Section: New Results

Resource allocation and Scheduling

Broadcasting on Large Scale Heterogeneous Platforms under the Bounded Multi-Port Model

Participants : Olivier Beaumont, Nicolas Bonichon, Lionel Eyraud-Dubois, Przemyslaw Uznanski.

In [17] , we consider the problem of broadcasting a large message in a large scale distributed network under the multi-port communication model. We are interested in building an overlay network, with the aim of maximizing the throughput and minimizing the degree of the participating nodes. We consider a classification of participating nodes into two parts: open nodes that stay in the open-Internet and "guarded" nodes that lie behind firewalls or NATs, with the constraint that two guarded nodes cannot communicate directly. Without guarded nodes, we prove that it is possible to reach the optimal throughput with a quasi-optimal (up to a small additive increase) degree of the participating nodes. In presence of guarded nodes, we provide a closed form formula for the optimal cyclic throughput and we observe that the optimal solution may require arbitrarily large degrees. In the acyclic case, we propose an algorithm that reaches the optimal acyclic throughput with low degree. Then, we prove a worst case 5/7 ratio between the optimal acyclic and cyclic throughput and show through simulations that this ratio is on average very close to 1, what makes acyclic solutions efficient both in terms of throughput maximization and degree minimization.

Non Linear Divisible Load Scheduling

Participants : Olivier Beaumont, Hubert Larchevêque.

Divisible Load Theory (DLT) has received a lot of attention in the past decade. A divisible load is a perfect parallel task, that can be split arbitrarily and executed in parallel on a set of possibly heterogeneous resources. The success of DLT is strongly related to the existence of many optimal resource allocation and scheduling algorithms, what strongly differs from general scheduling theory. Moreover, recently, close relationships have been underlined between DLT, that provides a fruitful theoretical framework for scheduling jobs on heterogeneous platforms, and MapReduce, that provides a simple and efficient programming framework to deploy applications on large scale distributed platforms. The success of both have suggested to extend their framework to non-linear complexity tasks. In [32] , we show that both DLT and MapReduce are better suited to workloads with linear complexity. In particular, we prove that divisible load theory cannot directly be applied to quadratic workloads, such as it has been proposed recently. We precisely state the limits for classical DLT studies and we review and propose solutions based on a careful preparation of the dataset and clever data partitioning algorithms. In particular, through simulations, we show the possible impact of this approach on the volume of communications generated by MapReduce, in the context of Matrix Multiplication and Outer Product algorithms. (Joint work with Loris Marchal from ENS Lyon)

Reliable Service Allocation in Clouds

Participants : Olivier Beaumont, Lionel Eyraud-Dubois, Hubert Larchevêque, Paul Renaud-Goud, Philippe Duchon.

In [30] , we consider several reliability problems that arise when allocating applications to processing resources in a Cloud computing platform. More specifically, we assume on the one hand that each computing resource is associated to a capacity constraint and to a probability of failure. On the other hand, we assume that each service runs as a set of independent instances of identical Virtual Machines, and that the Service Level Agreement between the Cloud provider and the client states that a minimal number of instances of the service should run with a given probability. In this context, given the capacity and failure probabilities of the machines, and the capacity and reliability demands of the services, the question for the cloud provider is to find an allocation of the instances of the services (possibly using replication) onto machines satisfying all types of constraints during a given time period. The goal of this work is to assess the impact of the reliability constraint on the complexity of resource allocation problems. We consider several variants of this problem, depending on the number of services and whether their reliability demand is individual or global. We prove several fundamental complexity results ($\#$P' and NP-completeness results) and we provide several optimal and approximation algorithms. In particular, we prove that a basic randomized allocation algorithm, that is easy to implement, provides optimal or quasi-optimal results in several contexts, and we show through simulations that it also achieves very good results in more general settings.

In [29] , we extend this work to an energy minimisation framework, by considering two energy consumption models based on DVFS techniques, where the clock frequency of physical resources can be changed with a Dynamic Voltage and Frequency Scaling (DVFS) method. For each allocation problem and each energy model, we prove deterministic approximation ratios on the consumed energy for algorithms that provide guaranteed probability failures, as well as an efficient heuristic, whose energy ratio is not guaranteed.

In [37] , we study the robustness of an allocation of Virtual Machines (VM) on a set of Physical Machines (PM) when the resource demand of the VMs can change over time. This may imply sometimes expensive “SLA violations”, corresponding to some VM's consumption not satisfied because of overloaded PMs. Thus, while optimizing the global resource utilization of the PMs, it is necessary to ensure that at any moment a VM's need evolves, a few number of migrations (moving a VM from PM to PM) is sufficient to find a new configuration in which all the VMs' consumptions are satisfied. We modelize this problem using a fully dynamic bin packing approach and we present an algorithm ensuring a global utilization of the resources of 66%. Moreover, each time a PM is overloaded at most one migration is necessary to fall back in a configuration with no overloaded PM, and only 3 different PMs are concerned by required migrations that may occur to keep the global resource utilization correct. This allows the platform to be highly resilient to a great number of changes.

Splittable Single Source-Sink Routing on CMP Grids: A Sublinear Number of Paths Suffice

Participants : Adrian Kosowski, Przemyslaw Uznanski.

In [44] , we study single chip multiprocessors (CMP) with grid topologies, where a significant part of power consumption is attributed to communications between the cores of the grid. We investigate the problem of routing communications between CMP cores using shortest paths, in a model in which the power cost associated with activating a communication link at a transmission speed of $f$ bytes/second is proportional to $f^{\alpha }$, for some constant exponent $\alpha >2$. Our main result is a trade-off showing how the power required for communication in CMP grids depends on the ability to split communication requests between a given pair of node, routing each such request along multiple paths. For a pair of cores in a $m\times n$ grid, the number of available communication paths between them grows exponentially with $n,m$. By contrast, we show that optimal power consumption (up to constant factors) can be achieved by splitting each communication request into $k$ paths, starting from a threshold value of $k=\Theta \left(n^{1/\left(\alpha -1\right)}\right)$. This threshold is much smaller than $n$ for typical values of $\alpha \approx 3$, and may be considered practically feasible for use in routing schemes on the grid. More generally, we provide efficient algorithms for routing multiple $k$-splittable communication requests between two cores in the grid, providing solutions within a constant approximation of the optimum cost. We support our results with algorithm simulations, showing that for practical instances, our approach using $k$-splittable requests leads to a power cost close to that of the optimal solution with arbitrarily splittable requests, starting from the stated threshold value of $k$.

Maximum matching in multi-interface networks

Participants : Adrian Kosowski, Dominik Pajak.

In [26] , we consider the standard matching problem in the context of multi-interface wireless networks. In heterogeneous networks, devices can communicate by means of multiple wireless interfaces. By choosing which interfaces to switch on at each device, several connections might be established. That is, the devices at the endpoints of each connection share at least one active interface. In the studied problem, the aim is to maximize the number of parallel connections without incurring in interferences. Given a network $G=(V,E)$, nodes $V$ represent the devices, edges $E$ represent the connections that can be established. If node $x$ participates in the communication with one of its neighbors by means of interface $i$, then another neighboring node of $x$ can establish a connection (but not with $x$) only if it makes use of interface $j\ne i$. The size of a solution for an instance of the outcoming matching problem, that we call Maximum Matching in Multi-Interface networks (MMMI for short), is always in between the sizes of the solutions for the same instance with respect to the standard matching and its induced version problems. However, we prove that MMMI is NP-hard even for proper interval graphs and for bipartite graphs of maximum degree $\Delta \ge 3$. We also show polynomially solvable cases of MMMI with respect to different assumptions.

Parallel scheduling of task trees with limited memory

Participant : Lionel Eyraud-Dubois.

In a paper submitted to ACM TOPC, we have investigated the execution of tree-shaped task graphs using multiple processors. Each edge of such a tree represents some large data. A task can only be executed if all input and output data fit into memory, and a data can only be removed from memory after the completion of the task that uses it as an input data. Such trees arise, for instance, in the multifrontal method of sparse matrix factorization. The peak memory needed for the processing of the entire tree depends on the execution order of the tasks. With one processor the objective of the tree traversal is to minimize the required memory. This problem was well studied and optimal polynomial algorithms were proposed. We have extended the problem by considering multiple processors, which is of obvious interest in the application area of matrix factorization. With multiple processors comes the additional objective to minimize the time needed to traverse the tree, i.e., to minimize the makespan. Not surprisingly, this problem proves to be much harder than the sequential one. We study the computational complexity of this problem and provide inapproximability results even for unit weight trees. We design a series of practical heuristics achieving different trade-offs between the minimization of peak memory usage and makespan. Some of these heuristics are able to process a tree while keeping the memory usage under a given memory limit. The different heuristics are evaluated in an extensive experimental evaluation using realistic trees.

Point-to-point and congestion bandwidth estimation: experimental evaluation on PlanetLab

Participants : Lionel Eyraud-Dubois, Przemyslaw Uznanski.

In large scale Internet platforms, measuring the available bandwidth between nodes of the platform is difficult and costly. However, having access to this information allows to design clever algorithms to optimize resource usage for some collective communications, like broadcasting a message or organizing master/slave computations. In [54] , we analyze the feasibility to provide estimations, based on a limited number of measurements, for the point-to-point available bandwidth values, and for the congestion which happens when several communications take place at the same time. We present a dataset obtained with both types of measurements performed on a set of nodes from the PlanetLab platform. We show that matrix factorization techniques are quite efficient at predicting point-to-point available bandwidth, but are not adapted for congestion analysis. However, a LastMile modeling of the platform allows to perform congestion predictions with a reasonable level of accuracy, even with a small amount of information, despite the variability of the measured platform.

Parallel Mining of Functional Dependencies

Participants : Sofian Maabout, Nicolas Hanusse.

The problem of extracting functional dependencies (FDs) from databases has a long history dating back to the 90’s. Still, efficient solutions taking into account both material evolution, namely the advent of multicore machines, and the amount of data that are to be mined, are still needed. In [46] we propose a parallel algorithm which, upon small modifications, extracts (i) the minimal keys, (ii) the minimal exact FDs, (iii) the minimal approximate FDs and (iv) the Conditional functional dependencies (CFDs) holding in a table. Under some natural conditions, we prove a theoretical speed up of our solution with respect to a baseline algorithm which follows a depth first search strategy. Since mining most of these dependencies require a procedure for computing the number of distinct values (NDV) which is a space consuming operation, we show how sketching techniques for estimating the exact value of NDV can be used for reducing both memory consumption as well as communications overhead when considering distributed data while guaranteeing a certain quality of the result. Our solution is implemented in both shared, using C++ and OpenMP, and distributed memory, using Hadoop implementation of Map-Reduce. The experimental results showthe efficiency and scalability of our proposal. Most notably, the theoretical speed ups are confirmed by the experiments.

Fast Skyline Query Evaluation with Skycuboids Materialization based on Functional Dependencies

Participants : Sofian Maabout, Nicolas Hanusse.

Ranking multidimensional data via different Skyline queries gives rise to the so called skycube structure. Most of previous work on optimizing the subspaces skyline queries have concentrated on full materialization of the skycube. Due to the exponential number of skylines one must pre-compute, the full materialization is unfeasible in practice. However, due to the non monotonic nature of skylines, there is no immediate inclusion relationship between the skycuboids when we have an inclusion of the dimensions. This makes the partial materialization harder. In this paper, we identify sufficient conditions for establishing inclusions between skycuboids thanks to the functional dependencies that hold in the underlying data. This leads to the characterization of a minimal set of skycuboids to be materialized in order to answer all the possible skyline queries without resorting to the underlying data. We conduct an extensive set of experiments showing that with the help of a small fraction of the skycube, we can efficiently answer all the possible skyline queries. In addition, our proposal turns to be helpful even in the full materialization setting. Indeed, thanks to the inclusions we identify, we devise a full materialization algorithm which outperforms state of the art skycube computation algorithms especially when data and dimensions get large. The results are reported in the technical report submitted to SIGMOD'14.