Practical Range Minimum Queries Revisited

Finding the position of the minimal element in a subarray A[i..j] of an array A of size n is a fundamental operation in many applications. In 2011, Fischer and Heun presented the first index of size 2n+o(n) bits which answers the operation in constant time for any subarray. The index can be computed in linear time and queries can be answered without consulting the original array. The most recent and currently fastest practical index is due to Ferrada and Navarro (DCC\u2716). It reduces the range minimum query (RMQ) to more fundamental and well studied queries on binary vectors, namely rank and select, and a RMQ query on an array of sublinear size derived from A. A range min-max tree is employed to solve this recursive RMQ call. In this paper, we review their practical design and suggest a series of changes which result in consistently faster query times. Specifically, we provide a customized select implementation, switch to two levels of recursion, and use the sparse table solution for the recursion base case instead of a range min-max tree. We provide an extensive empirical evaluation of our new implementation and also compare it to the state of the art. Our experimental study shows that our proposal significantly outperforms the previous solutions on established benchmarks (up to a factor of three) and furthermore accelerates real world applications such as traversing a succinct tree or listing all distinct elements in an interval of an array

The Quantile Index - Succinct Self-Index for Top-k Document Retrieval

One of the central problems in information retrieval is that of finding the k documents in a large text collection that best match a query given by a user. A recent result of Navarro & Nekrich (SODA 2012) showed that single term and phrase queries of length m can be solved in optimal O(m+k) time using a linear word sized index. While a verbatim implementation of the index would be at least an order of magnitude larger than the original collection, various authors incrementally improved the index to a point where the space requirement is currently within a factor of 1.5 to 2.0 of the text size for standard collections. In this paper, we propose a new time/space trade-off for different top-k indexes. This is achieved by sampling only a quantile of the postings in the original inverted file or suffix array-based index. For those queries that cannot be answered using the sampled version of the index we show how to compute the query results on the fly efficiently. As an example, we apply our method to the top-k framework by Navarro & Nekrich. Under probabilistic assumptions that hold for most standard texts, and for a standard scoring function called term frequency, our index can be represented with only sublinearly many bits plus the space needed for a compressed suffix array of the text, while maintaining poly-logarithmic query times. We evaluate our solution on real-world datasets and compare its practical space usage and performance against state-of-the-art implementations. Our experiments show that our index compresses below the size of the original text. To our knowledge it is the first suffix array-based text index that is able to break this bound in practice even for non-repetitive collections, while still maintaining reasonable query times of under half a millisecond on average for top-10 queries

Interactions between Connected Half-Sarcomeres Produce Emergent Mechanical Behavior in a Mathematical Model of Muscle

Most reductionist theories of muscle attribute a fiber's mechanical properties to the scaled behavior of a single half-sarcomere. Mathematical models of this type can explain many of the known mechanical properties of muscle but have to incorporate a passive mechanical component that becomes ∼300% stiffer in activating conditions to reproduce the force response elicited by stretching a fast mammalian muscle fiber. The available experimental data suggests that titin filaments, which are the mostly likely source of the passive component, become at most ∼30% stiffer in saturating Ca2+ solutions. The work described in this manuscript used computer modeling to test an alternative systems theory that attributes the stretch response of a mammalian fiber to the composite behavior of a collection of half-sarcomeres. The principal finding was that the stretch response of a chemically permeabilized rabbit psoas fiber could be reproduced with a framework consisting of 300 half-sarcomeres arranged in 6 parallel myofibrils without requiring titin filaments to stiffen in activating solutions. Ablation of inter-myofibrillar links in the computer simulations lowered isometric force values and lowered energy absorption during a stretch. This computed behavior mimics effects previously observed in experiments using muscles from desmin-deficient mice in which the connections between Z-disks in adjacent myofibrils are presumably compromised. The current simulations suggest that muscle fibers exhibit emergent properties that reflect interactions between half-sarcomeres and are not properties of a single half-sarcomere in isolation. It is therefore likely that full quantitative understanding of a fiber's mechanical properties requires detailed analysis of a complete fiber system and cannot be achieved by focusing solely on the properties of a single half-sarcomere

Tension Recovery in Permeabilized Rat Soleus Muscle Fibers after Rapid Shortening and Restretch

Permeabilized rat soleus muscle fibers were subjected to rapid shortening/restretch protocols (20% muscle length, 20 ms duration) in solutions with pCa values ranging from 6.5 to 4.5. Force redeveloped after each restretch but temporarily exceeded the steady-state isometric tension reaching a maximum value ∼2.5 s after relengthening. The relative size of the overshoot was <5% in pCa 6.5 and pCa 4.5 solutions but equaled 17% ± 4% at pCa 6.0 (approximately half-maximal Ca(2+) activation). Muscle stiffness was estimated during pCa 6.0 activations by imposing length steps at different time intervals after repeated shortening/restretch perturbations. Relative stiffness and relative tension were correlated (p < 0.001) during recovery, suggesting that tension overshoots reflect a temporary increase in the number of attached cross-bridges. Rates of tension recovery (k(tr)) correlated (p < 0.001) with the relative residual force prevailing immediately after restretch. Force also recovered to the isometric value more quickly at 5.7 ≤ pCa ≤ 5.9 than at pCa 4.5 (ANOVA, p < 0.05). These results show that k(tr) measurements underestimate the rate of isometric force development during submaximal Ca(2+) activations and suggest that the rate of tension recovery is limited primarily by the availability of actin binding sites

Crossbridge properties during force enhancement by slow stretching in single intact frog muscle fibres

The mechanism of force enhancement during lengthening was investigated on single frog muscle fibres by using fast stretches to measure the rupture tension of the crossbridge ensemble. Fast stretches were applied to one end of the activated fibre and force responses were measured at the other. Sarcomere length was measured by a striation follower device. Fast stretching induced a linear increase of tension that reached a peak and fell before the end of the stretch indicating that a sudden increase of fibre compliance occurred due to forced crossbridge detachment induced by the fast loading. The peak tension (critical tension, Pc) and the sarcomere length needed to reach Pc (critical length, Lc) were measured at various tensions during the isometric tetanus rise and during force enhancement by slow lengthening. The data showed that Pc was proportional to the tension generated by the fibre under both isometric and slow lengthening conditions. However, for a given tension increase, Pc was 6.5 times greater during isometric than during lengthening conditions. Isometric critical length was 13.04 ± 0.17 nm per half-sarcomere (nm hs−1) independently of tension. During slow lengthening critical length fell as the force enhancement increased. For 90% enhancement, Lc reduced to 8.19 ± 0.039 nm hs−1. Assuming that the rupture force of the individual crossbridge is constant, these data indicate that the increase of crossbridge number during lengthening accounts for only 15.4% of the total force enhancement. The remaining 84.6% is accounted for by the increased mean strain of the crossbridges

A cross-bridge mechanism can explain the thixotropic short-range elastic component of relaxed frog skeletal muscle

The passive tension and sarcomere length of relaxed frog skeletal muscle fibres were measured in response to imposed length stretches. The tension response to a constant-velocity stretch exhibited a clear discontinuity. Tension rose more rapidly during the initial ∼ 0.4%L0 of the stretch than during the latter stages (where L0 is the resting length of the fibre). This initial tension response is attributed to the short-range elastic component (SREC).The use of paired triangular stretches revealed that the maximum tension produced during the SREC response of the second stretch was significantly reduced by the first stretch. This history-dependent behaviour of the SREC reflects thixotropic stiffness changes that have been previously described in relaxed muscle.The biphasic nature of the SREC tension response to movement was most apparent during the first imposed length change after a period at a fixed length, irrespective of the direction of movement.If a relaxed muscle was subjected to an imposed triangular length change so that the muscle was initially stretched and subsequently shortened back to its original fibre length, the resting tension at the end of the stretch was reduced relative to its initial pre-stretch value. Following the end of the stretch, tension slowly increased towards its initial value but the tension recovery was not accompanied by a contemporaneous increase in sarcomere length. This finding suggests that the resting tension of a relaxed muscle fibre is not entirely due to passive elasticity. The results are compatible with the suggestion that a portion of the resting tension - the filamentary resting tension (FRT) - is produced by a low level of active force generation.If a second identical stretch was imposed on the muscle at a time when the resting tension was reduced by the previous stretch, the maximal tension produced during the second stretch was the same as that produced during the first, despite the second stretch commencing from a lower initial resting tension.In experiments using paired triangular length changes, an inter-stretch interval of zero did not produce a substantially greater thixotropic reduction in the second stretch elastic limit force than an inter-stretch interval in the range 0.5-1 s.A theoretical model was developed in which the SREC and FRT arise as manifestations of a small number of slowly cycling cross-bridges linking the actin and myosin filaments of a relaxed skeletal muscle. The predictions of the model are compatible with many of the experimental observations. If the SREC and FRT are indeed due to cross-bridge activity, the model suggests that the cross-bridge attachment rate must increase during interfilamentary movement. A mechanism (based on misregistration between the actin binding sites and the myosin cross-bridges) by which this might arise is presented