Travis minimizer muonline is the free tool for Mu online game. It's developed for people who want to minimize your muonline game and run other programs while playing ( i.e. Internet Browser, Chat Client, other MU software etc ).
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Many minimizers are based on the idea of gradient descent, i.e., they compute the local gradient of \(f(\vec\theta)\) and follow the function along the direction of steepest discent until the minimum. There are however also gradient-free algorithms, like for example COBYLA. While going into the details of how each algorithm works is beyond the scope, we illustrate here an example by using the Minuit algorithm.
Now we found the global minimum. This is a simple example to show that the solution find by a local minimizers can depend on the starting point, and might not be the global minimum. In practice, one can rarely be guaranteed that the likelihood function has only one minimum. This is especially true in many dimensions and in cases of data with poor statistic.
The idea behind this is very simple: the user defines a grid of values for the parameters, which are used as starting points for minimization performed by a local minimizers. At the end, the solution with the smallest value for \(-\logL\) will be used as final solution.
Of course the GRID minimizer can be used in multiple dimensions (simply define a grid for the other parameters as well). It is a simple brute force solution that works well in practice, especially when the likelihood function computation is not too time-consuming. When there are many parameters, you should choose carefully the parameters to use in the grid. For example, when looking for a spectral line in a spectrum, it makes sense to use the location of the line as parameter in the grid, butnot its normalization.
We study different implementation schemes based on these ideas and compare their space performance, as well as associated query time. Our results show that our algorithms are useful for both low and high entropy datasets. For large enough k (and large enough minimizers lengths), we are able to compress count values in less space than their empirical entropy while retaining fast query times. To the best of our knowledge, this is the first implementation proposing such a compact representation. We also study an extension of our algorithm to the approximate case for which we save additional space by allowing a pre-defined absolute error over queries.
Minimizers have been successfully applied to various data-intensive sequence analysis problems in bioinformatics, such as metagenomics (Kraken [29]) or minimizing cache misses in k-mer counting (KMC [7]), or mapping and assembling long single-molecule reads [30, 31]. Recently, there has been a series of works on both theoretical and practical aspects of designing efficient minimizers, see e.g. [32, 33] and references therein.
A key idea to reduce the computational burden of counting k-mers, is to use minimizers to bucket k-mers and split the counting process across multiple tables (cf e.g. [7]). Here we use the same principle to bucket count values instead of k-mers themselves. Let \(M_m(K)=\\mu _m(q)\,\) be the set of minimizers of all k-mers of K of a given length \(m
Our first implementation is named AMB (from AMBiguity). An extended version of AMB (explained below) is presented in Algorithm 3. For non-ambiguous minimizers u, AMB defines g(u) to be the unique value of the bucket. For ambiguous minimizers v, we set \(g(v)=0\), where 0 is viewed as a special value marking ambiguous buckets (k-mers with count 0 are not present in the input). This has the disadvantage of providing no information about the values of ambiguous buckets, and also of making g less compressible (because of an additional value). On the other hand, this has the advantage of distinguishing between ambiguous and non-ambiguous buckets and allows the query to immediately return the answer for k-mers hashing to non-ambiguous buckets. As a consequence, unambiguous k-mers are not propagated to the second layer, and if \(g(\mu _m(q)) \ne 0\) it can be immediately returned as f(q). We then have to store mapping f restricted only to k-mers from ambiguous buckets, which we denote \(\tildef\). Both mappings g and \(\tildef\) are stored using BCSFs.
The multi-layer scheme is particularly intuitive for the AMB implementation, where each layer stores a unique value for non-ambiguous minimizers and a special value 0 otherwise. In this case, \(K_i\) consists of those k-mers of \(K_i-1\) hashed to ambiguous buckets, and \(f_i\) is simply a restriction of f to those k-mers. Algorithm 3 shows a pseudo-code of multi-level AMB extended to the approximate case (see Sect. 3.4 below). The multi-layer version of the FIL scheme is shown in Appendix (Algorithm 4).
Unless stated otherwise, FIL and AMB were run on all possible combinations of two and three minimizer lengths for \(k \in [13, 15, 18, 21]\) with only the best combinations reported using the following naming convention:
With very skewed data, collisions of k-mer counts may happen between unrelated k-mers simply because one counter value strongly dominates the spectrum. In order to demonstrate the utility of minimizers in a more general setting other than whole genome count tables, we applied our methods to less skewed distributions. To this end, we compressed the k-mer count tables when using dataset SRR10211353 whose results are presented in Fig. 3. As opposed to fully assembled genomes, entropy in this case remains well above 1 even for larger values of k. Nonetheless, both AMB and FIL are able to produce representations more compact than both simple CSFs and BCSFs for all \(k > 13\), beating the entropy lower bound.
In order to show how the approximate algorithm achieves better compression ratios, k was chosen from [10, 11, 12, 13], a range of values which is particularly difficult for AMB (or FIL) with \(\delta =0\). Trying all possible minimizer combinations compatible with such ks, the best results are obtained for very short minimizer lengths (between 1 and 5). Building minimizer layers for such small values of m does not lead to better compression than simple (B)CSFs, with Fig. 7 showing no tangible differences between (B)CSFs and AMB (or FIL). For these reasons, minimizer lengths in Fig. 6 are equal to \(k-1\) (and \(k-2\)) for every choice of k (e.g. if \(k=10\), layers will be 8, 9, 10 for three-layer AMB). Using the same small lengths of the exact case would not allow meaningful bucketing of counts values.
In all reported cases, good minimizer lengths for the first layer (\(m_0\)) follow the rule: \(m_0 > m_s = (\log _4 G+ 2)\) with \(G\), the size in base pairs of the genome. Smaller \(m_0\), are no longer capable of partitioning k-mers in a meaningful way. Furthermore, space tends to first monotonically decrease to a minimum for increasing minimizer lengths, to increase again once the optimal value is passed. It is therefore possible to find the minimum by sequentially trying all possible minimizers greater than \(m_s\) and stop as soon as the compressed size starts to increase again.
If it is not possible to choose \(m_0 > m_s = (\log _4 G+ 2)\) because, e.g. k is already too small, approximation might be a viable option even for relatively small \(\delta \). The only caveat to pay attention to in this case is to check if a minimizer layer would be useful or not. If yes, \(\delta \) can be incremented without further adjustments compared to exact case. If not, minimizer lengths for the bucketing layers should be chosen as big as possible to allow meaningful bucketing of count values.
We validated our algorithms on four different types of count tables, two fully assembled genomes (E.Coli and C.Elegans) of different sizes, one dataset of E.Coli reads at 10x coverage and one document frequency table of 29 different E.Coli genomes, for different k-mer lengths showing how BCSF, AMB and FIL behave in different situations. AMB and FIL have a clear advantage when minimizers are long enough to bucket k-mers in a meaningful way, for both skewed and high entropy data. When it is not possible to define a long-enough minimizer length, the advantage of using intermediate minimizer layers vanishes, and simple CSF and its BCSF provide a better solution.
At query time, CSF and BCSF are the fastest methods requiring about 100ns on average for a single query. For a fixed number of layers, AMB is faster than FIL in all situations when minimizers are useful. FIL becomes faster than AMB only for those cases when both algorithms achieve worse compression ratios than simple (B)CSF.
All construction code is written in python, except for the CSF part which is handled by a simple Java program using Sux4J [24]. An utility written in C using the code provided by Sux4J for reading and querying its CSFs provides time measurements. We use xxHash to define an ordering over minimizers. All our code is available at
DB proposed the idea of using Compressed Static Functions with Bloom filters. YS proposed to use minimizers for count bucketing (AMB algorithm). GK proposed the FIL algorithm. YS developed and tested the software. YS, DB and GK analysed the data. YS wrote the manuscript, with editorial contribution and supervision from GK and DB. All authors read and approved the final manuscript.
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