Rafał Kapelko

How to Improve the Reliability of Chord

In this paper we focus on Chord P2P protocol and we study the process of unexpected departures of nodes from this system. Each of such departures may effect in losing any information and in classical versions of this protocol the probability of losing some information is proportional to the quantity of information put into this system. This effect can be partially solved by gathering in the protocol multiple copies (replicas) of information. The replication mechanism was proposed by many authors. We present a detailed analysis of one variant of blind replication and show that this solution only partially solves the problem. Next we propose two less obvious modifications of the Chord protocol. We call the first construction a direct sums of Chords and the second - a folded Chord. We discuss the recovery mechanisms of partially lost information in each of these systems and investigate their reliability. We show that our modification increases essentially the expected lifetime of information put into the system. Our modifications of the Chord protocol are very soft and require only a small interference in the programming code of the original Chord protocol.

On the displacement for covering a d-dimensional cube with randomly placed sensors

Consider n sensors placed randomly and independently with the uniform distribution in a d-dimensional unit cube (d ? 2). The sensors have identical sensing range equal to r, for some r > 0. We are interested in moving the sensors from their initial positions to new positions so as to ensure that the d-dimensional unit cube is completely covered, i.e., every point in the d - dimensional cube is within the range of a sensor. If the ith sensor is displaced a distance di, what is a displacement of minimum cost that ensures coverage? As cost measure for the displacement of the team of sensors we consider the a-total movement defined as the sum M a : = ? i = 1 n d i a , for some constant a > 0. We assume that r and n are chosen so as to allow full coverage of the d-dimensional unit cube. Motivation for using this cost metric arises from the fact that there might be a terrain affecting the movement of the sensors from their initial to their final destinations (e.g., a terrain surface which is either obstructing or speeding the movement). Therefore the a-total displacement is not more general but can be a more realistic metric than the one previously considered for a = 1 .The main contribution of this paper is to show the existence of a tradeoff in the d-dimensional cube between sensing radius and a-total movement. Omitting low order terms, the main results can be summarized as follows for the case of the d-dimensional unit cube.1.If the d-dimensional cube sensing radius is 1 2 n 1 / d and n = m d , for some m ? N , then we present an algorithm that uses ? ( n 1 - a 2 d ) total expected movement, when a ? 1 and O ( n 1 - a 2 d ) total expected movement, when a ? (0, 1) (see Algorithm?2, Theorem?5 and Theorem?6).2.If the the d-dimensional cube sensing radius is greater than 3 3 / d ( 3 1 / d - 1 ) ( 3 1 / d - 1 ) 1 2 n 1 / d and n is a natural number then the total expected movement is O ( n 1 - a 2 d ( ln n n ) a 2 d ) (see Algorithm?3 and Theorem?8).This sharp decline from O ( n 1 - a 2 d ) to O ( n 1 - a 2 d ( ln n n ) a 2 d ) in the a-total movement of the sensors to attain complete coverage of the d-dimensional unit cube indicates the presence of an interesting threshold on the sensing radius in a d-dimensional unit cube as it increases from 1 2 n 1 / d to 3 3 / d ( 3 1 / d - 1 ) ( 3 1 / d - 1 ) 1 2 n 1 / d . In addition, we simulate Algorithm?2 and discuss the results of our simulations.

On the displacement for covering a unit interval with randomly placed sensors

Consider n mobile sensors placed independently at random with the uniform distribution on a barrier represented as the unit line segment 0 , 1 . The sensors have identical sensing radius, say r. When a sensor is displaced on the line a distance equal to d it consumes energy (in movement) which is proportional to some (fixed) power a 0 of the distance d traveled. The energy consumption of a system of n sensors thus displaced is defined as the sum of the energy consumptions for the displacement of the individual sensors.We focus on the problem of energy efficient displacement of the sensors so that in their final placement the sensor system ensures coverage of the barrier and the energy consumed for the displacement of the sensors to these final positions is minimized in expectation. In particular, we analyze the problem of displacing the sensors from their initial positions so as to attain coverage of the unit interval and derive trade-offs for this displacement as a function of the sensor range. We obtain several tight bounds in this setting thus generalizing several of the results of 10 to any power a 0 . We prove that total movement to power a of the sensors with range r = 1 2 n is ( a 2 ) ! 2 a 2 ( 1 + a ) 1 n a 2 - 1 + O ( 1 n a 2 ) , when a is an even natural number.We give the Algorithm 1 that uses expected O ( 1 n a 2 - 1 ( ln ź n n ) a 2 ) , total movement to power a, where a 0 and r ź 6 2 n .

On the Displacement for Covering a Square with Randomly Placed Sensors

Consider

Towards Fault-Tolerant Chord P2P System: Analysis of Some Replication Strategies

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Towards efficient replication of documents in chord: case (r,s) erasure codes

sensors placed randomly and independently with the uniform distribution in a unit square. The sensors have identical sensing range equal to

On the Dynamics of Systems of Urns

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