Efficient Computation of Sequence Mappability

Sequence mappability is an important task in genome re-sequencing. In the $(k,m)$-mappability problem, for a given sequence $T$ of length $n$, our goal is to compute a table whose $i$th entry is the number of indices $j \ne i$ such that length-$m$ substrings of $T$ starting at positions $i$ and $j$ have at most $k$ mismatches. Previous works on this problem focused on heuristic approaches to compute a rough approximation of the result or on the case of $k=1$. We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that works in $\mathcal{O}(n \min\{m^k,\log^{k+1} n\})$ time and $\mathcal{O}(n)$ space for $k=\mathcal{O}(1)$. It requires a carefu l adaptation of the technique of Cole et al.~[STOC 2004] to avoid multiple counting of pairs of substrings. We also show $\mathcal{O}(n^2)$-time algorithms to compute all results for a fixed $m$ and all $k=0,\ldots,m$ or a fixed $k$ and all $m=k,\ldots,n-1$. Finally we show that the $(k,m)$-mappability problem cannot be solved in strongly subquadratic time for $k,m = \Theta(\log n)$ unless the Strong Exponential Time Hypothesis fails.Comment: Accepted to SPIRE 201

Efficient Algorithms for Scheduling Moldable Tasks

We study the problem of scheduling $n$ independent moldable tasks on $m$ processors that arises in large-scale parallel computations. When tasks are monotonic, the best known result is a $(\frac{3}{2}+\epsilon)$-approximation algorithm for makespan minimization with a complexity linear in $n$ and polynomial in $\log{m}$ and $\frac{1}{\epsilon}$ where $\epsilon$ is arbitrarily small. We propose a new perspective of the existing speedup models: the speedup of a task $T_{j}$ is linear when the number $p$ of assigned processors is small (up to a threshold $\delta_{j}$) while it presents monotonicity when $p$ ranges in $[\delta_{j}, k_{j}]$; the bound $k_{j}$ indicates an unacceptable overhead when parallelizing on too many processors. For a given integer $\delta\geq 5$, let $u=\left\lceil \sqrt[2]{\delta} \right\rceil-1$. In this paper, we propose a $\frac{1}{\theta(\delta)} (1+\epsilon)$-approximation algorithm for makespan minimization with a complexity $\mathcal{O}(n\log{\frac{n}{\epsilon}}\log{m})$ where $\theta(\delta) = \frac{u+1}{u+2}\left( 1- \frac{k}{m} \right)$ ($m\gg k$). As a by-product, we also propose a $\theta(\delta)$-approximation algorithm for throughput maximization with a common deadline with a complexity $\mathcal{O}(n^{2}\log{m})$

Vinogradov systems with a slice off

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\ldots +x_s^j=y_1^j+\ldots +y_s^j\quad (\text{$1\le j\le k$, $j\ne r$}),$$ with $1\le x_i,y_i\le X$ $(1\le i\le s)$. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for $I_{s,k,r}(X)$ for $1\le r\le k-1$. In particular, when $s,k\in \mathbb N$ satisfy $k\ge 3$ and $1\le s\le (k^2-1)/2$, we establish the essentially diagonal behaviour $I_{s,k,1}(X)\ll X^{s+\epsilon}$.Comment: 19 page

Efficient self-sustained pulsed CO laser

In this paper a simple sealed-off TEA CO laser is described with a self-sustained discharge without an external UV preionization source. At 77 K this system yields more than 600 mJ from a lasing volume of about 60 cm3 CO-N2-He mixture (45 J/ℓ atm. with 15.6% efficiency)

Cache-Oblivious Selection in Sorted X+Y Matrices

Let X[0..n-1] and Y[0..m-1] be two sorted arrays, and define the mxn matrix A by A[j][i]=X[i]+Y[j]. Frederickson and Johnson gave an efficient algorithm for selecting the k-th smallest element from A. We show how to make this algorithm IO-efficient. Our cache-oblivious algorithm performs O((m+n)/B) IOs, where B is the block size of memory transfers

Statistics of Partial Minima

Motivated by multi-objective optimization, we study extrema of a set of N points independently distributed inside the d-dimensional hypercube. A point in this set is k-dominated by another point when at least k of its coordinates are larger, and is a k-minimum if it is not k-dominated by any other point. We obtain statistical properties of these partial minima using exact probabilistic methods and heuristic scaling techniques. The average number of partial minima, A, decays algebraically with the total number of points, A ~ N^{-(d-k)/k}, when 1<=k<d. Interestingly, there are k-1 distinct scaling laws characterizing the largest coordinates as the distribution P(y_j) of the jth largest coordinate, y_j, decays algebraically, P(y_j) ~ (y_j)^{-alpha_j-1}, with alpha_j=j(d-k)/(k-j) for 1<=j<=k-1. The average number of partial minima grows logarithmically, A ~ [1/(d-1)!](ln N)^{d-1}, when k=d. The full distribution of the number of minima is obtained in closed form in two-dimensions.Comment: 6 pages, 1 figur