Worst-case Time Complexity Research Articles

Multi-agent deep reinforcement learning for user association and resource allocation in integrated terrestrial and non-terrestrial networks

Integrating the terrestrial network with non-terrestrial networks to provide radio access as anticipated in the beyond 5G networks calls for efficient user association and resource allocation strategies. In this work, a weighted sum single objective optimization problem that maximizes the total network data rate while minimizing the mobility-induced handoffs and prioritizing service provisioning of mission-critical users in the integrated terrestrial and non-terrestrial network is formulated. The problem’s complexity is reduced by solving the defined problem in two phases; the user association followed by the resource distribution phase. Several proposed approaches to the user association sub-problem utilize a central node that requires nearly-complete information, which may not be available in real time. On the contrary, this paper proposes a centralized training and distributed execution multi-agent dueling double deep Q network (MA3DQN) solution. Each user collects the channel state and access node loading information in this approach and makes an association decision that considers its quality of service requirements. The algorithm’s performance is validated through comparison with the genetic algorithm (GA), the integer linear programming (ILP), a heuristic approximation-based solution, the greedy approach, and the random user association (RUA) algorithm. Moreover, the paper simulates the multi-agent deep Q network solution as an additional benchmark algorithm. Simulation results reveal that as the number of users in the network increases, the acquired data rate of the MA3DQN is within 0.48% and 0.4% of that achieved by the GA and ILP, respectively, and outperforms all other algorithms. Notably, the proposed MA3DQN algorithm presents the best running time, attaining a gain of 99.9% over the GA algorithm, which performs the poorest among the algorithms characterized by polynomial worst-case time complexity. Besides, the MA3DQN approach maintains a handoff probability of zero, unlike the ILP, the approximation-based, greedy, and RUA solutions.

GriT-DBSCAN: A spatial clustering algorithm for very large databases

DBSCAN is a fundamental spatial clustering algorithm with numerous practical applications. However, a bottleneck of DBSCAN is its O(n2) worst-case time complexity. To address this limitation, we propose a new grid-based algorithm for exact DBSCAN in Euclidean space called GriT-DBSCAN, which is based on the following two techniques. First, we introduce grid tree to organize the non-empty grids for the purpose of efficient non-empty neighboring grids queries. Second, by utilizing the spatial relationships among points, we propose a technique that iteratively prunes unnecessary distance calculations when determining whether the minimum distance between two sets is less than or equal to a certain threshold. We theoretically demonstrate that GriT-DBSCAN has excellent reliability in terms of time complexity. In addition, we obtain two variants of GriT-DBSCAN by incorporating heuristics, or by combining the second technique with an existing algorithm. Experiments are conducted on both synthetic and real-world data sets to evaluate the efficiency of GriT-DBSCAN and its variants. The results show that our algorithms outperform existing algorithms.

(p,q)-biclique counting and enumeration for large sparse bipartite graphs.

In this paper, we study the problem of (, )-biclique counting and enumeration for large sparse bipartite graphs. Given a bipartite graph and two integer parameters p and q, we aim to efficiently count and enumerate all (, )-bicliques in G, where a (, )-biclique B(L,R) is a complete subgraph of G with , , , and . The problem of (, )-biclique counting and enumeration has many applications, such as graph neural network information aggregation, densest subgraph detection, and cohesive subgroup analysis. Despite the wide range of applications, to the best of our knowledge, we note that there is no efficient and scalable solution to this problem in the literature . This problem is computationally challenging, due to the worst-case exponential number of (, )-bicliques. In this paper, we propose a competitive branch-and-bound baseline method, namely BCList, which explores the search space in a depth-first manner, together with a variety of pruning techniques. Although BCList offers a useful computation framework to our problem, its worst-case time complexity is exponential to . To alleviate this, we propose an advanced approach, called BCList++. Particularly, BCList++ applies a layer-based exploring strategy to enumerate (, )-bicliques by anchoring the search on either U or V only, which has a worst-case time complexity exponential to either p or q only. Consequently, a vital task is to choose a layer with the least computation cost. To this end, we develop a cost model, which is built upon an unbiased estimator for the density of 2-hop graph induced by U or V. To improve computation efficiency, BCList++ exploits pre-allocated arrays and vertex labeling techniques such that the frequent subgraph creating operations can be substituted by array element switching operations. We conduct extensive experiments on 16 real-life datasets, and the experimental results demonstrate that BCList++ significantly outperforms the baseline methods by up to 3 orders of magnitude. We show via a case study that (, )-bicliques optimizes the efficiency of graph neural networks. In this paper, we extend our techniques to count and enumerate (, )-bicliques on uncertain bipartite graphs. An efficient method IUBCList is developed on the top of BCList++, together with a couple of pruning techniques, including common neighbor refinement and search branch early termination, to discard unpromising uncertain (, )-bicliques early. The experimental results demonstrate that IUBCList significantly outperforms the baseline method by up to 2 orders of magnitude.

Using Subobservers to Synthesize Opacity-Enforcing Supervisors

In discrete-event system control, the worst-case time complexity for computing a system's observer is exponential in the number of that system's states. This results in practical difficulties since some problems require calculating multiple observers for a changing system, e.g., synthesizing an opacity-enforcing supervisor. Although calculating these observers in an iterative manner allows us to synthesize an opacity-enforcing supervisor and although methods have been proposed to reduce the computational demands, room exists for a practical and intuitive solution. Here we extend the subautomaton relationship to the notion of a subobserver and demonstrate its use in reducing the computations required for iterated observer calculations. We then demonstrate the subobserver relationship's power by simplifying state-of-the-art synthesis approaches for opacity-enforcing supervisors under realistic assumptions.

Fast network k-function-based spatial analysis

Network K -function has been the de facto operation for analyzing point patterns in spatial networks, which is widely used in many communities, including geography, ecology, transportation science, social science, and criminology. To analyze a location dataset, domain experts need to generate a network K -function plot that involves computing multiple network K -functions. However, network K -function is a computationally expensive operation that is not feasible to support large-scale datasets, let alone to generate a network K -function plot. To handle this issue, we develop two efficient algorithms, namely count augmentation (CA) and neighbor sharing (NS), which can reduce the worst-case time complexity for computing network K -functions. In addition, we incorporate the advanced shortest path sharing (ASPS) approach into these two methods to further lower the worst-case time complexity for generating network K -function plots. Experiment results on four large-scale location datasets (up to 7.33 million data points) show that our methods can achieve up to 165.85x speedup compared with the state-of-the-art methods.

Optimizing Time Complexity of Searching Operation: A Noble Algorithm

By the passage of time , there are many searching algorithm and data structure evolved to make basic operations more efficient with respect to time and space complexity like storing, searching, deleting the data and they play very important role in real-world applications. Based on mechanism of searching, Searching algorithm can be classified into three types of algorithm: linear, binary, hashing, but excluding these, there are many searching techniques proposed by many researchers. The aim of all these proposed searching technique is to optimize the time complexity of searching operation generally compared to Binary Search, which have average and worst case time complexity of log(n), where n is number of elements. This paper proposes a noble searching algorithm that works for sorted array data structure and performs better in most of the cases than that of binary search. The proposed searching Technique will either find index of target element or find a small space within array space by using statistical operation, and then binary search will be applied to that small space, thus it will reduce the overall time complexity. This paper briefly discusses, proposed algorithm and it’s time complexity and comparing the efficiency of the algorithm with some of the well-known other algorithms, using graphs and tables and its practical applications. We will analyze the behavior of algorithm in different parameters like:-size of input data, Nature of elements in list, Different Case Scenario (worst, best, and avg. case) etc.

Speeding Up the RUL¯ Dynamic-Controllability-Checking Algorithm for Simple Temporal Networks with Uncertainty

A Simple Temporal Network with Uncertainty (STNU) includes real-valued variables, called time-points; binary difference constraints on those time-points; and contingent links that represent actions with uncertain durations. STNUs have been used for robot control, web-service composition, and business processes. The most important property of an STNU is called dynamic controllability (DC); and algorithms for checking this property are called DC-checking algorithms. The DC-checking algorithm for STNUs with the best worst-case time-complexity is the RUL¯ algorithm due to Cairo, Hunsberger and Rizzi. Its complexity is O(mn + k²n + kn log n), where n is the number of time-points, m is the number of constraints, and k is the number of contingent links. It is expected that this worst-case complexity cannot be improved upon. However, this paper provides a new algorithm, called RUL2021, that improves its performance in practice by an order of magnitude, as demonstrated by a thorough empirical evaluation.

Two metrics on rooted unordered trees with labels

BackgroundThe early development of a zygote can be mathematically described by a developmental tree. To compare developmental trees of different species, we need to define distances on trees. If children cells after a division are not distinguishable, developmental trees are represented by the space {mathcal {T}} of rooted trees with possibly repeated labels, where all vertices are unordered. If children cells after a division are partially distinguishable, developmental trees are represented by the space {mathcal {P}} of rooted trees with possibly repeated labels, where vertices can be ordered or unordered.ResultsOn {mathcal {T}}, the space of rooted unordered trees with possibly repeated labels, we define two metrics: the best-match metric and the left-regular metric, which show some advantages over existing methods. On {mathcal {P}}, the space of rooted labeled trees with ordered or unordered vertices, there is no metric, and we define a semimetric, which is a variant of the best-match metric. To compute the best-match distance between two trees, the expected time complexity and worst-case time complexity are both {mathcal {O}}(n^2), where n is the tree size. To compute the left-regular distance between two trees, the expected time complexity is {mathcal {O}}(n), and the worst-case time complexity is {mathcal {O}}(nlog n).ConclusionsFor rooted labeled trees with (fully/partially) unordered vertices, we define metrics (semimetric) that have fast algorithms to compute and have advantages over existing methods. Such trees also appear outside of developmental biology, and such metrics can be applied to other types of trees which have more extensive applications, especially in molecular biology.

A Class of Convex Quadratic Nonseparable Resource Allocation Problems with Generalized Bound Constraints

We study a convex quadratic nonseparable resource allocation problem that arises in the area of decentralized energy management (DEM), where unbalance in electricity networks has to be minimized. In this problem, the given resource is allocated over a set of activities that is divided into subsets, and a cost is assigned to the overall allocated amount of resources to activities within the same subset. We derive two efficient algorithms with [Formula: see text] worst-case time complexity to solve this problem. For the special case where all subsets have the same size, one of these algorithms even runs in linear time given the subset size. Both algorithms are inspired by well-studied breakpoint search methods for separable convex resource allocation problems. Numerical evaluations on both real and synthetic data confirm the theoretical efficiency of both algorithms and demonstrate their suitability for integration in DEM systems.

GO-MOCE: Greedy Order Method of Conditional Expectations for Max Sat

In this paper we present and study a new algorithm for the Maximum Satisfiability (Max Sat) problem. The algorithm is based on the Method of Conditional Expectations (MOCE, also known as Johnson’s Algorithm) and applies a greedy variable ordering to MOCE. Thus, we name it Greedy Order MOCE (GO-MOCE). We also suggest a combination of GO-MOCE with CCLS, a state-of-the-art solver. We refer to this combined solver as GO-MOCE-CCLS.We conduct a comprehensive comparative evaluation of GO-MOCE versus MOCE on random instances and on public competition benchmark instances. We show that GO-MOCE reduces the number of unsatisfied clauses by tens of percents, while keeping the runtime almost the same. The worst case time complexity of GO-MOCE is linear. We also show that GO-MOCE-CCLS improves on CCLS consistently by up to about 80%.We study the asymptotic performance of GO-MOCE. To this end, we introduce three measures for evaluating the asymptotic performance of algorithms for Max Sat. We point out to further possible improvements of GO-MOCE, based on an empirical study of the main quantities managed by GO-MOCE during its execution.

Design of Intelligent Customer Service Questioning and Answering a System for Power Business Scenario Based on AI Technology

In order to solve the problems of large demand for power business and small number of customer service, an intelligent customer service questioning and answering a system for power business scenario based on AI technology is designed. The approach first uses the particle swarm optimization algorithm to automatically classify the question attributes and then uses the fuzzy c-means clustering algorithm to match the answers with the highest similarity to the questions and return to the customers. The system collects the questions raised by customers through the acquisition module, uploads the question work order to the knowledge base through the information assistance module, and stores the preprocessed questions to the knowledge base. After completing the problem attribute classification through the particle swarm clustering algorithm classification model in the batch analysis and calculation module, the question answers are matched through fuzzy c-means clustering. At the same time, the similarity of different keywords is calculated to find a series of related questions. After the obtained data are analyzed in real time through the self-service customer service module, the answer is extracted and fed back to the customer, and the question answer is presented to the customer in the system interface. The experimental results show that the designed system has low worst-case time complexity, which is up to 0.35 only. The reason is that the system in this paper can use priority information to deal with the problems raised by customers, which is different from the past work where dealing with customers request via priority information is not used. The system can give the corresponding answers according to the customer’s options. It has convenient operation, high integrated control ability, and good information management performance. Compared with the traditional approach which could waste a lot of resource and data, the proposed approach can reduce the differences between problem data, eliminate invalid data, and simplify the data classification process. The application of the system can effectively accelerate the information transmission efficiency of the power company and can be used for power exchange platform automation in the future.

VERTEX ENUMERATION OF POLYHEDRA

The vertex enumeration problem of a polyhedron P in ℜn , given by m inequalities, is widely discussed in the literature. In this work it is introduced a new algorithm to solve it. The algorithm is based on lexicographic pivoting and the worst-case time complexity is Omm+n2×minm,n which is OmnVP for the case of non-degenerate polyhedra, where VP is the number of vertices of P. The proposed algorithm was coded in Matlab and numerical experiments performed for several randomly generated problems show its efficiency.

Efficient optimization of the Held–Karp lower bound

Given a weighted undirected graph G=(V,E), the Held–Karp lower bound for the Traveling Salesman Problem (TSP) is obtained by selecting an arbitrary vertex p ¯∈V, by computing a minimum cost tree spanning V∖{p ¯} and adding two minimum cost edges adjacent to p ¯. In general, different selections of vertex p ¯ provide different lower bounds. In this paper it is shown that the selection of vertex p ¯ can be optimized, to obtain the largest possible Held–Karp lower bound, with the same worst-case computational time complexity required to compute a single minimum spanning tree. Although motivated by the optimization of the Held–Karp lower bound for the TSP, the algorithm solves a more general problem, allowing for the efficient pre-computation of alternative minimum spanning trees in weighted graphs where any vertex can be deleted.

(p,q)-biclique counting and enumeration for large sparse bipartite graphs

In this paper, we study the problem of ( p , q)-biclique counting and enumeration for large sparse bipartite graphs. Given a bipartite G = ( U, V , E), and two integer parameters p and q, we aim to efficiently count and enumerate all (p, q)-bicliques in G , where a (p, q)-biclique B ( L, R ) is a complete subgraph of G with L ⊆ U, R ⊆ V , |L| = p, and |R| = q. The problem of (p, q)-biclique counting and enumeration has many applications, such as graph neural network information aggregation, densest subgraph detection, and cohesive subgroup analysis, etc. Despite the wide range of applications, to the best of our knowledge, we note that there is no efficient and scalable solution to this problem in the literature. This problem is computationally challenging, due to the worst-case exponential number of (p, q)-bicliques. In this paper, we propose a competitive branch-and-bound baseline method, namely BCList, which explores the search space in a depth-first manner, together with a variety of pruning techniques. Although BCList offers a useful computation framework to our problem, its worst-case time complexity is exponential to p + q. To alleviate this, we propose an advanced approach, called BCList++. Particularly, BCList++ applies a layer based exploring strategy to enumerate ( p, q )-bicliques by anchoring the search on either U or V only, which has a worst-case time complexity exponential to either p or q only. Consequently, a vital task is to choose a layer with the least computation cost. To this end, we develop a cost model, which is built upon an unbiased estimator for the density of 2-hop graph induced by U or V. To improve computation efficiency, BCList++ exploits pre-allocated arrays and vertex labeling techniques such that the frequent subgraph creating operations can be substituted by array element switching operations. We conduct extensive experiments on 16 real-life datasets, and the experimental results demonstrate that BCList++ significantly outperforms the baseline methods by up to 3 orders of magnitude. We show via a case study that (p, q)-bicliques optimize the efficiency of graph neural networks.

A New Algorithm for the Single Source Weber Problem with Limited Distances

The single source Weber problem with limited distances (SSWPLD) is a continuous optimization problem in location theory. The SSWPLD algorithms proposed so far are based on the enumeration of all regions of [Formula: see text] defined by a given set of n intersecting circumferences. Early algorithms require [Formula: see text] time for the enumeration, but they were recently shown to be incorrect in case of degenerate intersections, that is, when three or more circumferences pass through the same intersection point. This problem was fixed by a modified enumeration algorithm with complexity [Formula: see text], based on the construction of neighborhoods of degenerate intersection points. In this paper, it is shown that the complexity for correctly dealing with degenerate intersections can be reduced to [Formula: see text] so that existing enumeration algorithms can be fixed without increasing their [Formula: see text] time complexity, which is due to some preliminary computations unrelated to intersection degeneracy. Furthermore, a new algorithm for enumerating all regions to solve the SSWPLD is described: its worst-case time complexity is [Formula: see text]. The new algorithm also guarantees that the regions are enumerated only once.

Verifying Time Complexity of Binary Search using Dafny

Formal software verification techniques are widely used to specify and prove the functional correctness of programs. However, nonfunctional properties such as time complexity are usually carried out with pen and paper. Inefficient code in terms of time complexity may cause massive performance problems in large-scale complex systems. We present a proof of concept for using the Dafny verification tool to specify and verify the worst-case time complexity of binary search. This approach can also be used for academic purposes as a new way to teach algorithms and complexity.

A Novel Hilbert Curve for Cache-Locality Preserving Loops

Modern microprocessors offer a rich memory hierarchy including various levels of cache and registers. Some of these memories (like main memory, L3 cache) are big but slow and shared among all cores. Others (registers, L1 cache) are fast and exclusively assigned to a single core but small. Only if the data accesses have a high locality, we can avoid excessive data transfers between the memory hierarchy. In this paper we consider fundamental algorithms like matrix multiplication, K-Means, Cholesky decomposition as well as the algorithm by Floyd and Warshall typically operating in two or three nested loops. We propose to traverse these loops whenever possible not in the canonical order but in an order defined by a space-filling curve. This traversal order dramatically improves data locality over a wide granularity allowing not only to efficiently support a cache of a single, known size (cache conscious) but also a hierarchy of various caches where the effective size available to our algorithms may even be unknown (cache oblivious). We propose a new space-filling curve called Fast Unrestricted (FUR) Hilbert with the following advantages: (1) we overcome the usual limitation to square-like grid sizes where the side-length is a power of 2 or 3. Instead, our approach allows arbitrary loop boundaries for all variables. (2) FUR-Hilbert is non-recursive with a guaranteed constant worst case time complexity per loop iteration (in contrast to O(log(grid-size)) for previous methods). (3) Our non-recursive approach makes the application of our cache-oblivious loops in any host algorithm as easy as conventional loops and facilitates automatic optimization by the compiler. (4) We demonstrate that crucial algorithms like Cholesky decomposition as well as the algorithm by Floyd and Warshall by can be efficiently supported. (5) Extensive experiments on runtime efficiency, cache usage and energy consumption demonstrate the profit of our approach. We believe that future compilers could translate nested loops into cache-oblivious loops either fully automatic or by a user-guided analysis of the data dependency.

Incrementally Solving STNs by Enforcing Partial Path Consistency

Efficient management and propagation of temporal constraints is important for temporal planning as well as for scheduling. During plan development, new events and temporal constraints are added and existing constraints may be tightened; the consistency of the whole temporal network is frequently checked; and results of constraint propagation guide further search. Recent work shows that enforcing partial path consistency provides an efficient means of propagating temporal information for the popular Simple Temporal Network (STN). We show that partial path consistency can be enforced incrementally, thus exploiting the similarities of the constraint network between subsequent edge tightenings. We prove that the worst-case time complexity of our algorithm can be bounded both by the number of edges in the chordal graph (which is better than the previous bound of the number of vertices squared), and by the degree of the chordal graph times the number of vertices incident on updated edges. We show that for many sparse graphs, the latter bound is better than that of the previously best-known approaches. In addition, our algorithm requires space only linear in the number of edges of the chordal graph, whereas earlier work uses space quadratic in the number of vertices. Finally, empirical results show that when incrementally solving sparse STNs, stemming from problems such as Hierarchical Task Network planning, our approach outperforms extant algorithms.

Multi-Task Learning-Based Task Scheduling Switcher for a Resource-Constrained IoT System

In this journal, we proposed a novel method of using multi-task learning to switch the scheduling algorithm. With multi-task learning to change the scheduling algorithm inside the scheduling framework, the scheduling framework can create a scheduler with the best task execution optimization under the computation deadline. With the changing number of tasks, the number of types of resources taken, and computation deadline, it is hard for a single scheduling algorithm to achieve the best scheduler optimization while avoiding the worst-case time complexity in a resource-constrained Internet of Things (IoT) system due to the trade-off in computation time and optimization in each scheduling algorithm. Furthermore, different hardware specifications affect the scheduler computation time differently, making it hard to rely on Big-O complexity as a reference. With multi-task learning to profile the scheduling algorithm behavior on the hardware used to compute the scheduler, we can identify the best scheduling algorithm. Our benchmark result shows that it can achieve an average of 93.68% of accuracy in meeting the computation deadline, along with 23.41% of average optimization. Based on the results, our method can improve the scheduling of the resource-constrained IoT system.

Exact algorithms for scheduling programs with shared tasks

We study a scheduling problem where the jobs we have to perform are composed of one or more tasks. If two jobs sharing a non-empty subset of tasks are scheduled on the same machine, then these shared tasks have to be performed only once. This kind of problem is known in the literature under the names of VM-PACKING or PAGINATION. Our objective is to schedule a set of these objects on two parallel identical machines, with the aim of minimizing the makespan. This problem is NP-complete as an extension of the PARTITION problem. In this paper we present three exact algorithms with worst-case time-complexity guarantees, by exploring different branching techniques. Our first algorithm focuses on the relation between jobs sharing one or more symbols in common, whereas the two other algorithms branches on the shared symbols.