Optimal Deterministic Algorithm Research Articles

Optimal deterministic quantum algorithm for the promised element distinctness problem

The element distinctness problem is to determine whether a string of N elements, i.e. x=(x1,…,xN) where xi∈{1,…,M} and M≥N, contains two elements of the same value (a.k.a colliding pair), for which Ambainis proposed an optimal quantum algorithm. The idea behind Ambainis' algorithm is to first reduce the problem to the promised version in which x is promised to contain at most one colliding pair, and then design an algorithm A requiring O(N2/3) queries based on quantum walk search for the promise problem. However, A is probabilistic and may fail to give the right answer. We thus, in this work, design a deterministic quantum algorithm for the promise problem which never errs and requires O(N2/3) queries. This algorithm is proved optimal. Technically, we modify the quantum walk search operator on quasi-Johnson graph to have arbitrary phases, and then use Jordan's lemma as the analyzing tool to reduce the quantum walk search operator to the generalized Grover's operator. This allows us to utilize the recently proposed fixed-axis-rotation (FXR) method for deterministic quantum search, and hence achieve 100% success.

Heterogeneous Training Intensity for Federated Learning: A Deep Reinforcement Learning Approach

Federated learning (FL) has recently received considerable attention in Internet of Things, due to its capability of letting multiple clients collaboratively train machine learning models, without sharing their private information. However, in synchronous FL, the client with weak computing or communication capability may significantly drag down the model training process, which leads to very high waiting latency for other clients. Intuitively, to alleviate this straggler problem, the clients with lower (higher) training capabilities should be assigned with less (more) training intensity. Inspired by this observation, this paper formulates a novel Heterogeneous Training Intensity assignment problem for FL, named <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HTI_FL</i> , aiming at reducing the largest training latency gap among clients. To address HTI_FL problem, we first propose an optimal deterministic algorithm, which however is only suitable for a static FL context with stable network conditions and clients' computing capabilities. To consider a practical dynamic context, we propose a Deep Reinforcement Learning Approach to learning the network conditions and clients' capabilities, and furthermore adaptively assign training intensities to clients. Finally, simulation results demonstrate the effectiveness of the proposed scheme in reducing the waiting time and accelerating the convergence of FL.

Geometric Algorithm for Finding Time-Sensitive Data Gathering Path in Energy Harvesting Sensor Networks

To perform large-scale monitoring of sensitive events, energy harvesting wireless sensor network is considered where a mobile data sink <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$MS$ </tex-math></inline-formula> collects data while travelling on a fixed path <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{ms}$ </tex-math></inline-formula> . The sensor nodes sense environmental data continuously at a pre-specified rate. The sensors close to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{ms}$ </tex-math></inline-formula> are referred as gateways. Sensors forward their data to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$MS$ </tex-math></inline-formula> through the gateways. In practice, the usage of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$MS$ </tex-math></inline-formula> is not suitable for time-sensitive applications due to its long data gathering delay. Time-bound data gathering for path constrained environment is not accounted in literature. We aim at finding energy-efficient maximum data gathering sub-path for the MS for a given data gathering period <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> . To deal with the problem a novel optimal deterministic data collection sub-path finding algorithm is proposed which is based on the geometric properties of the sensors’ communication disks and the data gathering path. It maximizes the data collection and reduces the energy consumption by jointly optimizing the data gathering sub-path selection and the data forwarding path optimization. The performance of the proposed algorithm is compared with an existing baseline algorithm DDGA and a heuristic algorithm H-DGSPF. The simulation results show that our proposed algorithm outperforms DDGA and H-DGSPF in terms of data collection, data delivery success ratio, and energy consumption

An online optimization approach for post-disaster relief distribution with online blocked edges

Disasters can disrupt road networks by blocking some of the roads and consequently impeding accessibility to critical locations. In the immediate post-disaster response phase, while the blockage information is yet to be collected, relief distribution crews (RDCs) should dispatch from warehouses to supply critical locations with first aid items. The RDCs are not capable of unblocking damaged roads and should find a way to bypass them once such edges are observed in their routes. With the objective of minimizing total latency of the critical nodes, we study the problem that addresses the relief distribution operations with k non-recoverable online blocked edges. The online blocked edges are not known to the RDCs initially and the blockage of a blocked edge is revealed when one of the RDCs arrives at one of its end-nodes. Once one of the RDCs knows about a blocked edge, this information is communicated among the rest of the RDCs and they will all be informed about that blocked edge. We first investigate the worst-case performance of online algorithms against offline optimal solutions using competitive ratio. We then prove a lower bound on the competitive ratio of deterministic online algorithms. We also provide an upper bound on the competitive ratio of the optimal deterministic online algorithms by introducing a deterministic algorithm which achieves a bounded competitive ratio. We then develop three heuristic algorithms to solve this problem. One of our algorithms is based on solving an Integer Programming model to find the assignment of the nodes to the RDCs and then finding the routes dynamically. The other algorithms are not based on solving optimization models and hence are more efficient in terms of their computational time. We compare our proposed heuristic algorithms with the best known algorithms from the literature that are developed for a single RDC variation of the problem. Finally, we provide a through computational analysis of our algorithms on instances adopted from real-world road networks.

Optimal deterministic group testing algorithms to estimate the number of defectives

In this work, we study the problem of estimating the number of defectives d in a set of n items up to a multiplicative factor of Δ>1 using deterministic group testing algorithms. Particularly, given Δ>1 and D>d, we give upper and lower bounds on the number of tests required for the task of generating an estimation dˆ such that d/Δ≤dˆ≤dΔ.In the adaptive settings, we prove that any adaptive deterministic algorithm for estimating d must perform at least Ω((D/Δ2)log⁡(n/D)) tests. This extends the same lower bound achieved in [1] for non-adaptive algorithms. In addition, we show that our bound is tight up to a small additive term by constructing an adaptive algorithm for this task. Furthermore, we give a non-constructive proof of an upper bound of O((D/Δ2)(log⁡(n/D)+log⁡Δ)) for the non-adaptive settings. This bound is an improvement over the upper bound O((log⁡D)/(log⁡Δ))Dlog⁡n) from [1], and matches the lower bound up to a small additive term.Moreover, we use existing techniques for building expander regular bipartite graphs, extractors and condensers from [2,3] to construct two polynomial-time non-adaptive algorithms for estimating d. Using the construction from [2], a polynomial non-adaptive algorithm that makes O((D1+o(1)/Δ2)⋅log⁡n) tests are defined. The second algorithm exploits the construction from [3] to build another non-adaptive algorithm that performs (D/Δ2)⋅Quazipoly(log⁡n) tests. This is the first explicit construction with an almost optimal test complexity.

Deterministic Linear Time Constrained Triangulation Using Simplified Earcut.

Triangulation algorithms that conform to a set of non-intersecting input segments typically proceed in an incremental fashion, by inserting points first, and then segments. Inserting a segment amounts to: (1) deleting all the triangles it intersects; (2) filling the so generated hole with two polygons that have the wanted segment as shared edge; (3) triangulate each polygon separately. In this article we prove that these polygons are such that all their convex vertices but two can be used to form triangles in an earcut fashion, without the need to check whether other polygon points are located within each ear. The fact that any simple polygon contains at least three convex vertices guarantees the existence of a valid ear to cut, ensuring convergence. Not only this translates to an optimal deterministic linear time triangulation algorithm, but such algorithm is also trivial to implement. We formally prove the correctness of our approach, also validating it in practical applications and comparing it with prior art.

Weighted online minimum latency problem with edge uncertainty

In the minimum latency problem, an undirected connected graph and a root node together with non-negative edge distances are given to an agent. The agent looks for a tour starting at the root node and visiting all the nodes to minimise the sum of the latencies of the nodes, where the latency of a node is the distance from the root node to the node at its first visit on the tour by the agent. We study an online variant of the problem, where there are k blocked edges in the graph which are not known to the agent in advance. A blocked edge is learned online when the agent arrives at one of its end-nodes. Furthermore, we investigate another online variant of the minimum latency problem involving k blocked edges where each node is associated with a weight to express its priority and the objective is to minimise the summation of the weighted latency of the nodes. In this paper, we prove that the lower bound of 2k+1 on the competitive ratio of deterministic online algorithms is tight for both weighted and non-weighted variations by introducing an optimal deterministic online algorithm which meets this lower bound. We also present a lower bound of k+1 on the expected competitive ratio of randomized online algorithms for both problems. We then develop two polynomial time heuristic algorithms to solve these online problems. We test our algorithms on real life as well as randomly generated instances that are partially adopted from the literature.

Online Failure Diagnosis in Interdependent Networks

In interdependent networks, nodes are connected to each other with respect to their failure dependency relations. As a result of this dependency, a failure in one of the nodes of one of the networks within a system of several interdependent networks can cause the failure of the entire system. Diagnosing the initial source of the failure in a collapsed system of interdependent networks is an important problem to be addressed. We study an online failure diagnosis problem defined on a collapsed system of interdependent networks where the source of the failure is at an unknown node (v). In this problem, each node of the system has a positive inspection cost and the source of the failure is diagnosed when v is inspected. The objective is to provide an online algorithm which considers dependency relations between nodes and diagnoses v with minimum total inspection cost. We address this problem from worst-case competitive analysis perspective for the first time. In this approach, solutions which are provided under incomplete information are compared with the best solution that is provided in presence of complete information using the competitive ratio (CR) notion. We give a lower bound of the CR for deterministic online algorithms and prove its tightness by providing an optimal deterministic online algorithm. Furthermore, we provide a lower bound on the expected CR of randomized online algorithms and prove its tightness by presenting an optimal randomized online algorithm. We prove that randomized algorithms are able to obtain better CR compared to deterministic algorithms in the expected sense for this online problem.

Algorithms for Handoff Minimization in Wireless Networks

This study focuses on the problem of handoff minimization for a set of users moving in a wireless network. This problem is analyzed by considering two cases for the user’s movement under access point capacity constraints: 1) all users move together, and 2) each user can have their chosen path within the network. In the first case, we propose an optimal competitive ratio algorithm for the problem. However, in the second case, having the connectivity assumption, that is, “if a user is connected to an access point so long that the received signal strength of the access point is not less than a specified threshold, the user should continue his/her connection”, we prove that no approach can reduce the number of unnecessary handoffs in an offline setting. However, without connectivity assumption, we present an optimal deterministic algorithm with the competitive ratio of nΔ for this problem under online setting, where n is the number of users and Δ is the maximum number of access points which cover any single point in the environment. Also, we prove that the randomized version of the algorithm achieves an expected competitive ratio of O(log Δ).

Optimal deterministic distributed algorithms for maximal independent set in geometric graphs

Finding a maximal independent set (MIS) in a graph is one of the fundamental problems in distributed computing. Researchers in this community are trying to close the time complexity gap of computing a MIS in a graph, way back from Luby’s [STOC’85] O(logn) time (randomized) algorithm and Linial’s [SICOMP’92] Ω(log∗n) lower bound result (n is the number of nodes of the graph). Since then, an extensive research has been done on this problem, however, most of the results are randomized and we are lack of efficient deterministic solution. In fact, no polylogarithmic (in n) time deterministic algorithm is known so far in general graphs — an open question raised by Linial in 1993 [Comb. Probab. Comput.’93]. In this paper, we study the MIS problem in geometric graphs, a class of graphs which has both theoretical and practical importance. We present O(1)-time deterministic algorithms (hence, optimal) for computing MIS in unit interval graphs, unit square graphs and unit disk graphs. The same idea applies to develop optimal MIS algorithm for higher dimensional unit balls and unit hypercubes. We further extend the MIS algorithms to compute approximate maximum independent set (MaxIS) in the above geometric graph classes. The theoretical results are corroborated through extensive experiments to show the effectiveness and efficiency of our algorithms.Our algorithms are fully decentralized, scalable, and robust. In general, nodes exchange only O(logn) bits message through an edge per communication. Moreover, being local, our algorithms are also suitable in failure model, self-stabilizing and appropriate for dynamic environment.

Correction to: Optimal deterministic algorithm generation

Unfortunately, the copyright information of the article in Springerlink appears as “© Springer Science+Business Media, LLC, part of Springer Nature 2018” this has been corrected to “© The Author(s) 2018.”

Optimal deterministic algorithm generation

A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of functions describing the algorithmic steps. The optimization variables are the parameters—within this family of algorithms—that encode algorithm design: the computational steps of which the selected algorithms consist. The objective function of the optimization problem encodes the merit function of the algorithm, e.g., the computational cost (possibly also including a cost component for memory requirements) of the algorithm execution. The constraints of the optimization problem ensure convergence of the algorithm, i.e., solution of the problem at hand. The formulation is described prototypically for algorithms used in solving nonlinear equations and in performing unconstrained optimization; the parametrized algorithm family considered is that of monomials in function and derivative evaluation (including negative powers). A prototype implementation in GAMS is provided along with illustrative results demonstrating cases for which well-known algorithms are shown to be optimal. The formulation is a mixed-integer nonlinear program. To overcome the multimodality arising from nonconvexity in the optimization problem, a combination of brute force and general-purpose deterministic global algorithms is employed to guarantee the optimality of the algorithm devised. We then discuss several directions towards which this methodology can be extended, their scope and limitations.

An Optimal Randomized Online Algorithm for QoS Buffer Management

The QoS (Quality of Service) buffer management problem, with significant and diverse computer applications, e.g., in online cloud resource allocation problems, is a classic online admission control problem in the presence of resource constraints. In its basic setting, packets with different values according to their QoS requirements, arrive in online fashion to a switching node with limited buffer size. Then, the switch needs to make an immediate decision to either admit or reject the incoming packet based on the value of the packet and its buffer availability. The objective is to maximize the cumulative profit of the admitted packets, while respecting the buffer constraint. Even though the QoS buffer management problem was proposed more than a decade ago, no optimal online solution has been proposed in the literature. This paper contributes to this problem by proposing: 1) A fixed threshold-based online algorithm with smaller competitive ratio than the existing results; 2) an optimal deterministic online algorithm under fractional admission model in which a packet could be admitted partially; and 3) an optimal randomized online algorithm for the general problem. We consider the last result being the main contribution of this paper.

Efficient Integer Coefficient Search for Compute-and-Forward

Integer coefficient selection is an important decoding step in the implementation of compute-and-forward (C-F) relaying scheme. Choosing the optimal integer coefficients in C-F has been shown to be a shortest vector problem (SVP) which is known to be NP hard in its general form. Exhaustive search of the integer coefficients is only feasible in complexity for small number of users while approximation algorithms such as Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm only find a vector within an exponential factor of the shortest vector. An optimal deterministic algorithm was proposed for C-F by Sahraei and Gastpar specifically for the real valued channel case. In this paper, we adapt their idea to the complex valued channel and propose an efficient search algorithm to find the optimal integer coefficient vectors over the ring of Gaussian integers and the ring of Eisenstein integers. A second algorithm is then proposed that generalises our search algorithm to the Integer-Forcing MIMO C-F receiver. Performance and efficiency of the proposed algorithms are evaluated through simulations and theoretical analysis.

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (Comput Geom 2(3):169---186, 1992) and the optimal but randomized algorithm of Ramos (Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG'99, 1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, $$({\le }k)$$(≤k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, $$\varepsilon $$?-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (Discrete Comput Geom 6(1):385---406, 1991) and Chazelle (Discrete Comput Geom 9(1):145---158, 1993).

Online scheduling with equal processing times and machine eligibility constraints

We consider the online scheduling problem on m parallel machines with eligibility constraints. The jobs arrive over time and have equal processing times. The objective is to minimize the makespan. We develop optimal deterministic online algorithms for the nested processing set case and the inclusive processing set case with an arbitrary number of machines, as well as the tree-like processing set case with three machines.

Optimal deterministic algorithms for some variants of Online Quota Traveling Salesman Problem

This paper is concerned with the Online Quota Traveling Salesman Problem. Depending on the symmetry of the metric and the requirement for the salesman to return to the origin, four variants are analyzed. We present optimal deterministic algorithms for each variant defined on a general space, a real line, or a half-line. As a byproduct, an improved lower bound for a variant of Online TSP on a half-line is also obtained.

A sparsity preserving stochastic gradient methods for sparse regression

We propose a new stochastic first-order algorithm for solving sparse regression problems. In each iteration, our algorithm utilizes a stochastic oracle of the subgradient of the objective function. Our algorithm is based on a stochastic version of the estimate sequence technique introduced by Nesterov (Introductory lectures on convex optimization: a basic course, Kluwer, Amsterdam, 2003). The convergence rate of our algorithm depends continuously on the noise level of the gradient. In particular, in the limiting case of noiseless gradient, the convergence rate of our algorithm is the same as that of optimal deterministic gradient algorithms. We also establish some large deviation properties of our algorithm. Unlike existing stochastic gradient methods with optimal convergence rates, our algorithm has the advantage of readily enforcing sparsity at all iterations, which is a critical property for applications of sparse regressions.

Near-optimal deterministic algorithms for volume computation via M-ellipsoids

We give a deterministic algorithm for computing an M-ellipsoid of a convex body, matching a known lower bound. This leads to a nearly optimal deterministic algorithm for estimating the volume of a convex body and improved deterministic algorithms for fundamental lattice problems under general norms.

Duration automation in scheduling program for a cluster computer system

Optimal schedule for works running on machines, in a general case, is a hard problem and there is no complete optimal deterministic algorithm in polynomial time. Optimal and approximated solutions were issued for some specific cases with constraints. One can find the solutions for the cases 1 and 2 machines [4,16] as the initial algorithms. In [16], author solved late works problem using algorithm with setup times included. [4] solve the due works problem on two machines, and can be extended for the case of 3 machines with some conditions on works. Other authors looked at schedule problems in specific cases like [5,8,17]. In this paper, we apply duration automata [1,2] to solve the schedule problem dynamically for the works with uncertain processing time in a cluster computer, which is a system consisting of many computers (computing nodes) co-working together. We solve the schedule problem in a cluster with $m$ machines for two cases: all machines are the same and different in configurations (machine's resources are formally considered as a configuration information, represented by the time need to finish the works). Because of uncertainly processing time, tranditional algorithms can not be used effectively. By using DA model with DA's order criterion, we issue schedule algorithms and practically prove to be better in time consuming compare to FIFO (natural order), the fastest first (greedy) and the longest first (safety) methods without synchronized points.