This paper presents an efficient reversible algorithm for linear regression, both with and withou... more This paper presents an efficient reversible algorithm for linear regression, both with and without ridge regression. Our reversible algorithm matches the asymptotic time and space complexity of standard irreversible algorithms for this problem. Needed for this result is the expansion of the analysis of efficient reversible matrix multiplication to rectangular matrices and matrix inversion.
We study Snipperclips, a computer puzzle game whose objective is to create a target shape with tw... more We study Snipperclips, a computer puzzle game whose objective is to create a target shape with two tools. The tools start as constant-complexity shapes, and each tool can snip (i.e., subtract its current shape from) the other tool. We study the computational problem of, given a target shape represented by a polygonal domain of n vertices, is it possible to create it as one of the tools' shape via a sequence of snip operations? If so, how many snip operations are required? We consider several variants of the problem (such as allowing the tools to be disconnected and/or using an undo operation) and bound the number of operations needed for each of the variants.
Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a... more Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree. These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems); polynomial-time algorithms for simple graph classes such as cliques, stars, paths, and cycles; and constant-factor approximation algorithms in some cases. The two natural cases of sequenti...
In the Nikoli pencil-and-paper game Tatamibari, a puzzle consists of an m × n grid of cells, wher... more In the Nikoli pencil-and-paper game Tatamibari, a puzzle consists of an m × n grid of cells, where each cell possibly contains a clue among +, -, |. The goal is to partition the grid into disjoint rectangles, where every rectangle contains exactly one clue, rectangles containing + are square, rectangles containing - are strictly longer horizontally than vertically, rectangles containing | are strictly longer vertically than horizontally, and no four rectangles share a corner. We prove this puzzle NP-complete, establishing a Nikoli gap of 16 years. Along the way, we introduce a gadget framework for proving hardness of similar puzzles involving area coverage, and show that it applies to an existing NP-hardness proof for Spiral Galaxies. We also present a mathematical puzzle font for Tatamibari.
This paper shows NP-completeness for finding Hamiltonian cycles in induced subgraphs of the dual ... more This paper shows NP-completeness for finding Hamiltonian cycles in induced subgraphs of the dual graphs of semi-regular tessilations. It also shows NP-hardness for a new, wide class of graphs called augmented square grids. This work follows up on prior studies of the complexity of finding Hamiltonian cycles in regular and semi-regular grid graphs.
This paper proves that push-pull block puzzles in 3D are PSPACE-complete to solve, and push-pull ... more This paper proves that push-pull block puzzles in 3D are PSPACE-complete to solve, and push-pull block puzzles in 2D with thin walls are NP-hard to solve, settling an open question [ZR11]. Pushpull block puzzles are a type of recreational motion planning problem, similar to Sokoban, that involve moving a 'robot' on a square grid with 1 × 1 obstacles. The obstacles cannot be traversed by the robot, but some can be pushed and pulled by the robot into adjacent squares. Thin walls prevent movement between two adjacent squares. This work follows in a long line of algorithms and complexity work on similar problems [Wil91, DDO00, Hof00, DHH04, DH01, DO92, DHH02, Cul98, DZ96, Rit10]. The 2D push-pull block puzzle shows up in the video games Pukoban as well as The Legend of Zelda: A Link to the Past, giving another proof of hardness for the latter [ADGV14]. This variant of block-pushing puzzles is of particular interest because of its connections to reversibility, since any action (e.g., push or pull) can be inverted by another valid action (e.g., pull or push).
This paper looks at the recent field of relativistic quantum cryptography, which uses quantum mec... more This paper looks at the recent field of relativistic quantum cryptography, which uses quantum mechanics and relativity to produce guarantees about cryptographic security. We analyze some of their security assumptions in these protocols particularly those of Minkowski space-times and perfect knowledge of the communication path. We show how an attacker could use the gravitational bending of space-time to break these cryptographic protocols. We also discuss measures to make this more difficult and some situations in which these attacks are not feasible.
We prove the NP-completeness of finding a Hamiltonian path in an N × N × N cube graph with turns ... more 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.
International Journal of Computational Geometry & Applications, 2013
We consider two types of folding applied to equilateral plane graph linkages. First, under contin... more 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 origa...
Suppose an "escaping" player moves continuously at maximum speed 1 in the interior of a... more Suppose an "escaping" player moves continuously at maximum speed 1 in the interior of a region, while a "pursuing" player moves continuously at maximum speed r outside the region. For what r can the first player escape the region, that is, reach the boundary a positive distance away from the pursuing player, assuming optimal play by both players? We formalize a model for this infinitesimally alternating 2-player game that we prove has a unique winner in any region with locally rectifiable boundary, avoiding pathological behaviors (where both players can have "winning strategies") previously identified for pursuit-evasion games such as the Lion and Man problem in certain metric spaces. For some regions, including both equilateral triangle and square, we give exact results for the critical speed ratio, above which the pursuing player can win and below which the escaping player can win (and at which the pursuing player can win). For simple polygons, we giv...
We consider the computational complexity of winning this turn (mate-in-1 or "finding lethal&... more We consider the computational complexity of winning this turn (mate-in-1 or "finding lethal") in Hearthstone as well as several other single turn puzzle types introduced in the Boomsday Lab expansion. We consider three natural generalizations of Hearthstone (in which hand size, board size, and deck size scale) and prove the various puzzle types in each generalization NP-hard.
We build a general theory for characterizing the computational complexity of motion planning of r... more We build a general theory for characterizing the computational complexity of motion planning of robot(s) through a graph of "gadgets", where each gadget has its own state defining a set of allowed traversals which in turn modify the gadget's state. We study two families of such gadgets, one which naturally leads to motion planning problems with polynomially bounded solutions, and another which leads to polynomially unbounded (potentially exponential) solutions. We also study a range of competitive game-theoretic scenarios, from one player controlling one robot to teams of players each controlling their own robot and racing to achieve their team's goal. Under small restrictions on these gadgets, we fully characterize the complexity of bounded 1-player motion planning (NL vs. NP-complete), unbounded 1-player motion planning (NL vs. PSPACE-complete), and bounded 2-player motion planning (P vs. PSPACE-complete), and we partially characterize the complexity of unbounded...
This paper studies optimal-area visibility representations of n-vertex outer-1-plane graphs, i.e.... more This paper studies optimal-area visibility representations of n-vertex outer-1-plane graphs, i.e. graphs with a given embedding where all vertices are on the boundary of the outer face and each edge is crossed at most once. We show that any graph of this family admits an embedding-preserving visibility representation whose area is O(n) and prove that this area bound is worst-case optimal. We also show that O(n) area can be achieved if we represent the vertices as Lshaped orthogonal polygons or if we do not respect the embedding but still have at most one crossing per edge. We also extend the study to other representation models and, among other results, construct asymptotically optimal O(n pw(G)) area bar-1-visibility representations, where pw(G) ∈ O(logn) is the pathwidth of the outer-1-planar graph G.
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Papers by Jayson Lynch