From 66ea391a3e386a6ed3b47d5011977ca136f65ad2 Mon Sep 17 00:00:00 2001 From: =?utf8?q?Pierre-Cyrille=20H=C3=A9am?= <pheam@femto-st.fr> Date: Wed, 22 Jun 2016 14:45:04 +0200 Subject: [PATCH] pch --- main.aux | 27 +- main.blg | 2 +- main.log | 1260 +++++++++++++++++--------------------------------- main.pdf | Bin 613533 -> 615170 bytes stopping.tex | 27 +- 5 files changed, 462 insertions(+), 854 deletions(-) diff --git a/main.aux b/main.aux index 0bd4d61..a426c9e 100644 --- a/main.aux +++ b/main.aux @@ -38,13 +38,13 @@ \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{A Metric Space for PRNG Iterations}}{5}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{A metric on $\mathcal {X}_{\mathsf {N},\mathcal {P}}$}}{6}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.4}{$\Gamma _{\mathcal {P}}(f)$ as an extension of $\Gamma (f)$}}{8}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proofs of chaos}}{8}} 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uniform distribution of switches}}{13}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stopping Time}}{13}} -\newlabel{sec:hypercube}{{6}{13}} \citation{LevinPeresWilmer2006} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.3}{Toward a local uniform distribution of switches}}{14}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stopping Time}}{14}} +\newlabel{sec:hypercube}{{6}{14}} +\newlabel{thm-sst}{{6.1}{15}} \newlabel{eq:Markov:rairo}{{3}{15}} -\newlabel{lm:h}{{6.2}{15}} -\citation{proba} +\newlabel{lm:h}{{6.2}{16}} \newlabel{prop:stop}{{6.4}{16}} \newlabel{prop:lambda}{{6.5}{16}} +\citation{proba} \newlabel{lm:stopprime}{{6.6}{17}} \@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Experiments}}{18}} \newlabel{sec:prng}{{7}{18}} \@writefile{loa}{\contentsline {algocf}{\numberline {2}{\ignorespaces Pseudo Code of the $\chi _{\textit {15Rairo}}$ PRNG\relax }}{18}} \newlabel{CI Algorithm:2}{{2}{18}} -\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Functions with DSCC Matrix and smallest MT\relax }}{19}} -\newlabel{table:nc}{{1}{19}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Conclusion}}{19}} \bibstyle{alpha} \bibdata{biblio} \bibcite{Banks92}{BBCS92} -\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces NIST SP 800-22 test results ($\mathbb {P}_T$)\relax }}{20}} -\newlabel{The passing rate}{{2}{20}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Conclusion}}{20}} \bibcite{bcgr11:ip}{BCGR11} \bibcite{Nist10}{BR10} \bibcite{DBLP:journals/combinatorics/BhatS96}{BS96} +\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Functions with DSCC Matrix and smallest MT\relax }}{20}} +\newlabel{table:nc}{{1}{20}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{20}} \bibcite{Bykov2016}{Byk16} \bibcite{chgw14oip}{CHG{$^{+}$}14a} \bibcite{DBLP:conf/secrypt/CouchotHGWB14}{CHG{$^{+}$}14b} @@ -106,12 +106,13 @@ \bibcite{LEcuyerS07}{LS07} \bibcite{Marsaglia1996}{Mar96} \bibcite{proba}{MU05} +\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces NIST SP 800-22 test results ($\mathbb {P}_T$)\relax }}{21}} +\newlabel{The passing rate}{{2}{21}} \bibcite{Robinson:1981:CS}{RC81} \bibcite{915385}{SK01} \bibcite{915396}{SPK01} \bibcite{ZanSup04}{SZ04} \bibcite{wbg10ip}{WBGF10} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{21}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.77466pt} \newlabel{tocindent1}{17.77344pt} diff --git a/main.blg b/main.blg index 01af0c2..d3e8eb8 100644 --- a/main.blg +++ b/main.blg @@ -1,4 +1,4 @@ -This is BibTeX, Version 0.99d (TeX Live 2015/dev/Debian) +This is BibTeX, Version 0.99d (TeX Live 2013/Debian) Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 The top-level auxiliary file: main.aux The style file: alpha.bst diff --git a/main.log b/main.log index 24b84b6..633dfbf 100644 --- a/main.log +++ b/main.log @@ 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ABD: EveryShipout initializing macros (./intro.tex - -LaTeX Warning: Citation `915396' on page 1 undefined on input line 2. - - -LaTeX Warning: Citation `915385' on page 1 undefined on input line 2. - - -LaTeX Warning: Citation `5376454' on page 1 undefined on input line 2. - - -LaTeX Warning: Citation `915396' on page 1 undefined on input line 5. - - -LaTeX Warning: Citation `915385' on page 1 undefined on input line 5. - - -LaTeX Warning: Citation `5376454' on page 1 undefined on input line 5. - - -LaTeX Warning: Citation `Marsaglia1996' on page 1 undefined on input line 10. - - -LaTeX Warning: Citation `Nist10' on page 1 undefined on input line 10. - - -LaTeX Warning: Citation `LEcuyerS07' on page 1 undefined on input line 10. - - -LaTeX Warning: Citation `Devaney' on page 1 undefined on input line 19. - LaTeX Font Info: Try loading font information for OMS+cmr on input line 26. + (/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd -File: omscmr.fd 2014/09/29 v2.5h Standard LaTeX font definitions +File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <7> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 26. @@ -1261,35 +1165,7 @@ Non-PDF special ignored! 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(pdftex.def) Requested size: 230.36006pt x 170.13521pt. - - -LaTeX Warning: Citation `DBLP:conf/secrypt/CouchotHGWB14' on page 3 undefined o -n input line 66. - -) [3] (./chaos.tex - -LaTeX Warning: Citation `bcgr11:ip' on page 4 undefined on input line 2. - - -LaTeX Warning: Citation `bcgr11:ip' on page 4 undefined on input line 21. - -[4 <./images/iter_f0c.pdf>] - -LaTeX Warning: Citation `Devaney' on page 5 undefined on input line 62. - - -LaTeX Warning: Citation `Banks92' on page 5 undefined on input line 79. - + [3]) (./chaos.tex [4 <./images/iter_f0c.pdf>] [5] Overfull \hbox (12.18526pt too wide) in paragraph at lines 229--232 []\T1/cmr/m/n/10 The next $\OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ^^B [] []$ \T1/c @@ -1376,86 +1234,8 @@ Underfull \hbox (badness 10000) in paragraph at lines 472--477 ; :::; a[]; :::; a[];$ [] -[9 <./graphe1.pdf> <./graphe2.pdf>]) (./generating.tex - -LaTeX Warning: Citation `bcgr11:ip' on page 10 undefined on input line 2. - - -LaTeX Warning: Citation `DBLP:conf/secrypt/CouchotHGWB14' on page 10 undefined -on input line 9. - -[10] - -LaTeX Warning: Citation `bcgr11:ip' on page 11 undefined on input line 61. - -) (./hamilton.tex - -LaTeX Warning: Citation `Robinson:1981:CS' on page 11 undefined on input line 2 -. - - -LaTeX Warning: Citation `DBLP:journals/combinatorics/BhatS96' on page 11 undefi -ned on input line 2. - - -LaTeX Warning: Citation `ZanSup04' on page 11 undefined on input line 2. - - -LaTeX Warning: Citation `Bykov2016' on page 11 undefined on input line 2. - - -LaTeX Warning: Citation `DBLP:journals/combinatorics/BhatS96' on page 11 undefi -ned on input line 4. - - -LaTeX Warning: Citation `ZanSup04' on page 11 undefined on input line 4. - - -LaTeX Warning: Citation `Bykov2016' on page 11 undefined on input line 14. - - -LaTeX Warning: Citation `ZanSup04' on page 11 undefined on input line 27. - - -LaTeX Warning: Citation `DBLP:journals/combinatorics/BhatS96' on page 11 undefi -ned on input line 27. - - -LaTeX Warning: Citation `Bykov2016' on page 11 undefined on input line 28. - - -LaTeX Warning: Citation `ZanSup04' on page 11 undefined on input line 31. - -[11] - -LaTeX Warning: Citation `Robinson:1981:CS' on page 12 undefined on input line 3 -7. - - -LaTeX Warning: Citation `DBLP:journals/combinatorics/BhatS96' on page 12 undefi -ned on input line 37. - - -LaTeX Warning: Citation `ZanSup04' on page 12 undefined on input line 38. - - -LaTeX Warning: Citation `Robinson:1981:CS' on page 12 undefined on input line 4 -0. - - -LaTeX Warning: Citation `DBLP:journals/combinatorics/BhatS96' on page 12 undefi -ned on input line 43. - - -LaTeX Warning: Citation `ZanSup04' on page 12 undefined on input line 45. - - -LaTeX Warning: Citation `ZanSup04' on page 12 undefined on input line 62. - - -LaTeX Warning: Citation `ZanSup04' on page 12 undefined on input line 84. - -[12] +[9 <./graphe1.pdf> <./graphe2.pdf>]) (./generating.tex [10]) (./hamilton.tex +[11] [12] Overfull \hbox (104.68428pt too wide) in paragraph at lines 119--124 []\T1/cmr/m/it/10 Let now $\OML/cmm/m/it/10 L[] \OT1/cmr/m/n/10 = 0000\OML/cmm/ m/it/10 ; \OT1/cmr/m/n/10 0010\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0110\OML/cmm/m @@ -1479,170 +1259,13 @@ t/10 ; \OT1/cmr/m/n/10 0100\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1100\OML/cmm/m/it /10 ;$ [] -Runaway argument? -{2^{\mathsf {N} -n.a_{\mathsf {N}}}{2} \\ c_{\mathsf {N}} = \mathsf {\ETC. -! 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LaTeX Error: Something's wrong--perhaps a missing \item. - -See the LaTeX manual or LaTeX Companion for explanation. -Type H <return> for immediate help. - ... - -l.194 \subsection - {Toward a local uniform distribution of switches} -Try typing <return> to proceed. -If that doesn't work, type X <return> to quit. - - -! LaTeX Error: Something's wrong--perhaps a missing \item. - -See the LaTeX manual or LaTeX Companion for explanation. -Type H <return> for immediate help. - ... - -l.194 \subsection - {Toward a local uniform distribution of switches} -Try typing <return> to proceed. -If that doesn't work, type X <return> to quit. - -! Missing } inserted. -<inserted text> - } -l.194 ...a local uniform distribution of switches} - -I've inserted something that you may have forgotten. -(See the <inserted text> above.) -With luck, this will get me unwedged. But if you -really didn't forget anything, try typing `2' now; then -my insertion and my current dilemma will both disappear. - -! Missing \cr inserted. -<inserted text> - \cr -l.194 ...a local uniform distribution of switches} - -I'm guessing that you meant to end an alignment here. - -! Missing $ inserted. -<inserted text> - $ -l.194 ...a local uniform distribution of switches} - -I've inserted a begin-math/end-math symbol since I think -you left one out. Proceed, with fingers crossed. - -) -! Missing $ inserted. -<inserted text> - $ -l.122 - -I've inserted a begin-math/end-math symbol since I think -you left one out. Proceed, with fingers crossed. - -! Missing \endgroup inserted. -<inserted text> - \endgroup -l.122 - -I've inserted something that you may have forgotten. -(See the <inserted text> above.) -With luck, this will get me unwedged. But if you -really didn't forget anything, try typing `2' now; then -my insertion and my current dilemma will both disappear. - -! Missing \right. inserted. -<inserted text> - \right . -l.122 - -I've inserted something that you may have forgotten. -(See the <inserted text> above.) -With luck, this will get me unwedged. But if you -really didn't forget anything, try typing `2' now; then -my insertion and my current dilemma will both disappear. - -! Display math should end with $$. -<to be read again> - \par -l.122 - -The `$' that I just saw supposedly matches a previous `$$'. -So I shall assume that you typed `$$' both times. - - -Overfull \hbox (482.90009pt too wide) detected at line 122 -[] \OMS/cmsy/m/n/10 , [] - [] - -(./stopping.tex [13] - -LaTeX Warning: Citation `LevinPeresWilmer2006' on page 14 undefined on input li -ne 85. - -[14] [15] - -LaTeX Warning: Reference `lm:h' on page 16 undefined on input line 301. - - -LaTeX Warning: Citation `proba' on page 16 undefined on input line 314. - -[16] - -LaTeX Warning: Reference `prop:stop' on page 17 undefined on input line 360. - - -LaTeX Warning: Reference `prop:stop' on page 17 undefined on input line 367. - - -LaTeX Warning: Reference `prop:lambda' on page 17 undefined on input line 368. - - -LaTeX Warning: Reference `lm:stopprime' on page 17 undefined on input line 368. - - -) [17] (./prng.tex - -LaTeX Warning: Reference `CI Algorithm:2' on page 18 undefined on input line 13 -. - - -LaTeX Warning: Reference `sec:hypercube' on page 18 undefined on input line 44. - - - -LaTeX Warning: Reference `eq:Markov:rairo' on page 18 undefined on input line 5 -9. - - -LaTeX Warning: Reference `table:nc' on page 18 undefined on input line 62. - +? +) (./stopping.tex [13] [14] [15] [16] [17]) (./prng.tex [18] LaTeX Warning: Command \textcircled invalid in math mode on input line 64. @@ -1657,11 +1280,6 @@ LaTeX Warning: Command \textcircled invalid in math mode on input line 67. LaTeX Warning: Command \textcircled invalid in math mode on input line 67. -[18] - -LaTeX Warning: Reference `sec:hypercube' on page 19 undefined on input line 76. - - LaTeX Warning: Command \textcircled invalid in math mode on input line 89. @@ -1693,14 +1311,6 @@ LaTeX Warning: Command \textcircled invalid in math mode on input line 172. LaTeX Warning: Command \textcircled invalid in math mode on input line 172. -LaTeX Warning: Reference `The passing rate' on page 19 undefined on input line -214. - - -LaTeX Warning: Reference `The passing rate' on page 19 undefined on input line -217. - - LaTeX Warning: Command \textcircled invalid in math mode on input line 231. @@ -1738,38 +1348,24 @@ Overfull \hbox (4.26556pt too wide) in paragraph at lines 229--250 ) (./conclusion.tex Missing character: There is no p in font dsrom8! Missing character: There is no p in font dsrom8! - [19]) (./main.bbl -Underfull \vbox (badness 10000) has occurred while \output is active [] - - [20] +) [19] (./main.bbl [20] Underfull \hbox (badness 10000) in paragraph at lines 81--84 []\T1/cmr/m/n/8 G. Marsaglia. Diehard: a bat-tery of tests of ran-dom-ness. [] - -Underfull \vbox (badness 10000) has occurred while \output is active [] - - [21]) -[22] (./main.aux) - -LaTeX Warning: There were undefined references. - - -LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. - - ) +[21]) [22] (./main.aux) ) (\end occurred inside a group at level 1) ### simple group (level 1) entered at line 10 ({) ### bottom level Here is how much of TeX's memory you used: - 22192 strings out of 493105 - 431459 string characters out of 6137072 - 566154 words of memory out of 5000000 - 24950 multiletter control sequences out of 15000+600000 + 20136 strings out of 493304 + 370159 string characters out of 6139871 + 525134 words of memory out of 5000000 + 22909 multiletter control sequences out of 15000+600000 28957 words of font info for 91 fonts, out of 8000000 for 9000 - 1302 hyphenation exceptions out of 8191 - 54i,17n,75p,539b,695s stack positions out of 5000i,500n,10000p,200000b,80000s + 1118 hyphenation exceptions out of 8191 + 48i,17n,75p,539b,695s stack positions out of 5000i,500n,10000p,200000b,80000s {/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc}</us r/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/shar e/texlive/texmf-dist/fonts/type1/public/amsfonts/cmextra/cmex7.pfb></usr/share/ @@ -1810,7 +1406,7 @@ fonts/type1/public/cm-super/sfti0800.pfb></usr/share/texmf/fonts/type1/public/c m-super/sfti1000.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/stmaryrd /stmary10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/stmaryrd/stmary 7.pfb> -Output written on main.pdf (22 pages, 613533 bytes). +Output written on main.pdf (22 pages, 615170 bytes). 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zO+vc<RAs~7qhd8@0^e(v)5IS5h^WHbJH2Uu3C{C2RYqn#GW8Vs0!#o(h}Vr9&cfv$ z$Nnpt0A>Fw5nWFIL8=hl=#o_<y7f|4gXqN}SuLW`<FYzLACAfDk;#w98jv}BQZ@sb zv!`V<k-3%rwpqx$Pk(PC3d;YLHKE{-bgmiM{1e%1WcL@!T9AF*CTm6ZQ>APUa*JNc z+K{`Gp0pit``@w-#9dcpbD3u#6W)AT$qo<?o*BFIEobb1@Ds;am)T+o9t$!wIW!6- IB}Gq03Q=J;AOHXW diff --git a/stopping.tex b/stopping.tex index 3a07e06..d72f8bb 100644 --- a/stopping.tex +++ b/stopping.tex @@ -144,7 +144,7 @@ A stopping time $\tau$ is a {\emph strong stationary time} if $X_{\tau}$ is independent of $\tau$. -\begin{thrm} +\begin{thrm}\label{thm-sst} If $\tau$ is a strong stationary time, then $d(t)\leq \max_{X\in\Bool^{\mathsf{N}}} \P_X(\tau > t)$. \end{thrm} @@ -362,16 +362,27 @@ One can now prove Theorem~\ref{prop:stop}. \begin{proof} Since $\ts^\prime$ is the time used to obtain $\mathsf{N}-1$ fair bits. Assume that the last unfair bit is $\ell$. One has -$\ts=\ts^\prime+S_{X_\tau,\ell}$, and therefore -$E[\ts] = E[\ts^\prime]+E[S_{X_\tau,\ell}]$. Therefore, -Theorem~\ref{prop:stop} is a direct application of -lemma~\ref{prop:lambda} and~\ref{lm:stopprime}. +$\ts=\ts^\prime+S_{X_\tau,\ell}$, and therefore $E[\ts] = +E[\ts^\prime]+E[S_{X_\tau,\ell}]$. Therefore, Theorem~\ref{prop:stop} is a +direct application of lemma~\ref{prop:lambda} and~\ref{lm:stopprime}. \end{proof} +Now using Markov Inequality, one has $\P_X(\tau > t)\leq \frac{E[\tau]}{t}$. +With $t=32N^2+16N\ln (N+1)$, one obtains: $\P_X(\tau > t)\leq \frac{1}{4}$. +Therefore, using the defintion of $t_{\rm mix)}$ and +Theorem~\ref{thm-sst}, it follows that +$t_{\rm mix}\leq 32N^2+16N\ln (N+1)=O(N^2)$. + + Notice that the calculus of the stationary time upper bound is obtained under the following constraint: for each vertex in the $\mathsf{N}$-cube there are one ongoing arc and one outgoing arc that are removed. The calculus does not consider (balanced) Hamiltonian cycles, which -are more regular and more binding than this constraint. -In this later context, we claim that the upper bound for the stopping time -should be reduced. +are more regular and more binding than this constraint. Moreover, the bound +is obtained using Markov Inequality which is frequently coarse. For the +classical random walkin the $\mathsf{N}$-cube, without removing any +Hamiltonian cylce, the mixing time is in $\Theta(N\ln N)$. +We conjecture that in our context, the mixing time is also in $\Theta(N\ln +N)$. +%In this later context, we claim that the upper bound for the stopping time +%should be reduced. -- 2.39.5