Removed notes.
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Michael Whittaker
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{\Large Quorum Systems}
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\today{}
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{\input{sections/quorum_systems.tex}}
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@article{ibaraki1993theory,
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title={A theory of coteries: Mutual exclusion in distributed systems},
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author={Ibaraki, Toshihide and Kameda, Tiko},
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journal={IEEE Transactions on Parallel and Distributed Systems},
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volume={4},
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number={7},
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pages={779--794},
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year={1993},
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publisher={IEEE}
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}
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@article{naor1998load,
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title={The load, capacity, and availability of quorum systems},
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author={Naor, Moni and Wool, Avishai},
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journal={SIAM Journal on Computing},
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volume={27},
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number={2},
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pages={423--447},
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year={1998},
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publisher={SIAM}
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}
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@article{neilsen1991general,
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title={A general method to define quorums},
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author={Neilsen, Mitchell and Mizuno, Masaaki and Raynal, Michel},
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year={1991}
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}
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@article{vukolic2013origin,
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title={The origin of quorum systems},
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author={Vukoli{\'c}, Marko and others},
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journal={Bulletin of EATCS},
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volume={2},
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number={101},
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year={2013}
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}
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\begin{itemize}
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\item
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Recursive quorum systems are all non-dominated bicoteries. We can view the
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tree as a way to generate an arbitrary set of read quorums. Then we just
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bubble down dual to compute write quorums.
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\item
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The theory of coteries and decomposition has been very well studied. The
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one thing that seems a little less studied is the relationship between load
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and bicoteries. Can we find a nice way to find an optimal load bicoterie?
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This is probably hard, but maybe something about bicoteries makes this
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easier than for coteries.
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\item
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If we wanted to develop a library for bicoteries, we would like to (a)
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allow people to form arbitrary bicoteries, (b) compute an optimal strategy
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for a bicotier, (c) efficiently pick quorums based on the strategy, and (d)
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efficiently determine if something is a superset of a quorum. We can do (a)
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with a library of dual-able operators like or and and along with some
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builtins for things like grids. (b) we can use LP with specializations. We
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might have to do something smart to get ``good' strategies (e.g.,
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balanced). For (d), we can just evaluate the tree or its dual or maybe use
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something more efficient like BDDs? For (c) I'm not completely sure. For
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some quorum systems, the number of quorums is exponential, so we wouldn't
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want to represent them explicitly. Ideally we could also find an optimal
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load bicoterie, but that's probably not going to happen. The library can
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also help compute fault tolerance and maybe load under failure and whatnot.
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How to compute fault tolerance with duplicates?
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\item
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Maybe we can run some practical experiments to show how important a good
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bicoterie is. Maybe we'll find it's not that important.
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\item
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Is there anything smart about reconfiguring or changing coteries on the
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fly? For example, grids are not good to go from 6 to 7 nodes. With
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bicoteries, we probably want to adjust on the fly as the read and write
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ratios change.
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\item
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Maybe we can introduce the idea that not all nodes have the same capacity.
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Some have higher load than others. We could bake that into the nodes like,
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a = Node("a", cap=1000).
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\item
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Maybe some nodes are also co-located, so the fault tolerance is a little
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different. How can we compute fault tolerance with duplicates anyway?
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\item
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Just for fun, we could probably cross compile to javascript and show how to
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subsume other quorum systems with our library. We should be able to subsume
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all if we use duplicates.
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\end{itemize}
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\TODO[michael]{Prove that for every read-write quorum system, there exists a coterie with at least as good load.}
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\TODO[michael]{The above shows that the optimal load quorum system is a non-dominated coterie. Maybe this is useful? We can generate every NDC?}
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\TODO[michael]{Understand how domination relates to subsumption.}
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\TODO[michael]{Understand dual-major, dual-minor, and self-dual.}
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\TODO[michael]{Extend these notions to read-write quorums.}
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\TODO[michael]{Enumerate all read-write quorums on four nodes and see the ones we can and can't subsume.}
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\TODO[michael]{Prove that if I have the read quorums R, then the write quorums are bar(R).}
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\section{Quorum Systems}
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Given a set $X = \set{x_1, \ldots, x_n}$, a \defword{quorum system} over $X$ is
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a set $Q = \set{q_1, \ldots, q_m}$ of subsets of $X$, called \defword{quorums},
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such that every pair of quorums intersect. That is, for every $q_1, q_2 \in
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Q$, $q_1 \cap q_2 \neq \emptyset$.
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%
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A quorum system $Q$ is a \defword{coterie} if there does not exist quorums
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$q_1, q_2 \in Q$ such that $q_1 \subset q_2$. In other words, a coterie is a
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quorum system that does not contain some quorum $q_1$ that is a strict subset
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of some other quorum $q_2$.
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%
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Let $P, Q$ be two quorum systems over the same set $X$. $P$ \defword{dominates}
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$Q$, denoted $P > Q$, if $P \neq Q$ and for every $q \in Q$, there exists some
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$p \in P$ such that $p \subseteq q$. A quorum system $Q$ is \defword{dominated}
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if there exists some quorum system $P$ that dominates it.
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We can associate every quorum system $Q$ with a monotone boolean function
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$f_Q$. For example, the majority quorum system $Q = \set{\set{a, b}, \set{b,
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c}, \set{a, c}}$ corresponds to the function $f_Q = ab + bc + ac$. The prime
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implicants of the boolean function correspond to the minimal sets of the quorum
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system. We say $f \leq g$ if for every $\vec{x}$, $f(\vec{x}) \implies
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g(\vec{x})$. In other words, $f \leq g$ if $\setst{\vec{x}}{g(\vec{x})}
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\subseteq \setst{\vec{x}}{f(\vec{x})}$. Consider two quorum systems $Q_1$ and
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$Q_2$, $Q_1$ dominates $Q_2$ if and only if $f_{Q_2} < f_{Q_1}$.
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Let the dual $\dual{f}$ of $f$ be $\dual{f}(x) = \bar{f}(\bar{x})$. $\dual{f}$
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is the function we get if we swap and with or. Note that $\dual{f}$ corresponds
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to the sets that intersect every set in $f$. We say a monotone function $f$
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is \defword{dual-minor} if $f \leq \dual{f}$, \defword{dual-major} if $f \geq
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\dual{f}$, and \defword{self-dual} if $f = \dual{f}$. A function $f$
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corresponds to a quorum system if and only if $f$ is dual-minor. A quorum
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system $Q$ is non-dominated if and only if $f_Q$ is self-dual. Every
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non-dominated coterie can be represented as a composition of the simple
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majority function (with duplicates). Every coterie can be represented by a
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composition of and, or, and majority. \cite{ibaraki1993theory} also talks about
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when a function can be decomposed without duplicates as well as other things
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involving decomposition. \cite{neilsen1991general} talks about composition as
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well, and even includes examples for things like grids and trees.
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Let $\sigma: Q \to [0, 1]$ be a discrete probability distribution over the
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quorums of $Q$ (i.e., $\sum_{q \in Q} \sigma(q) = 1$). We call $\sigma$ a
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\defword{strategy}. Intuitively, $\sigma$ is a strategy to pick quorums at
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random. We have the following definitions.
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\begin{align*}
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l_\sigma(x) &\defeq \sum_{\setst{q \in Q}{x \in q}} \sigma(q) \\
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L_\sigma(Q) &\defeq \max_{x \in X} l_\sigma(x) \\
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L(Q) &\defeq \min_\sigma L_\sigma(Q)
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\end{align*}
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$l_\sigma(x)$ is the load on $x$ given some strategy $\sigma$. $L_\sigma(Q)$ is
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the load on most loaded element $x$. $L(Q)$ is the \defword{load} of the best
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possible strategy. Intuitively, the lower the load of a quorum system, the
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higher the throughput it can support.
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\begin{example}
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Consider the majority quorum system $Q = \set{\set{a, b}, \set{a, c}, \set{b,
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c}}$ on elements $X = \set{a, b, c}$. Let $\sigma(\set{a, b}) =
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\sigma(\set{a, c}) = 0.5$ and $\sigma(\set{b, c}) = 0$.
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\begin{align*}
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l_\sigma(a) &= \sigma(\set{a, b}) + \sigma(\set{a, c}) = 0.5 + 0.5 = 1 \\
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l_\sigma(b) &= \sigma(\set{a, b}) + \sigma(\set{b, c}) = 0.5 + 0 = 0.5 \\
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l_\sigma(c) &= \sigma(\set{a, c}) + \sigma(\set{b, c}) = 0.5 + 0 = 0.5 \\
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L_\sigma(Q) &= \max(1, 0.5, 0.5) = 1
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\end{align*}
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The load of $Q$ with respect to $\sigma$ is 1, but the load $L(Q)$ is not 1
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because $\sigma$ is not an optimal strategy. If we instead choose
|
||||
$\sigma_{\text{opt}}(-) = \frac{1}{3}$, the load is reduced to $\frac{2}{3}$
|
||||
which is optimal.
|
||||
\end{example}
|
||||
|
||||
\cite{naor1998load} shows how to use a linear program to compute load.
|
||||
\cite{naor1998load} also proves that if a quorum system dominates another, it
|
||||
has lower or equal load. This shows that there always is a non-dominated
|
||||
coterie that has the lowest possible load. If we're trying to optimize for
|
||||
load, this tells us that we limit ourselves to non-dominated coteries.
|
||||
|
||||
The \defword{resilience} or \defword{fault tolerance} of a quorum system $Q$ is
|
||||
the largest number $f$ such that for every subset $F \subseteq X$ with $|F| =
|
||||
f$, there still exists some quorum $q \in Q$ such that $q \cap F = \emptyset$.
|
||||
Intuitively, a quorum system has fault tolerance $f$ if we can fail an
|
||||
arbitrary set of $f$ elements and still have some quorum left.
|
||||
|
|
@ -1,50 +0,0 @@
|
|||
\section{Read-Write Quorum Systems}
|
||||
Given a set $X = \set{x_1, \ldots, x_n}$, a \defword{read-write quorum system}
|
||||
over $X$ is a pair of sets $Q = (R, W)$ of subsets of $X$ such that every $r
|
||||
\in R$ intersects every $w \in W$. The elements $r \in R$ are called
|
||||
\defword{read quorums}, and the elements $w \in W$ are called \defword{write
|
||||
quorums}. Read-write quorum systems are also called bicoteries.
|
||||
%
|
||||
$(R, W)$ is a coterie if both $R$ and $W$ are minimal. $(R_1, W_1)$ dominates
|
||||
$(R_2, W_2)$ if $R_1$ dominates $R_2$ and $W_1$ dominates $W_2$. Again, we can
|
||||
view a read-write quorum system $(R, W)$ as a pair of monotone functions $f_R$
|
||||
and $f_W$. $(f_R, f_W)$ is a read-write quorum sytem if $f_R \leq \dual{f_W}$.
|
||||
It is a non-dominated coterie if $f_R = \dual{f_W}$. Thus, to generate a
|
||||
non-dominated bicoterie, let $f_R$ be an arbitrary set of quorums and let $f_W$
|
||||
be $\dual{f_R}$.
|
||||
|
||||
Let $\sigma_R: R \to [0, 1]$ and $\sigma_W: W \to [0, 1]$ be a discrete
|
||||
probability distribution over the read and write quorums of $Q$. We call
|
||||
$\sigma$ a \defword{strategy}. Let $0 \leq p_r \leq 1$ be the probability of
|
||||
performing a read and $p_w = 1 - p_r$ be the probability of performing a write.
|
||||
We have the following definitions.
|
||||
\begin{align*}
|
||||
l_{\sigma,p_r,p_w}(x)
|
||||
&\defeq p_r \cdot \parens*{\sum_{\setst{r \in R}{x \in r}} \sigma_R(r)} +
|
||||
p_w \cdot \parens*{\sum_{\setst{w \in W}{x \in w}} \sigma_W(w)} \\
|
||||
L_{\sigma,p_r,p_w}(Q) &\defeq \max_{x \in X} l_{\sigma,p_r,p_w}(x) \\
|
||||
L_{p_r,p_w}(Q) &\defeq \min_\sigma L_{\sigma,p_r,p_w}(Q)
|
||||
\end{align*}
|
||||
$l_{\sigma,p_r,p_w}(x)$ is the load on $x$ given some strategy $\sigma$ and
|
||||
some workload $p_r,p_w$. $L_{\sigma,p_r,p_w}(Q)$ is the load on most loaded
|
||||
element $x$. $L{p_r,p_w}(Q)$ is the \defword{load} of the best possible
|
||||
strategy.
|
||||
|
||||
The \defword{read resilience} or \defword{read fault tolerance} of a quorum
|
||||
system $Q$ is the largest number $f_r$ such that for every subset $F \subseteq
|
||||
X$ with $|F| = f_r$, there still exists some read quorum $r \in R$ such that $r
|
||||
\cap F = \emptyset$.
|
||||
%
|
||||
The \defword{write resilience} or \defword{write fault tolerance} of a quorum
|
||||
system $Q$ is the largest number $f_w$ such that for every subset $F \subseteq
|
||||
X$ with $|F| = f_w$, there still exists some write quorum $w \in W$ such that $w
|
||||
\cap F = \emptyset$.
|
||||
%
|
||||
The \defword{resilience} or \defword{fault tolerance} of a quorum system is the
|
||||
minimum of its read resilience and write resilience. Intuitively, a quorum
|
||||
system has read fault tolerance $f_r$ if we can fail an arbitrary set of $f_r$
|
||||
elements and still have some read quorum left; a quorum system has write fault
|
||||
tolerance $f_w$ if we can fail an arbitrary set of $f_w$ elements and still
|
||||
have some write quorum left; and a quorum system has fault tolerance $f$ if we
|
||||
can fail an arbitrary set of $f$ elements and still have some read quorum and
|
||||
some write quorum left.
|
||||
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Reference in a new issue