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A Fault-Contained Spanning Tree Protocol for Arbitrary Networks

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Academic year: 2021

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(26)   . ∧ ¬Lmax(i)    "!   #$% j ∈ N eighi  reqi = i; f romi = i; toi = j; diri = ask;. &. rootj ;. Sat(i) ∧ ¬∃j : F orw(i, j) ∧ ¬Idle(i) reqi = f romi = toi = diri = ⊥; Sat(i) ∧ Idle(i) ∧ Asks(j, i) reqi = reqj ; f romi = j; toi = pari ; diri = ask; Sat(i) ∧ Root(i) ∧ F orw(i, j) ∧ diri = ask diri = grant;. ‘“ ‘’ “‘ž’‡‰ŠžžŸ‰–  ۑ®¢¦˜ª–—ÀZ¡/ë'‡|ŠŒŠž–—t†x £nø ‡‰‰“ & ”'”Œ©†x‰‰’ ‰“kŠžž–—–†xˆ ‹ †x £ †xž–'‘‰ˆ ¤k–Ç|–]‡|Šž– £ ›¥¤†v‰ £ – ¬’ ŠžŸ Šžž‡‰–‘§ ©5†(¥”Œ‘?’ ‰?+“‘˜Y¬ˆ†x–'–‡‰¯Šžˆ ŠŒ –—‹Ž†x]¡‘‘”Œ¦\’ –—Š ˜ª–—†x\ÀZ’Š‰¡|˜Y ë'†x‡‰–—‡|\ˆ ŠŒŠ]Š?Šž§D–—˜ª½Ž–]†+†xŠž˜Y‡‰†‘£vž`–]Šø ” ˜Y‡‰†x£ ‰‡‰‡‰“ˆ žŠ &’ ”¢”‰“k†x†xžˆ ž“‘– –]‘£ ž ’–Šž¯ŠžŸ‰–]–— zž`¤(Šž– Ÿ˜ª£ ž†‘‘”Dˆ x  † ‹ “‘’—†x †xˆ Šž©‰Ÿ‰¤Ÿ,–›‰”Œ‡|Ÿ‰‘Š'+¬”šŠž‘,žž‡ž’ ,–]`Šž`«–Šž–£“‘£‡‰”Œž©›¥–D†x¤ù‰‰“‘ÅY‰†\ˆ ’ ‘È`‰§›“¢†xÁʞˆÉžŸ‰‘–—–©5–‘”Œ¡+–—‘¬ ˆ †+’ ’ £ŠžŠž’Ÿ–B‘ˆÉŠž’ Ÿ‰‰§ ––]D”†x‘”ŒžŸ‰ž+”Œ+’¬?¬"£ ”É–—– ’ £  Šž“‘£ Ÿ‰–]’ –”‹ xˆ—’ †+˜<‰Šž–]–]’ ”‰†‘ “'”ŒŸnŸ‰Šž+Ÿ‰‰¬¿–? £©– –v†x£ ž“‘†x–—–]ž\”ÖŠ<’ ”ŒxŸ‰˜

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(28) ˆ ’ ¯t†x†ž’¼£‰†x£ ›‰’ ˆ Šž–]’ ”—‘¡<p†xŠžˆ¼”ŒoŸŸ† †‘¯ ”’ ‰†p“žž–]–]†† ££ ‹‹ †‘—–]”ž”DŠžvŠžŸ‰–,ˆ |—†xˆ ¯ & †xž’¼†x›‰ˆ –]”x˜#‰–—’ “‘Ÿ¥›5‘ ” & ‰–—’ “‘Ÿ¥›5‘ ”—§ ”ˆŒ|’ ‘—»ÁÆxˆ‰Ÿ‰x’ –(˜?’Šž‘£ Ÿ‰–]–n†”Œ’¼†x”šxŠžˆ ˜“‘–—‘ŠžžŸ‰’ ¤‘–vŠž¡‘Ÿ‰˜Y’ %†xŠ ‡‰£ ˆ’¼¥ŠŒ”‹Ž–]Šž”<‘Ÿ‰\†+xŠ Š†xŠD’¬‰‰Ÿ‰–]– £–—–]•”ž”Œ”ž©†o†x†xž‰‰’ ˆ‰ ¤’£ ‰ (– “†x£ ŠžÀ‘–ž–?Šž–—–]–†,`Š ¯”‘–—ŒŒ† ‹‹ ŠžŠžžŸ‰Ÿ–]`–†+ŠžŠ¢’—‘Šž†x•Ÿ‰‡–v”Œ –,ˆ | x¯—˜#†x–‘Šžˆ#¡Ÿ‰’ ’’¼”?”šˆ‘Šž|–]—† ”Œ†x£’¼ˆ/”šŠž’ ’–—Šk«‘¤ ”š”Œ (Š’¼”š†IŠž†+–—¤ŠŒŠž–—›5 ¤‘– §©|£ ½lŠ ‡‰˜D”(–†ŠžŠž ‰ £ †(£ –– Šž˜Y†x–—ž‡‰ «ˆ Š, ’ ‰£†+–Š” †xˆ ‘‰–‘¡ÉŠžŸ‰–—*’ Šv‘žž–]`Š ”'’ Š ”ˆ |—†xˆD”šŠ †+Šž–k†x©‰©‰ž‘©‰ž’¼†+Šž–—ˆ ¤‘§ ?ŠžŸ‰–—ž¬’¼”Œ–‘¡’ ‘”Œ’¼”šŠž–—¤" (†I¤"›5– £ ‡‰–(Šž"†˜Y†x‡‰ˆ Š†+Šv† ˜Y‰†x–—‡‰’ “‘ˆ Š]Ÿ¥§ ›5‘ ¡ÉŠžŸ‰–—oŠžŸ‰–(‰ £ – ¬Ã†x’ Š ”e˜ª‘ Šžn‘žž–]`Š'’ Š ” x†x˜ £ ê"–DŠž‰Ÿ+†+¬tŠ £ ©‰–—ž›5‰–]–(x’¼Šž”ŒŠž–]–#Ÿ‰”–ŠžŠžŸ‰Ÿ‰”Œ’¼––”/Š’ ”Œ£ –x–]Š¢˜Ã†‰x‰§ ˜#–—É’’ “‘–£ ŠŸ¥–—›5\Šž‘’  Šž”'’ ›5–]x–”?˜ †Ã’ «§Šž‰Ÿ‰½Ž’ tŠž––D‰Šž’ Ÿ‰\–ŠžŠš–k¬–—ž‘¯–]ž†x”šÀ Š ˆ ŠžxŸ˜#†+ŠžŠŸ‰’¼ (”e†x©’ †x\©5Š †x–—’]¡5”¬Šš– ¬£–—“‘‰žx‘Šž‡‰–©›¥”¤ x˜ ¯ Šž†xŸ‰ž–’¼†x’ ›‰£ ˆ–—–]\”eŠž’ ® Šš¤x˜D†(‰ £ – ۑ§Á߉– ¯ †xž’¼†x›‰ˆŸ‰–]‘”ˆ £ž–—”kˆ¼†+ŠžŠžŸ‰– –o£ ’Šž£ –—\ŠžŸ‰Šž’–'Šš¤²Šžž–—x–,˜,”šŠžŠžŸ‰ž–"‡`žŠž¥‡‰xžŠ–‘® x˜,ŠžŸ‰– ŠžŠžžk–—–”Œ‘Šž –'¬‰Ÿ‰ ’¼£ Ÿp–'’‰  £ – ›5–—ˆ ‘‰“\”—§ ’¼”¢–]­\‡†xˆ ½lŠ?’¼”?†x’¼”Êž’ \Ÿ‰Šž– –—“‘£ –—’¼”šÃŠ †x’ Šž–'Ÿ‰–,˜ªž ‘†x ú‰“‘‰–  £ – Šž(’ Š ”?ž¥xŠ]§ (a). T ree(i) ∧ ¬Lmax(i) ∧ Grant(toi , i) rooti = rootj ; pari = j; disti = distj + 1; reqi = f romi = toi = diri = ⊥;. i. (r,1) j. (r,2) k. disti = distk + 1. (r,1). (r,2). i

(29)    Tree(i)   '. (r,1) j. /. 0. (b). $1. i. i. i. j. i. j. 2-. INf. IN N eighi. i. i. i. 3 rooti 3 disti. i INf 4. rooti. i [1 . . . |INf |] 4.

(30) ’¼”Œ”¢‘ ŠžŸ‰–'–’¼”?‰© Šž†xŸ‰£ ž–'––—\’’ Š'£ –—x\˜ Šž’ Šš¤’ oxŠž˜DŸ‰†(–v‰Šž–—ž’–—§ “‘–‘Ÿ¥§›5½l‘Š,¢’¼x”e˜ –]­\‡¬†xŸ‰ˆ#’¼ ŠžŸ ù|§Á߉– ¯ †x ž’¼–]†x”ž›‰”žˆ †x–]“‘”?–]ž”—–—® ˆ¼†+Šž– £ Šž©†‘”ž”Œ’ ‰“(ŠžŸ‰– †x £ ‡‰›5ž–—žˆ–—‘’¼\”‰Šž“\ˆ–—¤'”—’ Šž¡‰žŸ‰‘–]–—­\‡‰–]Šž–]­\Ÿ‰”š‡Šž–k†x’ ‰ˆZ’ “?£ Šž–— Šž\DŠžÕš’ ’Šš‘˜ ¤"’ x’Šž˜?Š Ÿ‰”Œ†––—ˆ Šž˜B‰ž’¼–—”–<£ Šž–(Šžž'¤¥Šž’Ÿ¬‰†+Ÿ‰“Šv’¼ Šž’¼Ÿ ” ՚’ ‘’ *†x‰xŠžŸ‰–—'Šžž–—–‘§k½lŠ’¼”,–]­\‡†xˆ#Šžp”Œ‘ –(‰ £ – ˜ª’ ˜ž‘ %Ÿ†‘¬”?’¼Ÿ‰”¢’ ’¼‰ –—’ Ÿ Šž’ Šž’¼†+Ÿ‰Šž–—–e£‘Šž©‰Ÿ‰†k’ ––ž£ –]’ £ ­\Šž–—‡‰Ÿ‰\–]– Šž”š’ Š¢¯Šš¤n†x’ oˆ ‡‰x–˜B†xoŠžxŸ‰˜ †+–vŠŒŠž‰–—–— ’ “‘¡B©|Ÿ¥‘Še›5 Šž‘  ՚‘’ p†(‰–—¬&Šžž–—–‘§½lŠe’¼”¢–]­\‡†xˆ

(31) Šž”Œ‘ –‰ £ –’  ¬ Ÿ‰–]’¼”ž ’¼”ž”Ÿ»†x“‘Šž’ –‘Ÿ‰Šv§–’¼”’ £ Šž½l–—žŠ6\¤¥Šž’ ‰’’¼Šš”Æ“p¤*–]Šžxo­\˜¢‡©‰‰†xž–—ˆt‘’ “‘©ŠžŸ¥¾†x›5“\‘†+”ŒŠž‘– x˜Šž– Ÿ‰–‰ŠžžŸ‰ –]£ž­\‘–߇‰‡‰–]“‘’”šŸ Š ¬ŠžžŸ‰–—–‘\¡x”Œ’¼–(‘” ’ –—£ ’ –—Šž\Ÿ‰Šž–—’ Šš¤tŠž,’¼””Œ’¡B’“‘ Šž‰*’ ˜ª¤'”Œ’ Šž“‘Ÿ‰¬†+’ ˜ªŠB’¼¤t”ŒŠžŸ‰Ÿ‰Šž–]Ÿ’¼”,”#†+žŠžŠ–]k­\ŠžŸ‰Õš‡‰‘––]’ ”štŠ#‰ŸŠžŸ‰£†‘––” ›5–—–—H“‘ †x\Šž– £ §"§½lŠ’¼”–]­\‡†xˆDŠžt”Œ‘ – ¯ †xˆ ‡‰–’  ˆÁ/+ ¬’£ ‰–]“”ž©‰žž’ ›5– £ –(’¼—Šž†+Ÿ‰Šž––]”—Šž® ž–—–”šŠžž‡`Šž‡‰ž–‘¡#¬–’ \Šžž £ ‡–(ŠžŸ‰–˜ª‘ˆ ‹ 3 pari. i. i N eighi ∪ {i}. request. grant 3 reqi i. i. i. INf ∪ {⊥} 4. 3 f romi. i. reqi. i. i. N eighi ∪ {i} ∪ {⊥} 4. 3 toi. i. F orwA(i, j) ≡ j ∈ N eighi ∪ {i} ∧ toi = j ∧ pari = j ∧ diri = grant ∧ Sat(i) ∧ f romi 6= i ∧ reqi 6= i ∧ reqi 6= ⊥ ∧ reqi = reqf romi ∧ ∀k ∈ Child(i) : (rootk = rooti ⇒ distk = disti + 1) Rgcorrect(i) ≡ Idle(i) ∨ ∃j : F orwN a(i, j) ∨ (∃j : F orwA(i, j) ∧ dirf romi = ask). ž‡‰’  ŠžŸ‰£– –—˜ª¡‘†ž¬–”Œ–p’ ¬‰’ “‘–ˆ ¯ –“‘–]’”šŠž¯ Šž –’†x“\†+† ”ŒŠž’ –£–—\–ŠžŠ Š'Ÿ‰†x˜Y–o’ †xˆ – ‡‰›5£•ˆ –—Š]Ÿ§£ † –]ɯ ”ž’–‘žŠk’ ©|†xŠž’˜,ˆ ‘–—H“‘ŠžŸ‰’ Šžx–o’ ˜¢ (†xŠž†+ˆ Ÿ‰“‘Šž–n–(‘ž”š†x’ ŠžŠ ˆ Ÿ‰“‘†+Šžx  – ‹ xx˜e˜DŠžŠžŸ‰Ÿ‰–k–p”Œ”Œ¤|¤|”š”šŠžŠž–—–— ¾ î’¼›5”,– †£ ˆ –—–«“‘’‰Šž–’  (£ †+†‘Šž”–(˜ª”š‘Š ˆ†+Šž+–‘¬?¡É”—’ §Ù˜†x¦  £ “‘ˆ‘‘‰›ˆ †x¤pˆ?’ ”š˜Š Šž†+Ÿ‰Šž–– Šž‰Ÿ–—†+’ “‘ŠŸ¥’¼›5” ‘žŸ‰¥ £ x˜–]†‘ Ÿ‰ £ – ’ ŠžŸ‰–,“‘ †x©‰Ÿ’¼”‘Ÿ‰–—ž–—\Š]¡. Grants(i, j) ≡ f romi = j ∧ diri = grant. -. i. Coherent(i) ≡ T ree(i) ∧ ∀k ∈ Child(i) : distk = disti + 1 ∧ ∀k ∈ N eighi : rootk = rooti. ½l†+Š<Š<’¼†e”B‰–]†‘ ”Œ£ ¤– Šž†x”Œ Ÿ‰£ +¬»ŠžŸ‰Šž’¼”BŸ†+‰ŠD £’ ˜

(32) –†,’¼””Œ’Šž‰Ÿ‰“‘–ˆ –Ýž¥Šžx †xŠBx”Œ˜5’ –—Šž\Ÿ‰Š#–ØYŠž†xž‡‰–—–‘ˆ ŠD¡+|ŠžŸ‰—–—(‡‰ † ” ”š”ŒŠ –†xŠŒŠž›‰’ ˆ‰–Ó'”šŠ ’ Š †+”#Šž–é†x’¼”Bž–—ž\–]ŠB†‘ †xŸ‰ – £ £ ž†x¥“\x†xŠ ’ ¯ †x›¥ž¤’¼†x†'›‰ˆ ”Œ–]’ ”‰“‘¯ †xˆ –È ‡‰ –]” ¯Šž'–x’ Š ˜5”Œ‰–—ˆ ˜š£¡‘–’ Š ” £¯ †x’¼”šžŠ ’¼†x†x›‰ˆ –]– ”—§¯ †xž?’¼†xŠž›‰Ÿ‰ˆ–—– ž¬¯ †x’¼”Œˆ –‘‡‰¡Ã–,‰Šž £ – †x £Šžž’ž–]–]”(”Œ–ŠžŠŒ Šž’ ‰£ “k–Šž’ –—Š ž”? ž–]’ ‰­\–n‡‰–]Šž”šŸ‰Š –” ž–]†‘”Œ‘”<x˜ÉŠžŸ‰–'ˆ |—†xˆZ’ ¬‘’Ÿ‰Šž–—Ÿ ž–—Šž–¢Ÿ‰–(›¥¤k‡‰ žŸ‰ž–]–— \À¥Š ’ ‰¯ “†xˆ ©‰‡‰ž–v– £x˜D’¼—Šž†+Ÿ‰Šž–– ©‰£ ’¼†x”šž£ Š –—†x– \Š(–’¼¬”¢¯’ †(Šž†xŸbžˆ –]’¼†xŠž†+Ÿ‰›‰˜B–ˆ Šž–Ÿ‰¯ x–—†x˜nˆ ‡‰ (¡– †x¥†¤‰–—‘’ ‡‰“‘’ ©‰Ÿ¥›5ˆ –]‘” ex˜ §?†‘†x”exžŠž–,¬–n–—©5ˆŠž\BŸ”ž†‘†+”Œ’”eŠ]›‰¡’ˆ Š –‘’ ”˜§ Šž?Ÿ‰ Šž–,Ÿ‰Ÿ‰–]–—¥ ž‡‰¬À| ”e’¼”Œ›5’–‘˜–—¡/†x‘ˆx‰#˜Bˆ ’ ¤oŠ ‘”,‘‡‰‰‰©‰–—– ˆ ’ –]“‘¯” Ÿ¥†x›5ˆ ‡‰‘–k ”¢xŸ˜ ’¼”† ¯ ž––]’¼”,”šŠžŠžŸ‰©5 –k†x\’ ”ž‰”ž”Œ†x–’ ›‰ £ ˆ §#––v †xž’ ¥ x”š£ Š]Š ¡‰Ÿ‰¯ ‰–—†xˆ ‡‰£ ––– Šž©5»¡Ã\”Œ¬Šž’ ŠžŸ‰Ÿ’ –—¯ †+’ –Š(n£ ’¼’ ”’¼£ ”š–—Š “‘\†xžŠž–]’ †+Šž–’Šž–]–—”—¯ v¡e†xŠžˆ †H‡‰Ÿ–‘†x©§# †x½lž˜/–—ž\†x¥Š  x£ŠD£ ’‰“‘Z ž–—£–]ž†+––—Šž\–—Š’¼”˜ªŠžžŸ†‘†x  ‰²–—’ ’ ‘“‘Šp(Ÿ¥†x›5–]‘­\£D‡x†x† ˜ˆ Á/£ ‡ –'£ –]Šž”žŸ‰–'ž’ ˜ª›5‘–nˆ +Šž¬Ÿ‰–o’ ‰“vž–]©‰­\ž‡‰– –]£ ”š’¼Š —I†+žŠž–—–]©‰”'ˆ ¤»®  –] Ÿ†x‰’¼”Œ ¡¬–p’ \Šžžx‹ Šž¡|Šž‘Ÿ‰–—†xˆ ‡ ’£ Še£ –] †o”'Ÿ‰–]©Šž Ÿ†xÀ|†+ž”Ö—Š\¬’ Š Ÿ‰–‘ŠžŸ‰Ÿ‰–]–—–—­\ž –—‡†xˆŸ¤oŠž†‘ ” († †I¤p§H£ ’¼›5½Ž”š»Š – †xŠž£ Ÿ‰‡‰’¼– –(”(¯ Šž—†xn†‘ˆ ”Œ‡‰–—–‘–,’ ¡<Šž–]Ÿ‰‰­\–— ‡£ †x†– ˆ ˜Y’ »†xˆ¼”Œ’ –Š ” ¯ £†x’¼ˆ ”š‡‰Š –†x’ »– ’ Š ¯ ”v†xžž’¼¥†xx›‰Š ˆ –‘¯ ¡†xŠžž+’¼†x¬Ã›‰†xˆ –k£ ‘”,v’ Š †x”»©’†xž–—‘\žŠvž–]‘`Š]¡ ‘¯‰†x–kˆ ‡‰x– ˜ ’©‰Š ž”–  £ Ÿ‰’¼—’ ˆ†+£ Šžž–¢–—É˜ª‘§<ˆ ¦&+¬?˜ª”—‘§ ž (†xˆÉ†x ££ –Š †x’ ˆ – ££ –]”žž’ ©|Šž’ ‘x˜ŠžŸ‰’¼” N eighi ∪ {i} ∪ {⊥} 4. 3 diri. ask. reqi. i. grant. i. {ask, grant, ⊥}. Child(i) ≡ {j ∈ N eighi : parj = i}. T ree(i) ≡ (rooti = i ∧ disti = 0 ∧ pari = i) ∨ (pari 6= i ∧ rooti = rootpari > i ∧ disti = distpari + 1). Ownrmax(i) ≡ if (∀k ∈ N eighi : rooti ≥ rootk ) then i else j = {M ink∈N eighi : rootj ≥ rootk }. 0. i. Onelocalf ault(i, d, j) d i j d−1 i. i distj (d, j). d. (d, j). i r. i. i. r. Sat(i) ≡ T ree(i) ∧ Ownrmax(i) = i. .. r. 0. id. r. i. Idle(i) ≡ reqi = ⊥ ∧ f romi = ⊥ ∧ toi = ⊥ ∧ diri = ⊥ Asks(i, j) ≡ j ∈ N eighi ∧ reqi = i ∧ f romi = i ∧ toi = j ∧ diri = ask ∧ Sat(j) ∧ rooti = i < rootj ∧ pari = j ∧ ∀k ∈ N eighi : (rootk = rootj ⇒ distk = distj + 2). F orwN a(i, j) ≡ j ∈ N eighi ∪ {i} ∧ toi = j ∧ pari = j ∧ diri = ask ∧ Sat(i) ∧ f romi 6= i ∧ reqi 6= i ∧ reqi 6= ⊥ ∧ reqi = reqf romi ∧ ∀k ∈ Child(i) : (rootk = rooti ⇒ distk = disti + 1). Incoherent(i) ≡ ¬Coherent(i). ‘ù|Û §§. ÅÅ+I‘‘‰Ÿ‰–e–—˜Yž†x–—‡‰\ˆŠÃŠ¢‰†+–—Š ’ “‘Ÿ¥x›5È ‘žŸ‰¥ £ xÈ. Onelocalf ault(i, d, j) ≡ j ∈ N eighi ∧ ∃r > i : Incoherent(i) ∀k ∈ N eighi :. i. • rootk = r ∧ k ≤ r ∧ (k = r ⇒ distr = 0 ∧ parr = r).

(33) ׉§ ›5‘ xÈ. ï|§,˜YņxI‡‰–—ˆˆ ’Š¢ †+’ Š?†+†Šž ’ ‘Ÿ‰H’ ˆ £ xx˜?È ˜Y†xˆ¼”Œ– £ –Šž–]`Šž’ ‘ £ ‡‰–Šžt†o‡‰‰’¼­\‡‰–. Å+—†‘”Œ–(x˜†‡‰‰’¼­\‡‰–k‰–—’ “‘Ÿ|‹. • k 6= r ⇒ park 6= k ∧ distk > 0. ∃k : N eighi = {k} ⇒. (rootpari = r ∧ distpari = distparpari + 1 ∧ Idle(pari )) ⇒. • ∀k 0 ∈ N eighk \ {i} : rootk0 = r ∧ k 0 ≤ r ∧ (k 0 = r ⇒ distr = 0 ∧ parr = r) • (k 6= r ∧ park 6= i) ⇒ distk = distpark + 1 • ∀k 0 6= i ∈ Child(k) : distk0 = distk + 1. Ú§ ÅI– ¯ –—\Šž‡†xˆÉ‘žž–]`Šž’ ‘k¬’ ŠžŸŠžŸ‰–,Ÿ‰–—ˆ ©x˜ xÈ ï|§,˜YÅ †xI‡‰–—ˆˆ ’Š¢ †+’ Š?†+†(Šž ’ ‘Ÿ‰H’ ˆ £ xx˜?È ˜Y†xˆ¼”Œ– £ –Šž–]`Šž’ ‘ £ ‡‰–Šžt†o‡‰‰’¼­\‡‰–. • |Child(pari )| > 1 ∨ • Child(pari ) = {i} ∧ ∃k 0 6= pari ∈ Child(i) ∧ (∀k 0 6= pari ∈ Child(i) : distk0 = disti + 1 ∨ parpari = i). ð‰§,ÅI–—ˆ ’  ’ †+Šž’ ‘x˜#ž\”ž”©†xž–—\Š xÈ. ∀k ∈ Child(i) : distk = d + 1 ∧ distj = d − 1 j. rooti = r ∧ disti = distpari + 1 ∧ Idle(i) ⇒. • |Child(i)| > 1 ∨ • ∃k : Child(i) = {k} ∧ ∃k 0 6= i ∈ Child(k) ∧ (∀k 0 6= i ∈ Child(k) : distk0 = distk + 1 ∨ pari = k). ð‰§,˜YņxI‡‰–—ˆˆ ’Š¢ †+’ ŠÃ†+ŠžŸ‰Šž’–,‘H©†xxž˜?–—\˜YŠ†xxˆ¼”ŒÈ – £ –Šž–]`Šž’ ‘ £ ‡‰–Šžt†o‡‰‰’¼­\‡‰–. (rooti = r ∧ ∀k ∈ Child(i) : distk = disti + 1 ∧ Idle(i) ∧ parpari 6= i) ⇒ (distpari = distparpari +1∧∀k 0 6= i ∈ Child(pari ) : distk0 = distpari + 1). ¥ñ §,ÅI–—ˆ ’  ’ †+Šž’ ‘x˜Bž\”ž”©†xž–—\Š xÈ ½Ž†(”Œ‘’ ‰ “‘£ ˆ–—–'#ŠžŠž ,†x†‘ ”ŒŸ‰’ –—’ \– ¯Š–˜Y†x˜Y†x‡‰‡‰ˆ Š¢ˆ ŠD|—‘\‡‰Š ž†xž–’ ‰£  †+–—Še\†Š]¡‘‰’ ( £ †'– ”šŠ ¡‰†+Šž¬––,¬ Ÿ‰‡–—ž”š–Š ’‰‰ Ÿ‰£ ’ –'›‰’ xŠ¢˜/–]Šž†‘Ÿ‰ –Ÿn”Œ ¤|Ÿ‰”š’ Šžˆ –—£  úx˜ †x £ Šž(Ÿ‰©‰–—ž–]”Œ–e‡‰Šž –'ˆ –]Šž† Ÿ¯ †+–¢Š¢Šž’ Ÿ‰Š¢–'’¼”ŠžžŠž–—Ÿ‰–'–,›¥˜Y†x¤(‡‰›5ˆ Šš–¤ ‹ ‘ ’ ‰“¢†ž¥xŠ]§/Á߉’¼”/†xˆ +¬?”

(34) ‡‰‡ ”É£ Šž–—,£ ¬–Ÿ‰«’¼‰ –Ÿo†–]†‘‰ –—Ÿo¬t ©‰Ÿ‰ž’ – ˆ ££ ’¼—x†+˜Šž†– ‰‰ ££ ––, (x˜/†IŠž¤vŸ‰– Ÿ‰”Œ–]¤| ”šÀŠž–—’ ˜

(35)  ’ Š § ”B©†xž–—\Š<–]”šŠž’  (†+Šž–]”Šž,›5–ŠžŸ‰–˜Y†x‡‰ˆ Šš¤ Ûù|‘§§ Å I‘‰–e˜Y†x‡‰ˆ Š¢†+Š xÈ Å+‘Ÿ‰–—ž–—\ŠÃ‰–—’ “‘Ÿ¥›5‘žŸ‰¥ £ xÈ. (rooti = r ∧ Idle(i) ∧ parpari = i ∧ pari 6= i) ⇒ ∀ k ∈ Child(pari ) \ {i} : distk = distpari + 1 i. i. Onelocalparf ault(i, d, j). Onelocalparf ault(i, d, j) ≡ pari 6= i ∧ j ∈ N eighpari ∧ ∃r > pari : Incoherent(pari ). pari. ∀k ∈ V oisinspari :. • rootk = r ∧ k ≤ r ∧ (k = r ⇒ distr = 0 ∧ parr = r) • k 6= r ⇒ park 6= k ∧ distk > 0. ׉§ ›5‘ xÈ. V oisinspari = {i} ⇒. Å+—†‘”Œ–x˜'†"‡‰‰’¼­\‡‰–‰–—’ “‘Ÿ|‹. • ∀k 0 ∈ N eighi \{pari } : rootk0 = r ∧ k 0 ≤ r ∧ (k 0 = r ⇒ distr = 0 ∧ parr = r) • ∀k 0 6= pari ∈ Child(i) : distk0 = disti + 1. Ú§ ÅI– ¯ –—\Šž‡†xˆÉ‘žž–]`Šž’ ‘k¬’ ŠžŸŠžŸ‰–,Ÿ‰–—ˆ ©x˜ x È. ∀k ∈ Child(pari ) : distk = d + 1 ∧ distj = d − 1 j. (rootpari = r ∧ Idle(pari ) ∧ parpari = i ∧ pari 6= i) ⇒ ∀k ∈ Child(i) \ {pari } : distk = disti + 1. ?–Ç¥Š]¡ ¬– ’ \Šžž £ ‡– † ¬Ÿ‰©‰–—ž–ž–°£ ’¼—Šž†+Ÿ‰Šž–-– ©†x†xž†x–—ˆ\‘Šz“‘‘x‡˜¿” ŠžŸ‰Šž– ‰ £ – ’¼”ÊžŸ‰–'˜Y†x‡‰ˆ Šš¤k‰ £ –†x £ ŠžŸ‰–,ž¥xŠx˜/ŠžŸ‰–'Šžž–—– . Onelocalparf ault(i, d, j) i. Onerootparf ault(i) ≡ pari 6= i ∧ ∀k ∈ N eighpari : (k < pari ) ∧ (rootk = pari ) ∧ ∀k ∈ N eighi \ {pari } : k < pari ∧ rootk = pari ∧. žŠÁ߉Ÿ‰–e–Šž˜ªž‘–—ˆ –,+’ ¬n’ ‰—†‘“e”Œ©‰–ežx– ˜B£ ’¼†—†+˜Y†xŠž–ljˆ©‰Ššž¤k– ¯©–—†x\ž–—Š ”#\Š†e‰ £ – ˜ªž‘ Âˆ –]† ¯ ’ ‰“ ”šŠ ’ †+“‘¦Šž‡‰–—ž –˜ª‘–—׉ž\ (§?Š ”v†x¦Âˆ ˜ª‘ £ ˆ –]‘+”žž¬?– ž”—’ £§¿©|–ŠžŠ ’¦‘†x²˜™’ Šžˆ – –—x£pk˜e£ŠžŠžŸ‰–]Ÿ‰–n”ž–pž|†x’ —©|ˆ “‘Šž‡‰‘’ ‘žžž’ Šž–—Ÿ‰x U˜B–ŠžŸ‰’¼x”(–˜,“‘“‘†"’‡¯ †x˜Y–—†x²£‡‰–ˆ ’ Š]£ ¡ ’‰˜' †o£ –'‰’ ‘|‹lŠžžŸ‰¥–'x‰Šv–‰Šš¬ £‘–žÀZ¡¥’–]Š?”šŠž–’ Ç| (–]†+‡|Šž–]Šž”–]”›5Šž–—Ÿ‰’ ‰–“o”šŠ Šž†+Ÿ‰Šž––— ‘–—‰\ˆ ¤tŠ ˜Y†x‡‰ˆ Šš¤ ¡ ”ŠžšžŠž–—–—›5–'©–—”—’”š‰§¢Šž“"ž ‡’’` Š Šž”š”v‡‰Š]ž¡5©–‘’†x˜<¡|ž’–—‰Š¢\ †+Š]£ ŠŒ§ – Šž–—$‘ ‘–]’©|”š‰ŠžŠ ’ ’”‰ (“Šž†+Šž–]–]& ””š”¢Š Šž†xŸž›‰† –—ˆ¯ –’¼’”Œ‰Ÿ ’¼”“k£†xp‘‰’ –’ ‘»Ÿ‰›¥Šž¤k–—Ÿ‰ž›5–—ž\–—––Š‹ ’†‘Š ‘`”<Šž ’©‘’ †x‰•ž“k–—\†‘ŠDž’¼”Œ¥”<’¼x”šŠžŠ ŠŸ‰”–¢ÅY†‘x‘`˜e‰Šž›5ˆ’ ¤‘–]  ˜Y†x‘' ‡‰ ˆ Šš’ÈЉ¤v’“*˜B‰’†"Š £ £ž–¢¥¥’ x(–]Š(”ŒŠž †xŸ‰& Š?–¢£ –]‰”š–ŠžžŠš’–]¬ (”Œ–‘†+ŠŒžŠžŠžÀZ–]’ ‰§”“tÁʞŸŸ‰ž†+–’¼Š”‹ ­\”šŠž‡‰ž–]‡”š`Š Šž‡‰¯ ž†x–?ž’¼†x†x ›‰£ ˆ –]†”—§ÆÁ߉–—¯ ¿†xˆ ‡‰‰–¢ ”Œ£  (– †xˆ ¡–—¬<’ŠžŠžŸ‰ŸÙ–—k†»†x¥‘¤ Ÿ‰–—ž–—\Š(¯ †xŠžžˆ ‡‰–—–– Šžx†xŸ‰˜< –£’ Š  ”e†‘Ÿ‰”Œ‰’ ˆÀ¥–—£ ’’ ‰“‘x“tŸ¥˜#›5’’ ‘Š Šk” ”—˜ª‰‘¡Z–—†+’ ŠŒ“‘†ŠžŸ¥–—Õš›5 ‘‘’ ©|²Š ”¢©5¬Šž–—ž’ Šž ՚Ÿ‘’¼”ž’Šž"”ŒŸ‰’ ‘–,†(ÆŸ‰Šž’žÅY“‘–—†‘Ÿ‰–`–]Šž›¥”š’ ‘Š¤ ž›5¥)–]xŠ ‘È`¯ §O†x’ˆ ‰‡‰½Ž“– Šž‘’ Š Ÿ ”,£†+Š–— Ÿ‰?’ Šžˆ £’¼”?ž––—ŠžÇ|oŸ‰–]–¬‡|‘’ Šž‰ŠžŸo–ˆ ¤ŠžŠžŸ‰˜YŸ‰†x’¼–k”'‡‰ˆ”ž†‘Šš†x¤` Šž’‰‘–  É£ ¡5–‰’ n£ ¯ –Šž†xŸ‰ˆ ‡‰––( ‰‡†‘–Šš””š¬Še‘‰ž xÀZŠ,‡¡5”š†‘†xŠ,”ž ”ŒŸ£ ‡‰†  †x¯ ˆ –– ¬†vŸ‰‘–—žž–]`“‘Š  £ †x’¼\”šŠ Š ”D†xŠžŸ‰– –,¯ ©5†x–—ˆ ž‡‰ –‘§B’¼”ž ”Œ’ ’ ‘t†xˆ ÅY¤‘†‘¡|`Šž‰’ ‘(£ – * È`՚§‘’ ”ŠžŸ‰–eŠžž–—– ©‰‡|ˆ –—ŠžÁà– £Ÿ‰–—–\›¥Špž¤"–—Šž (†nŸ‰–H†x‰’ ‰ž£’–]‰­\– “‡‰–]†‘”š¬`Š ŠžI’ ’Šž‘žŸ –—©‰”ˆ ¤Ù-Å ,  ¡ –]. ŠžŸž†x‡‰‰–‘’¼§”Œ¡   0¦ß†x‰†x   ££ £ – †x1 ž–»ÈÈ ––]’ Ç|†x v –”‹‹ ∀k ∈ Child(pari ) : distk = 1. i pari 6= i. Oneparf ault(i) ≡ Onerootparf ault(i) ∨ ∃(d, j) : Onelocalparf ault(i, d, j). i. i (j). . j. j. i. T ree(i). i. i. root. root. j. i. i. j. root. j. i. j. i. .. i. i. Sat(i). i (j). i. i. i.

(36) $ &  #$   "    

(37)  "%    i(j) 

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(39)   . . ∀(k, d) : ¬Onelocalf ault(i, d, k) ∧ ¬T ree(i)∧ ∀k : ¬Asks(i, k) ∧ ¬Oneparf ault(i) rooti = pari = i; disti = 0; reqi = f romi = toi = diri = ⊥;. "         i 

(40)    '. .      i 

(41) ∃j : Asks(i, j) ∧ Grants(j, i) ∧ distj + 1 ∈ INf    rooti = rootj ; pari = j; disti = distj + 1; reqi = f romi = toi = diri = ⊥;.   . . ∀(k, d) : ¬Onelocalf ault(i, d, j) ∧ Sat(i)∧ ¬Rgcorrect(i) reqi = f romi = toi = diri = ⊥;. &%. '. (. -. .-0/. =<. .-. *       #$    i 

(42). Sat(i) ∧ i = root i ∧ F orwN a(i, i)    diri = grant;. ’ “‘‡‰ž–'׉®<Á߉–e˜Y†x‡‰ˆ ŠŒ‹Ž‘\Š †x’ ‰– £ †xˆ “‘‘ž’ ŠžŸ‰ . :9;1. 8/O1. K/. ML. -. P+. 21 ?/. .-0/ QSRT=<U6. WV. X<. ?<UZ. Y/. 21. 8V. ). diri = grant;. 8/. ?/. ?DFE GIHJC. >A. [6.          i 

(43) Sat(i) ∧ ∃j : (F orwN a(i, j) ∧ Grants(j, i))   . 76. >6 8/. @) BA=C. reqi = reqj ; f romi = j; toi = pari ; diri = ask;. *) ,+. 21 43 215) /. N 21,). )     #$    i (j) 

(44)  Sat(i) ∧ Idle(i) ∧ (Asks(j, i) ∨ F orwN a(j, i))   . +. #"$". i. T ree(i) ∧ Ownrmax(i) 6= i ∧ Sat(Ownrmax(i))∧ ∀k ∈ Child(i) : (rootk = rootOwnrmax(i) ⇒ distk = distOwnrmax(i) + 2) reqi = i; f romi = i; toi = Ownrmax(i); diri = ask; rooti = i; pari = Ownrmax(i). (     #$  !  $ . i 

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