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3f50ea0bce3946f4a22b2df4ebf49bcc88bc9961 | subsection | 45 | 83 | Wired spanning forest oriented toward a root | Then, for
\pi -almost
every initial rotor
configuration,
the scaled walk (\frac{1}{\sqrt{n}} X_{\lfloor nt\rfloor } )_{t \ge 0} converges weakly on D_{\mathbb {R}^d}[0,\infty ) to a Brownian motion with diffusion matrix \Gamma .
\end{}
}}As a corollary of
Theorem~\ref {scaling limit for elliptic rotor walk},
we derive ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.... |
99ca79bede64bbc082d3cb11b59d53402f0c1b23 | subsection | 46 | 83 | Wired spanning forest oriented toward a root | A stationary distribution \pi of M is \emph {ergodic} if {\pi }[{B}]\in \lbrace 0,1\rbrace for any invariant set {B}.
}}}}}}}}Let \Omega ^{\mathbb {N}} be the \emph {trajectory space} of M,
\Omega ^{\mathbb {N}}:=\lbrace (\omega _i)_{i \ge 0} \mid \omega _i \in \Omega \rbrace ,
equipped with the product \sigma -algebr... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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a0bd60354b6ff7e89c819601e7966ca0c4559512 | subsection | 47 | 83 | Wired spanning forest oriented toward a root | Now note that,
for any n \ge 1,
\begin{align*}
\pi [{B}] &= \int _{\Omega } P^{(n)}(x, {B}) \ d\pi (x) \qquad \text{(by the stationarity of $\pi $)}\\
&=\int _{{B}} P^{(n)}(x,{B}) \ d\pi (x) + \int _{{B}^{\prime } \setminus {B}} P^{(n)}(x, {B}) \ d\pi (x) \qquad \text{(as $P^{(n)}(x, {B}) = 0$ for $x \notin {B}^{\prime... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0a3c977980c0dcca3b194a32c9db7a4841da16f8 | subsection | 48 | 83 | Wired spanning forest oriented toward a root | We write
{{{C}}:= \lbrace \rho \in \text{Rot}(G) \mid \exists ~\rho ^{\prime } \in {{{B}}\text{ s.t. } \rho \text{ and } \rho ^{\prime } \text{ differ at finitely many vertices}\rbrace .
Note that {{{C}} is a tail event that contains {{{B}}.
It then suffices to show that \pi [{{{C}}\setminus {{{B}}]=0.
}}Let \rho be an... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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6086505b09bf677484e0b1cf549213c6aea92127 | subsection | 49 | 83 | Wired spanning forest oriented toward a root | (d) The final rotor configuration of the RWLM at the end of this process, which is the same regardless of whether the initial configuration is (a) or (b).
}
\end{}
}}}Write
”:=xn-1(n)=xn-1(n').
Note that ” is the rotor configuration at the n-th step of the scenery process if the walker of the RWLM follows x0,...,xn a... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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adc307f02842c801e042be5bd677c0be8c296536 | subsection | 50 | 83 | Wired spanning forest oriented toward a root | Recall that X_n denotes the walker^{\prime }s location and \rho _n denotes the rotor configuration at the n-th step of the RWLM.
}}}\begin{}
Let (G,c) be a weighted lattice graph in \mathbb {R}^d, and
suppose that the mechanism of the RWLM satisfies \ref {item: TR} and \ref {item: ELL}.
Let \pi be any tail trivial stat... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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bc2dc532fba7094c2ca231bcc3073178c5ef89a0 | subsection | 51 | 83 | Wired spanning forest oriented toward a root | It follows from Definition REF and that
V_i=& X_{i+1}- X_{i}
= \sum _{\mathbf {y}\in {N}(\operatorname{\mathbf {0}})} \mathbb {1}\lbrace \rho _{i}(X_i)-X_i=\mathbf {y}\rbrace \, Y_{\mathbf {y},i},
where
Y_{\mathbf {y},i} is a random variable on neighbors of the origin sampled from p_{\operatorname{\mathbf {0}}}(\math... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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6b58ffb0ac8781de15ced4e8a6700500f6458415 | subsection | 52 | 83 | Wired spanning forest oriented toward a root | Since \mu _{\operatorname{\mathbf {0}}} is a stationary distribution of the mechanism at \operatorname{\mathbf {0}} by ,
it then follows that:
\lim _{n \rightarrow \infty } \frac{1}{n}\sum _{i=0}^{n-1} \mathbb {E}\mathopen {}\mathclose {\left[V_i V_i^\top \mid {F}_i\right]}
=&\sum _{\mathbf {y}^{\prime } \in {N}(\oper... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1002/rsa.20747",
"end": 1940,
"openalex_id": "https://openalex.org/W2962799388",
"raw": "Wilfried Huss, Lionel Levine, and Ecaterina Sava-Huss, Interpolating between random walk and rotor walk, Random structures & algorithms 52 (2018), n... | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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f10281a1d32c98bf9eed9b401171b7062702db1b | subsection | 53 | 83 | Wired spanning forest oriented toward a root | Let {{{F}}:={{{F}}(G) \subseteq 2^{{{E}(G)} be the \sigma -algebra on the set of oriented subgraphs of G generated by sets of the form
\lbrace {{H}\in 2^{{{E}(G)} \mid {{{B}}\subseteq {{H}\rbrace , where {{{B}} is a finite subset of {{E}(G).
The oriented wired spanning forest will be a probability distribution on the m... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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1f2f5707bee49d0ab720ca5bc3e5d98e010440c8 | subsection | 54 | 83 | Wired spanning forest oriented toward a root | The \emph {$r$-oriented wired spanning forest}, denoted ${{\operatorname{\mathsf {{WSF}}}}_r:={{\operatorname{\mathsf {{WSF}}}}_r(G,c),
is a probability distribution on oriented subgraphs of G
such that, for any wired exhaustion
and any
finite {{{B}}\subseteq {{E}(G),
\begin{equation}
{{\operatorname{\mathsf {{WSF}}}}_... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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053d14ebe64f47b734eaac9121b24c2dc48cc30b | subsection | 55 | 83 | Wired spanning forest oriented toward a root | However, it is always an \emph {r-oriented spanning forest} of G:
the underlying graph of {{F} is a spanning forest,
every vertex in V(G)\setminus \lbrace r\rbrace has outdegree 1 in {{F}, and r has outdegree 0 in {{F}.
The first condition follows from Lemma~\ref {lemma: wsf and oriented wsf}, and the others can be ver... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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e214c6c6b428a7637e266826e6628a9393e4399a | subsection | 56 | 83 | Wired spanning forest oriented toward a root | \item Suppose that
{{{T}}(i)
has been generated.
Start an independent network random walk at x_{i+1} and stop it at the first time it hits
{{{T}}(i)
(note that the random walk hits {{{T}}(i) a.s.\ by recurrence).
Let \langle y_0,\ldots , y_m \rangle be the loop erasure of this random walk.
\item Set
{{{T}}(i+1)
to be t... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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5f5018a8c9661e8a8b25543f2903bb02877f9cb5 | subsection | 57 | 83 | Wired spanning forest oriented toward a root | Then for any finite subset {{{B}} of {{E}(G), any ordering of V(G) \setminus \lbrace r\rbrace , and any wired exhaustion of G,
\mathbb {P}[{{{B}}\subseteq {{{T}}] =\lim _{n \rightarrow 0} \ {{\mu _{r,n}}[{{{B}}\subseteq {{{T}}_n],
where {{{T}} is a random tree of G generated using Wilson^{\prime }s method,
with root ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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ac3e93285e237ccc2d8d84be9ed5769c4885f0c0 | subsection | 58 | 83 | Wired spanning forest oriented toward a root | Start a network random walk at x_{i+1}.
Stop the walk the first time it hits {{{F}}(i); if it never hits {{{F}}(i) then let it run indefinitely.
This walk is locally finite a.s.\ by transience.
Let \langle y_0^{\prime },y_1^{\prime },\ldots \rangle be the loop erasure of this random walk.
}\item Set {{{F}}(i+1) to be t... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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d1d6638d0d293f5e1aa2f8453ef09cf5d3516072 | subsection | 59 | 83 | Wired spanning forest oriented toward a root | That is, if \operatorname{{\mathsf {LE}}}\langle x_i \mid i \le I\rangle =\langle y_{I,i} \mid i \le m_I\rangle and \operatorname{{\mathsf {LE}}}\langle x_i \mid i \ge 0 \rangle =\langle y_{i} \mid i \ge 0\rangle , then for every i and all sufficiently large I we have y_{I,i}=y_i.
Since G is transient,
it follows that ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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4ad9fc1def2c63bf5b410fb032abcfe9a491c5f3 | subsection | 60 | 83 | Wired spanning forest oriented toward a root | It then follows from definition of oriented wired spanning forest for finite graphs (Definition~\ref {definition: wsf finite graphs}) that
{{{T}}_n has the law of {{\operatorname{\mathsf {{WSF}}}}_{r}(G_n,c_n).
}Let \tau ^j_n be the first time that \langle X_i^j \mid i \ge 0\rangle reaches the portion of the spanning t... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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965da278e15f42e28da20e32883cabd9949508b4 | subsection | 61 | 83 | Wired spanning forest oriented toward a root | Starting an RWLM from a native environment allows us to use ergodic theory in Section~\ref {SLLN}.
The main result of this section is Theorem~\ref {stationarity theorem},
which gives an explicit distribution as a native environment for the RWLM.
}\end{equation}In this section the underlying graph of the RWLM will be a ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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94bd2749027dd2bda13142971cb9b290e1fe07a5 | subsection | 62 | 83 | Wired spanning forest oriented toward a root | Note that the measure \mu _{\operatorname{\mathnormal {o}}} is symmetric (i.e., \mu _{\operatorname{\mathnormal {o}}}(x)=\mu _{\operatorname{\mathnormal {o}}}(x^{-1})) as a consequence of c: \mathcal {S}\rightarrow \mathbb {R}_{>0} being symmetric.
}}{{}{\bfseries {Definition 5.16 (Transitive mechanism)}}
Let $(G,c)$ ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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b58e1a1eeac3c968275724ea5023901bcea63026 | subsection | 63 | 83 | Wired spanning forest oriented toward a root | \end{}}The scenery process $(\widehat{\rho }_n)_{n \ge 0}$ is a Markov chain with state space the set of rotor configurations of $G$ and with transition rule
\begin{equation}
\widehat{\rho }_{n+1}(x) :=
{\mathopen {}\mathclose {\left\lbrace \begin{array}{ll}\operatorname{\mathnormal {o}}&\text{ if } x= Y_{n}^{-1};\\
Y_... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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6a92af2fb838ff71ddcd66d419b57d3c318e9ea4 | subsection | 64 | 83 | Wired spanning forest oriented toward a root | The \emph {$r$-oriented wired spanning forest plus one edge}, denoted ${{\operatorname{\mathsf {{WSF}}}}_r^+:={{\operatorname{\mathsf {{WSF}}}}_r^+(G,c), is the law of the random subgraph {{{F}}\sqcup \lbrace (r, Y)\rbrace , where {{{F}} is a random r-oriented forest of G sampled from {{\operatorname{\mathsf {{WSF}}}}_... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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f7f9b2fe517db2bd2e368dc78a532a414218b099 | subsection | 65 | 83 | Wired spanning forest oriented toward a root | Each unicycle {{{U}} is picked with probability proportional to
{\Xi ({{{U}})}, where \Xi ({{{U}}) is as in Definition~\ref {definition: weight of a directed tree}.
This implies that
{{F_{Y}} \, \sqcup \, \lbrace (Y, r) \rbrace is distributed as {{\operatorname{\mathsf {{WSF}}}}^+_r, as desired.
}
}}\begin{}[Proof of T... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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4ada44daa7795e8b8dbadc02efe3d7d83012eee2 | subsection | 66 | 83 | Wired spanning forest oriented toward a root | It now follows from Lemma~\ref {lemma: wsfplus sum}
that \widehat{\rho }_1 is distributed according to {{\operatorname{\mathsf {{WSF}}}}^+_{\operatorname{\mathnormal {o}}}, and the proof is complete.
}
}An important property of {{\operatorname{\mathsf {{WSF}}}}_r^+ is that it is a tail trivial measure, defined below.
T... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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2f6f913a22ef70a9cab662efd2b2908aa6371f1c | subsection | 67 | 83 | Wired spanning forest oriented toward a root | Also note that {{\operatorname{\mathsf {{WSF}}}}_r[g({{{B}})]=\operatorname{\mathsf {{WSF}}}[f\circ g({{{B}})] by
Lemma~\ref {lemma: wsf and oriented wsf}.
Finally, note that
the set f \circ g({{{B}}) is a tail event in {F} since {{{B}} is a tail event in {{{F}}.
The conclusion of the proposition now follows from the t... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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9dce6df3864fd14d9641d30de830edd1400460b8 | subsection | 68 | 83 | Wired spanning forest oriented toward a root | \end{}
We will further assume that the initial rotor configuration of the RWLM is sampled from a {tail trivial} native environment~(Definitions~\ref {definition: native environment} and \ref {definition: tail trivial}).
}}Several remarks are in order.
Condition \ref {item: ELL} is known as an \emph {ellipticity} condit... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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ee97da6b93ecc187e8271b39b9710bb33d8bd2b1 | subsection | 69 | 83 | Wired spanning forest oriented toward a root | Then, for
\pi -almost
every initial rotor
configuration,
the scaled walk (\frac{1}{\sqrt{n}} X_{\lfloor nt\rfloor } )_{t \ge 0} converges weakly on D_{\mathbb {R}^d}[0,\infty ) to a Brownian motion with diffusion matrix \Gamma .
\end{}
}}As a corollary of
Theorem~\ref {scaling limit for elliptic rotor walk},
we derive ... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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4ff9a65dd79dd8760a5a5e89ead0961b034a5d56 | subsection | 70 | 83 | Wired spanning forest oriented toward a root | A stationary distribution \pi of M is \emph {ergodic} if {\pi }[{B}]\in \lbrace 0,1\rbrace for any invariant set {B}.
}}}}}}}}Let \Omega ^{\mathbb {N}} be the \emph {trajectory space} of M,
\Omega ^{\mathbb {N}}:=\lbrace (\omega _i)_{i \ge 0} \mid \omega _i \in \Omega \rbrace ,
equipped with the product \sigma -algebr... | {
"cite_spans": []
} | 10.1007/s10955-021-02791-5 | 1809.04710 | Random walks with local memory | [
"Swee Hong Chan",
"Lila Greco",
"Lionel Levine",
"Peter Li"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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d99bddf1951b6ecb59df5ab52d46537853d3b853 | subsection | 71 | 83 | Wired spanning forest oriented toward a root | Now note that,
for any n \ge 1,
\begin{align*}
\pi [{B}] &= \int _{\Omega } P^{(n)}(x, {B}) \ d\pi (x) \qquad \text{(by the stationarity of $\pi $)}\\
&=\int _{{B}} P^{(n)}(x,{B}) \ d\pi (x) + \int _{{B}^{\prime } \setminus {B}} P^{(n)}(x, {B}) \ d\pi (x) \qquad \text{(as $P^{(n)}(x, {B}) = 0$ for $x \notin {B}^{\prime... | {
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5dd165ae15b225984e0d63d76478300ca61b9b00 | subsection | 72 | 83 | Wired spanning forest oriented toward a root | We write
{{{C}}:= \lbrace \rho \in \text{Rot}(G) \mid \exists ~\rho ^{\prime } \in {{{B}}\text{ s.t. } \rho \text{ and } \rho ^{\prime } \text{ differ at finitely many vertices}\rbrace .
Note that {{{C}} is a tail event that contains {{{B}}.
It then suffices to show that \pi [{{{C}}\setminus {{{B}}]=0.
}}Let \rho be an... | {
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ffe2a4eca5b04dd1b7da917355093c2d3d7c576a | subsection | 73 | 83 | Wired spanning forest oriented toward a root | (d) The final rotor configuration of the RWLM at the end of this process, which is the same regardless of whether the initial configuration is (a) or (b).
}
\end{}
}}}Write
”:=xn-1(n)=xn-1(n').
Note that ” is the rotor configuration at the n-th step of the scenery process if the walker of the RWLM follows x0,...,xn a... | {
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a165bca56220a3370ff0669d19cb053a6345a00b | subsection | 74 | 83 | Wired spanning forest oriented toward a root | Recall that X_n denotes the walker^{\prime }s location and \rho _n denotes the rotor configuration at the n-th step of the RWLM.
}}}\begin{}
Let (G,c) be a weighted lattice graph in \mathbb {R}^d, and
suppose that the mechanism of the RWLM satisfies \ref {item: TR} and \ref {item: ELL}.
Let \pi be any tail trivial stat... | {
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04034fce00fa5e18bcb4abf158ec9bdbb9e00752 | subsection | 75 | 83 | Wired spanning forest oriented toward a root | It follows from Definition REF and that
V_i=& X_{i+1}- X_{i}
= \sum _{\mathbf {y}\in {N}(\operatorname{\mathbf {0}})} \mathbb {1}\lbrace \rho _{i}(X_i)-X_i=\mathbf {y}\rbrace \, Y_{\mathbf {y},i},
where
Y_{\mathbf {y},i} is a random variable on neighbors of the origin sampled from p_{\operatorname{\mathbf {0}}}(\math... | {
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ccd85d0eda2ab18f920a3c389d997c83ffe64e65 | subsection | 76 | 83 | Wired spanning forest oriented toward a root | Since \mu _{\operatorname{\mathbf {0}}} is a stationary distribution of the mechanism at \operatorname{\mathbf {0}} by ,
it then follows that:
\lim _{n \rightarrow \infty } \frac{1}{n}\sum _{i=0}^{n-1} \mathbb {E}\mathopen {}\mathclose {\left[V_i V_i^\top \mid {F}_i\right]}
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90708ffb09dfea983a895bf3aad18bf90a64375c | subsection | 77 | 83 | Unoriented wired spanning forest | We begin by defining the unoriented wired spanning forest and
refer to and
for a detailed discussion on this topic.Recall that G:=(V(G),E(G)) is a simple, connected,
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6b8cee8a01d9d5192d7b83fbe01380149b138b27 | subsection | 78 | 83 | Unoriented wired spanning forest | The graph G_n is the undirected graph obtained from G by identifying all the vertices of V(G)\setminus W_n to a single vertex z_n and removing loops and extra multiple edges that are formed.
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1fc0e2b1e2f8e4cc859cc01b362022c6ee0bc48f | subsection | 79 | 83 | Unoriented wired spanning forest | Then, for any tail event {B}\in {F}, we have \operatorname{\mathsf {{WSF}}}[{B}] \in \lbrace 0,1\rbrace .This tail triviality will be useful for us in proving ergodicity of the scenery process for random walks with local memory later in Section .We begin by defining the unoriented wired spanning forest and
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451cbc2324c4fd558183ad329f3a920ab7968ee1 | subsection | 80 | 83 | Unoriented wired spanning forest | Let (W_n)_{n \ge 0} be a sequence of finite, connected subsets of V(G) such that\bigcup _{n \ge 0}W_n=V(G); and
W_n \subseteq W_{n+1} for all n \ge 0.The wired exhaustion of G is the sequence of electrical networks (G_n, c_n)_{n \ge 0} defined as follows.
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b4a4b011c7f637a199162094d8cd583f531b9784 | subsection | 81 | 83 | Unoriented wired spanning forest | For any subset K \subseteq E(G), let {F}(K) \subseteq {F} denote the \sigma -algebra of events that depend only on K.
An event {B}\in {F} is a tail event if {B}\in {F}(E \setminus K) for all finite K \subseteq E.Theorem 5.6 ()
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368180be92ba4e66eb671230d764bb05323d3a38 | subsection | 82 | 83 | Further Questions | We conclude with a few natural questions.By Theorem , if the RWLM starts at the origin
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28b7e93b6d57653af0c143dd7d38fa1dc24ed93c | abstract | 0 | 53 | Abstract | It is commonly believed that datacenter networking software must sacrifice
generality to attain high performance. The popularity of specialized
distributed systems designed specifically for niche technologies such as RDMA,
lossless networks, FPGAs, and programmable switches testifies to this belief.
In this paper, we s... | {
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53aeb7cc287510ccc75dbb1708675d1196116e29 | subsection | 1 | 53 | Body | 0pt1.5ex plus 1ex minus .2ex1.3ex plus .2exitemsep=0pt,parsep=0pt,nosepdetect-alllanguage=C++,
basicstyle=,
keywordstyle=,
breaklines=true,
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59cedd1df26eef93eef3237be0685c14a67083e4 | subsection | 2 | 53 | Introduction | “Using performance to justify placing functions in a low-level
subsystem must be done carefully. Sometimes, by examining the problem thoroughly,
the same or better performance can be achieved at the high level.”— End-to-end Arguments in System DesignSqueezing the best performance out of modern, high-speed datacenter ne... | {
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194c7aa4d27e3a2c7d992706cdd01d38e96b98a5 | subsection | 3 | 53 | Introduction | For example, in our two-layer testbed
that resembles real deployments, each switch has 12 of dynamic buffer,
while the BDP is only 19.eRPC (efficient RPC) is available at
https://github.com/efficient/eRPC. Our research contributions are:We describe the design and implementation of a high-performance RPC
library for dat... | {
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e419470f518060c17ad6fdc3b9074f0db99504a6 | subsection | 4 | 53 | Background and motivation | We first discuss aspects of modern datacenter networks relevant to eRPC. Next,
we discuss limitations of existing networking software that underlie the need
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3226c44573c1978254f78f70cc13be6157af0592 | subsection | 5 | 53 | High-speed datacenter networking | Modern datacenter networks provide tens of Gbps per-port bandwidth and a few
microseconds round-trip latency . They support
polling-based network I/O from userspace, eliminating interrupts and system
call overhead from the datapath , .
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netwo... | {
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c1f97310fc582713bd1dbe0cf8362fb9a218341e | subsection | 6 | 53 | Lossless fabrics. | Lossless packet delivery is a link-level feature
that prevents congestion-based packet drops. For example, PFC for Ethernet
prevents a link's sender from overflowing the receiver's buffer by using pause
frames. Some datacenter operators, including Microsoft, have deployed PFC at
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4d37448f76c3e42611cdac86386ef1214477e27d | subsection | 7 | 53 | Switch buffer | The increase in datacenter bandwidth has
been accompanied by a corresponding decrease in round-trip time (RTT),
resulting in a small BDP. Switch buffers have
grown in size, to the point where “shallow-buffered” switches that use SRAM
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30a8675c2c7834eba5be7efff76cfccc9acd0cab | subsection | 8 | 53 | Limitations of existing options | Two reasons underlie our choice to design a new general-purpose RPC system for
datacenter networks: First, existing datacenter networking software options sacrifice
performance or generality, preventing unmodified applications from using the
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6ba0cf5a6c7fe4f5c7c9b297ce87f001d114aa4f | subsection | 9 | 53 | Drawbacks of specialization | Co-designing distributed systems with network hardware is a well-known
technique to improve performance. Co-design with RDMA is popular, with numerous
examples from key-value stores , , , , , state machine
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28634e8da42f02877ece61bcc0dfa7d3c5620a42 | subsection | 10 | 53 | eRPC overview | We provide an overview of eRPC's API and threading model below. In these
aspects, eRPC is similar to existing high-performance RPC systems like
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767c5f4026d055c9aa18b685904607a885c6b1dc | subsection | 11 | 53 | RPC API | RPCs execute at most once, and are asynchronous to avoid stalling on network
round trips; intra-thread concurrency is provided using an event loop. RPC
servers register request handler functions with unique request types; clients
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ea046c1d73775d622c99a3bc10100a69d00e8b4d | subsection | 12 | 53 | Client control flow: | rpc->enqueue_request() queues a
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edc051d74fbed5e302a520e11570c6b2ee880f51 | subsection | 13 | 53 | Server control flow: | The event loop of the Rpc that owns
the server session invokes (or dispatches) a request handler on receiving a
request. We allow nested RPCs, i.e., the handler need not enqueue a
response before returning. It may issue its own RPCs and call
enqueue_response() for the first request later when all dependencies
complete. | {
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9df0ef7f30f9f92bc95b95569929c07b6572db53 | subsection | 14 | 53 | Worker threads | A key design decision for an RPC system is which thread runs an RPC handler.
Some RPC systems such as RAMCloud use dispatch threads for only network I/O.
RAMCloud's dispatch threads communicate with worker threads that run
request handlers. At datacenter network speeds, however, inter-thread
communication is expensive:... | {
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6a9aa8e499442352175d43e7619432119f2f3afb | subsection | 15 | 53 | Evaluation clusters | Table REF shows the clusters used in this paper. They include
two types of networks (lossy Ethernet, and lossless InfiniBand), and three
generations of NICs released between 2011 (CX3) and 2017 (CX5). eRPC works well
on all three clusters, showing that our design is robust to NIC and network
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40be41dcde6dff49296543c55e7a3a9ee0aa58c9 | subsection | 16 | 53 | eRPC design | Achieving eRPC's performance goals requires careful design and implementation.
We discuss three aspects of eRPC's design in this section: scalability of our
networking primitives, the challenges involved in supporting zero-copy, and the
design of sessions. The next section discusses eRPC's wire protocol and
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73d430ffeb8b1abe9ddf4d7f3ddf17ca5fe69b76 | subsection | 17 | 53 | Scalability considerations | We chose plain packet I/O instead of RDMA writes , , to send messages in eRPC. This decision is
based on prior insights from our design of FaSST: First, packet I/O provides
completion queues that can scalably detect received packets. Second, RDMA
caches connection state in NICs, which does not scale to
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6a6e178eb08b6b136f2944f784ce02e4b84fb0df | subsection | 18 | 53 | Packet I/O scales well | RPC systems that use RDMA writes have a fundamental scalability
limitation. In these systems, clients write requests directly to per-client
circular buffers in the server's memory; the server must poll these buffers to
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9403d51f9087b64e7e6bab7944462c66562c9883 | subsection | 19 | 53 | Scalability limits of RDMA | RDMA requires NIC-managed connection state. This limits scalability because
NICs have limited SRAM to cache connection state. The number of in-NIC
connections may be reduced by sharing them among CPU cores, but doing so
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115a3ad599c590c18ac472ab36155be2f6c14956 | subsection | 20 | 53 | Challenges in zero-copy transmission | eRPC uses zero-copy packet I/O to provide performance comparable to low-level
interfaces such as DPDK and RDMA. This section describes the challenges
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912b4fc5e4daa48f74e107671cfa112e882a5bcb | subsection | 21 | 53 | Message buffer layout | eRPC provides DMA-capable message buffers to applications for zero-copy
transfers. A msgbuf holds one, possibly multi-packet message. It consists of
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de8655e040b8b4fe79c133627cf0eaab24fa8f8e | subsection | 22 | 53 | Message buffer ownership | Since eRPC transfers packets directly from application-owned msgbufs, msgbuf
references must never be used by eRPC after msgbuf ownership is returned
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71f0ca270ada6773c9e64aafa19d91ea2d70fc60 | subsection | 23 | 53 | Zero-copy request processing | Zero-copy reception is harder than transmission: To provide a contiguous
request msgbuf to the request handler at the server, we must strip headers from
received packets, and copy only application data to the target msgbuf. However,
we were able to provide zero-copy reception for our common-case workload
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150a008138067dff5c440b96cef9b9b7d67ee26c | subsection | 24 | 53 | Sessions | Each session maintains multiple outstanding requests to keep the network pipe
full. Concurrently requests on a session can complete out-of-order with
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ceb9a1df18122e7ab8bde0abfa34b7b797920d9f | subsection | 25 | 53 | Session credits | eRPC limits the number of unacknowledged packets on a session for two reasons.
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cf070b86af2a839a1dd1565f5c8cf2a33ab28b9f | subsection | 26 | 53 | Session scalability | eRPC's scalability depends on the user's desired value of C, and the number
and size of RQs that the NIC and host can effectively support. Lowering C
increases scalability, but reduces session throughput by restricting the
session's packet window. Small values of C (e.g., C = 1) should be used in
applications that (a) ... | {
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49bb2bd57ca484c4fa12464253615f183332f3b7 | subsection | 27 | 53 | Wire protocol | We designed a wire protocol for eRPC that is optimized for small RPCs and
accounts for per-session credit limits. For simplicity, we chose a simple
client-driven protocol, meaning that each packet sent by the server is
in response to a client packet. A client-driven protocol has fewer “moving
parts” than a protocol in ... | {
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} | 1806.00680 | Datacenter RPCs can be General and Fast | [
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44675c5d353c82a146b18fba240a7387c1d8af1f | subsection | 28 | 53 | Protocol messages | Figure REF shows the packets sent with C = 2 for a small
single-packet RPC, and for an RPC whose request and response require three
packets each. Single-packet RPCs use the fewest packets possible. The client
begins by sending a window of up to C request data packets. For each request
packet except the last, the server... | {
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2251118ac576261fc969e29b300bf9d1f4a19044 | subsection | 29 | 53 | Congestion control | Congestion control for datacenter networks aims to reduce switch queueing,
thereby preventing packet drops and reducing RTT. Prior high-performance
RPC implementations such as FaSST do not implement congestion control, and some
researchers have hypothesized that doing so will substantially reduce
performance . Can effe... | {
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200ad4d14d7ffd4a2874f219bba03bca1cd46419 | subsection | 30 | 53 | Available options | Congestion control for high-speed datacenter networks is an evolving area of
research, with two major approaches for commodity hardware: RTT-based
approaches such as Timely , and ECN-based approaches
such as DCQCN . Timely and DCQCN have been deployed at
Google and Microsoft, respectively. We wish to use these protocol... | {
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"raw": "R. Mittal, T. Lam, N. Dukkipati, E. Blem, H. Wassel, M. Ghobadi, A. Vahdat, Y. Wang, D. Wetherall, and D. Zats. TIMELY: RTT-based congestion control for the datacenter. In Proc. ACM SIGCOMM, London,... | 1806.00680 | Datacenter RPCs can be General and Fast | [
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0cbfa255ee57638d658bae07674762ab8c6c188b | subsection | 31 | 53 | Common-case optimizations | We use three optimizations for our common-case workloads. Our evaluation shows
that these optimizations reduce the overhead of congestion control from 20% to
9%, and that they do not reduce the effectiveness of congestion control. The
first two are based on the observation that datacenter networks are typically
unconge... | {
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caa3f01727cdb0d026140d071a32ea25dbf25a85 | subsection | 32 | 53 | Comparison with IRN | IRN is a new RDMA NIC architecture designed for lossy
networks, with two key improvements. First, it uses BDP flow control to limit
the outstanding data per RDMA connection to one BDP. Second, it uses
efficient selective acks instead of simple go-back-N for packet loss recovery.IRN was evaluated with simulated switche... | {
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a896b483d882a4b1c7a8981f72e94ba6058ebe29 | subsection | 33 | 53 | Handling packet loss | For simplicity, eRPC treats reordered packets as losses by dropping them. This
is not a major deficiency because datacenter networks typically use ECMP for
load balancing, which preserves intra-flow ordering , , except during rare route churn events. Note
that current RDMA NICs also drop reordered packets .On suspectin... | {
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83d0b39cfd238f3896d60c04e6c648b7bc853216 | subsection | 34 | 53 | Microbenchmarks | eRPC is implemented in 6200 SLOC of C++, excluding tests and
benchmarks. We use static polymorphism to create an Rpc class that works with
multiple transport types without the overhead of virtual function calls. In
this section, we evaluate eRPC's latency, message rate, scalability, and
bandwidth using microbenchmarks.... | {
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0aaf0b5fea74051cadd43d3c94f56aa07bb998d9 | subsection | 35 | 53 | Small RPC latency | How much latency does eRPC add? Table REF compares the
median latency of 32 RPCs and RDMA reads between two nodes connected to
the same ToR switch. Across all clusters, eRPC is at most 800 slower than
RDMA reads.eRPC's median latency on CX5 is only 2.3, showing that latency with
commodity Ethernet NICs and software net... | {
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f8f3daf63fcbbc4257235b6fb0f5fc3618388b48 | subsection | 36 | 53 | Small RPC rate | What is the CPU cost of providing generality in an RPC system? We compare
eRPC's small message performance against FaSST RPCs, which outperform other RPC
systems such as FaRM . FaSST RPCs are specialized for
single-packet RPCs in a lossless network, and they do not handle congestion.We mimic FaSST's experiment setting:... | {
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2492d212f354db61460f0d07392255232856ce08 | subsection | 37 | 53 | Factor analysis. | How important are eRPC's common-case
optimizations? Table REF shows the performance impact of
disabling some of eRPC's common-case optimizations on CX4; other
optimizations such as our single-DMA msgbuf format and unsignaled transmissions
cannot be disabled easily. For our baseline, we use B = 3 and enable
congestion c... | {
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} | 1806.00680 | Datacenter RPCs can be General and Fast | [
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322726f7824ef5b7390aead2c257841acb37446a | subsection | 38 | 53 | Large RPC bandwidth | We evaluate eRPC's bandwidth using a client thread that sends large messages to
a remote server thread. The client sends R-byte requests and keeps one
request outstanding; the server replies with a small 32 response. We use
up to 8 requests, which is the largest message size supported by
eRPC. We use 32 credits per ses... | {
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7aba26d44f137c47519cb3f657d5030495c43f1c | subsection | 39 | 53 | Effectiveness of congestion control | We evaluate if our congestion control is successful at reducing switch
queueing. We create an incast traffic pattern by increasing the number of
client nodes in the previous setup (R = 8). The one server node acts as the
incast victim. During an incast, queuing primarily
happens at the victim's ToR switch. We use per-p... | {
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64e21e1617edbb1b31be3672790cb23e5a103d9d | subsection | 40 | 53 | Incast with background traffic. | Next, we augment the setup above to
mimic an experiment from Timely : we create
one additional thread at each node that is not the incast victim. These threads
exchange latency-sensitive RPCs (64 request and response), keeping one
RPC outstanding. During a 100-way incast, the 99th percentile latency of these
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5661e5cb6bda592a964b3702c396622c669709de | subsection | 41 | 53 | Full-system benchmarks | In this section, we evaluate whether eRPC can be used in real applications with
unmodified existing storage software: We build a state machine replication
system using an open-source implementation of Raft , and a
networked ordered key-value store using Masstree . | {
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c8df634a3dd0760c8577c54bc46af97757a403b3 | subsection | 42 | 53 | Raft over eRPC | State machine replication (SMR) is used to build fault-tolerant services. An
SMR service consists of a group of server nodes that receive commands from
clients. SMR protocols ensure that each server executes the same sequence of
commands, and that the service remains available if servers fail.
Raft is such a protocol ... | {
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1b752c87a87c1fc03378e94189b3972cca62e301 | subsection | 43 | 53 | Workloads. | We mimic NetChain and ZabFPGA's experiment setups for
latency measurement: we implement a 3-way replicated in-memory key-value store,
and use one client to issue PUT requests. The replicas' command logs and
key-value store are stored in DRAM. NetChain and ZabFPGA use 16 keys,
and 16–64 values; we use 16 keys and 64 val... | {
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53e77a9caf88264134853574e57b6ec79d496604 | subsection | 44 | 53 | Comparison with NetChain | NetChain's key assumption is that software networking adds 1–2 orders of
magnitude more latency than switches .
However, we have shown that eRPC adds 850, which is only around 2x higher
than latency added by current programmable switches
(400 ).Raft's latency over eRPC is 5.5, which is substantially lower than
NetChain... | {
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2ff26922b92ac222e0e2e3b63f7f00894d1b663b | subsection | 45 | 53 | Comparison with ZabFPGA | Although ZabFPGA's SMR servers are FPGAs, the clients are commodity
workstations that communicate with the FPGAs over slow kernel-based TCP. For
a challenging comparison, we compare against ZabFPGA's commit latency measured
at the leader, which involves only FPGAs. In addition, we consider its “direct
connect” mode, wh... | {
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b6516ef0342097120279ec4c46567a698ca2358b | subsection | 46 | 53 | Masstree over eRPC | Masstree is an ordered in-memory key-value store. We
use it to implement a single-node database index that supports low-latency
point queries in the presence of less performance-critical longer-running
scans. This requires running scans in worker threads. We use CX3 for
this experiment to show that eRPC works well on ... | {
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993f01541a867239a79fbca83b68a833d1b8b601 | subsection | 47 | 53 | RPCs. | There is a vast amount of literature on RPCs. The practice of
optimizing an RPC wire protocol for small RPCs originates with
, who introduce the idea of an implicit-ACK. Similar to
eRPC, the Sprite RPC system directly uses raw datagrams and
performs retransmissions only at clients. The Direct Access File
System was o... | {
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dd41f8c0c516710e3c22f23b2d652deeabddbbd6 | subsection | 48 | 53 | Co-design. | There is a rapidly-growing list of projects that
co-design distributed systems with the network. This includes key-value
stores , , , , distributed databases and transaction processing
systems , , , , state machine replication , , and graph-processing systems . We believe
the availability of eRPC will motivate research... | {
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3b81c28fb20b0f773f0468d126c37453d2e9e6bc | subsection | 49 | 53 | Conclusion | eRPC is a fast, general-purpose RPC system that provides an attractive
alternative to putting more functions in network hardware, and specialized
system designs that depend on these functions. eRPC's speed comes from
prioritizing common-case performance, carefully combining a wide range of old
and new optimizations, an... | {
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2a2228623034787d6475f1aefaa2e2b56ac4ac48 | subsection | 50 | 53 | eRPC's NIC memory footprint | Primarily, four on-NIC structures contribute to eRPC's NIC memory footprint:
the TX and RX queues, and their corresponding completion queues. The TX queue
must allow sufficient pipelining to hide PCIe latency; we found that 64 entries
are sufficient in all cases. eRPC's TX queue and TX completion queue have 64
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63769,
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150,
47,
1274,
112,
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13,
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11... | [
0.012969970703125,
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0.09979248046875,... | |
2c8623a55fac24322a0578ac4d8cc9dc15d54cae | subsection | 51 | 53 | Handling node failures | eRPC launches a session management thread that handles sockets-based management
messaging for creating and destroying sessions, and detects failure of remote
nodes with timeouts. When the management thread suspects a remote node failure,
each dispatch thread with sessions to the remote node acts as follows. First,
it f... | {
"cite_spans": []
} | 1806.00680 | Datacenter RPCs can be General and Fast | [
"Anuj Kalia",
"Michael Kaminsky",
"David G. Andersen"
] | [
"cs.OS"
] | 2,018 | en | Computer Science | [
28,
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0.08837890625,
0.131103515625,
0.1842041015625,
... | |
ee071cdc21669366bd5b61a17d28afe985a753d4 | subsection | 52 | 53 | Rate limiting with zero-copy | Recall the request retransmission example discussed in
§ REF : On receiving the response for the first
copy of a retransmitted request, we wish to ensure that the rate limiter does
not contain a reference to the retransmitted copy. Unlike eRPC's NIC DMA queue
that holds only a few tens of packets, the rate limiter trac... | {
"cite_spans": []
} | 1806.00680 | Datacenter RPCs can be General and Fast | [
"Anuj Kalia",
"Michael Kaminsky",
"David G. Andersen"
] | [
"cs.OS"
] | 2,018 | en | Computer Science | [
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0.03936767578125,
0.04440307617187... | |
5c017091042bd3301b707a1a3a52d4036bbde4fa | abstract | 0 | 63 | Abstract | We analyze binary data, available for a relatively large number (big data) of
families (or households), which are within small areas, from a population-based
survey. Inference is required for the finite population proportion of
individuals with a specific character for each area. To accommodate the binary
data and impo... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
1401,
7968,
53,
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0.1... | |
145aa0a57076d715f0bd034ac1a0475420e132a1 | subsection | 1 | 63 | -5pt | 12pt plus 4pt minus 2pt0pt plus 2pt minus 2pt12pt plus 4pt minus 2pt0pt plus 2pt minus 2pt12pt plus 4pt minus 2pt0pt plus 2pt minus 2pt2017]
Bayesian Logistic Regression for Small Areas with Numerous Households
BALGOBIN NANDRAM, LU CHEN, SHUTING FU AND BINOD MANANDHAR[Vol. .., Nos. 1&2firstpage
Statistics and Applic... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.181884765625,... | |
300f3590e098647d88eab57c8b09b04d20496b0b | subsection | 2 | 63 | Introduction | In the second Nepal Living standards Survey (NLSS II), there are data from households. One question of interest is health status (good versus poor health), a binary variable, and there are several covariates that can explain the binary outcomes. Our interest is to provide smoothed
estimates of the household proportions... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
70,
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bc00b2e2d12f633743a0979d17ed5e7e1a858168 | subsection | 3 | 63 | Introduction | Roberts, Rao and Kumar (1987) discussed logistic regression for sample survey data (not small area estimation). Nandram and Chen (1996) show how to accelerate the Gibbs sampler
for a model with latent variables introduced earlier by Albert and Chib (1993) for Bayesian probit analysis.Albert and Chib (1993) started an i... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.09588623046875,
0.111206... | |
2ddece716e5c2980808d54a6ba77fd66d217105c | subsection | 4 | 63 | Introduction | Yet
INLA has found many useful applications. See, for example, Fong, Rue and Wakefield (2010) for an application on Poisson
regression, and Illian, Sørbye and Rue (2012) for a realistic application on spatial point pattern data. We note
that INLA can be problematic especially for logistic and Poisson hierarchical regre... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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0.1531982421875,
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0.0960693359375,
0.051208496... | |
bb70ef6d7008bf8fb0cc6c328b85884fba5a0fd6 | subsection | 5 | 63 | Approximate Theory and Method | The method we developed here for many small areas can be applied to any generalized linear model in the same manner. Of course, the specific models will be different. For example, for the model for Poisson regression is different from the model for logistic regression. However, note that for logistic regression model, ... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
55300,
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186,
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0.00204467... | |
5019bf353a773b1fa15ba1523e932cf8587f6f75 | subsection | 6 | 63 | Approximate Theory and Method | Let y_{ij} and {x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij} = (1,x_{ij1},\dots ,x_{ijp-1} )^T, denote the responses and the p vector of covariates with an intercept (x_{ij0}=1).A standard hierarchical Bayesian logistic regression model isy_{ij}|{\beta ,\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}\nu _i \sta... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
10842,
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454,
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0.26806640625,
0.02142333... | |
1f816493b8553700937b5be226be8d5df2f354b5 | subsection | 7 | 63 | Approximate Theory and Method | Omitting the intercept term from the covariate x_x\tilde{}ij, we
have
y_{ij}|\mu _i,{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}\vss }}{(0)} \stackrel{ind}{\sim } \mbox{Bernoulli} \left\lbrace \frac{e^{{x_\crcr \vbox to.2ex{\hbox{$x\tilde{}$}\vss }}{ij}^{\prime }{\beta _\crcr \vbox to.2ex{\hbox{$\beta \tilde{}$}... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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552,
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0.066... | |
a39cbafd7330498fd249ee6943298ae90cd9bea9 | subsection | 8 | 63 | Approximate Theory and Method | Because we are not linking
the census to the NLSS, we do not have the covariates and the number of members in
each nonsampled households, both being obtained using a Bayesian bootstrap (Rubin 1981) of the
original samples.To develop the approximate methodology, we will work with the no-intercept model. Then, using Baye... | {
"cite_spans": []
} | 1806.00446 | Bayesian Logistic Regression for Small Areas with Numerous Households | [
"Balgobin Nandram",
"Lu Chen",
"Shuting Fu",
"Binod Manandhar"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
88949,
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