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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Itai-Rodeh-Algorithmus</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Itai-Rodeh-Algorithmus</b> ist ein <a href="Algorithmus" title="Algorithmus">Algorithmus</a> der <a href="Las-Vegas-Algorithmus" title="Las-Vegas-Algorithmus">Las-Vegas Klasse</a> zur Auswahl anonymer <a href="Unidirektional" title="Unidirektional">unidirektionale</a> <a href="Topologie_(Rechnernetz)" title="Topologie (Rechnernetz)">Ringe</a> und baut auf dem <a href="Nachrichtenausl%C3%B6schung_nach_Chang_und_Roberts" title="Nachrichtenauslöschung nach Chang und Roberts">Chang- und Roberts-Algorithmus</a> auf.
</p>
<div class="mw-heading mw-heading2"><h2 id="Voraussetzungen">Voraussetzungen</h2></div>
<ul><li>unidirektionaler Ring</li>
<li>Ringgröße (Anzahl der <a href="Netzwerkknoten" title="Netzwerkknoten">Knoten</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> bekannt</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Ablauf">Ablauf</h2></div>
<p>Der Algorithmus läuft in Phasen (Wahlgängen) ab.
</p>
<div class="mw-heading mw-heading3"><h3 id="Erste_Phase">Erste Phase</h3></div>
<p>In der ersten Phase wählen alle Knoten eine zufällige <a href="Identifikationsnummer" class="mw-redirect" title="Identifikationsnummer">Identifikationsnummer</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ID} >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">I</mi>
<mi mathvariant="normal">D</mi>
</mrow>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {ID} >0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5db7a718e07dbc5eac2c734ea761a9a207d8e7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.876ex; height:2.176ex;" alt="{\displaystyle \mathrm {ID} >0}" loading="lazy"></span>. Dann schickt jeder Knoten eine <a href="Nachricht" title="Nachricht">Nachricht</a> bestehend aus eigener ID <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, Sprungzähler <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> (<i>hopcount</i>, gibt an, wie oft die Nachricht weitergeleitet wurde), einem Merker <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> (<i>flag</i>) und der aktuellen Phase <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>. Initial gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=1,f=1,p=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mi>f</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mi>p</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h=1,f=1,p=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d1b8d030da163004b4383263397f8854f7a2daf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.638ex; height:2.509ex;" alt="{\displaystyle h=1,f=1,p=1}" loading="lazy"></span>.
</p>
<ul><li>wenn eine Nachricht <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle i,h,f,p\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>i</mi>
<mo>,</mo>
<mi>h</mi>
<mo>,</mo>
<mi>f</mi>
<mo>,</mo>
<mi>p</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle i,h,f,p\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da0d4bb1807f88baa5df70d98e6b87bb192ae8f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.501ex; height:2.843ex;" alt="{\displaystyle \langle i,h,f,p\rangle }" loading="lazy"></span> empfangen wird:
<ul><li>falls <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> kleiner ist als die aktuelle Phase beim Empfänger, wird die Nachricht nicht weitergeleitet („verschluckt“ nach <a href="Nachrichtenausl%C3%B6schung_nach_Chang_und_Roberts" title="Nachrichtenauslöschung nach Chang und Roberts">Chang und Roberts</a>)</li>
<li>falls <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i>\mathrm {ID} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">I</mi>
<mi mathvariant="normal">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i>\mathrm {ID} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86fa9e937f3d89e42fa3c51b22da543d64056018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.516ex; height:2.176ex;" alt="{\displaystyle i>\mathrm {ID} }" loading="lazy"></span> wird die Nachricht weitergeleitet als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle i,h+1,f,p\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>i</mi>
<mo>,</mo>
<mi>h</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>f</mi>
<mo>,</mo>
<mi>p</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle i,h+1,f,p\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22d9918f5b08a5dd35d2de4d66fc46d989b01d6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.503ex; height:2.843ex;" alt="{\displaystyle \langle i,h+1,f,p\rangle }" loading="lazy"></span></li>
<li>falls <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<\mathrm {ID} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo><</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">I</mi>
<mi mathvariant="normal">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i<\mathrm {ID} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/174a83c2995a803e8e2cca6e1e5f9276ebc969cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.516ex; height:2.176ex;" alt="{\displaystyle i<\mathrm {ID} }" loading="lazy"></span> wird die Nachricht nicht weitergeleitet</li>
<li>falls <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=\mathrm {ID} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">I</mi>
<mi mathvariant="normal">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i=\mathrm {ID} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44845d7c9dc7a217e0330e16114545f7ef05529a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.516ex; height:2.176ex;" alt="{\displaystyle i=\mathrm {ID} }" loading="lazy"></span>
<ul><li>wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\neq n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>≠<!-- ≠ --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h\neq n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b297d8241101bba3d0f92e75c36629cd91562b2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.676ex;" alt="{\displaystyle h\neq n}" loading="lazy"></span> wird <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> gesetzt (der Merker merkt sich, dass die ID mehrfach vergeben ist) und die Nachricht als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle i,h+1,0,p\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>i</mi>
<mo>,</mo>
<mi>h</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mi>p</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle i,h+1,0,p\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb8de8490ebd531aa48442492ad18d45864c22d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.387ex; height:2.843ex;" alt="{\displaystyle \langle i,h+1,0,p\rangle }" loading="lazy"></span> weitergeleitet</li>
<li>wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>=</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h=n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca144ffa4c586c3cfac786662f905fd70fe9e103.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.832ex; height:2.176ex;" alt="{\displaystyle h=n}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60eab5895ca5fe74555e9d59e499c94e1d2ef3f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=1}" loading="lazy"></span> hat der Knoten die Auswahl gewonnen (Mitteilung an alle anderen durch eine spezielle Nachricht)</li>
<li>wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>=</mo>
<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle h=n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca144ffa4c586c3cfac786662f905fd70fe9e103.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.832ex; height:2.176ex;" alt="{\displaystyle h=n}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mn>0</mn>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle f=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}" loading="lazy"></span> gibt es mehrere Gewinner.</li></ul></li></ul></li></ul>
<div class="mw-heading mw-heading3"><h3 id="Weitere_Phasen">Weitere Phasen</h3></div>
<p>Falls es mehrere Gewinner der ersten bzw. vorherigen Phase gibt, dann startet diese Gruppe einen weiteren Durchlauf des Algorithmus mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=p+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi>p</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p=p+1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/298bfc906bb04d03b41ea3004b5d8d9863015699.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.53ex; height:2.509ex;" alt="{\displaystyle p=p+1}" loading="lazy"></span>. Der Ablauf ist genau wie in der ersten Phase, jedoch mit verringerter Anzahl der Teilnehmer. Passive Knoten leiten Nachrichten lediglich weiter; lediglich der Sprungzähler <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> wird dabei erhöht.
</p>
<div class="mw-heading mw-heading2"><h2 id="Nachrichtenkomplexität"><span id="Nachrichtenkomplexit.C3.A4t"></span>Nachrichtenkomplexität</h2></div>
<p>Für die erste Phase werden <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> Nachrichten benötigt. Da die Anzahl der Phasen theoretisch unbegrenzt ist, geht die Nachrichtenkomplexität gegen unendlich. Praktisch ist dieser Fall aber sehr unwahrscheinlich. So kommen für jede weitere Phase weniger als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> Nachrichten hinzu.
</p><p>Der <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> für die Anzahl der Wahlgänge (wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall ID:ID\in [1,...,n]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>I</mi>
<mi>D</mi>
<mo>:</mo>
<mi>I</mi>
<mi>D</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<mn>1</mn>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<mi>n</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall ID:ID\in [1,...,n]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbbe7c1447994606e691626854653bb56b9b59f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.283ex; height:2.843ex;" alt="{\displaystyle \forall ID:ID\in [1,...,n]}" loading="lazy"></span>): <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\leq e\left({\frac {n}{n-1}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo>≤<!-- ≤ --></mo>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>n</mi>
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
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</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E\leq e\left({\frac {n}{n-1}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54c84b20826d80027b639e48db92d28f592dc8c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16ex; height:6.176ex;" alt="{\displaystyle E\leq e\left({\frac {n}{n-1}}\right)}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>e</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> ist die <a href="Eulersche_Zahl" title="Eulersche Zahl">Eulersche Zahl</a>)
</p>
<div class="mw-heading mw-heading2"><h2 id="Quellen">Quellen</h2></div>
<ul><li>Vorlesung <i>Verteilte Systeme</i> an der <a href="TU-Berlin" class="mw-redirect" title="TU-Berlin">TU-Berlin</a></li>
<li>A. Itai and M. Rodeh. Symmetry breaking in distributed networks, In <i>Proceedings of the 22nd IEEE Symposium on Science</i>, pages 150-158. IEEE Press, 1981.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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