Tight Bounds for Double Coverage Against Weak Adversaries
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Abstract
We study the Double Coverage (DC) algorithm for the kserver problem in tree metrics in the (h, k)setting, i.e., when DC with k servers is compared against an offline optimum algorithm with h ≤ k servers. It is wellknown that in such metric spaces DC is kcompetitive (and thus optimal) for h = k. We prove that even if k > h the competitive ratio of DC does not improve; in fact, it increases slightly as k grows, tending to h + 1. Specifically, we give matching upper and lower bounds of \(\frac {k(h+1)}{k+1}\) on the competitive ratio of DC on any tree metric.
Keywords
kserver Weak adversaries Resource augmentation Double coverage1 Introduction
We consider the kserver problem defined as follows. There are k servers in a given metric space. In each step, a request arrives at some point of the metric space and must be served by moving some server to that point. The goal is to minimize the total distance traveled by the servers.
The kserver problem was introduced by Manasse et al. [8] as a far reaching generalization of various online problems. The most wellstudied of those is the paging (caching) problem, which corresponds to kserver problem on a uniform metric space. Sleator and Tarjan [9] gave several kcompetitive algorithms for paging and showed that this is the best possible ratio for any deterministic algorithm.
Interestingly, the kserver problem does not seem to get harder on more general metrics. The celebrated kserver conjecture states that a kcompetitive deterministic algorithm exists for every metric space. In a breakthrough result, Koutsoupias and Papadimitriou [7] showed that the work function algorithm (WFA) is (2k − 1)competitive for every metric space, almost resolving the conjecture. The conjecture has been settled for several special metrics (an excellent reference is [2]). In particular for the line metric, Chrobak et al. [3] gave an elegant kcompetitive algorithm called Double Coverage (DC). This algorithm was later extended and shown to be kcompetitive for all tree metrics [4]. Additionally, in [1] it was shown that WFA is kcompetitive for some special metrics, including the line.
The (h, k)Server Problem
In this paper, we consider the (h, k)setting, where the online algorithm has k servers, but its performance is compared to an offline optimal algorithm with h ≤ k servers. This is also known as the weak adversaries model [6], or the resource augmentation version of kserver. It is a salient point whether the algorithm knows the value of h. We assume that it does not, as the DC algorithm that we analyze does not utilize this value (and the same is true of WFA). Moreover, this assumption is more in the spirit of resource augmentation. Note that in general this distinction matters, as knowing h, an algorithm might decide to limit the number of servers it will use to serve the requests. The (h, k)server setting turns out to be much more intriguing and is much less understood.
For the uniform metric (the (h, k)paging problem), k/(k − h + 1)competitive algorithms are known [9] and no deterministic algorithm can achieve a better ratio. Note that this guarantee equals k for h = k, and tends to 1 as the ratio of the number of online to offline servers k/h becomes arbitrarily large. This shows that the weak adversaries model could give more accurate interpretation on the performance of online algorithms: The competitive ratio of k obtained in the classical setting grows with the number of servers, which could possibly mean that more servers worsen the performance of an algorithm. On the other hand, the ratio obtained in the (h, k) setting shows that the performance improves substantially when the number of servers grows. The same competitive ratio can also be achieved for the weighted caching problem [10] (and even the more general file caching problem [11], which is not a special case of the (h, k)server problem).
However, unlike classical kserver, the underlying metric space seems to play an important role in the (h, k)setting. BarNoy and Schieber (cf. [2]) showed that, for the (2,k)server problem on a line metric, no deterministic algorithm can be better than 2competitive for any k. In particular, the ratio does not tend to 1 as k increases.
In fact, there is huge gap in our understanding of the problem, even for very special metrics. For example, for the line no guarantee better than h is known even when k/h → ∞. On the other hand, the only lower bounds known are the result of BarNoy and Schieber mentioned above and a general lower bound of k/(k − h + 1) for any metric space with at least k + 1 points (cf. [2] for both results). In particular, no lower bound better than 2 is known for any metric space and any h > 2, if we let k/h → ∞. The only general upper bound is due to Koutsoupias [6], who showed that WFA is 2hcompetitive^{1} for the (h, k)server problem on any metric. It is worth stressing that this is an upper bound for WFA that is oblivious of h and uses all of its k servers, and that ratio 2h − 1 can be attained by running WFA with h servers when this value is known to the algorithm.
The DC Algorithm
This motivates us to consider the (h, k)server problem on the line and more generally on trees. In particular, we consider the DC algorithm [3] originally defined for a line, and its generalization to trees [4]. We refer to both as DC, since the latter specializes to the former when the underlying tree is in fact a line. As understanding both the algorithm and its analysis for the line may be simpler and more insightful, we provide definitions of both variants. In both, we call an algorithm’s server s adjacent to the request r if there are no algorithm’s servers on the unique path between the locations of r and s, excluding the point where s is located. Note that there may be multiple servers in one location, satisfying this requirement — in such case, one of them is chosen arbitrarily as the adjacent server for this location, and the others are considered nonadjacent.
DCLine
If the current request r lies outside the convex hull of current servers, serve it with the nearest server. Otherwise, we move the two servers adjacent to r towards it with equal speed until some server reaches r.
DCTree
Repeat the following until a server reaches the request r, constantly updating the set of adjacent servers: move all the servers adjacent to r towards r at equal speed. Note that the set of servers adjacent to r can change only when one of them reaches either a vertex of the tree or the request itself, which ends the move. We call the parts of the move between updates of the set of adjacent servers elementary moves.
There are several natural reasons to consider DC for line and trees. For paging (and weighted paging), all known kcompetitive algorithms also attain the optimal ratio for the (h, k) version. This suggests that a kcompetitive algorithm for the kserver problem might attain the “right” ratio in the (h, k)setting. The only algorithm that satisfies this condition for a nontrivial metric is DC for trees, as well as WFA for the simpler case of a line. Of the two, DC has the advantage that it attains the optimum k/(k − h + 1)competitive ratio for the (h, k)paging problem, when it is modelled as a star graph where requests appear in leaves, since it is equivalent to FlushWhenFull algorithm, as pointed out by Chrobak and Larmore [4]; see Appendix A for an explicit proof. As for WFA, all known upper bounds, including [6], bound the extended cost instead of the actual cost of the algorithm. Using this approach we can easily show that WFA is (h + 1)competitive for the line (cf. Appendix B).
Our Results
We show that the exact competitive ratio of DC on lines and trees in the (h, k)setting is \(\frac {k(h+1)}{(k+1)}\).
Theorem 1
The competitive ratio of DC is at least \(\frac {k(h+1)}{(k+1)}\) , even for a line.
Note that for a fixed h, the competitive ratio worsens slightly as the number of online servers k increases. In particular, it equals h for k = h and it approaches h + 1 as k → ∞.
Consider the seemingly trivial case of h = 1. If k = 1, clearly DC is 1competitive. However, for k = 2 it becomes 4/3 competitive, as we now sketch. Consider the instance where all servers are at x = 0 initially. A request arrives at x = 2, upon which both DC and offline move a server there and pay 2. Then a request arrives at x = 1. DC moves both servers there and pays 2 while offline pays 1. All servers are now at x = 1, and the instance repeats.
Generalizing this example to (1,k) already becomes quite involved. Our lower bound in Theorem 1 for general h and k is based on an adversarial strategy obtained by a careful recursive construction.
We also give a matching upper bound.
Theorem 2
For any tree, the competitive ratio of DC is at most \(\frac {k(h+1)}{(k+1)}\).
This generalizes the previous results for h = k [3, 4]. Our proof also follows similar ideas, but our potential function is more involved (it has three terms instead of two), and the analysis is more subtle. To keep the main ideas clear, we first prove Theorem 2 for the simpler case of a line in Section 3. The proof for trees is analogous but more involved, and is described in Section 4.
2 Lower Bound for the Line Metric
We now prove Theorem 1. We will describe an adversarial strategy S _{ k } for the setting where DC has k servers and the offline optimum (adversary) has h servers, whose analysis establishes that the competitive ratio of DC is at least k(h + 1)/(k + 1).
Roughly speaking (and ignoring some details), the strategy S _{ k } works as follows. Let I = [0,b _{ k }] be the working interval associated with S _{ k }. Let L = [0,𝜖 b _{ k }] and R = [(1−𝜖)b _{ k }, b _{ k }] denote the (tiny) left front and right front of I. Initially, all offline and online servers are located in L. The adversary moves all its h servers to R and starts requesting points in R, until DC eventually moves all its servers to R. The strategy inside R is defined recursively depending on the number of DC servers currently in R: if DC has i servers in R, the adversary executes the strategy S _{ i } repeatedly inside R, until another DC server arrives there, at which point it switches to the strategy S _{ i + 1}. When all DC servers reach R, the adversary moves all its h servers back to L and repeats the symmetric version of the above instance until all servers move from R to L. This defines a phase. To show the desired lower bound, we recursively bound the online and offline costs during a phase of S _{ k } in terms of costs incurred by strategies S _{1}, S _{2}, …,S _{ k − 1}.
Formal Description

d _{ i }, lower bound for the cost of DC inside the working interval.

A _{ i }, upper bound for the cost of the adversary.

p _{ i }, P _{ i }, lower resp. upper bound for the “pull” exerted on any external DC servers located to the left of the working interval of S _{ i }. Note that, as will be clear later, by symmetry the same pull is exerted to the right.
Strategies S _{ i } for i ≤ h
For i ≤ h, strategies S _{ i } are performed in a typeh interval (recall this has length h). Let Q be h + 1 points in such an interval, with distance 1 between consecutive points.
As for the cost incurred by the adversary, we have A _{ i } = 0, for i<h, as the offline servers do not have to move at all. For i = h, the offline can serve the sequence with cost 2, by using the server in q _{ h } to serve request in q _{ h + 1} and then moving it back to q _{ h }, therefore A _{ h } = 2.
For strategy \(\overset {\leftarrow }{S_{i}}\), we just number the points of Q in the opposite direction (q _{1} will be leftmost and q _{ h + 1} rightmost). The request sequence, analysis, and assumptions about initial position are the same.
Strategies S _{ i } for i > h
We define the strategy S _{ i } for i > h, assuming that S _{1}, …,S _{ i − 1} are already defined. Let I denote the working interval for S _{ i }. We assume that, initially, all offline and DC servers lie in the leftmost (or analogously rightmost) type (i − 1) interval of I. Indeed, for S _{ k } this is achieved by the initial configuration, and for i<k we will ensure this condition before applying strategy S _{ i }. In this case our phase consists of lefttoright step followed by righttoleft step (analogously, if all servers start in the rightmost interval, we apply first righttoleft step followed by lefttoright step to complete the phase).
 1.Adversary moves all its servers from L _{ i − 1} to R _{ h }, specifically to the points q _{1}, …,q _{ h } to prepare for the strategy \(\overrightarrow {S_{1}}\). Next, point q _{1} is requested, which forces DC to move one server to q _{1}, thus satisfying the initial conditions of \(\overrightarrow {S_{1}}\). The figure below illustrates the servers’ positions after these moves are performed.
 2.For j = 1 to h: keep applying \(\overset {\rightarrow }{S}_{j}\) to interval R _{ h } until the (j + 1)th server arrives at the point q _{ j + 1} of R _{ h }. (Recall that Fig. 3 illustrates Strategy \(\overrightarrow {S_{j}}\) for j ≤ h.) Once it arrives there, complete the request sequence \(\overset {\rightarrow }{S_{j}}\), so that DC servers will reside in points q _{ j + 1}, …,q _{1}, ready for strategy \(\overset {\rightarrow }{S_{j+1}}\). The figure below illustrates the servers’ positions after all those moves (i.e., the whole outer loop, for j = 1…,h) are performed.
 3.For j = h + 1 to i − 1: keep applying S _{ j } to interval R _{ j } until the (j + 1)th server arrives in R _{ j }. To clarify, S _{ j } stands for either \(\overrightarrow {S_{j}}\) or \(\overleftarrow {S_{j}}\), depending on the locations of servers within R _{ j }. In particular, the first S _{ j } for any j is \(\overleftarrow {S_{j}}\). Note that there is exactly one DC server in the working interval of S _{ i } moving toward R _{ j } from the left: the other servers in that working interval are either still in L _{ i − 1} or not moving, since they are not adjacent to the request, or already in R _{ j }. Since R _{ j } is the rightmost interval of R _{ j + 1} and L _{ i − 1}∩R _{ j + 1} = ∅, the resulting configuration is ready for strategy \(\overleftarrow {S_{j+1}}\). The figure below illustrates the very beginning of this sequence of moves, for j = h + 1, right after the execution of the first step (of this threestep description) of \(\overleftarrow {S_{j+1}}\).
Bounding Costs
We begin with a simple but useful observation that follows directly from the definition of DC. For any subset X of i ≤ k consecutive DC servers, let us call center of mass of X the average position of servers in X. We call a request external with respect to X, when it is outside the convex hull of X and internal otherwise.
Lemma 1
For any sequence of internal requests with respect to X, the center of mass of X remains the same.
Proof
Follows trivially since for any internal request, DC moves precisely two servers towards it, by an equal amount in opposite directions. □
Now we are ready to prove Theorem 1.
Proof of Theorem 1
Induction Base (i = h ) For the base case we have a _{ h } = 2, d _{ h } = 2h, and p _{ h } = P _{ h } = 1, so \(\frac {d_{h}}{P_{h}} = 2h\) and \(\frac {A_{h}}{p_{h}} = 2\), i.e., (4) holds.
3 Upper Bound
In this section, we give an upper bound on the competitive ratio of DC that matches the lower bound from the previous section.
We begin by introducing some notation. We denote the optimal offline algorithm by OPT. For the current request r at time t, we let X and Y denote the configurations (i.e. the multisets of points in which their servers are located) of DC and OPT respectively before serving request r. Similarly, X ^{′} and Y ^{′} denote their corresponding configurations after serving r.
Note this generalizes the potential considered in [3, 4] for the case of h = k. In that setting, all the online servers are matched and hence D _{ M } = D _{ X } and is independent of M, and thus the potential above becomes k times that minimum cost matching between X and Y plus D _{ x }. On the other hand in our setting, we need to select the right set M of DC servers to be matched to the offline servers based on minimizing Ψ_{ M }(X, Y).
Let us first give a useful property concerning minimizers of Ψ, which will be crucial later in our analysis. Note that Ψ_{ M }(X, Y) is not simply the best matching between X and Y, but also includes the term D _{ M } which makes the argument slightly subtle.
Lemma 2
Let X and Y be the configurations of DC and OPT and consider some fixed offline server at location y ∈ Y. There exists a minimizer M of Ψ that contains some DC server x which is adjacent to y. Moreover, there is a minimum cost matching \(\mathcal {M}\) between M and Y that matches x to y.
We remark that the adjacency in the lemma statement and the proof is defined as for the DC algorithm (cf. Section 1); specifically, as if there was a request at y’s position. Moreover, we tote that the statement does not necessarily hold simultaneously for every offline server, but only for a single fixed offline server y.
Proof Proof of Lemma 2
Let M ^{′} be some minimizer of Ψ_{ M }(X, Y) and \(\mathcal {M}^{\prime }\) be some associated minimum cost matching between M ^{′} and Y. Let x ^{′} denote the online server currently matched to y in \(\mathcal {M}^{\prime }\) and suppose that x ^{′} is not adjacent to y. Let x denote the server in X adjacent to y on the path from y to x ^{′}.
 1.
If x ∈ M ^{′}: Let y ^{′} denote the offline server which is matched to x in \(\mathcal {M}^{\prime }\). To create new matching \(\mathcal {M}\), we swap the edges and match x to y and x ^{′} to y ^{′}, see Fig. 4. The cost of the edge connecting y in the matching reduces by exactly d(x ^{′}, y)−d(x, y) = d(x ^{′}, x). On the other hand, the cost of the matching edge for y ^{′} increases by d(x ^{′}, y ^{′})−d(x, y ^{′})≤d(x, x ^{′}), due to triangle inequality. Thus, the new matching has no larger cost. Moreover, the set of matched servers does not change, i.e., M = M ^{′}, and hence \(D_{M} = D_{M^{\prime }}\), which implies that \(\Psi _{M}(X,Y) \leq \Psi _{M^{\prime }}(X,Y)\).
 2.If x ∉ M ^{′}: In this case, we set M = M ^{′}∖{x ^{′}}∪{x} and we form \(\mathcal {M}\), where y is matched to x and all other offline servers are matched to the same server as in \(\mathcal {M}^{\prime }\). Now, the cost of the matching reduces by d(x ^{′}, y)−d(x, y) = d(x, x ^{′}). Moreover, \(D_{M} \leq D_{M^{\prime }} + (h1) \cdot d(x,x^{\prime })\), as the distance of each server in M ^{′}∖{x ^{′}} to x can be greater than the distance to x ^{′} by at most d(x, x ^{′}). This givesand hence Ψ_{ M }(X, Y) is strictly smaller than \(\Psi _{M^{\prime }}(X,Y)\).$$\begin{array}{@{}rcl@{}} \Psi_{M}(X,Y)  \Psi_{M^{\prime}}(X,Y) &\leq&  \frac{(h+1)k}{k+1} \cdot d(x,x^{\prime}) + \frac{k(h1)}{k+1} \cdot d(x,x^{\prime}) \\ &=&  \frac{2k}{k+1} \cdot d(x,x^{\prime}) <0 \quad, \end{array} $$
We are now ready to prove Theorem 2 for the line.
Proof
Recall, that we are at time t and request r is arriving. We divide the analysis into two steps: (i) OPT serves r, and then (ii) DC serves r. As a consequence, whenever a server of DC serves r, we can assume that a server of OPT is already there.
In all the steps considered, M is the minimizer of Ψ_{ M }(X, Y) in the beginning of the step. It might happen that, after change of X, Y during the step, a better minimizer can be found. However, an upper bound for ΔΨ_{ M }(X, Y) is sufficient to bound the change in the first term of the potential function. □
OPT Moves
DC Moves
 1.Suppose DC moves its rightmost server (the leftmost server case is identical) by distance d. Let y denote the offline server at r. By Lemma 2 we can assume that y is matched to the rightmost server of DC. Thus, the cost of the minimum cost matching between M and Y decreases by d. Moreover, D _{ M } increases by exactly (h − 1)d (as the distance to rightmost server increases by d for all servers of DC). Thus, Ψ_{ M }(X, Y) changes by$$\frac{k(h+1)}{k+1} \cdot d + \frac{k(h1)}{k+1} \cdot d = \frac{2k}{k+1} \cdot d \quad. $$Similarly, D _{ X } increases by exactly (k − 1)d. This gives us that$$\Delta \Phi(X,Y) \leq \frac{2k}{k+1} \cdot d + \frac{k1}{k+1} \cdot d = d \quad. $$
As D C(t) = d, this implies that (5) holds.
 2.
We now consider the case when DC moves 2 servers x and x ^{′}, each by distance d. Let y denote the offline server at the request r. By Lemma 2 applied to y, we can assume that M contains at least one of x or x ^{′}, and that y is matched to one of them (say x) in some minimum cost matching \(\mathcal {M}\) of M to Y.
We note that D _{ X } decreases by precisely 2d. In particular, the distance between x and x ^{′} decreases by 2d, and for any other server of X∖{x, x ^{′}} its total distance to other servers does not change. Moreover, D C(t) = 2d. Hence, to prove (5), it suffices to showTo this end, we consider two subcases.$$ \Delta \Psi_{M}(X,Y) \leq \frac{k}{k+1} \cdot 2d \quad. $$(6) (a)
Both x and x ^{ ′ } are matched: In this case, the cost of the matching \(\mathcal {M}\) does not increase as the cost of the matching edge (x, y) decreases by d and the move of x ^{′} can increase the cost of the matching by at most d. Moreover, D _{ M } decreases by precisely 2d (due to x and x ^{′} moving closer). Thus, \(\Delta \Psi _{M}(X,Y) \leq \frac {k}{k+1} \cdot 2d\), and hence (6) holds.
 (b)Only x is matched (to y) and x ^{ ′ } is unmatched: In this case, the cost of the matching \(\mathcal {M}\) decreases by d. Moreover, D _{ M } can increase by at most (h − 1)d, as x can move away from each server in M∖{x} by distance at most d. Soi.e., (6) holds.$$\Delta \Psi_{M}(X,Y) \leq \frac{(h+1)k}{k+1} \cdot d + \frac{k(h1)}{k+1} \cdot d =  \frac{2k}{k+1} \cdot d \quad, $$
 (a)
4 Extension to Trees
Proof Proof of Theorem 2
As in the analysis for the line, we split the analysis in two parts: (i) OPT serves r, and then (ii) DC serves r. As a consequence, whenever a server of DC serves r, we can assume that a server of OPT is already there. □
OPT Moves
DC Moves
Instead of focusing on the whole move done by DC to serve request r, we prove that (7) holds for each elementary move.
In order to calculate the change in D _{ X } and D _{ M }, it is convenient to consider the moves of active servers sequentially rather than simultaneously.
Thus, (7) holds, as D C(t) = q⋅d.
Footnotes
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