An O(n^2 log^2 n) Time Algorithm for Minmax Regret Minsum Sink on Path Networks

We model evacuation in emergency situations by dynamic flow in a network. We want to minimize the aggregate evacuation time to an evacuation center (called a sink) on a path network with uniform edge capacities. The evacuees are initially located at the vertices, but their precise numbers are unknown, and are given by upper and lower bounds. Under this assumption, we compute a sink location that minimizes the maximum "regret." We present the first sub-cubic time algorithm in n to solve this problem, where n is the number of vertices. Although we cast our problem as evacuation, our result is accurate if the "evacuees" are fluid-like continuous material, but is a good approximation for discrete evacuees

Asymptotically Optimal Approximation Algorithms for Coflow Scheduling

Many modern datacenter applications involve large-scale computations composed of multiple data flows that need to be completed over a shared set of distributed resources. Such a computation completes when all of its flows complete. A useful abstraction for modeling such scenarios is a {\em coflow}, which is a collection of flows (e.g., tasks, packets, data transmissions) that all share the same performance goal. In this paper, we present the first approximation algorithms for scheduling coflows over general network topologies with the objective of minimizing total weighted completion time. We consider two different models for coflows based on the nature of individual flows: circuits, and packets. We design constant-factor polynomial-time approximation algorithms for scheduling packet-based coflows with or without given flow paths, and circuit-based coflows with given flow paths. Furthermore, we give an $O(\log n/\log \log n)$-approximation polynomial time algorithm for scheduling circuit-based coflows where flow paths are not given (here $n$ is the number of network edges). We obtain our results by developing a general framework for coflow schedules, based on interval-indexed linear programs, which may extend to other coflow models and objective functions and may also yield improved approximation bounds for specific network scenarios. We also present an experimental evaluation of our approach for circuit-based coflows that show a performance improvement of at least 22% on average over competing heuristics.Comment: Fixed minor typo

Minsum sink location and evacuation problem on dynamic cycle networks

In this thesis, we study 1-sink location and k-sink evacuation problem on dynamic cycle networks. We consider the 1-sink location problem is to find the optimal location of the 1 sink, while the k-sink evacuation problem is to find the optimal evacuation protocol for the given locations of the k sinks. Both results minimize the sum of the evacuation times of all the supply located at the vertices to the sink/s of a given cycle network of n vertices. We present an efficient algorithm with a useful data structure that finds the optimal location of the 1 sink in O(n) time when the capacity of the edges are uniform. If the edges have arbitrary capacities, we solve the problem in O(nlogn)$ time by an extension of the data structure. We also propose an O(n) time algorithm to solve the k-sink evacuation problem with uniform edge capacity

On the capacity provisioning on dynamic networks

In this thesis, we consider the development of algorithms suitable for designing evacuation procedures in sparse or remote communities. The works are extensions of sink location problems on dynamic networks, which are motivated by real-life disaster events such as the Tohoku Japanese Tsunami, the Australian wildfire and many more. The available algorithms in this context consider the location of the sinks (safe-havens) with the assumptions that the evacuation by foot is possible, which is reasonable when immediate evacuation is needed in urban settings. However, for remote communities, emergency vehicles may need to be dispatched or situated strategically for an efficient evacuation process. With the assumption removed, our problems transform to the task of allocating capacities on the edges of dynamic networks given a budget capacity c. We first of all consider this problem on a dynamic path network of n vertices with the objective of minimizing the completion time (minmax criterion) given that the position of the sink is known. This leads to an O(nlogn + nlog(c/ξ)) time, where ξ is a refinement or precision parameter for an additional binary search in the worst case scenario. Next, we extend the problem to star topologies. The case where the sink is located at the middle of the star network follows the same approach for the path network. However, when the sink is located on a leaf node, the problem becomes more complicated when the number of links (edges) exceeds three. The second phase of this thesis focuses on allocating capacities on the edges of dynamic path networks with the objective of minimizing the total evacuation time (minsum criterion) given the position of the sink and the budget (fixed) capacity. In general, minsum problems are more difficult than minmax problems in the context of sink location problems. Due to few combinatorial properties discovered together with the possibility of changing objective. function configuration in the course of the optimization process, we consider the development of numerical procedure which involves the use of sequential quadratic programming (SQP). The sequential quadratic programming employed allows the specification of an arbitrary initial capacities and also helps in monitoring the changing configuration of the objective function. We propose to consider these problems on more complex topolgies such as trees and general graph in future.NSERC Discovery Grants program. University of Lethbridge Graduate Research Award. Alberta Innovates Awar

Sublinear Computation Paradigm

This open access book gives an overview of cutting-edge work on a new paradigm called the “sublinear computation paradigm,” which was proposed in the large multiyear academic research project “Foundations of Innovative Algorithms for Big Data.” That project ran from October 2014 to March 2020, in Japan. To handle the unprecedented explosion of big data sets in research, industry, and other areas of society, there is an urgent need to develop novel methods and approaches for big data analysis. To meet this need, innovative changes in algorithm theory for big data are being pursued. For example, polynomial-time algorithms have thus far been regarded as “fast,” but if a quadratic-time algorithm is applied to a petabyte-scale or larger big data set, problems are encountered in terms of computational resources or running time. To deal with this critical computational and algorithmic bottleneck, linear, sublinear, and constant time algorithms are required. The sublinear computation paradigm is proposed here in order to support innovation in the big data era. A foundation of innovative algorithms has been created by developing computational procedures, data structures, and modelling techniques for big data. The project is organized into three teams that focus on sublinear algorithms, sublinear data structures, and sublinear modelling. The work has provided high-level academic research results of strong computational and algorithmic interest, which are presented in this book. The book consists of five parts: Part I, which consists of a single chapter on the concept of the sublinear computation paradigm; Parts II, III, and IV review results on sublinear algorithms, sublinear data structures, and sublinear modelling, respectively; Part V presents application results. The information presented here will inspire the researchers who work in the field of modern algorithms

Optimizing the sensor movement for barrier coverage in a sink-based deployed mobile sensor network

Barrier coverage is an important coverage model for intrusion detection. Clearly energy consumption of sensors is a critical issue to the design of a sensor deployment scheme. In mobile sensor network, it costs the sensors much energy to move. In this paper, we study how to optimize the sensor movement while scheduling the mobile sensors to achieve barrier coverage. Given a line barrier and $n$ sink stations that can supply a required number of mobile sensors, we study how to find the mobile sensors' final positions on the line barrier so that the barrier is covered and the total sensor movement is minimized. We first propose a fast algorithm for determining the nearest sink for the given point on the barrier. We then propose a greedy algorithm and an optimal polynomial-time algorithm for calculating the optimal sensor movement. To obtain an optimal algorithm, we first introduce a notion of the virtual-cluster which represents a subset of sensors covering a specified line segment of the barrier and their sensor movements are minimized. Then we construct a weighted barrier graph with the virtual-clusters modeled as vertexes and the weight of each vertex as the total sensor movements of the virtual-cluster. We also prove that the minimum total sensor movements for achieving barrier coverage is the minimum total weights of the path between the two endpoints of the line barrier in this graph. We also solve this barrier coverage problem for the case when the barrier is a cycle by extending the techniques used for the line barrier. Finally, we demonstrate the effectiveness and efficiency of our algorithms by simulations.Shuangjuan Li, Hong Shen, Qiong Huang, Longkun Gu

Optimal Embedding of Functions for In-Network Computation: Complexity Analysis and Algorithms

We consider optimal distributed computation of a given function of distributed data. The input (data) nodes and the sink node that receives the function form a connected network that is described by an undirected weighted network graph. The algorithm to compute the given function is described by a weighted directed acyclic graph and is called the computation graph. An embedding defines the computation communication sequence that obtains the function at the sink. Two kinds of optimal embeddings are sought, the embedding that---(1)~minimizes delay in obtaining function at sink, and (2)~minimizes cost of one instance of computation of function. This abstraction is motivated by three applications---in-network computation over sensor networks, operator placement in distributed databases, and module placement in distributed computing. We first show that obtaining minimum-delay and minimum-cost embeddings are both NP-complete problems and that cost minimization is actually MAX SNP-hard. Next, we consider specific forms of the computation graph for which polynomial time solutions are possible. When the computation graph is a tree, a polynomial time algorithm to obtain the minimum delay embedding is described. Next, for the case when the function is described by a layered graph we describe an algorithm that obtains the minimum cost embedding in polynomial time. This algorithm can also be used to obtain an approximation for delay minimization. We then consider bounded treewidth computation graphs and give an algorithm to obtain the minimum cost embedding in polynomial time

Faster Algorithms for Semi-Matching Problems

We consider the problem of finding \textit{semi-matching} in bipartite graphs which is also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted case. For the weighted case, we give an $O(nm\log n)$-time algorithm, where $n$ is the number of vertices and $m$ is the number of edges, by exploiting the geometric structure of the problem. This improves the classical $O(n^3)$ algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi [Communications of the ACM 1974]. For the unweighted case, the bound could be improved even further. We give a simple divide-and-conquer algorithm which runs in $O(\sqrt{n}m\log n)$ time, improving two previous $O(nm)$-time algorithms by Abraham [MSc thesis, University of Glasgow 2003] and Harvey, Ladner, Lov\'asz and Tamir [WADS 2003 and Journal of Algorithms 2006]. We also extend this algorithm to solve the \textit{Balance Edge Cover} problem in $O(\sqrt{n}m\log n)$ time, improving the previous $O(nm)$-time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC 2008].Comment: ICALP 201