Lower bounds for the broadcast problem in mobile radio networks

Summary.  In this paper, we prove a lower bound on the number of rounds required by a deterministic distributed protocol for broadcasting a message in radio networks whose processors do not know the identities of their neighbors. Such an assumption captures the main characteristic of mobile and wireless environments [3], i.e., the instability of the network topology. For any distributed broadcast protocol Π, for any n and for any D≦n/2, we exhibit a network G with n nodes and diameter D such that the number of rounds needed by Π for broadcasting a message in G is Ω(D log n). The result still holds even if the processors in the network use a different program and know n and D. We also consider the version of the broadcast problem in which an arbitrary number of processors issue at the same time an identical message that has to be delivered to the other processors. In such a case we prove that, even assuming that the processors know the network topology, Ω(n) rounds are required for solving the problem on a complete network (D=1) with n processors. Received: August 1994 / Accepted: August 1996     

Fast deterministic broadcast and gossiping algorithms for mobile ad hoc networks

We present a mobility resilient deterministic broadcast algorithm with worst-case time complexity of O(nlogn)$O(nlogn)$ for ad hoc networks where the nodes possess collision detection capabilities; n$n$ is the total number of nodes in the network. The algorithm is based on a breadth-first traversal of the network and allows multiple simultaneous transmissions by the nodes. The idea of this broadcast algorithm is then extended to develop a mobility resilient deterministic gossiping algorithm having O(Dnlogn)$O(Dnlogn)$ worst-case run time (D$D$ is the diameter of the network graph), which is an improvement over the existing algorithms. Simulation results show that on an average, the time for completing the broadcast or gossiping is significantly lower than the theoretical worst-case time requirement.     

Upper and lower bounds for approximation in the gap metric

Upper and lower bounds for the closest approximant of degree k <n in the gap metric to a plant of degree n are obtained. The bounds are expressed in terms of the singular values of two Hankel operators defined in terms of the symbol of the graph of the plant. The question of robust stability and performance of feedback systems is examined in the context of approximation of plant and controller in the gap metric     

Upper and lower covariance bounds for perturbed linear systems

Both upper and lower bounds are established for state covariance matrices under parameter perturbations of the plant. The motivation for this study lies in the fact that many robustness properties of linear systems are given explicitly in terms of the state covariance matrix. Moreover, there exists a theory for control by covariance assignment. The results provide robustness properties of these covariance controllers     

Upper and lower bounds for finding connected motifs in vertex-colored graphs

We study the problem of finding occurrences of motifs in vertex-colored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices whose multiset of colors equals the motif. This problem is a natural graph-theoretic pattern matching variant where we are not interested in the actual structure of the occurrence of the pattern, we only require it to preserve the very basic topological requirement of connectedness. We give two positive results and three negative results that together give an extensive picture of tractable and intractable instances of the problem.     

Upper and lower bounds via penalty functions for control problems

A close relationship between penalty functions and duality is shown to exist for a certain class of convex optimal control problems with state and control variable constraints. This relationship can be used to generate upper and lower bounds on the value of the solution to a control problem of the prescribed class.     

Upper and lower bounds for some depth-3 circuit classes

We investigate the complexity of depth-3 threshold circuits with majority gates at the output, possibly negated AND gates at level two, and MOD _m gates at level one. We show that the fan-in of the AND gates can be reduced toO(logn) in the case wherem is unbounded, and to a constant in the case wherem is constant. We then use these upper bounds to derive exponential lower bounds for this class of circuits. In the unboundedm case, this yields a new proof of a lower bound of Grolmusz; in the constantm case, our result sharpens his lower bound. In addition, we prove an exponential lower bound if OR gates are also permitted on level two andm is a constant prime power.Dedicated to the memory of Roman Smolensky     

Distributed selectsort sorting algorithms on broadcast communication networks

In this paper, a distributed selectsort algorithm and a parameterized selectsort algorithm are presented to be applied on distributed systems for cases when N P where N is the number of elements to be sorted and P is the number of processors in the system. The distributed system considered in this paper uses a broadcasting channel for communication between processors. We show that the number of messages required for the parameterized selectsort algorithm is independent of N and is of complexity O(P), which is optimal in a distributed system with P processors. Furthermore, the amount of communication required in terms of elements is N + O(P³) and the computation time complexity is O((N/P)lgN + P²lg(N/P)). Hence, when N P³, the computation time complexity is O((N/P)lgN), which is optimal using P processors. In addition, this parameterized algorithm provides us with a parameter K such that by choosing the value of K allows us to trade among processing requirement, memory requirement, and communication requirement. It is shown that this parameterized algorithm can reduce the communication requirements significantly while only slightly increasing the computation requirements.     

Upper and lower bounds for dynamic cluster assignment for multi-target tracking in heterogeneous WSNs

In this paper, we consider the problem of cluster task assignment to maximize total utilities of nodes for target coverage in heterogeneous Wireless Sensor Networks. We define this problem as assigning the tasks of Cluster Head (CH) and Cluster Members (CM) to nodes for each target and requiring communication connectivity between every CH with its members. The utility of each node for each target is defined as a function of its distance to the target and its remaining energy. We propose an upper bound based on Lagrangian Relaxation (LR) and a lower bound by Linear Programming (LP) relaxation with a combination of Randomized Rounding (RR) and a greedy-based heuristic. Furthermore, we propose a distributed heuristic algorithm based on matching and a general assignment problem. Dynamic movements of targets are taken into account by intra/inter-cluster task reassignments. Simulation results, compared with optimal values, reveal that the LR upper bound performs better than the bound reached by pure LP relaxation. The lower bound obtained by LP relaxation combined with the RR technique provides close results in comparison with the distributed heuristic algorithm. Furthermore, the results of the distributed heuristic algorithm remain between the upper and lower bounds and close to optimal values.     

Upper and lower matrix bounds of the solution for the discreteLyapunov equation

New matrix bounds of the solution for the discrete algebraic matrix Lyapunov equation are established. The upper and lower eigenvalue bounds such as each eigenvalue including the extreme ones, the trace, and the determinant are also determined by these matrix bounds for the same solution. The present schemes are tighter as compared with the majority of existing results     

Hadamard tensors and lower bounds on multiparty communication complexity

We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the “Number on the Forehead” model of Chandra, Furst, and Lipton. We define an analog of the Hadamard property of matrices for tensors in multiple dimensions and show that any k-party communication problem represented by a Hadamard tensor must have Ω(n/2 ^k ) multiparty communication complexity. We also exhibit constructions of Hadamard tensors. This allows us to obtain Ω(n/2 ^k ) lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions.     

Reliable broadcasts and communication models: tradeoffs and lower bounds

Reliable Broadcast is a mechanism by which a processor in a distributed system disseminates a value to all other processors in the presence of both communication and processor failures. Protocols to achieve Reliable Broadcast are at the heart of most fault-tolerant applications. We characterize the execution time of Reliable Broadcast protocols as a function of the communication model. This model includes familiar communication structures such as fully-connected point-to-point graphs, linear chains, rings, broadcast networks (such as Ethernet) and buses. We derive a parameterized protocol that implements Reliable Broadcast for any member within this class. We obtain lower bound results that show the optimality of our protocols. The lower bound results identify a time complexity gap between systems where processors may only fail to send messages, and systems where processors may fail both to send and to receive messages. The tradeoffs that our results reveal between performance, resiliency and network cost offer many new alternatives previously not considered in designing fault-tolerant systems. Özalp Babaoglu is Associate Professor in the Department of Computer Science at Cornell University, Ithaca, New York. His research interests include distributed systems, fault tolerance, performance evaluation and modeling. He received a BS in electrical engineering from George Washington University, Washington, D.C. in 1976. From the University of California, Berkeley, he received a MS in 1977 and a PhD in 1981, both in computer science. While at Berkeley, he designed and implemented the virtual memory extensions to VAX Unix that came to be known as 3. Obsd. Pat Stephenson is a Doctoral Candidate in the Computer Science Department at Cornell University, Ithaca, New York. His research interests include distributed systems and fault tolerance. In 1983, he received a B.A. (Mod.) in computer science from Trinity College, Dublin, Ireland. He received his MS in computer science from Cornell in 1986. He is currently working on new tradeoffs in the design of fault-tolerant algorithms. Rogério Drummond is Assistant Professor in the Computer Science Department at the Universidade Estadual de Campinas (UNICAMP), São Paulo, Brazil. His interests include distributed computing, fault tolerance and operating systems. He received a BS and a MS in computer science from the Universidade Estadual de Campinas in 1978 and 1980, respectively. In 1986 he received a PhD in computer science from Cornell University. He is currently working on integrated environments for the development of software and hardware.Partial support for this work was provided by the National Science Foundation under Grants DCR-86-01864 and MCS-82-10356 and AT&T under a Foundation GrantSupported partially by the Defense Advanced Research Projects Agency (DoD) under ARPA order 5378, Contract MDA 903-85-C-0124, and partially by an IBM graduate fellowship. The views, opinions and findings contained in this report are solely those of the authors and should not be construed as an official Department of Defense position, policy, or decisionSupported partially by the CAPES and CNPq agencies of the Ministry of Education of Brazil     

Branching programs provide lower bounds on the areas of multilective deterministic and nondeterministic VLSI-circuits

Each (nondeterministic) multilective VLSI-circuit C of area A can be simulated by an oblivious (disjunctive) branching program of width exp(O(A)) which has the same multiplicity of reading as C. That is why exponential lower bounds on the width of (disjunctive) oblivious branching programs of linear depth provide lower bounds of order Ω(n^1–2α), , on the area of (nondeterministic) multilective VLSI-circuits computing explicitly defined one-output Boolean functions, if the multiplicity of reading is bounded by O(log^αn). Lower bounds are derived for the sequence equality problem (SEQ) and the graph accessibility problem (GAP).     

Super-logarithmic depth lower bounds via the direct sum in communication complexity

Is it easier to solve two communication problems together than separately? This question is related to the complexity of the composition of boolean functions. Based on this relationship, an approach to separatingNC ¹ fromP is outlined. Furthermore, it is shown that the approach provides a new proof of the separation of monotoneNC ¹ from monotoneP.     

Upper bounds for approximation of continuous-time dynamics using delayed outputs and feedforward neural networks

The problem of approximation of unknown dynamics of a continuous-time observable nonlinear system is considered using a feedforward neural network, operating over delayed sampled outputs of the system. Error bounds are derived that explicitly depend upon the sampling time interval and network architecture. The main result of this note broadens the class of nonlinear dynamical systems for which adaptive output feedback control and state estimation problems are solvable.     

Proof techniques for port-directed communication and broadcast

This paper presents proof rules for port-directed communication and broadcast. The proof method is an extension of the proof technique proposed by Misra and Chandy in which input/output sequences are used to describe the state of a process or a subsystem. Various examples are presented to illustrate the use of the proof technique.     

Constructive lower bounds on model complexity of shallow perceptron networks

Limitations of shallow (one-hidden-layer) perceptron networks are investigated with respect to computing multivariable functions on finite domains. Lower bounds are derived on growth of the number of network units or sizes of output weights in terms of variations of functions to be computed. A concrete construction is presented with a class of functions which cannot be computed by signum or Heaviside perceptron networks with considerably smaller numbers of units and smaller output weights than the sizes of the function’s domains. A subclass of these functions is described whose elements can be computed by two-hidden-layer perceptron networks with the number of units depending on logarithm of the size of the domain linearly.

     

Performing work in broadcast networks

We consider the problem of how to schedule t similar and independent tasks to be performed in a synchronous distributed system of p stations communicating via multiple-access channels. Stations are prone to crashes whose patterns of occurrence are specified by adversarial models. Work, defined as the number of the available processor steps, is the complexity measure. We consider only reliable algorithms that perform all the tasks as long as at least one station remains operational. It is shown that every reliable algorithm has to perform work Ω(t+p√t) even when no failures occur. An optimal deterministic algorithm for the channel with collision detection is developed, which performs work (t+p√t). Another algorithm, for the channel without collision detection, performs work (t+p √t+ p min {f,t}), where f < p is the number of failures. This algorithm is proved to be optimal, provided that the adversary is restricted in failing no more than f stations. Finally, we consider the question if randomization helps against weaker adversaries for the channel without collision detection. A randomized algorithm is developed which performs the expected minimum amount (t+p√t) of work, provided that the adversary may fail a constant fraction of stations and it has to select failure-prone stations prior to the start of an execution of the algorithm. The work of the first author is supported by the NSF Grant 0310503. A preliminary version of this paper appeared as “The do-all problem in broadcast networks,” in Proceedings, 20th ACM Symposium on Principles of Distributed Computing, Newport, Rhode Island, 2001, pp. 117–126.