arXiv subjects

Evangelos Kranakis

Publications and source records attributed to Evangelos Kranakis.

50 records · Page 3Linked to original sources

arXiv2016

Searching with Advice: Robot Fence-Jumping

We study a new search problem on the plane involving a robot and an immobile treasure, initially placed at distance $1$ from each other. The length $β$ of an arc (a fence) within the perimeter of the corresponding circle, as well as the promise that the treasure is outside the fence, is given as part of the input. The goal is to device movement trajectories so that the robot locates the treasure in minimum time. Notably, although the presence of the fence limits searching uncertainty, the location of the fence is unknown, and in the worst case analysis is determined adversarially. Nevertheless, the robot has the ability to move in the interior of the circle. In particular the robot can attempt a number of chord-jump moves if it happens to be within the fence or if an endpoint of the fence is discovered. The optimal solution to our question can be obtained as a solution to a complicated optimization problem, which involves trigonometric functions, and trigonometric equations that do not admit closed form solutions. For the 1-Jump Algorithm, we fully describe the optimal trajectory, and provide an analysis of the associated cost as a function of $β$. Our analysis indicates that the optimal k-Jump Algorithm requires that the robot has enough memory and computation power to compute the optimal chord-jumps. Motivated by this, we give an abstract performance analysis for every k-Jump Algorithm. Subsequently, we present a highly efficient Halving Heuristic k-Jump Algorithm that can effectively approximate the optimal k-Jump Algorithm, with very limited memory and computation requirements.

cs.CG↗

arXiv2016

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 $a >0$.

cs.DS↗

arXiv2016

Know When to Persist: Deriving Value from a Stream Buffer

We consider \textsc{Persistence}, a new online problem concerning optimizing weighted observations in a stream of data when the observer has limited buffer capacity. A stream of weighted items arrive one at a time at the entrance of a buffer with two holding locations. A processor (or observer) can process (observe) an item at the buffer location it chooses, deriving this way the weight of the observed item as profit. The main constraint is that the processor can only move {\em synchronously} with the item stream; as a result, moving from the end of the buffer to the entrance, it crosses paths with the item already there, and will never have the chance to process or even identify it. \textsc{Persistence}\ is the online problem of scheduling the processor movements through the buffer so that its total derived value is maximized under this constraint. We study the performance of the straight-forward heuristic {\em Threshold}, i.e., forcing the processor to "follow" an item through the whole buffer only if its value is above a threshold. We analyze both the optimal offline and Threshold algorithms in case the input stream is a random permutation, or its items are iid valued. We show that in both cases the competitive ratio achieved by the Threshold algorithm is at least $2/3$ when the only statistical knowledge of the items is the median of all possible values. We generalize our results by showing that Threshold, equipped with some minimal statistical advice about the input, achieves competitive ratios in the whole spectrum between $2/3$ and $1$, following the variation of a newly defined density-like measure of the input. This result is a significant improvement over the case of arbitrary input streams, since in this case we show that no online algorithm can achieve a competitive ratio better than $1/2$.

cs.DS↗

arXiv2015

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 ($d\ge 2$). 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 $i$-th sensor is displaced a distance $d_i$, what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider the $a$-total movement defined as the sum $M_a:= \sum_{i=1}^n d_i^a$, 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 and $a > 0$. The main contribution of the paper is to show the existence of a tradeoff between the $d-$dimensional cube, sensing radius and $a$-total movement. The main results can be summarized as follows for the case of the $d-$dimensional cube. If the $d-$dimensional cube sensing radius is $\frac{1}{2n^{1/d}}$ and $n=m^d$, for some $m\in N$, then we present an algorithm that uses $O\left(n^{1-\frac{a}{2d}}\right)$ total expected movement (see Algorithm 2 and Theorem 5). If the $d-$dimensional cube sensing radius is greater than $\frac{3^{3/d}}{(3^{1/d}-1)(3^{1/d}-1)}\frac{1}{2n^{1/d}}$ and $n$ is a natural number then the total expected movement is $O\left(n^{1-\frac{a}{2d}}\left(\frac{\ln n}{n}\right)^{\frac{a}{2d}}\right)$ (see Algorithm 3 and Theorem 7). In addition, we simulate Algorithm 2 and discuss the results of our simulations.

cs.DS↗

arXiv2015

On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols

In this work we focus on a natural class of population protocols whose dynamics are modelled by the discrete version of Lotka-Volterra equations. In such protocols, when an agent $a$ of type (species) $i$ interacts with an agent $b$ of type (species) $j$ with $a$ as the initiator, then $b$'s type becomes $i$ with probability $P\_{ij}$. In such an interaction, we think of $a$ as the predator, $b$ as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an $n$-agent population converges to some absorbing state in time polynomial in $n$, w.h.p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, even the Rock-Paper-Scissors protocol requires exponential time to converge. We then study threshold effects exhibited by Lotka-Volterra-type protocols with 3 and more species under interactions between any pair of agents. We start by presenting a simple 4-type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but "control" convergence. We then prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a distributed system. Some of our techniques may be of independent value.

cs.DS↗

arXiv2014

Broadcasting in Networks of Unknown Topology in the Presence of Swamping

In this paper, we address the problem of broadcasting in a wireless network under a novel communication model: the {\em swamping} communication model. In this model, nodes communicate only with those nodes at geometric distance greater than $s$ and at most $r$ from them. Communication between nearby nodes under this model can be very time consuming, as the length of the path between two nodes within distance $s$ is only bounded above by the diameter $D$, in many cases. For the $n$-node lattice networks, we present algorithms of optimal time complexity, respectively $O(n/r + r/(r-s))$ for the lattice line and $O(\sqrt{n}/r + r/(r-s))$ for the two-dimensional lattice. We also consider networks of unknown topology of diameter $D$ and of a parameter $g$ ({\em granularity}). More specifically, we consider networks with $γ$ the minimum distance between any two nodes and $g = 1/γ$. We present broadcast algorithms for networks of nodes placed on the line and on the plane with respective time complexities $O(D/l + g^2)$ and $O(Dg/l + g^4)$, where $l \in Θ(\max\{(1-s),γ\})$.

cs.DC↗

arXiv2014

Excuse Me! or The Courteous Theatregoers' Problem

Consider a theatre consisting of $m$ rows each containing $n$ seats. Theatregoers enter the theatre along aisles and pick a row which they enter along one of its two entrances so as to occupy a seat. Assume they select their seats uniformly and independently at random among the empty ones. A row of seats is narrow and an occupant who is already occupying a seat is blocking passage to new incoming theatregoers. As a consequence, occupying a specific seat depends on the courtesy of theatregoers and their willingness to get up so as to create free space that will allow passage to others. Thus, courtesy facilitates and may well increase the overall seat occupancy of the theatre. We say a theatregoer is courteous if (s)he will get up to let others pass. Otherwise, the theatregoer is selfish. A set of theatregoers is $p$-courteous if each theatregoer in the set is courteous with probability $p$, randomly and independently. It is assumed that the behaviour of a theatregoer does not change during the occupancy of the row. In this paper, we are interested in the following question: what is the expected number of occupied seats as a function of the total number of seats in a theatre, $n$, and the probability that a theatregoer is courteous, $p$? We study and analyze interesting variants of this problem reflecting behaviour of the theatregoers as entirely selfish, and $p$-courteous for a row of seats with one or two entrances and as a consequence for a theatre with $m$ rows of seats with multiple aisles. We also consider the case where seats in a row are chosen according to the geometric distribution and the Zipf distibrution (as opposed to the uniform distribution) and provide bounds on the occupancy of a row (and thus the theatre) in each case.

cs.DM↗

arXiv2013

The Beachcombers' Problem: Walking and Searching with Mobile Robots

We introduce and study a new problem concerning the exploration of a geometric domain by mobile robots. Consider a line segment $[0,I]$ and a set of $n$ mobile robots $r_1,r_2,..., r_n$ placed at one of its endpoints. Each robot has a {\em searching speed} $s_i$ and a {\em walking speed} $w_i$, where $s_i <w_i$. We assume that each robot is aware of the number of robots of the collection and their corresponding speeds. At each time moment a robot $r_i$ either walks along a portion of the segment not exceeding its walking speed $w_i$ or searches a portion of the segment with the speed not exceeding $s_i$. A search of segment $[0,I]$ is completed at the time when each of its points have been searched by at least one of the $n$ robots. We want to develop {\em mobility schedules} (algorithms) for the robots which complete the search of the segment as fast as possible. More exactly we want to maximize the {\em speed} of the mobility schedule (equal to the ratio of the segment length versus the time of the completion of the schedule). We analyze first the offline scenario when the robots know the length of the segment that is to be searched. We give an algorithm producing a mobility schedule for arbitrary walking and searching speeds and prove its optimality. Then we propose an online algorithm, when the robots do not know in advance the actual length of the segment to be searched. The speed $S$ of such algorithm is defined as $S = \inf_{I_L} S(I_L)$ where $S(I_L)$ denotes the speed of searching of segment $I_L=[0,L]$. We prove that the proposed online algorithm is 2-competitive. The competitive ratio is shown to be better in the case when the robots' walking speeds are all the same.

cs.DS↗

arXiv2013

Power Strip Packing of Malleable Demands in Smart Grid

We consider a problem of supplying electricity to a set of $\mathcal{N}$ customers in a smart-grid framework. Each customer requires a certain amount of electrical energy which has to be supplied during the time interval $[0,1]$. We assume that each demand has to be supplied without interruption, with possible duration between $\ell$ and $r$, which are given system parameters ($\ell\le r$). At each moment of time, the power of the grid is the sum of all the consumption rates for the demands being supplied at that moment. Our goal is to find an assignment that minimizes the {\it power peak} - maximal power over $[0,1]$ - while satisfying all the demands. To do this first we find the lower bound of optimal power peak. We show that the problem depends on whether or not the pair $\ell, r$ belongs to a "good" region $\mathcal{G}$. If it does - then an optimal assignment almost perfectly "fills" the rectangle $time \times power = [0,1] \times [0, A]$ with $A$ being the sum of all the energy demands - thus achieving an optimal power peak $A$. Conversely, if $\ell, r$ do not belong to $\mathcal{G}$, we identify the lower bound $\bar{A} >A$ on the optimal value of power peak and introduce a simple linear time algorithm that almost perfectly arranges all the demands in a rectangle $[0, A /\bar{A}] \times [0, \bar{A}]$ and show that it is asymptotically optimal.

cs.DS↗

arXiv2011

Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

Given a set $P$ of $n$ points in the plane, we solve the problems of constructing a geometric planar graph spanning $P$ 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set $P$, respectively. First, we construct in $O(n\log{n})$ time a geometric planar graph of minimum degree 2 and max edge length bounded by 2 times the optimal. This is then used to construct in $O(n\log n)$ time a 2-edge connected geometric planar graph spanning $P$ with max edge length bounded by $\sqrt{5}$ times the optimal, assuming that the set $P$ forms a connected Unit Disk Graph. Second, we prove that 2 times the optimal is always sufficient if the set of points forms a 2 edge connected Unit Disk Graph and give an algorithm that runs in $O(n^2)$ time. We also show that for $k \in O(\sqrt{n})$, there exists a set $P$ of $n$ points in the plane such that even though the Unit Disk Graph spanning $P$ is $k$-vertex connected, there is no 2-edge connected geometric planar graph spanning $P$ even if the length of its edges is allowed to be up to 17/16.

cs.DM↗

arXiv2011

Computing Majority with Triple Queries

Consider a bin containing $n$ balls colored with two colors. In a $k$-query, $k$ balls are selected by a questioner and the oracle's reply is related (depending on the computation model being considered) to the distribution of colors of the balls in this $k$-tuple; however, the oracle never reveals the colors of the individual balls. Following a number of queries the questioner is said to determine the majority color if it can output a ball of the majority color if it exists, and can prove that there is no majority if it does not exist. We investigate two computation models (depending on the type of replies being allowed). We give algorithms to compute the minimum number of 3-queries which are needed so that the questioner can determine the majority color and provide tight and almost tight upper and lower bounds on the number of queries needed in each case.

cs.DS↗

arXiv2010

A Tight Bound on the Maximum Interference of Random Sensors in the Highway Model

Consider $n$ sensors whose positions are represented by $n$ uniform, independent and identically distributed random variables assuming values in the open unit interval $(0,1)$. A natural way to guarantee connectivity in the resulting sensor network is to assign to each sensor as its range, the maximum of the two possible distances to its two neighbors. The interference at a given sensor is defined as the number of sensors that have this sensor within their range. In this paper we prove that the expected maximum interference of the sensors is $Θ(\sqrt{\ln n})$.

cs.CG↗

arXiv2009

Evaluation of Anonymized ONS Queries

Electronic Product Code (EPC) is the basis of a pervasive infrastructure for the automatic identification of objects on supply chain applications (e.g., pharmaceutical or military applications). This infrastructure relies on the use of the (1) Radio Frequency Identification (RFID) technology to tag objects in motion and (2) distributed services providing information about objects via the Internet. A lookup service, called the Object Name Service (ONS) and based on the use of the Domain Name System (DNS), can be publicly accessed by EPC applications looking for information associated with tagged objects. Privacy issues may affect corporate infrastructures based on EPC technologies if their lookup service is not properly protected. A possible solution to mitigate these issues is the use of online anonymity. We present an evaluation experiment that compares the of use of Tor (The second generation Onion Router) on a global ONS/DNS setup, with respect to benefits, limitations, and latency.

cs.CR↗

arXiv2009

Asymptotics of Canonical and Saturated RNA Secondary Structures

It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is $1.104366 n^{-3/2} 2.618034^n$. In this paper, we study combinatorial asymptotics for two special subclasses of RNA secondary structures - canonical and saturated structures. Canonical secondary structures were introduced by Bompfünewerer et al., who noted that the run time of Vienna RNA Package is substantially reduced when restricting computations to canonical structures. Here we provide an explanation for the speed-up. Saturated secondary structures have the property that no base pairs can be added without violating the definition of secondary structure (i.e. introducing a pseudoknot or base triple). Here we compute the asymptotic number of saturated structures, we show that the asymptotic expected number of base pairs is $0.337361 n$, and the asymptotic number of saturated stem-loop structures is $0.323954 1.69562^n$, in contrast to the number $2^{n-2}$ of (arbitrary) stem-loop structures as classically computed by Stein and Waterman. Finally, we show that the density of states for [all resp. canonical resp. saturated] secondary structures is asymptotically Gaussian. We introduce a stochastic greedy method to sample random saturated structures, called quasi-random saturated structures, and show that the expected number of base pairs of is $0.340633 n$.

q-bio.QM↗