arXiv subjects
Publications and source records attributed to Erik D. Demaine.
We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.
A convex surface that is flat everywhere but on finitely many smooth curves (or "seams") and points is a seam form. We show that the only creases through the flat components of a seam form are either between vertices or tangent to the seams. As corollaries we resolve open problems about certain special seam forms: the flat components of a D-form have no creases at all, and the flat component of a pita-form has at most one crease, between the seam's endpoints.
In general, the games are played on a host graph, where each node is a selfish independent agent (player) and each edge has a fixed link creation cost α. Together the agents create a network (a subgraph of the host graph) while selfishly minimizing the link creation costs plus the sum of the distances to all other players (usage cost). In this paper, we pursue two important facets of the network creation game. First, we study extensively a natural version of the game, called the cooperative model, where nodes can collaborate and share the cost of creating any edge in the host graph. We prove the first nontrivial bounds in this model, establishing that the price of anarchy is polylogarithmic in n for all values of α in complete host graphs. This bound is the first result of this type for any version of the network creation game; most previous general upper bounds are polynomial in n. Interestingly, we also show that equilibrium graphs have polylogarithmic diameter for the most natural range of α(at most n polylg n). Second, we study the impact of the natural assumption that the host graph is a general graph, not necessarily complete. This model is a simple example of nonuniform creation costs among the edges (effectively allowing weights of αand \infty). We prove the first assemblage of upper and lower bounds for this context, stablishing nontrivial tight bounds for many ranges of α, for both the unilateral and cooperative versions of network creation. In particular, we establish polynomial lower bounds for both versions and many ranges of α, even for this simple nonuniform cost model, which sharply contrasts the conjectured constant bounds for these games in complete (uniform) graphs.
The Carpenter's Rule Theorem states that any chain linkage in the plane can be folded continuously between any two configurations while preserving the bar lengths and without the bars crossing. However, this theorem applies only to strictly simple configurations, where bars intersect only at their common endpoints. We generalize the theorem to self-touching configurations, where bars can touch but not properly cross. At the heart of our proof is a new definition of self-touching configurations of planar linkages, based on an annotated configuration space and limits of nontouching configurations. We show that this definition is equivalent to the previously proposed definition of self-touching configurations, which is based on a combinatorial description of overlapping features. Using our new definition, we prove the generalized Carpenter's Rule Theorem using a topological argument. We believe that our topological methodology provides a powerful tool for manipulating many kinds of self-touching objects, such as 3D hinged assemblies of polygons and rigid origami. In particular, we show how to apply our methodology to extend to self-touching configurations universal reconfigurability results for open chains with slender polygonal adornments, and single-vertex rigid origami with convex cones.
Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.
We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma.
We introduce staged self-assembly of Wang tiles, where tiles can be added dynamically in sequence and where intermediate constructions can be stored for later mixing. This model and its various constraints and performance measures are motivated by a practical nanofabrication scenario through protein-based bioengineering. Staging allows us to break through the traditional lower bounds in tile self-assembly by encoding the shape in the staging algorithm instead of the tiles. All of our results are based on the practical assumption that only a constant number of glues, and thus only a constant number of tiles, can be engineered, as each new glue type requires significant biochemical research and experiments. Under this assumption, traditional tile self-assembly cannot even manufacture an n*n square; in contrast, we show how staged assembly enables manufacture of arbitrary orthogonal shapes in a variety of precise formulations of the model.
We give a counterexample to a conjecture of Poon [Poo06] that any orthogonal tree in two dimensions can always be flattened by a continuous motion that preserves edge lengths and avoids self-intersection. We show our example is locked by extending results on strongly locked self-touching linkages due to Connelly, Demaine and Rote [CDR02] to allow zero-length edges as defined in [ADG07], which may be of independent interest. Our results also yield a locked tree with only eleven edges, which is the smallest known example of a locked tree.
We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years. Our result generalizes and indeed builds upon the result from 1814 that polygons have common dissections (without hinges). We also extend our common dissection result to edge-hinged dissections of solid 3D polyhedra that have a common (unhinged) dissection, as determined by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are constructive, giving explicit algorithms in all cases. For a constant number of planar polygons, both the number of pieces and running time required by our construction are pseudopolynomial. This bound is the best possible, even for unhinged dissections. Hinged dissections have possible applications to reconfigurable robotics, programmable matter, and nanomanufacturing.
We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erdős in the plane.
The {\em edit distance} between two ordered trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this paper, we present a worst-case $O(n^3)$-time algorithm for this problem, improving the previous best $O(n^3\log n)$-time algorithm~\cite{Klein}. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems (which is interesting in its own right), together with a deeper understanding of the previous algorithms for the problem. We also prove the optimality of our algorithm among the family of \emph{decomposition strategy} algorithms--which also includes the previous fastest algorithms--by tightening the known lower bound of $Ω(n^2\log^2 n)$~\cite{Touzet} to $Ω(n^3)$, matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds of $Θ(n m^2 (1 + \log \frac{n}{m}))$ when the two trees have different sizes $m$ and~$n$, where $m < n$.
We develop dynamic dictionaries on the word RAM that use asymptotically optimal space, up to constant factors, subject to insertions and deletions, and subject to supporting perfect-hashing queries and/or membership queries, each operation in constant time with high probability. When supporting only membership queries, we attain the optimal space bound of Theta(n lg(u/n)) bits, where n and u are the sizes of the dictionary and the universe, respectively. Previous dictionaries either did not achieve this space bound or had time bounds that were only expected and amortized. When supporting perfect-hashing queries, the optimal space bound depends on the range {1,2,...,n+t} of hashcodes allowed as output. We prove that the optimal space bound is Theta(n lglg(u/n) + n lg(n/(t+1))) bits when supporting only perfect-hashing queries, and it is Theta(n lg(u/n) + n lg(n/(t+1))) bits when also supporting membership queries. All upper bounds are new, as is the Omega(n lg(n/(t+1))) lower bound.
This paper gives processor-allocation algorithms for minimizing the average number of communication hops between the assigned processors for grid architectures, in the presence of occupied cells. The simpler problem of assigning processors on a free grid has been studied by Karp, McKellar, and Wong who show that the solutions have nontrivial structure; they left open the complexity of the problem. The associated clustering problem is as follows: Given n points in Re^d, find k points that minimize their average pairwise L1 distance. We present a natural approximation algorithm and show that it is a 7/4-approximation for 2D grids. For d-dimensional space, the approximation guarantee is 2-(1/2d), which is tight. We also give a polynomial-time approximation scheme (PTAS) for constant dimension d, and report on experimental results.
We give the first algorithmic study of a class of ``covering tour'' problems related to the geometric Traveling Salesman Problem: Find a polygonal tour for a cutter so that it sweeps out a specified region (``pocket''), in order to minimize a cost that depends mainly on the number of em turns. These problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems, as well as arc routing (``postman'') problems with turn penalties. We prove the NP-completeness of minimum-turn milling and give efficient approximation algorithms for several natural versions of the problem, including a polynomial-time approximation scheme based on a novel adaptation of the m-guillotine method.
We develop a new technique for proving cell-probe lower bounds on dynamic data structures. This technique enables us to prove an amortized randomized Omega(lg n) lower bound per operation for several data structural problems on n elements, including partial sums, dynamic connectivity among disjoint paths (or a forest or a graph), and several other dynamic graph problems (by simple reductions). Such a lower bound breaks a long-standing barrier of Omega(lg n / lglg n) for any dynamic language membership problem. It also establishes the optimality of several existing data structures, such as Sleator and Tarjan's dynamic trees. We also prove the first Omega(log_B n) lower bound in the external-memory model without assumptions on the data structure (such as the comparison model). Our lower bounds also give a query-update trade-off curve matched, e.g., by several data structures for dynamic connectivity in graphs. We also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update.
We consider the problem of searching for an object on a line at an unknown distance OPT from the original position of the searcher, in the presence of a cost of d for each time the searcher changes direction. This is a generalization of the well-studied linear-search problem. We describe a strategy that is guaranteed to find the object at a cost of at most 9*OPT + 2d, which has the optimal competitive ratio 9 with respect to OPT plus the minimum corresponding additive term. Our argument for upper and lower bound uses an infinite linear program, which we solve by experimental solution of an infinite series of approximating finite linear programs, estimating the limits, and solving the resulting recurrences. We feel that this technique is interesting in its own right and should help solve other searching problems. In particular, we consider the star search or cow-path problem with turn cost, where the hidden object is placed on one of m rays emanating from the original position of the searcher. For this problem we give a tight bound of (1+(2(m^m)/((m-1)^(m-1))) OPT + m ((m/(m-1))^(m-1) - 1) d. We also discuss tradeoff between the corresponding coefficients, and briefly consider randomized strategies on the line.
In this paper we extend the theory of bidimensionality to two families of graphs that do not exclude fixed minors: map graphs and power graphs. In both cases we prove a polynomial relation between the treewidth of a graph in the family and the size of the largest grid minor. These bounds improve the running times of a broad class of fixed-parameter algorithms. Our novel technique of using approximate max-min relations between treewidth and size of grid minors is powerful, and we show how it can also be used, e.g., to prove a linear relation between the treewidth of a bounded-genus graph and the treewidth of its dual.
We present a nondeterministic model of computation based on reversing edge directions in weighted directed graphs with minimum in-flow constraints on vertices. Deciding whether this simple graph model can be manipulated in order to reverse the direction of a particular edge is shown to be PSPACE-complete by a reduction from Quantified Boolean Formulas. We prove this result in a variety of special cases including planar graphs and highly restricted vertex configurations, some of which correspond to a kind of passive constraint logic. Our framework is inspired by (and indeed a generalization of) the ``Generalized Rush Hour Logic'' developed by Flake and Baum. We illustrate the importance of our model of computation by giving simple reductions to show that several motion-planning problems are PSPACE-hard. Our main result along these lines is that classic unrestricted sliding-block puzzles are PSPACE-hard, even if the pieces are restricted to be all dominoes (1x2 blocks) and the goal is simply to move a particular piece. No prior complexity results were known about these puzzles. This result can be seen as a strengthening of the existing result that the restricted Rush Hour puzzles are PSPACE-complete, of which we also give a simpler proof. Finally, we strengthen the existing result that the pushing-blocks puzzle Sokoban is PSPACE-complete, by showing that it is PSPACE-complete even if no barriers are allowed.