The Complexity of Snake

Snake and Nibbler are two well-known video games in which a snake slithers through a maze and grows as it collects food. During this process, the snake must avoid any collision with its tail. Various goals can be associated with these video games, such as avoiding the tail as long as possible, or collecting a certain amount of food, or reaching some target location. Unfortunately, like many other motion-planning problems, even very restricted variants are computationally intractable. In particular, we prove the NP--hardness of collecting all food on solid grid graphs; as well as its PSPACE-completeness on general grid graphs. Moreover, given an initial and a target configuration of the snake, moving from one configuration to the other is PSPACE-complete, even on grid graphs without food, or with an initially short snake. Our results make use of the nondeterministic constraint logic framework by Hearn and Demaine, which has been used to analyze the computational complexity of many games and puzzles. We extend this framework for the analysis of puzzles whose initial state is chosen by the player

Using yeast two-hybrid system and molecular dynamics simulation to detect venom protein-protein interactions

Proteins and peptides are major components of snake venom. Venom protein transcriptomes and proteomes of many snake species have been reported; however, snake venom complexity (i.e., the venom protein-protein interactions, PPIs) remains largely unknown. To detect the venom protein interactions, we used the most common snake venom component, phospholipase A2s (PLA2s) as a “bait” to identify the interactions between PLA2s and 14 of the most common proteins in Western diamondback rattlesnake (Crotalus atrox) venom by using yeast two-hybrid (Y2H) analysis, a technique used to detect PPIs. As a result, we identified PLA2s interacting with themselves, and lysing-49 PLA2 (Lys49 PLA2) interacting with venom cysteine-rich secretory protein (CRISP). To reveal the complex structure of Lys49 PLA2-CRISP interaction at the structural level, we first built the three-dimensional (3D) structures of Lys49 PLA2 and CRISP by a widely used computational program-MODELLER. The binding modes of Lys49 PLA2-CRISP interaction were then predicted through three different docking programs including ClusPro, ZDOCK and HADDOCK. Furthermore, the most likely complex structure of Lys49 PLA2-CRISP was inferred by molecular dynamic (MD) simulations with GROMACS software. The techniques used and results obtained from this study strengthen the understanding of snake venom protein interactions and pave the way for the study of animal venom complexity

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

Snake: a Stochastic Proximal Gradient Algorithm for Regularized Problems over Large Graphs

A regularized optimization problem over a large unstructured graph is studied, where the regularization term is tied to the graph geometry. Typical regularization examples include the total variation and the Laplacian regularizations over the graph. When applying the proximal gradient algorithm to solve this problem, there exist quite affordable methods to implement the proximity operator (backward step) in the special case where the graph is a simple path without loops. In this paper, an algorithm, referred to as "Snake", is proposed to solve such regularized problems over general graphs, by taking benefit of these fast methods. The algorithm consists in properly selecting random simple paths in the graph and performing the proximal gradient algorithm over these simple paths. This algorithm is an instance of a new general stochastic proximal gradient algorithm, whose convergence is proven. Applications to trend filtering and graph inpainting are provided among others. Numerical experiments are conducted over large graphs

Parallel matrix inversion techniques

In this paper, we present techniques for inverting sparse, symmetric and positive definite matrices on parallel and distributed computers. We propose two algorithms, one for SIMD implementation and the other for MIMD implementation. These algorithms are modified versions of Gaussian elimination and they take into account the sparseness of the matrix. Our algorithms perform better than the general parallel Gaussian elimination algorithm. In order to demonstrate the usefulness of our technique, we implemented the snake problem using our sparse matrix algorithm. Our studies reveal that the proposed sparse matrix inversion algorithm significantly reduces the time taken for obtaining the solution of the snake problem. In this paper, we present the results of our experimental work

Inventing an arsenal: adaptive evolution and neofunctionalization of snake venom phospholipase A(2 )genes

BACKGROUND: Gene duplication followed by functional divergence has long been hypothesized to be the main source of molecular novelty. Convincing examples of neofunctionalization, however, remain rare. Snake venom phospholipase A(2 )genes are members of large multigene families with many diverse functions, thus they are excellent models to study the emergence of novel functions after gene duplications. RESULTS: Here, I show that positive Darwinian selection and neofunctionalization is common in snake venom phospholipase A(2 )genes. The pattern of gene duplication and positive selection indicates that adaptive molecular evolution occurs immediately after duplication events as novel functions emerge and continues as gene families diversify and are refined. Surprisingly, adaptive evolution of group-I phospholipases in elapids is also associated with speciation events, suggesting adaptation of the phospholipase arsenal to novel prey species after niche shifts. Mapping the location of sites under positive selection onto the crystal structure of phospholipase A(2 )identified regions evolving under diversifying selection are located on the molecular surface and are likely protein-protein interactions sites essential for toxin functions. CONCLUSION: These data show that increases in genomic complexity (through gene duplications) can lead to phenotypic complexity (venom composition) and that positive Darwinian selection is a common evolutionary force in snake venoms. Finally, regions identified under selection on the surface of phospholipase A(2 )enzymes are potential candidate sites for structure based antivenin design

Determination of the Topology of a Directed Network

We consider strongly-connected directed networks of identical synchronous, finite-state processors with in- and out-degree uniformly bounded by a network constant. Via a straightforward extension of Ostrovsky and Wilkerson's Backwards Communication Algorithm in [OW], we exhibit a protocol which solves the Global Topology Determination Problem, the problem of having the root processor map the global topology of a network of unknown size and topology, with running time O(ND) where N represents the number of processors and D represents the diameter of the network. A simple counting argument suffices to show that the Global Topology Determination Problem has time-complexity Omega(N logN) which makes the protocol presented asymptotically time-optimal for many large networks.Comment: 9 pages, no figures, accepted to appear in IPDPS 2002 (unable to attend), (journal version to appear in Information Processing Letters