# Serving Online Requests with Mobile Servers

We study an online problem in which a set of mobile servers have to be moved in order to efficiently serve a set of requests that arrive in an online fashion. More formally, there is a set of $n$ nodes and a set of $k$ mobile servers that are placed at some of the nodes. Each node can potentially host several servers and the servers can be moved between the nodes. There are requests $1,2,\ldots$ that are adversarially issued at nodes one at a time. An issued request at time $t$ needs to be served at all times $t' \geq t$. The cost for serving the requests is a function of the number of servers and requests at the different nodes. The requirements on how to serve the requests are governed by two parameters $\alpha\geq 1$ and $\beta\geq 0$. An algorithm needs to guarantee at all times that the total service cost remains within a multiplicative factor of $\alpha$ and an additive term $\beta$ of the current optimal service cost. We consider online algorithms for two different minimization objectives. We first consider the natural problem of minimizing the total number of server movements. We show that in this case for every $k$, the competitive ratio of every deterministic online algorithm needs to be at least $\Omega(n)$. Given this negative result, we then extend the minimization objective to also include the current service cost. We give almost tight bounds on the competitive ratio of the online problem where one needs to minimize the sum of the total number of movements and the current service cost. In particular, we show that at the cost of an additional additive term which is roughly linear in $k$, it is possible to achieve a multiplicative competitive ratio of $1+\varepsilon$ for every constant $\varepsilon>0$.

**Comments:**25 pages

## Similar Publications

In this work, we introduce an online model for communication complexity. Analogous to how online algorithms receive their input piece-by-piece, our model presents one of the players Bob his input piece-by-piece, and has the players Alice and Bob cooperate to compute a result it presents Bob with the next piece. This model has a closer and more natural correspondence to dynamic data structures than the classic communication models do and hence presents a new perspective on data structures. Read More

We revisit the range $\tau$-majority problem, which asks us to preprocess an array $A[1..n]$ for a fixed value of $\tau \in (0,1/2]$, such that for any query range $[i,j]$ we can return a position in $A$ of each distinct $\tau$-majority element. Read More

We study the problem of clustering sequences of unlabeled point sets taken from a common metric space. Such scenarios arise naturally in applications where a system or process is observed in distinct time intervals, such as biological surveys and contagious disease surveillance. In this more general setting existing algorithms for classical (i. Read More

In this paper, we examine the hash functions expressed as scalar products,
i.e., $f(x)=

Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of $1-\frac{1}{e}$ on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is $\frac{1}{1+1/e} \approx 0. Read More

We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest(EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF we are given an edge-weighted graph with a degree bound for every vertex. Read More

A sum where each of the $N$ summands can be independently chosen from two choices yields $2^N$ possible summation outcomes. There is an $\mathcal{O}(K^2)$-algorithm that finds the $K$ smallest/largest of these sums by evading the enumeration of all sums. Read More

We consider the problem of implementing a space-efficient dynamic trie, with an emphasis on good practical performance. For a trie with $n$ nodes with an alphabet of size $\sigma$, the information-theoretic lower bound is $n \log \sigma + O(n)$ bits. The Bonsai data structure is a compact trie proposed by Darragh et al. Read More

Suffix trees have recently become very successful data structures in handling large data sequences such as DNA or Protein sequences. Consequently parallel architectures have become ubiquitous. We present a novel alphabet-dependent parallel algorithm which attempts to take advantage of the perverseness of the multicore architecture. Read More

In this paper, we first remodel the line coverage as a 1D discrete problem with co-linear targets. Then, an order-based greedy algorithm, called OGA, is proposed to solve the problem optimally. It will be shown that the existing order in the 1D modeling, and especially the resulted Markov property of the selected sensors can help design greedy algorithms such as OGA. Read More