Tetris is Hard, Even to Approximate

In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p^(1-epsilon), when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of (2 - epsilon), for any epsilon>0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets.Comment: 56 pages, 11 figure

Increasing the simulation performance of large-scale evacuations using parallel computing techniques based on domain decomposition

Evacuation simulation has the potential to be used as part of a decision support system during large-scale incidents to provide advice to incident commanders. To be viable in these applications, it is essential that the simulation can run many times faster than real time. Parallel processing is a method of reducing run times for very large computational simulations by distributing the workload amongst a number of processors. This paper presents the development of a parallel version of the rule based evacuation simulation software buildingEXODUS using domain decomposition. Four Case Studies (CS) were tested using a cluster, consisting of 10 Intel Core 2 Duo (dual core) 3.16 GHz CPUs. CS-1 involved an idealised large geometry, with 20 exits, intended to illustrate the peak computational speed up performance of the parallel implementation, the population consisted of 100,000 agents; the peak computational speedup (PCS) was 14.6 and the peak real-time speedup (PRTS) was 4.0. CS-2 was a long area with a single exit area with a population of 100,000 agents; the PCS was 13.2 and the PRTS was 17.2. CS-3 was a 50 storey high rise building with a population of 8000/16,000 agents; the PCS was 2.48/4.49 and the PRTS was 17.9/12.9. CS-4 is a large realistic urban area with 60,000/120,000 agents; the PCS was 5.3/6.89 and the PRTS was 5.31/3.0. This type of computational performance opens evacuation simulation to a range of new innovative application areas such as real-time incident support, dynamic signage in smart buildings and virtual training environments

Spinal afferent neurons projecting to the rat lung and pleura express acid sensitive channels

BACKGROUND: The acid sensitive ion channels TRPV1 (transient receptor potential vanilloid receptor-1) and ASIC3 (acid sensing ion channel-3) respond to tissue acidification in the range that occurs during painful conditions such as inflammation and ischemia. Here, we investigated to which extent they are expressed by rat dorsal root ganglion neurons projecting to lung and pleura, respectively. METHODS: The tracer DiI was either injected into the left lung or applied to the costal pleura. Retrogradely labelled dorsal root ganglion neurons were subjected to triple-labelling immunohistochemistry using antisera against TRPV1, ASIC3 and neurofilament 68 (marker for myelinated neurons), and their soma diameter was measured. RESULTS: Whereas 22% of pulmonary spinal afferents contained neither channel-immunoreactivity, at least one is expressed by 97% of pleural afferents. TRPV1(+)/ASIC3(- )neurons with probably slow conduction velocity (small soma, neurofilament 68-negative) were significantly more frequent among pleural (35%) than pulmonary afferents (20%). TRPV1(+)/ASIC3(+ )neurons amounted to 14 and 10% respectively. TRPV1(-)/ASIC3(+ )neurons made up between 44% (lung) and 48% (pleura) of neurons, and half of them presumably conducted in the A-fibre range (larger soma, neurofilament 68-positive). CONCLUSION: Rat pleural and pulmonary spinal afferents express at least two different acid-sensitive channels that make them suitable to monitor tissue acidification. Patterns of co-expression and structural markers define neuronal subgroups that can be inferred to subserve different functions and may initiate specific reflex responses. The higher prevalence of TRPV1(+)/ASIC3(- )neurons among pleural afferents probably reflects the high sensitivity of the parietal pleura to painful stimuli