Sarah Eisenstat

Two Hands Are Better Than One (up to constant factors): Self-Assembly In The 2HAM vs. aTAM

We study the dierence between the standard seeded model (aTAM) of tile self-assembly, and the “seedless” two-handed model of tile self-assembly (2HAM). Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model. 1998 ACM Subject Classification F.1.2

Remarks on separating words

The separating words problem asks for the size of the smallest DFA needed to distinguish between two words of length ≤ n (by accepting one and rejecting the other). In this paper we survey what is known and unknown about the problem, consider some variations, and prove several new results.

One-dimensional staged self-assembly

We introduce the problem of staged self-assembly of one-dimensional nanostructures, which becomes interesting when the elements are labeled (e.g., representing functional units that must be placed at specific locations). In a restricted model in which each operation has a single terminal assembly, we prove that assembling a given string of labels with the fewest stages is equivalent, up to constant factors, to compressing the string to be uniquely derived from the smallest possible context-free grammar (a well-studied O(log n)-approximable problem). Without this restriction, we show that the optimal assembly can be substantially smaller than the optimal context-free grammar, by a factor of Ω(√n/ log n) even for binary strings of length n. Fortunately, we can bound this separation in model power by a quadratic function in the number of distinct glues or tiles allowed in the assembly, which is typically small in practice.

Finding the limit: examining the potential and complexity of compilation scheduling for JIT-based runtime systems

This work aims to find out the full potential of compilation scheduling for JIT-based runtime systems. Compilation scheduling determines the order in which the compilation units (e.g., functions) in a program are to be compiled or recompiled. It decides when what versions of the units are ready to run, and hence affects performance. But it has been a largely overlooked direction in JIT-related research, with some fundamental questions left open: How significant compilation scheduling is for performance, how good the scheduling schemes employed by existing runtime systems are, and whether a great potential exists for improvement. This study proves the strong NP-completeness of the problem, proposes a heuristic algorithm that yields near optimal schedules, examines the potential of two current scheduling schemes empirically, and explores the relations with JIT designs. It provides the first principled understanding to the complexity and potential of compilation scheduling, shedding some insights for JIT-based runtime system improvement.

Algorithms for Designing Pop-Up Cards

We prove that every simple polygon can be made as a (2D) pop-up card/book that opens to any desired angle between 0 and 360°. More precisely, given a simple polygon attached to the two walls of the open pop-up, our polynomial-time algorithm subdivides the polygon into a single-degree-of-freedom linkage structure, such that closing the pop-up flattens the linkage without collision. This result solves an open problem of Hara and Sugihara from 2009. We also show how to obtain a more efficient construction for the special case of orthogonal polygons, and how to make 3D orthogonal polyhedra, from pop-ups that open to 90°, 180°, 270°, or 360°.

Variations on Instant Insanity

In one of the first papers about the complexity of puzzles, Robertson and Munro [14] proved that a generalized form of the then-popular Instant Insanity puzzle is NP-complete. Here we study several variations of this puzzle, exploring how the complexity depends on the piece shapes and the allowable orientations of those shapes.

FOLDING EQUILATERAL PLANE GRAPHS

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

Flattening fixed-angle chains is strongly NP-hard

Planar configurations of fixed-angle chains and trees are well studied in polymer science and molecular biology. We prove that it is strongly NP-hard to decide whether a polygonal chain with fixed edge lengths and angles has a planar configuration without crossings. In particular, flattening is NP-hard when all the edge lengths are equal, whereas a previous (weak) NP-hardness proof used lengths that differ in size by an exponential factor. Our NP-hardness result also holds for (nonequilateral) chains with angles in the range [60° - e, 180°], whereas flattening is known to be always possible (and hence polynomially solvable) for equilateral chains with angles in the range (60°, 150°) and for general chains with angles in the range [90°, 180°]. We also show that the flattening problem is strongly NP-hard for equilateral fixed-angle trees, even when every angle is either 90° or 180°. Finally, we show that strong NP-hardness carries over to the previously studied problems of computing the minimum or maximum span (distance between endpoints) among non-crossing planar configurations.

Finding a Hamiltonian Path in a Cube with Specified Turns is Hard

We prove the NP-completeness of finding a Hamiltonian path in an N ×N ×N cube graph with turns exactly at specified lengths along the path. This result establishes NP-completeness of Snake Cube puzzles: folding a chain of N 3 unit cubes, joined at face centers (usually by a cord passing through all the cubes), into an N × N × N cube. Along the way, we prove a universality result that zig-zag chains (which must turn every unit) can fold into any polycube after 4×4×4 refinement, or into any Hamiltonian polycube after 2 × 2 × 2 refinement.

Solving the Rubik's Cube Optimally is NP-complete

In this paper, we prove that optimally solving an