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Dimostrando che la colorazione è corretta, tutte le sue istanze, di conseguenza, lo saranno. -> [!example]- Esempio Colorazione +> [!example]- Esempio Colorazione (hi-DPI, espandi) > ![[Minimum Spanning Tree-20240123013937896.png]] Durante l'esecuzione del processo di colorazione vale la: @@ -105,16 +105,42 @@ Caso: il passo $k+1$ colora di $\textcolor{royalblue}{blu}$ l'arco $e=(u,v)$ che - Se $e \in T, T$ verifica l'invariante del colore anche al passo $k+1$. - Se $e \notin T$, si consideri il cammino in $T$ da $u$ a $v$. - - Sia $(u,v_{1})$ un arco su tale cammino che attraversa il taglio $(V_{1},V_{2})$. Esso è **non colorato**. + - Sia $e'=(u_{1},v_{1})$ un arco su tale cammino che attraversa il taglio $(V_{1},V_{2})$. Esso è **non colorato**. - Si consideri ora $T'$ tale che: - $\mathcal{E}[T']=(\mathcal{E}[T]\backslash\{(u_{1},v_{1})\})\cup\{(u,v)\}$ -- Poichè $w(u,v)\leq w(u_{1},v_{1})$ si ha $w(T')\leq w(T)\leq w \implies \textbf{w(T')\ =\ w(T)}$ +- Poichè $w(u,v)\leq w(u_{1},v_{1})$ si ha $w(T')\leq w(T)\leq w(T') \implies \textbf{w(T')\ =\ w(T)}$   Cioè $T'$ è un MST contenente **tutti** gli archi blu al passo $k+1$.  +In altre parole:  + +L'arco $e$, appena colorato di blu al passo $k+1$, nasce da un taglio. Supponiamo di essere nel caso in cui $e$ non appartiene all'MST $T$.  + +Considerando un arco $e'=(u_{1},v_{1})$ sul cammino in $T$ da $u$ a $v$. Questo arco: +- Attraversa il taglio; +- Non può essere rosso perché non avevamo archi rossi al passo precedente; +- Non può essere blu perché abbiamo appena colorato di blu l'arco $e$. +Quindi è non colorato.  + + +A questo punto vale la relazione $w(e)\leq w(e')$ perché T è un MST.  + +Ma dato che T è un MST,vllora vale anche $w(T') \leq w(T)$.  + +Questo ci fa concludere che $T$ e $T'$ sono uguali, quindi anche $T'$ è un MST. + +Due domande che potremmo porci a questo punto sono: +- "ma $T'$ è ancora un albero?" $\implies$ Sì, perché la cardinalità degli archi è uguale +- "Ma dopo che esce $e'$ l'albero è ancora connesso?" $\implies$ Sì, perché un albero è un grafo connesso minimale. + + +> [!example]- Visualizzazione dimostrazione +> ![[Minimum Spanning Tree-20240123124251171.png]] + + Passo k+1 $\textcolor{red}{rosso}$  Caso: il passo k+1 colora di $\textcolor{red}{rosso}$ l'arco $e=(u_{1},u_{2})$ relativo al ciclo semplice $(u_{1},u_{2},\dots,u_{r},u_{r+1}=u_{1})$  -- Se WORKINPROGRESS \ No newline at end of file +- Se \ No newline at end of file