Online Scheduling on Identical Machines using SRPT

Due to its optimality on a single machine for the problem of minimizing average flow time, Shortest-Remaining-Processing-Time (\srpt) appears to be the most natural algorithm to consider for the problem of minimizing average flow time on multiple identical machines. It is known that \srpt achieves the best possible competitive ratio on multiple machines up to a constant factor. Using resource augmentation, \srpt is known to achieve total flow time at most that of the optimal solution when given machines of speed $2- \frac{1}{m}$. Further, it is known that \srpt's competitive ratio improves as the speed increases; \srpt is $s$-speed $\frac{1}{s}$-competitive when $s \geq 2- \frac{1}{m}$. However, a gap has persisted in our understanding of \srpt. Before this work, the performance of \srpt was not known when \srpt is given (1+\eps)-speed when 0 < \eps < 1-\frac{1}{m}, even though it has been thought that \srpt is (1+\eps)-speed $O(1)$-competitive for over a decade. Resolving this question was suggested in Open Problem 2.9 from the survey "Online Scheduling" by Pruhs, Sgall, and Torng \cite{PruhsST}, and we answer the question in this paper. We show that \srpt is \emph{scalable} on $m$ identical machines. That is, we show \srpt is (1+\eps)-speed O(\frac{1}{\eps})-competitive for \eps >0. We complement this by showing that \srpt is (1+\eps)-speed O(\frac{1}{\eps^2})-competitive for the objective of minimizing the $\ell_k$-norms of flow time on $m$ identical machines. Both of our results rely on new potential functions that capture the structure of \srpt. Our results, combined with previous work, show that \srpt is the best possible online algorithm in essentially every aspect when migration is permissible.Comment: Accepted for publication at SODA. This version fixes an error in a preliminary versio

Online Scheduling on Identical Machines Using SRPT

Due to its optimality on a single machine for the problem of minimizing average flow time, Shortest-Remaining-Processing-Time (SRPT) appears to be the most natural algorithm to consider for the problem of minimizing average flow time on multiple identical machines. It is known that SRPT achieves the best possible competitive ratio on multiple machines up to a constant factor. Using resource augmentation, SRPT is known to achieve total flow time at most that of the optimal solution when given machines of speed $2- 1/m$. Further, it is known that SRPT's competitive ratio improves as the speed increases; SRPT is $s$-speed $1/s$-competitive when $s \geq 2 - 1/m$. However, a gap has persisted in our understanding of SRPT. Before this work, we did not know the performance of SRPT when given machines of speed 1+\eps for any 0 < \eps < 1 - 1/m. We answer the question in this thesis. We show that SRPT is scalable on $m$ identical machines. That is, we show SRPT is (1+\eps)-speed O(1/\eps)-competitive for any \eps > 0. We also show that SRPT is (1+\eps)-speed O(1/\eps^2)-competitive for the objective of minimizing the $l_k$ norms of flow time on $m$ identical machines. Both of our results rely on new potential functions that capture the structure of SRPT. Our results, combined with previous work, show that SRPT is the best possible online algorithm in essentially every aspect when migration is permissible

A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection

Given an undirected graph with edge costs and node weights, the minimum bisection problem asks for a partition of the nodes into two parts of equal weight such that the sum of edge costs between the parts is minimized. We give a polynomial time bicriteria approximation scheme for bisection on planar graphs. Specifically, let $W$ be the total weight of all nodes in a planar graph $G$. For any constant $\varepsilon > 0$, our algorithm outputs a bipartition of the nodes such that each part weighs at most $W/2 + \varepsilon$ and the total cost of edges crossing the partition is at most $(1+\varepsilon)$ times the total cost of the optimal bisection. The previously best known approximation for planar minimum bisection, even with unit node weights, was $O(\log n)$. Our algorithm actually solves a more general problem where the input may include a target weight for the smaller side of the bipartition.Comment: To appear in STOC 201

An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles

We study a path-planning problem amid a set $\mathcal{O}$ of obstacles in $\mathbb{R}^2$, in which we wish to compute a short path between two points while also maintaining a high clearance from $\mathcal{O}$; the clearance of a point is its distance from a nearest obstacle in $\mathcal{O}$. Specifically, the problem asks for a path minimizing the reciprocal of the clearance integrated over the length of the path. We present the first polynomial-time approximation scheme for this problem. Let $n$ be the total number of obstacle vertices and let $\varepsilon \in (0,1]$. Our algorithm computes in time $O(\frac{n^2}{\varepsilon ^2} \log \frac{n}{\varepsilon})$ a path of total cost at most $(1+\varepsilon)$ times the cost of the optimal path.Comment: A preliminary version of this work appear in the Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithm

Minimum cycle and homology bases of surface embedded graphs

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional ($\mathbb{Z}_2$)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the $1$-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^\omega+2^{2g}n^2+m)$-time algorithm for graphs embedded on an orientable surface of genus $g$. The best known existing algorithms for surface embedded graphs are those for general graphs: an $O(m^\omega)$ time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2 m)$ time algorithm. For the minimum homology basis problem, we give a deterministic $O((g+b)^3 n \log n + m)$-time algorithm for graphs embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of $g$ and $n$. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$.Comment: A preliminary version of this work was presented at the 32nd Annual International Symposium on Computational Geometr

Approximating Dynamic Time Warping and Edit Distance for a Pair of Point Sequences

We give the first subquadratic-time approximation schemes for dynamic time warping (DTW) and edit distance (ED) of several natural families of point sequences in $\mathbb{R}^d$, for any fixed $d \ge 1$. In particular, our algorithms compute $(1+\varepsilon)$-approximations of DTW and ED in time near-linear for point sequences drawn from k-packed or k-bounded curves, and subquadratic for backbone sequences. Roughly speaking, a curve is $\kappa$-packed if the length of its intersection with any ball of radius $r$ is at most $\kappa \cdot r$, and a curve is $\kappa$-bounded if the sub-curve between two curve points does not go too far from the two points compared to the distance between the two points. In backbone sequences, consecutive points are spaced at approximately equal distances apart, and no two points lie very close together. Recent results suggest that a subquadratic algorithm for DTW or ED is unlikely for an arbitrary pair of point sequences even for $d=1$. Our algorithms work by constructing a small set of rectangular regions that cover the entries of the dynamic programming table commonly used for these distance measures. The weights of entries inside each rectangle are roughly the same, so we are able to use efficient procedures to approximately compute the cheapest paths through these rectangles

Approximating the Geometric Edit Distance

Edit distance is a measurement of similarity between two sequences such as strings, point sequences, or polygonal curves. Many matching problems from a variety of areas, such as signal analysis, bioinformatics, etc., need to be solved in a geometric space. Therefore, the geometric edit distance (GED) has been studied. In this paper, we describe the first strictly sublinear approximate near-linear time algorithm for computing the GED of two point sequences in constant dimensional Euclidean space. Specifically, we present a randomized O(n log^2n) time O(sqrt n)-approximation algorithm. Then, we generalize our result to give a randomized alpha-approximation algorithm for any alpha in [1, sqrt n], running in time O~(n^2/alpha^2). Both algorithms are Monte Carlo and return approximately optimal solutions with high probability