A $3+\varOmega (1)$ Lower Bound for Page Migration

Abstract

We address the page migration problem, one of the most classical online problems. In this problem, we are given online requests from nodes in a network for accessing a single page, i.e., a data set stored in a node, and asked to determine a node for the page to be stored in after each request. Serving a request costs the distance between the request and the page at the point of the request, and migrating the page costs the migration distance multiplied by the page size. The objective is to minimize the total sum of the service and migration costs. This problem is motivated by efficient cache management in multiprocessor systems. In this paper, we prove that no deterministic online page migration algorithm is $(3+o(1))$-competitive, where the o-notation is with respect to the page size. Our lower bound first breaks the barrier of 3 by an additive constant for an arbitrarily large page size and disproves Black and Sleator’s conjecture even in the asymptotic sense.

Appendix

We prove the following fact used in the proofs of Lemmas 9 and 10.

Lemma 14

If there exist values $x>0$ and X, nodes p, q, and u, and a sequence $\chi$ of clients such that

$$\begin{aligned} (2+x){\textsc { opt}}_p(u,\chi )+{\textsc { opt}}_q(u,\chi )<{\textsc { alg}}(u,\chi )+X+(2-x)D\text {d}_{{p}{q}}, \end{aligned}$$

and if we have recurrences

$$\begin{aligned} {\textsc { alg}}(u,\chi )+X+({\tilde{k}}_i+{\tilde{\ell }}_{i-1}+D(2i-1))\text {d}_{{p}{q}}&\le (3+x)({\textsc { opt}}_p(u,\chi )+{\tilde{\ell }}_{i-1}\text {d}_{{p}{q}}), \text { and}\\ {\textsc { alg}}(u,\chi )+X+({\tilde{k}}_i+{\tilde{\ell }}_{i}+2Di)\text {d}_{{p}{q}}&\le (3+x)({\textsc { opt}}_q(u,\chi )+{\tilde{k}}_{i}\text {d}_{{p}{q}}) \end{aligned}$$

for $i\ge 1$ and ${\tilde{\ell }}_0 = 0$, then the sequence $({\tilde{k}}_i)_{i\ge 1}$ is not monotonically increasing.

Proof

The recurrences yield the inequalities

$$\begin{aligned} {\tilde{k}}_i&\le (2+x){\tilde{\ell }}_{i-1}-D(2i-1)+A,\ \text {and} \end{aligned}$$

(41)

$$\begin{aligned} {\tilde{\ell }}_i&\le (2+x){\tilde{k}}_{i}-2Di+B\nonumber \end{aligned}$$

for $i\ge 1$, where $A:=((3+x){\textsc { opt}}_p(u,\chi )-{\textsc { alg}}(u,\chi )-X)/\text {d}_{{p}{q}}$ and $B:=((3+x){\textsc { opt}}_q(u,\chi )-{\textsc { alg}}(u,\chi )-X)/\text {d}_{{p}{q}}$. By combining these inequalities, we obtain the recurrence

$$\begin{aligned} {\tilde{k}}_i\le (2+x)^2 {\tilde{k}}_{i-1}-2(3+x) Di+(5+2x)D+A+(2+x)B\quad \text {for } i\ge 2. \end{aligned}$$

This is equivalent to

$$\begin{aligned} {\tilde{k}}_i-F(i)\le & {} \left( {\tilde{k}}_{i-1}-F(i-1)\right) (2+x)^2\quad \text {for } i\ge 2, \hbox { where}\\ F(i):= & {} \tfrac{2Di}{1+x}+\tfrac{\frac{3+x}{1+x}D-A-(2+x)B}{(3+x)(1+x)}. \end{aligned}$$

Therefore, it follows that

$$\begin{aligned} \begin{aligned} {\tilde{k}}_i&\le \left( {\tilde{k}}_1-F(1)\right) (2+x)^{2(i-1)}+F(i)\quad \text {for } i\ge 2. \end{aligned} \end{aligned}$$

Because ${\tilde{k}}_1\le A-D$ by (41),

$$\begin{aligned} \begin{aligned} {\tilde{k}}_i&\le \left( A-D-F(1)\right) (2+x)^{2(i-1)}+F(i)\\&= \left\{ -\tfrac{(3+x)(2+x)D}{1+x}+(2+x)A+B\right\} \tfrac{(2+x)^{2i-1}}{(3+x)(1+x)}+F(i)\\&=\left\{ -\tfrac{(3+x)(2+x)D}{1+x}+(2+x)A+B\right\} \cdot \varTheta \left( (2+x)^{2i}\right) +O(i). \end{aligned} \end{aligned}$$

The factor of $\varTheta ((2+x)^{2i})$ can be estimated as

$$\begin{aligned} -\tfrac{(3+x)(2+x)D}{1+x}+(2+x)A+B =\tfrac{3+x}{\text {d}_{{p}{q}}}\left\{ (2+x){\textsc { opt}}_p(\chi )+{\textsc { opt}}_q(\chi ) -{\textsc { alg}}(\chi )-X-\tfrac{2+x}{1+x}D\text {d}_{{p}{q}}\right\} , \end{aligned}$$

which is negative as $-\frac{2+x}{1+x}<-(2-x)$ for $x>0$ and by the assumption of the lemma. Therefore, ${\tilde{k}}_i$ decreases as i grows sufficiently large. $\square$