Divide And Conquer Version 5 Download !!HOT!!

I would like to transform a recursive algorithm on an array into an iterative function. It is not a tail recursive algorithm and has two recursive calls followed by some operation.The algorithm is a divide-and-conquer algorithm where at each step the array is split into two subarrays and some function f is applied to the two previous outcomes. In practice f is complicated, so the iterative algorithm should use the function f, for a minimal working example I have used a simple addition.

Edit : The real problem I am interested in has as input an array of tensors of different sizes, and the operation f solves a linear program involving these tensors and outputs a new tensor. I cannot iterate simply over the initial array since the size of the output of f grows exponentially in this case. This is why I use this divide and conquer approach, which reduces this size. The recursive program works fine, but slows down dramatically for large size, possibly due to the frames that python opens and keeps track of.

Divide And Conquer Version 5 Download

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The Master Theorem is a tool used to solve recurrence relations that arise in the analysis of divide-and-conquer algorithms. The Master Theorem provides a systematic way of solving recurrence relations of the form:

The above algorithm divides the problem into a subproblems, each of size n/b and solve them recursively to compute the problem and the extra work done for problem is given by f(n), i.e., the time to create the subproblems and combine their results in the above procedure.

This paper proposes a new approach to writing and verifying divide-and-conquer programs in Coq. Extending the rich line of previous work on algebraic approaches to recursion schemes, we present an algebraic approach to divide-and-conquer recursion: recursions are represented as a form of algebra, and from outer recursions, one may initiate inner recursions that can construct data upon which the outer recursions may legally recurse. Termination is enforced entirely by the typing discipline of our recursion schemes. Despite this, our approach requires little from the underlying type system, and can be implemented in System F plus a limited form of positive-recursive types. Our implementation of the method in Coq does not rely on structural recursion or on dependent types. The method is demonstrated on several examples, including mergesort, quicksort, Harper's regular-expression matcher, and others. An indexed version is also derived, implementing a form of divide-and-conquer induction that can be used to reason about functions defined via our method.

Scaling scRNA-seq to profile millions of cells is crucial for constructing high-resolution maps of transcriptional manifolds. Current analysis strategies, in particular dimensionality reduction and two-phase clustering, offer only limited scaling and sensitivity to define such manifolds. We introduce Metacell-2, a recursive divide-and-conquer algorithm allowing efficient decomposition of scRNA-seq datasets of any size into small and cohesive groups of cells called metacells. Metacell-2 improves outlier cell detection and rare cell type identification, as shown with human bone marrow cell atlas and mouse embryonic data. Metacell-2 is implemented over the scanpy framework for easy integration in any analysis pipeline.

Here, we introduce a new and greatly improved Metacell algorithm (MC2) that supports practically unlimited scaling, using an iterative divide-and-conquer approach. In addition to the divide-and-conquer scheme, the algorithm uses a new graph partition score to avoid time-consuming resampling and directly control metacell sizes, implements a new adaptive outlier detection module for better robustness, and employs a rare-gene-module detector ensuring very high sensitivity for detecting transcriptional states that are present in as little as 0.01% of the data. Our efficient implementation of the MC2 algorithm [21] can quickly compute metacells from any matrix to power quantitative and robust downstream analysis using scanpy [22], Seurat [23] or other toolsets [24, 25], or interactive visualization with the metacell viewer/annotator [26].

Schematics of the MC2 algorithm. The input is a large UMI matrix, and the output is a partition of cell into metacells and final outlier cells. MC2 is deriving a solution using recursive two-phase process. It first divides the data into random piles and generates low-quality metacells and outliers from them. It then (recursively) groups low-quality metacells into coherent piles and repartitions these piles to generate high-quality metacells. The algorithm is ensuring high sensitivity of rare behavior detection by identifying rare metacell through a pre-process, as well as through regrouping of outlier cells that are being pooled from all piles at both phases of the algorithm

Robustness of the divide-and-conquer metacell algorithm. A Distribution of metacell normalized inner variance for the PBMC dataset, using the Baran et al. algorithm (orange) vs. MC2 two-sided stability score optimization, working on the entire data in a single pile (i.e., no divide and conquer, green). B Distribution of normalized inner variance for the PMBC dataset using the full MC2 algorithms (blue) vs. the single-pile algorithm (green). C Metacell graph derived by MC2 on the PMBC dataset. Annotation as in Baran et al. D Distribution of metacell normalized inner variance for HSC and MPP cells, when using full MC2 on the HCA bone marrow data set (orange) or when restricting analysis to MPP/HSC cells alone (blue). E Metacell graph for the full HCA BM data set and for metacells computed on the zoomed-in HSC/MPP subset

When using a divide-and-conquer algorithm, we face a tradeoff between the size of the piles used and the quality of the results (Fig. 3C). Increasing the pile size provides diminishing quality returns, at the cost of a rapidly increasing run-time. For the mouse atlas, we have chosen a pile size that would generate (on average) 50 metacells per pile (this parameter is adjustable by the user).

One of the major challenges in analyzing very large-scale scRNA-seq is the sensitive detection of rare behaviors. Such behaviors may be lost when sub-sampling data and can require more statistical power for detection within vast samples of less informative recurrent states. MC2 uses two mechanisms for detecting rare behaviors: the first involving a pre-process that searches for rare gene modules and the second using the MC2 divide-and-conquer algorithm outlier detection and the recursive analysis of detected outliers for regrouping and inference of rare metacells. We screened the organogenesis metacell cover to identify genes expressed in one metacell at least 8-fold higher than in 99.8% of all other metacells. This resulted in identifying 260 genes spanning over 30 clusters with two or more genes, each representing a defined rare cell state (Fig. 4A). 47 of these genes were detected during the pre-process stage of MC2, while all others were detected within piles of common outliers or as specific metacell state within a coherent pile.

The implementation of MC2 and the original metacell algorithm is tuned for the typical distributions observed in scRNA-seq, and their application to other single-cell genomics data (e.g., scATAC-seq, scBIS-seq) is recommended only if adequate similarity metrics and feature selection strategies are developed. Such adaptation is not described here. It is however natural to use the divide-and-conquer strategy introduced here for scaling analysis of large-scale single-cell omics of multiple types.

Metacell-2 is an effective and scalable solution for transforming sparse large-scale single-cell RNA-seq dataset into quantitative metacell models. The algorithm divide-and-conquer approach is implementing a strategy for detecting rare behaviors that maintain very high sensitivity even for large datasets. The tool can be easily incorporated into pipelines performing additional downstream modeling of transcriptional manifolds and atlases.

MC2 can be thought of as a combination of (i) an improvement of the previous version of Metacell algorithm (MC1) and (ii) overall divide and conquer and outlier cell routing scheme that is running parallel independent instances of the improved basic algorithm. Overall, this allows processing large data sets efficiently. The description below will focus on the novel components of MC2: the detection of rare modules, graph partition goal function and optimization, outlier filtering scheme, and the overall divide-and-conquer design. We will also outline briefly components that are based more heavily on MC1, including mainly the feature gene selection and balanced K-nn graph construction.

To handle such cases, MC2 implements a rare gene module detector that efficiently pre-processes the entire UMI matrix and enhances sparse gene features. This stage detects rare gene modules, collects the cells that express these modules, and invokes the divide-and-conquer algorithm on the cells of each such module separately from the rest of the cell population. The resulting metacells are passed to the final output. The overall flow is working as follows (default main parameters are also specified in Additional File 1: Table T1):

All the threshold parameters used above are tuned to maximize complementarity between this rare gene module detection pre-processing, and phases of outlier cell detection and re-clustering within the main MC2 divide-and-conquer algorithm. We do not anticipate scenarios requiring adjustment of these parameters.

MC2 uses the same feature selection method as MC1, but feature selection is performed independently in each divide-and-conquer pile. Therefore, in the later phases of the algorithm, when all the cells in a pile are similar, MC2 can focus on genes that distinguish sub-types of the common cell type, providing additional sensitivity over a global algorithm.

After removing the deviant cells, we may end with some metacells that are too small. We completely dissolve any such too-small metacells, using the same method as described in step 4 (min-size control) above. We mark all the deviant and dissolved cells as outliers and move them to later re-analysis within the hierarchical divide-and-conquer scheme we next describe. ff782bc1db