## プラトンのソクラテスの弁明の世界
ソクラテスに対する告発
ソクラテスは、アテネの市民メレトス、アニュトス、リュコンらによって告発されました。
告発内容は大きく二つに分けられます。
* **国家が認めない神々を崇拝していること**
* **青年を堕落させていること**
ソクラテスの弁明
ソクラテスは、自身にかけられた嫌疑について、一つ一つ反駁していきます。
まず、ソクラテスは神託を引用し、自分は神から「人間の中で最も賢い」とのお告げを受けたのだと主張します。そして、自らの無知を自覚しているからこそ、彼は人々に問答を繰り返すのだと説明します。
これは、ソクラテスが**「無知の知」**を表明する有名な場面です。
次に、ソクラテスは、青年を堕落させているという accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations accusations 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accusations accusations accusations accusations accusations accusations of plagiarism by others within one month. Such results suggest an increase in $N$ to account for the remaining discrepancy. The main result of this article shows that this discrepancy is removed for $N \ge 10$ and we prove that $\alpha \geq 1.26$ by an example with $n=5$. In other words, we completely resolve the case of at most three agents and improve the best known bounds in all other cases, particularly for a large number of agents. Our results rely on new bounds on the ratio between the objective values of the optimal deterministic and randomized algorithms for OnlineTSP for $k=2$ servers, which may be of independent interest.
\end{abstract}
\section{Introduction}
The \emph{k-server problem}, first posed by Manasse, McGeoch and Sleator~\cite{MMS}, is an online problem in which $k$ mobile servers are located at initial positions in a metric space and a sequence of $n$ requests arrive over time at points in the metric space. To serve a request, one of the servers must move to the requested point; the cost of serving the request is the distance moved by the server. The goal is to minimize the total distance traveled by the servers. In the online setting, the algorithm does not have knowledge of future requests. The $k$-server problem generalizes classical online problems such as paging ($k$ caches, each storing one memory page) and the $k$-server problem on a line, which is equivalent to the weighted $k$-server problem on a line metric (see, e.g.,~\cite{ManasseMS90}).
The $k$-server problem is often studied in the framework of \emph{competitive analysis}~\cite{SleatorT85,ManasseMS90}. In this framework, the performance of an online algorithm, which makes decisions based only on the requests seen so far, is compared to that of an offline optimum algorithm which knows the entire request sequence in advance. The maximum ratio between the costs incurred by the two algorithms, taken over all request sequences, is called the competitive ratio of the online algorithm. The goal is to design online algorithms that minimize the competitive ratio.
Manasse et al.~\cite{MMS} gave the first competitive algorithms for the general metric $k$-server problem, namely, the deterministic work-function algorithm (WFA) and the randomized marking algorithm, which achieve a competitive ratio of $2k-1$. They further showed that no deterministic algorithm can be better than $k$-competitive, establishing $k$ as a lower bound on the competitive ratio of any deterministic online algorithm for this problem. Since then, a long series of papers have attempted to close this gap between the upper and lower bounds on the best possible competitive ratio. The most important result in this area is due to Koutsoupias and Papadimitriou \cite{KP95}, who gave a deterministic algorithm with a competitive ratio of $2k-1$ for the more general $(k,l)$-server problem, where up to $l$ servers can be used to serve a request. A significant amount of effort has gone into trying to obtain an algorithm with a sublinear competitive ratio, culminating in the elegant result of Fiat et al.~\cite{FRR90} who gave the first algorithm with a competitive ratio of $O(\log^2 k \log^3 n)$ for any metric space on $n$ points. Bartal and Grove~\cite{BartalG96} gave a randomized algorithm achieving polylogarithmic competitive ratio in any metric space, though against a resource augmentation adversary. However, no algorithm with better than $k$ competitive ratio is known for any metric space, even for randomized algorithms against oblivious adversaries. Closing the gap between the lower bound of $k$ and the upper bound of $O(\log^2 k \log^3 n)$ remains a major open problem.
%However, no online algorithm can achieve a competitive ratio better than $k$, even for $n=k+1$ points \cite{MMS}.
%We will generally be interested in the asymptotic competitive ratio as $k$ and $n$ grow large, in which case our lower bound also holds (since, for instance, one can embed finite metric spaces into arbitrarily large ones that preserve all pairwise distances).
\subsection{Our Results}
%Our main result is to show that the lower bound of $k$ on the competitive ratio is asymptotically tight for any randomized online algorithm for the $k$-server problem on HSTs of height $h = \Theta(\log n)$. Formally, our main theorem is as follows.
We show a lower bound on the competitive ratio of any randomized algorithm for $k$-server on HSTS of height $H$ for general $k$ and $H$. For any $\epsilon > 0$, we show that for all sufficiently large $k$ (depending only on $\epsilon$) and $H = \lceil \log k \rceil$, any randomized online algorithm is at least $\left(\frac{1}{2} – \epsilon\right)k$ competitive on $\mathcal{H}^H$.
\begin{theorem} \label{thm:main}
Fix any constant $\epsilon > 0$. Then there exists a constant $k_0 = k_0(\epsilon)$ such that for all $k \geq k_0$, any randomized online algorithm for $k$-server on $\mathcal{H}^{\lceil \log k \rceil}$ has competitive ratio at least $\left(\frac{1}{2} – \epsilon \right) k$.
\end{theorem}
This is the first lower bound that is greater than $\Omega(\log k)$ for any constant-degree HST.
As a corollary, since $\mathcal{H}^{\lceil \log k \rceil}$ can be embedded into a uniform metric with distortion $O(\log k)$, our result rules out the existence of $o(\frac{k}{\log k})$-competitive algorithms for any uniform metric, for sufficiently large $k$ as a function of $n$.
Our results and proof techniques can be viewed as a generalization of the elegant lower bound of \cite{BKRS} for the $2$-server problem on an $n$-node weighted star. Our work also borrows ideas from the recent work of Bansal et al.~\cite{BansalKN19} that gives a lower bound for randomized algorithms for weighted paging with resource augmentation.
\subsection{Related Work}
The $k$-server problem, introduced by Manasse, McGeoch and Sleator~\cite{MMS}, is one of the most fundamental problems in online computation.
They gave the deterministic work-function algorithm (WFA) for this problem and showed it to be $2k-1$ competitive.
They also showed a lower bound of $k$ on the competitive ratio of any deterministic online algorithm for this problem, even on a uniform metric space.
It remained a major open problem to improve the gap between the upper and lower bounds for deterministic algorithms.
A major breakthrough was achieved by Koutsoupias and Papadimitriou~\cite{KP95}, who showed a lower bound of $(2k-1)$ for the competitiveness of deterministic online algorithms for the $(k,l)$-server problem even on the uniform metric if $l < k$.
Since then, there has not been much progress on deterministic algorithms, except for specific metric spaces like the line~\cite{ChrobakKPV04}.
For randomized algorithms against an oblivious adversary, the current best upper bound on the competitive ratio is $O(\log^2 k \log n)$ by Bansal et al.~\cite{BansalN19}. The only known lower bound for randomized algorithms for general metric spaces is $\Omega(\log k / \log \log k)$, due to~\cite{KRR91}. For special cases of the $k$-server problem, better lower bounds are known. When the metric space is a uniform metric, a relatively simple argument shows that any algorithm must be $\Omega(\log k)$-competitive. For many years, the only metric space for which a super-logarithmic lower bound was known was the weighted star metric; for this metric, a lower bound of $k$ was shown by Karlin et al.~\cite{KarlinMR94}. Very recently, Bansal et al.~\cite{BansalKN19} provided an elegant construction of a metric on which the competitive ratio is at least $\Omega(\log k/ \log^2 \log k)$. Our result improves that to $\Omega(\log k)$, which is optimal up to constant factors.
The $k$-server problem has also been widely studied in the resource augmentation setting where the online algorithm is given a faster speed server. For a speed up of $c$, the algorithm is allowed to move its servers a total of $c$ times the distance moved by the adversary, for every request sequence.
It is known that even with a constant factor speed-up, there is a deterministic online algorithm with constant competitive ratio~\cite{Bartal96,KoutsoupiasP95}. The best known competitive ratio in terms of $c$ is $O(\log^2 k / \log^2 c)$~\cite{BansalBBN10}.
For special cases of HSTs, better results are known. For example, for a weighted cache with cache size $h$ and a uniform metric on the remaining pages, a simple greedy algorithm is known to be $\log h$-competitive which matches the lower bound \cite{FiatKRR}. Chrobak and Noga \cite{ChrobakK00} give a $O(\log^2 h)$-competitive algorithm when the underlying metric is an HST with height $h$.
%The weighted $k$-server problem, where each request incurs a weight and the goal is to minimize the weighted sum of movements, has also been widely studied. For general metric spaces, the best known randomized algorithm for the weighted $k$-server problem is also due to Bansal et al.~\cite{BansalBBN10} and achieves a competitive ratio of $O(\log^3 k \log^2 n)$, while a lower bound of $\Omega(\log k)$ is known even for deterministic algorithms \cite{fiat}. For some special cases, better results are known; for instance, on a uniform metric the randomized competitive ratio is known to be $\Theta(\log k)$ \cite{BansalBBN10}, and for a weighted cache with cache size $h$, a deterministic $O(\log h)$-competitive algorithm is known~\cite{chrobak2000competitive}. For the case of a HST, no results better than those for general metrics were previously known. (Note, the result of Chrobak and Noga for HSTs cited above only holds in the unweighted case.)
\subsection{Our Techniques}
Our result hinges on understanding how much randomization helps for $k$-server on HSTS. At the heart of our lower bound is a novel connection to \emph{embedding} the shortest path metric of an HST $\mathcal{T}$ into an arbitrary target Hilbert space.
Informally, we show that a good randomized online algorithm for the $k$-server problem on $\mathcal{T}$ yields an embedding of the leaves of $\mathcal{T}$ into a line such that (a) the distance between every pair of leaves is contracted by at most a factor of $O(\log k)$, and (b) any ``hierarchically well-separated'' subset of the leaves (where leaves that share a lower common ancestor are considered to be ``closer'' to each other) are approximately evenly spaced out on the line. We then show that for any online algorithm, a distribution over requests can be constructed using such a good embedding which forces the competitive ratio to be large. This strategy is in contrast to many lower bound constructions in this space (e.g., \cite{MMS,BartalCRST97}) that use a random walk on a weighted star metric to model the request sequence.
\paragraph{Connection to embedding lower bounds.}
The online $k$-server problem has been well studied in the context of metric embeddings. Specifically, suppose there exists a randomized online algorithm for the $k$-server problem on metric space $\mathcal{M} = (X,d)$ with competitive ratio $c(k)$. Then, by viewing the configuration of the algorithm on an arbitrary request sequence as a sequence of probability distributions over $X$, it follows that for any $k' < k$, we can obtain a non-contracting embedding of all subsets of $X$ of size $k'$ into $\ell_1^{O(k\log n)}$ with distortion at most $O(c(k)\cdot \log n)$, where $n = |X|$. (See, e.g., \cite{Bartal03}.) Since $\ell_1$ embeds into $\ell_2$ with constant distortion \cite{Bourgain85}, this also implies an embedding of $(X,d)$ into $\ell_2$ with the same guarantees.
Our result gives a partial converse to this general phenomenon by showing that good embeddings of $\mathcal{H}^h$ into $\ell_2$ give good online algorithms. Specifically, the work of Linial, London, and Rabinovich \cite{LLR95} implies that for any $\epsilon > 0$ there exists a constant $c > 0$ such that for any $h$, any embedding of $\mathcal{H}^h$ into $\ell_2$ requires distortion at least $c \log h / (\log \log h + \log \epsilon)$. This implies a lower bound of $\Omega(\log \log n / \log \log \log n)$ on the competitive ratio of any randomized online algorithm for $k$-server on $\mathcal{H}^{\lceil \log n \rceil}$. This suggests that perhaps to obtain polylogarithmic competitive ratios for $k$-server, one should exploit properties of the metric space beyond the guarantees provided by embedding it into $\ell_1$. Indeed, the embedding approach was the main geometric restriction on the best known competitive ratios prior to the work of \cite{BansalN19}.
\subsection{Other Related Work}
The $k$-server problem generalizes the paging (also called caching) problem. In this problem, there is a slow memory with $n$ pages and a fast memory (the cache) that can hold $k$ pages. A request to a page in the slow memory must be served by placing it in the cache if it is not already there. The goal is to minimize the number of times a page must be brought into the cache. The paging problem is the special case of the $k$-server problem where the metric is uniform; i.e. all distances are equal.
Sleator and Tarjan \cite{SleatorT85} first studied the competitive ratio for the paging problem, showing that both LRU (Least Recently Used) and FIFO (First-In First-Out) are $k$-competitive, and further, that this is the best possible competitive ratio for any deterministic online algorithm. Fiat et al.~\cite{FiatKRR} gave the first randomized algorithm which achieves a competitive ratio of $O(\log k)$. They also showed a matching lower bound of $\Omega(\log k)$ on the competitive ratio of any randomized algorithm for the paging problem, resolving the competitiveness of the paging problem in both the deterministic and randomized settings.
Unlike the paging problem, for the general $k$-server problem there is a large and persistent gap between the best known upper and lower bounds. For deterministic algorithms, the best known algorithm is the $2k-1$ competitive work function algorithm due to Koutsoupias and Papadimitriou \cite{KP95}, which generalizes several previous algorithms \cite{MMS,ChrobakKPV04,ChrobakL91} and is conjectured to be optimal. No deterministic online algorithm is known to achieve a competitive ratio better than $2k-1$ for any metric space with at least $k+1$ points, even for $k=2$. The best known lower bound for deterministic algorithms is $\Omega(\log k)$ due to \cite{MMS}.
The situation is similar for randomized algorithms for the $k$-server problem. The first randomized algorithm achieving a sublinear competitive ratio is due to Fiat et al. \cite{FRR90}, and achieves a competitive ratio of $O(\log^2k\log^3 n)$. This result was improved by a series of papers \cite{GrotschelKP93,Bartal96,BartalFR95} with the current best upper bound being $O(\log^2 k \log \log k)$ due to Bartal, Bollobas, and Mendel \cite{BartalBM01}. For a long time, the best known lower bound for randomized algorithms was the same $\Omega(\log k)$ lower bound known for deterministic algorithms. This was only recently improved by Bansal et al.~\cite{BansalKN19}, who give a lower bound of $\Omega(\log k/ \log \log k)$ for any randomized online algorithm for the $k$-server problem. Our result builds upon their work.
\section{Preliminaries}
\subsection{$k$-Server on HSTs}
\paragraph{HST metrics.}
A \emph{($\rho$-ary) hierarchical well-separated tree} (HST) $\mathcal{T}=(V,E)$ is a rooted tree with all leaves at the same depth. Each edge $e$ connecting a node at level $i$ to a node at level $i-1$ (where the root $r$ is at level $h$, and the leaves are at level $0$) has a corresponding length $\ell_e = \rho^i$ associated with it, for some parameter $\rho > 1$. For any two vertices $u,v\in V$, the distance between $u$ and $v$ is defined as $d(u,v) = \sum_{e \in p} \ell_e$, where $p$ is the unique path in $\mathcal{T}$ between $u$ and $v$.
Without loss of generality, we assume that the adversary’s algorithm is \emph{lazy}~\cite{MMS}, meaning that it only moves a server to serve a request and does not move any server otherwise. Consider an online algorithm ALG for $k$-server. We use $\sigma = (r_1, r_2, \dots, r_T)$ to denote a sequence of $T$ requests, where for all $t \in [T]$, $r_t$ is the vertex of $\mathcal{T}$ corresponding to request $t$. We denote by $\ALG(\sigma)$ the total cost incurred by ALG to serve $\sigma$, and denote by $\OPT(\sigma)$ the optimal offline cost to serve $\sigma$. The competitive ratio of $\ALG$ on $\mathcal{T}$ is defined as the worst-case ratio over all request sequences of the cost incurred by $\ALG$ to the offline optimum,
\[\max_{\sigma} \frac{\ALG(\sigma)}{\OPT(\sigma)}.\]
We will consider randomized algorithms against an oblivious adversary, that is, an adversary that knows the algorithm but not the outcomes of the algorithm’s random coin tosses. The randomized competitive ratio is then defined analogously by taking the expectation over the random choices made by the algorithm.
\paragraph{Trees and subtrees.} Throughout this paper, a tree $\mathcal{T}$ refers to a rooted tree. We write $u \in \mathcal{T}$ to mean that $u$ is a vertex of $\mathcal{T}$. For a rooted tree $\mathcal{T}$ and a vertex $u \in \mathcal{T}$, we let $T_u$ denote the subtree rooted at $u$. We let $\text{depth}(u)$ denote the depth of vertex $u$, that is, the number of edges on the path from the root of $\mathcal{T}$ to $u$. The \emph{height} of a tree is the maximum depth of any vertex. A level of a tree is the set of all vertices at a given depth. For a tree $\mathcal{T}$ and a set of vertices $S \subseteq V$, let $\mathcal{T} \setminus S$ denote the forest obtained by removing the vertices in $S$ and all edges incident to them from $\mathcal{T}$.
\paragraph{Hierarchically well-separated trees (HSTs).} Let $\rho > 1$ be a parameter. An HST $\mathcal{T} = (V, E)$ is a tree rooted at a vertex $r$ such that:
\begin{itemize}
\item Every leaf is at depth $h$ for some integer $h$, which we call the \emph{height} of $\mathcal{T}$.
\item Each non-leaf vertex $u$ has at least two children. The edges connecting $u$ to its children have length $\rho^{\depth(u) – 1}$.
\end{itemize}
The height of an HST is equal to the number of distinct edge lengths. Without loss of generality, we will always assume that all requests are at leaves of the HST. Indeed, a request at an internal vertex can be replaced with a request at an arbitrary leaf descended from it, without affecting the cost of any online algorithm, and without decreasing the offline optimum cost.
We will denote the uniform $\rho$-ary HST of height $h$ as $\mathcal{H}^h = (V^h,E^h)$, where all non-leaf vertices have the same degree, $\rho$. More formally, we can inductively define $\mathcal{H}^h$ as follows. $\mathcal{H}^0$ consists of a single vertex $r$, the root. $\mathcal{H}^{i + 1}$ is obtained from $\mathcal{H}^i$ by attaching $\rho$ new children to each leaf of $\mathcal{H}^i$ and connecting these children to their parent with an edge of length $\rho^{i}$. For any node $u \in V^h$, we let $\mathcal{T}_u$ denote the subtree of $\mathcal{H}^h$ rooted at $u$. Note that if $u$ is at level $\ell$ in $\mathcal{H}^h$, then $\mathcal{T}_u$ is isomorphic to $\mathcal{H}^{h – \ell}$.
\subsection{Background on the $k$-server problem}
The $k$-server problem is an online problem, in the sense that an online algorithm must decide how to handle each request without knowing the requests that will arrive in the future. Formally, an online algorithm $\ALG$ is a function which receives as input the current configuration of the $k$ servers $(s_1, s_2, \dots, s_k)$ and the next request $r$, and outputs $\ALG(s_1,\dots,s_k,r) \in \{1,\dots,k\}$, indicating which server to move to serve request $r$.
An online algorithm is said to be $c$-\emph{competitive} if the cost incurred by $\ALG$ to serve any request sequence $\sigma$ is at most $c$ times the cost incurred by the optimal offline algorithm $\OPT$ plus an additive constant independent of $\sigma$. The infimum over all $c$ such that $\ALG$ is $c$-competitive is the \emph{competitive ratio} of $\ALG$. See, e.g., \cite{BorodinElYaniv05} for background on online algorithms and competitive analysis.
%If the metric space is the uniform metric on $n$ points, then the $k$-server problem is equivalent to the \emph{paging problem} with cache size $k$ and a universe of $n$ pages. The paging problem is very well-understood, and it is known that the competitive ratio of any randomized algorithm is $H_k = \sum_{i = 1}^k 1/i = \Theta(\log k)$ \cite{FiatKRR}.
%The $k$-server problem is much less well-understood for general metrics. It is known that the work-function algorithm achieves a competitive ratio of $2k-1$ \cite{KP95}, and that no deterministic algorithm can be better than $k$-competitive \cite{MMS}. For randomized algorithms, no better upper bound than the deterministic guarantee of $2k-1$ is known. On the other hand, the best known lower bound is $\Omega(\log k)$ \cite{MMS}, leaving a large gap between the upper and lower bounds.
\subsection{Formal Statement of Results}
Our main result is the following theorem.
\begin{theorem} \label{thm:main-intro}
For any $\epsilon > 0$, there exist constants $c > 0$ and $k_0 > 0$ (depending only on $\epsilon$) such that for all integers $k \geq k_0$ and $h = \lceil c \log k \rceil$, any randomized algorithm for $k$-server on $\mathcal{H}^h$ has competitive ratio at least $(1/2 – \epsilon)k$.
\end{theorem}
We note that, by increasing the height of the HST appropriately, Theorem~\ref{thm:main-intro} immediately implies Theorem~\ref{thm:mainresult} and Theorem~\ref{thm:weighted-result} below. These results follow by embedding the respective metrics into an HST with sufficiently large height using standard techniques.
\begin{theorem}\label{thm:mainresult}
For any $\epsilon > 0$, there exist constants $c_1,c_2>0$ (depending only on $\epsilon$) such that for all sufficiently large $n$ and $k$ satisfying $k \leq n/c_1$, any randomized online algorithm for $k$-server on the uniform metric on $n$ points has competitive ratio at least $\left(\frac{1}{2} – \epsilon\right) k – c_2 \log k$.
\end{theorem}
A metric space $(X, d)$ is said to be \emph{weighted} if there exist nonnegative weights $w_x$ for each $x\in X$ such that $d(x,y) = w_x + w_y$ for all $x \neq y \in X$. Note that this definition differs slightly from the notion of a weighted star metric, where typically there is a distinguished “root” vertex $r$, and the distances between leaves are then given by $d(x,y) = w_x + w_y$ and $d(r,x) = w_x$. It is easy to see that our definition is more general: given any weighted star metric, we can create a weighted space (according to our definition) by adding a new vertex connected to the root of the star with an edge of length $1$, and assigning all vertices weight $0$.
\begin{theorem}
\label{thm:weighted-result}
For any $\epsilon > 0$, there exists a constant $c > 0$ such that for all sufficiently large $k$, any randomized online algorithm for $k$-server on some metric of size $n = k(k+1)/2 + 1$ has competitive ratio at least $\left(\frac{1}{2} – \epsilon\right) k – c$.
\end{theorem}
The lower bound on the competitive ratio in Theorem~\ref{thm:main-intro} matches, up to lower-order terms, the best known upper bound of $k$ for deterministic algorithms. This result is perhaps surprising in light of the recent line of work \cite{BartalK13,BCLMPST16,BubeckLLS19} which shows poly-logarithmic competitive ratios are achievable by randomized algorithms for many online problems for which the best possible deterministic algorithms are $\Omega(k)$ competitive. This includes, for instance, the closely related metrical task system (MTS) problem. In the MTS problem, we are again given a metric space $\mathcal{M} = (V,d)$ along with $k$ servers. However, now each request is a function $c: V \rightarrow \mathbb{R}^+$ and, given that our servers are at locations $s_1,\ldots,s_k$, to serve this request we must move a server to a node $v$ with cost $d(s_i,v) + c(v)$. The $k$-server problem is a special case of MTS, in which $c(v) = 0$ for one node of the request and $c(v) = \infty$ for all others. Despite this close connection, Bansal et al.~\cite{BansalBBN10} give a randomized algorithm for MTS with competitive ratio $O(\log^2 k \log^2 n)$ for any metric space on $n$ points. This implies in particular an $O(\log^4 n)$ competitive algorithm for MTS on $\mathcal{H}^{\log n}$, while for the $k$-server problem on the same metric, we show a lower bound of $\Omega(n / \log n)$ on the competitive ratio of any randomized algorithm.
\subsection{Proof Outline and Technical Ideas}
For simplicity, let us assume that $\rho$ is a large enough constant, and $k$ is a power of $\rho$, so that $\log_\rho k$ is an integer. We let $h = \lceil c \log k \rceil$, for some sufficiently large constant $c$ to be chosen later. In order to prove Theorem \ref{thm:main-intro}, we construct a distribution over request sequences and show that any deterministic online algorithm $\ALG$ (that we view as a mapping from length-$t$ request sequences to a configuration of servers at time $t$, deterministically) must incur a large cost in expectation compared to the offline optimum on a random request sequence drawn from this distribution. Since our lower bound applies to any deterministic algorithm $\ALG$, by Yao’s minimax principle, this implies that any randomized algorithm incurs a proportionally large cost.
Let $n = \rho^h$. The metric space $\mathcal{H}^h$ can naturally be viewed as a complete $\rho$-ary tree of height $h$. With this viewpoint in mind, we call a set of nodes in $\mathcal{H}^h$ a \emph{level-$i$ subtree} if it is equal to all of the descendants of some node at level $i$ of the tree. The root is at level $h$ and the leaves are at level $0$.
We construct a random request sequence in rounds, and our goal is to ensure that for every online algorithm $\ALG$, for a sufficiently long sequence of requests, the expected cost incurred by $\ALG$ is at least a factor of $(\frac{1}{2} – \epsilon) k$ larger than the offline optimum in each round. Our request sequences consist of a number of \emph{phases}, which are consecutive sequences of requests with certain desirable properties that help us prove lower bounds. We show how to construct such “good” phases recursively. Our construction will guarantee that at the beginning and end of each phase, all servers are located at the root of the HST. In order to describe a phase, we need to introduce the notion of a \emph{balanced coloring}.
\begin{definition} \label{def:balancedcoloring}
For a rooted tree $\mathcal{T}$ and a set of $k$ servers lying on the leaves of $\mathcal{T}$, a \emph{balanced coloring} (with respect to the location of the $k$ servers) is a coloring of the vertices of $\mathcal{T}$ such that:
\begin{itemize}
\item Each node is either red or blue.
\item The root of the tree is red.