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Erik Demaine

Publications and source records attributed to Erik Demaine.

10 recordsLinked to original sources
arXiv2022

Traversability, Reconfiguration, and Reachability in the Gadget Framework

Consider an agent traversing a graph of "gadgets", each with local state that changes with each traversal by the agent. We characterize the complexity of universal traversal, where the goal is to traverse every gadget at least once, for DAG gadgets, one-state gadgets, and reversible deterministic gadgets. We also study the complexity of reconfiguration, where the goal is to bring the system of gadgets to a specified state, proving many cases PSPACE-complete, and showing in some cases that reconfiguration can be strictly harder than reachability (where the goal is for the agent to reach a specified location), while in other cases, reachability is strictly harder than reconfiguration.

cs.CC↗
arXiv2021

Multidimensional Scaling: Approximation and Complexity

Metric Multidimensional scaling (MDS) is a classical method for generating meaningful (non-linear) low-dimensional embeddings of high-dimensional data. MDS has a long history in the statistics, machine learning, and graph drawing communities. In particular, the Kamada-Kawai force-directed graph drawing method is equivalent to MDS and is one of the most popular ways in practice to embed graphs into low dimensions. Despite its ubiquity, our theoretical understanding of MDS remains limited as its objective function is highly non-convex. In this paper, we prove that minimizing the Kamada-Kawai objective is NP-hard and give a provable approximation algorithm for optimizing it, which in particular is a PTAS on low-diameter graphs. We supplement this result with experiments suggesting possible connections between our greedy approximation algorithm and gradient-based methods.

cs.LG↗
arXiv2021

Edge-Unfolding Prismatoids: Tall or Rectangular Base

We show how to edge-unfold a new class of convex polyhedra, specifically a new class of prismatoids (the convex hull of two parallel convex polygons, called the top and base), by constructing a nonoverlapping "petal unfolding" in two new cases: (1) when the top and base are sufficiently far from each other; and (2) when the base is a rectangle and all other faces are nonobtuse triangles. The latter result extends a previous result by O'Rourke that the petal unfolding of a prismatoid avoids overlap when the base is a triangle (possibly obtuse) and all other faces are nonobtuse triangles. We also illustrate the difficulty of extending this result to a general quadrilateral base by giving a counterexample to our technique.

cs.CG↗
arXiv2020

Recursed is not Recursive: A Jarring Result

Recursed is a 2D puzzle platform video game featuring treasure chests that, when jumped into, instantiate a room that can later be exited (similar to function calls), optionally generating a jar that returns back to that room (similar to continuations). We prove that Recursed is RE-complete and thus undecidable (not recursive) by a reduction from the Post Correspondence Problem. Our reduction is "practical": the reduction from PCP results in fully playable levels that abide by all constraints governing levels (including the 15x20 room size) designed for the main game. Our reduction is also "efficient": a Turing machine can be simulated by a Recursed level whose size is linear in the encoding size of the Turing machine and whose solution length is polynomial in the running time of the Turing machine.

cs.AI↗
arXiv2017

Minimal forcing sets for 1D origami

This paper addresses the problem of finding minimum forcing sets in origami. The origami material folds flat along straight lines called creases that can be labeled as mountains or valleys. A forcing set is a subset of creases that force all the other creases to fold according to their labels. The result is a flat folding of the origami material. In this paper we develop a linear time algorithm that finds minimum forcing sets in one dimensional origami.

cs.DM↗
arXiv2016

Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement

We show that every orthogonal polyhedron of genus at most 2 can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal polyhedra beyond genus-0. Our unfolding algorithm relies on the existence of at most 2 special leaves in what we call the "unfolding tree" (which ties back to the genus), so unfolding polyhedra of genus 3 and beyond requires new techniques.

cs.CG↗
arXiv2014

Fast Dynamic Pointer Following via Link-Cut Trees

In this paper, we study the problem of fast dynamic pointer following: given a directed graph $G$ where each vertex has outdegree $1$, efficiently support the operations of i) changing the outgoing edge of any vertex, and ii) find the vertex $k$ vertices `after' a given vertex. We exhibit a solution to this problem based on link-cut trees that requires $O(\lg n)$ time per operation, and prove that this is optimal in the cell-probe complexity model.

cs.DS↗
arXiv2011

Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Theta(n^2) additional coordinate planes between every two such grid planes.

cs.CG↗
arXiv2000

On Reconfiguring Tree Linkages: Trees can Lock

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an $N$-link tree can have $2^{Ω(N)}$ equivalence classes of configurations.

cs.CG↗