13,285 research outputs found

    Silent MST approximation for tiny memory

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    In network distributed computing, minimum spanning tree (MST) is one of the key problems, and silent self-stabilization one of the most demanding fault-tolerance properties. For this problem and this model, a polynomial-time algorithm with O(log2 ⁣n)O(\log^2\!n) memory is known for the state model. This is memory optimal for weights in the classic [1,poly(n)][1,\text{poly}(n)] range (where nn is the size of the network). In this paper, we go below this O(log2 ⁣n)O(\log^2\!n) memory, using approximation and parametrized complexity. More specifically, our contributions are two-fold. We introduce a second parameter~ss, which is the space needed to encode a weight, and we design a silent polynomial-time self-stabilizing algorithm, with space O(logns)O(\log n \cdot s). In turn, this allows us to get an approximation algorithm for the problem, with a trade-off between the approximation ratio of the solution and the space used. For polynomial weights, this trade-off goes smoothly from memory O(logn)O(\log n) for an nn-approximation, to memory O(log2 ⁣n)O(\log^2\!n) for exact solutions, with for example memory O(lognloglogn)O(\log n\log\log n) for a 2-approximation

    Lock-in Problem for Parallel Rotor-router Walks

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    The rotor-router model, also called the Propp machine, was introduced as a deterministic alternative to the random walk. In this model, a group of identical tokens are initially placed at nodes of the graph. Each node maintains a cyclic ordering of the outgoing arcs, and during consecutive turns the tokens are propagated along arcs chosen according to this ordering in round-robin fashion. The behavior of the model is fully deterministic. Yanovski et al.(2003) proved that a single rotor-router walk on any graph with m edges and diameter DD stabilizes to a traversal of an Eulerian circuit on the set of all 2m directed arcs on the edge set of the graph, and that such periodic behaviour of the system is achieved after an initial transient phase of at most 2mD steps. The case of multiple parallel rotor-routers was studied experimentally, leading Yanovski et al. to the conjecture that a system of k \textgreater{} 1 parallel walks also stabilizes with a period of length at most 2m2m steps. In this work we disprove this conjecture, showing that the period of parallel rotor-router walks can in fact, be superpolynomial in the size of graph. On the positive side, we provide a characterization of the periodic behavior of parallel router walks, in terms of a structural property of stable states called a subcycle decomposition. This property provides us the tools to efficiently detect whether a given system configuration corresponds to the transient or to the limit behavior of the system. Moreover, we provide polynomial upper bounds of O(m4D2+mDlogk)O(m^4 D^2 + mD \log k) and O(m5k2)O(m^5 k^2) on the number of steps it takes for the system to stabilize. Thus, we are able to predict any future behavior of the system using an algorithm that takes polynomial time and space. In addition, we show that there exists a separation between the stabilization time of the single-walk and multiple-walk rotor-router systems, and that for some graphs the latter can be asymptotically larger even for the case of k=2k = 2 walks

    Fast and Compact Distributed Verification and Self-Stabilization of a DFS Tree

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    We present algorithms for distributed verification and silent-stabilization of a DFS(Depth First Search) spanning tree of a connected network. Computing and maintaining such a DFS tree is an important task, e.g., for constructing efficient routing schemes. Our algorithm improves upon previous work in various ways. Comparable previous work has space and time complexities of O(nlogΔ)O(n\log \Delta) bits per node and O(nD)O(nD) respectively, where Δ\Delta is the highest degree of a node, nn is the number of nodes and DD is the diameter of the network. In contrast, our algorithm has a space complexity of O(logn)O(\log n) bits per node, which is optimal for silent-stabilizing spanning trees and runs in O(n)O(n) time. In addition, our solution is modular since it utilizes the distributed verification algorithm as an independent subtask of the overall solution. It is possible to use the verification algorithm as a stand alone task or as a subtask in another algorithm. To demonstrate the simplicity of constructing efficient DFS algorithms using the modular approach, We also present a (non-sielnt) self-stabilizing DFS token circulation algorithm for general networks based on our silent-stabilizing DFS tree. The complexities of this token circulation algorithm are comparable to the known ones

    Self-stabilizing tree algorithms

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    Designers of distributed algorithms have to contend with the problem of making the algorithms tolerant to several forms of coordination loss, primarily faulty initialization. The processes in a distributed system do not share a global memory and can only get a partial view of the global state. Transient failures in one part of the system may go unnoticed in other parts and thus cause the system to go into an illegal state. If the system were self-stabilizing, however, it is guaranteed that it will return to a legal state after a finite number of state transitions. This thesis presents and proves self-stabilizing algorithms for calculating tree metrics and for achieving mutual exclusion on a tree structured distributed system
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