An Efficient Algorithm For Chinese Postman Walk on Bi-directed de Bruijn Graphs

Sequence assembly from short reads is an important problem in biology. It is known that solving the sequence assembly problem exactly on a bi-directed de Bruijn graph or a string graph is intractable. However finding a Shortest Double stranded DNA string (SDDNA) containing all the k-long words in the reads seems to be a good heuristic to get close to the original genome. This problem is equivalent to finding a cyclic Chinese Postman (CP) walk on the underlying un-weighted bi-directed de Bruijn graph built from the reads. The Chinese Postman walk Problem (CPP) is solved by reducing it to a general bi-directed flow on this graph which runs in O(|E|2 log2(|V |)) time. In this paper we show that the cyclic CPP on bi-directed graphs can be solved without reducing it to bi-directed flow. We present a ?(p(|V | + |E|) log(|V |) + (dmaxp)3) time algorithm to solve the cyclic CPP on a weighted bi-directed de Bruijn graph, where p = max{|{v|din(v) - dout(v) > 0}|, |{v|din(v) - dout(v) < 0}|} and dmax = max{|din(v) - dout(v)}. Our algorithm performs asymptotically better than the bidirected flow algorithm when the number of imbalanced nodes p is much less than the nodes in the bi-directed graph. From our experimental results on various datasets, we have noticed that the value of p/|V | lies between 0.08% and 0.13% with 95% probability

The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks

It is challenging to design large and low-cost communication networks. In this paper, we formulate this challenge as the prize-collecting Steiner Tree Problem (PCSTP). The objective is to minimize the costs of transmission routes and the disconnected monetary or informational profits. Initially, we note that the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to improve suboptimal solutions to PCSTP. Based on these techniques, we propose two fast heuristic algorithms: the first one is a quasilinear time heuristic algorithm that is faster and consumes less memory than other algorithms; and the second one is an improvement of a stateof-the-art polynomial time heuristic algorithm that can find high-quality solutions at a speed that is only inferior to the first one. We demonstrate the competitiveness of our heuristic algorithms by comparing them with the state-of-the-art ones on the largest existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we generate new instances that are even larger (1 000 000 vertices and 10 000 000 edges) to further demonstrate their advantages in large networks. The state-ofthe-art algorithms are too slow to find high-quality solutions for instances of this size, whereas our new heuristic algorithms can do this in around 6 to 45s on a personal computer. Ultimately, we apply our post-processing techniques to update the bestknown solution for a notoriously difficult benchmark instance to show that they can improve near-optimal solutions to PCSTP. In conclusion, we demonstrate the usefulness of our heuristic algorithms and post-processing techniques for designing large and low-cost communication networks

Principles of Dataset Versioning: Exploring the Recreation/Storage Tradeoff

The relative ease of collaborative data science and analysis has led to a proliferation of many thousands or millions of $versions$ of the same datasets in many scientific and commercial domains, acquired or constructed at various stages of data analysis across many users, and often over long periods of time. Managing, storing, and recreating these dataset versions is a non-trivial task. The fundamental challenge here is the $storage-recreation\;trade-off$: the more storage we use, the faster it is to recreate or retrieve versions, while the less storage we use, the slower it is to recreate or retrieve versions. Despite the fundamental nature of this problem, there has been a surprisingly little amount of work on it. In this paper, we study this trade-off in a principled manner: we formulate six problems under various settings, trading off these quantities in various ways, demonstrate that most of the problems are intractable, and propose a suite of inexpensive heuristics drawing from techniques in delay-constrained scheduling, and spanning tree literature, to solve these problems. We have built a prototype version management system, that aims to serve as a foundation to our DATAHUB system for facilitating collaborative data science. We demonstrate, via extensive experiments, that our proposed heuristics provide efficient solutions in practical dataset versioning scenarios

A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and $k$-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems

On the Complexity of Spill Everywhere under SSA Form

Compilation for embedded processors can be either aggressive (time consuming cross-compilation) or just in time (embedded and usually dynamic). The heuristics used in dynamic compilation are highly constrained by limited resources, time and memory in particular. Recent results on the SSA form open promising directions for the design of new register allocation heuristics for embedded systems and especially for embedded compilation. In particular, heuristics based on tree scan with two separated phases -- one for spilling, then one for coloring/coalescing -- seem good candidates for designing memory-friendly, fast, and competitive register allocators. Still, also because of the side effect on power consumption, the minimization of loads and stores overhead (spilling problem) is an important issue. This paper provides an exhaustive study of the complexity of the ``spill everywhere'' problem in the context of the SSA form. Unfortunately, conversely to our initial hopes, many of the questions we raised lead to NP-completeness results. We identify some polynomial cases but that are impractical in JIT context. Nevertheless, they can give hints to simplify formulations for the design of aggressive allocators.Comment: 10 page