact or communicate with each other before projects are given to printers to be printed. Many assumptions are made throughout the workflow. Finishers think designers know the correct finishing information or are informed by their printers/printing salespersons. Printing salespersons often lack the knowledge about finishing operations and procedures. Sometimes printing salespersons have knowledgeable suggestions, but it is often too late in the workflow for designers to make changes to the digital files. The lack of knowledge and communication in the graphic arts workflow results in the loss of time, money and resources for everyone involved. Each person in the graphic arts workflow needs to play his/her part in providing proper education and closing the communication gap. In college Graphic Design programs, designers need to be educated about the entire workflow, from start to finish, including finishing materials, processes, limitations, key terms, financial ramifications and proper digital file setup. Designers need to know IX the right questions to ask printers/finishers about each of the finishing processes. Designers should tour a finishing facility and acquire hands-on finishing experience. Design professionals need to ask printers/finishers questions about finishing requirements for proper digital file setup. Before submitting final designs to printers, designers should go over physical mock-ups with printers and finishers to ensure the designs meet printing/finishing requirements. Designers need to be open to feedback and suggestions made by printers/finishers and need to build time into their workflow so the suggested changes can be made. Designers need to educate clients paying for the projects on the importance of this step in the workflow and educate clients about production timelines. Printers and printing salespersons need to be knowledgeable about finishing so they can answer designers\u27 questions. Printers/finishers need to provide designers with examples of good design decisions. They should also show designers examples of poor design and the results of incorrect digital file-preparation decisions. Printers/finishers should create guide books or other resources about their specific printing/finishing requirements in order to help designers understand their processes

Gray Codes and Symmetric Chains

We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1-l,n+l], where 1 = 12

Space-Optimal Quasi-Gray Codes with Logarithmic Read Complexity

A quasi-Gray code of dimension n and length l over an alphabet Sigma is a sequence of distinct words w_1,w_2,...,w_l from Sigma^n such that any two consecutive words differ in at most c coordinates, for some fixed constant c>0. In this paper we are interested in the read and write complexity of quasi-Gray codes in the bit-probe model, where we measure the number of symbols read and written in order to transform any word w_i into its successor w_{i+1}. We present construction of quasi-Gray codes of dimension n and length 3^n over the ternary alphabet {0,1,2} with worst-case read complexity O(log n) and write complexity 2. This generalizes to arbitrary odd-size alphabets. For the binary alphabet, we present quasi-Gray codes of dimension n and length at least 2^n - 20n with worst-case read complexity 6+log n and write complexity 2. This complements a recent result by Raskin [Raskin \u2717] who shows that any quasi-Gray code over binary alphabet of length 2^n has read complexity Omega(n). Our results significantly improve on previously known constructions and for the odd-size alphabets we break the Omega(n) worst-case barrier for space-optimal (non-redundant) quasi-Gray codes with constant number of writes. We obtain our results via a novel application of algebraic tools together with the principles of catalytic computation [Buhrman et al. \u2714, Ben-Or and Cleve \u2792, Barrington \u2789, Coppersmith and Grossman \u2775]

Gray codes and symmetric chains

We consider the problem of constructing a cyclic listing of all bitstrings of length~$2n+1$ with Hamming weights in the interval $[n+1-\ell,n+\ell]$, where $1\leq \ell\leq n+1$, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (the case~$\ell=1$). We provide a solution for the case~$\ell=2$ and solve a relaxed version of the problem for general values of~$\ell$, by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the $n$-dimensional hypercube for any~$n\geq 12$