MST.jl

### A Pluto.jl notebook ###
# v0.19.9

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 7532e879-36ae-4a3c-bdac-3b3524f802d6
using Graphs, Plots, PlutoUI, GraphPlot, Colors, DataFrames, Compose, DataStructures, SimpleWeightedGraphs

# ╔═╡ 2278b12a-62b0-45cc-abe2-50312fcfcea7
using Printf, TypedTables

# ╔═╡ c639f755-3d8d-4a25-a6f5-057016e79656
begin
	using ImageShow, FileIO, ImageIO
	
	spacing_image = download("https://image-hosting.zhangjc.tech/ghost/content/images/2019/06/1870653438.png")
end

# ╔═╡ 8eccfa40-1592-11ed-2c77-8f1d323c1330
md"""
# Minimum Spanning Tree (MST)

**Problema** : determinare come interconnettere diversi elementi fra loro minimizzando certi vincoli sulle connessioni.

Questo problema prende il nome di albero ricoprente di peso minimo (**MST**), o minimo albero ricoprente.

Più formalmente, dato un grafo non diretto, connesso e pesato $G(V,E,w)$, dove $V$ è l'insieme dei nodi, $E ⊆ V×V$ è l'insieme di archi e $w$ è la funzione peso degli archi $w:E \to \mathbb{R}^+$, si vuole trovare un *albero* $T ⊆ E$, che è uno *spanning tree* di $G$, di **costo minimo**.

Dove il *costo di un albero $T$* è dato dalla somma dei pesi dei suoi archi :

$C(T) = \sum_{e \in T} \ w(e)$

"""

# ╔═╡ 0f119f11-f4eb-4363-89be-21c66233ed96
md"""
#### Applicazioni
I minimi alberi ricoprenti hanno applicazioni dirette nella progettazione di reti, tra cui reti di computer, reti di telecomunicazioni, reti di trasporto, reti di approvvigionamento idrico, reti elettriche...

Un'altra importante applicazione è il **k-Clustering**, di cui vedremo un esempio più avanti.

"""

# ╔═╡ d2538f31-63dd-4c47-bdf9-c2a4eca85d6b
md"""
# Approccio generale : Greedy
Un algoritmo *greedy* è un paradigma algoritmico, dove l'algoritmo cerca una soluzione ammissibile da un punto di vista globale attraverso la scelta della soluzione più appetibile ( "greedy", definita in precedenza dal programmatore) ad ogni passo locale.
Di seguito introduciamo due algoritmi greedy per il calcolo del *minimum spanning tree*: 

- **Algoritmo di Prim** : Parte da un qualunque nodo $s$ e fa una visita dove ogni volta si scelgono gli archi di peso minimo locali e uscenti dal nodo che si sta visitando.
	
- **Algoritmo di Kruskal** : L'algoritmo di Kruskal si basa sulla seguente semplice idea: ordina gli archi in ordine non decrescente di costo e successivamente li analizzia singolarmente, inserisce l'arco nella soluzione se non forma cicli con gli archi precedentemente selezionati.
"""

# ╔═╡ 26f298b1-3850-4406-b969-ef9a2826f43f
md"""
 ##### Imposta il tipo di grafo su cui lavoreremo : 
$@bind tipo_grafo Select([
:bull,
:chvatal,	
:cubical,	
:desargues,	
:diamond,	
:dodecahedral,	
:frucht,	
:heawood,	
:house,	
:housex,	
:icosahedral,	
:karate,	
:krackhardtkite,	
:moebiuskantor,	
:octahedral,
:pappus,	
:petersen,	
:sedgewickmaze,	
:tetrahedral,	
:truncatedcube,	
:truncatedtetrahedron,	
:truncatedtetrahedron_dir,	
:tutte])
"""

# ╔═╡ 1924f2fc-2c2a-449b-a035-b0346b77a244
md"""
### Algoritmo di Prim
---
L'albero di spanning minimo viene costruito gradualmente aggiungendo gli archi uno alla volta. All'inizio lo spanning tree è costituito solo da un singolo vertice (scelto arbitrariamente). Quindi l'arco di peso minimo in uscita da questo vertice viene selezionato e aggiunto all'albero di spanning. Dopodiché l'albero di spanning è già costituito da due vertici. Successivamente seleziona e aggiunge l'arco con il peso minimo che ha un'estremità in un vertice già selezionato (cioè un vertice che fa già parte dell'albero di spanning) e l'altra estremità in un vertice non selezionato. E così via, cioè ogni volta che selezioniamo e aggiungiamo l'arco con il minimo peso che collega un vertice selezionato con un vertice non selezionato. 

Il processo viene ripetuto fino a quando l'albero di spanning contiene tutti i vertici (o in modo equivalente fino a quando non abbiamo lati).


Per mantenere traccia degli dei pesi degli archi si può utilizzare una coda con 
priorità in modo tale che:

$priorità(v) = \min_{(u,v):u ∈ S} \ peso(u,v)$ $u,v ∈ V$ e $S$ è l'insieme dei nodi visitati. 


Più in dettaglio il funzionamento, dato un grafo $G$ :
- crea un grafo vuoto $T$ che sarà l'**MST** di $G$.
- crea un insieme vuoto per i **nodi visitati** $S$.
- crea una **coda con priorità** $Q$. Inizialmente le priorità dei nodi sono ∞.
- finche la coda $Q$ non risulta vuota:
   - rimuove l'elemento $u$ con priorità minima da $Q$, con la funzione $deleteMin$, e lo inserisce in $S$
   - per ogni vicino $v$ di $u$ :
      - se $v$ ∉ $S$ e il peso $w$ dell'arco $(u,v)$ è minore della priorità di $v$:
         - aggiorna la priorità di $v$ : priorità($v$) = $w$  con la funzione $decreaseKey$
         - rende u *nuovo padre* di v in $T$

"""

# ╔═╡ 29e0ce06-92dc-4daa-b304-32bbc5018ccb
md"""
##### MST calcolato con Prim : 

sistema nodi $(@bind locs CheckBox(default=true)) 
"""

# ╔═╡ 9706a014-4360-4947-84ef-7d95cc5d9a57
md"""
##### Simulazione passo per passo
---
"""

# ╔═╡ 0e78835d-2625-45ce-8322-75aaf4e1f948
md"""
|       colore              | significato  |
|:-------------------------:|:------------:|
| $(colorant"plum3")        |nodo corrente |
| $(colorant"lightskyblue1")| nodo visto   |
| $(colorant"palegreen3")   | nodo visitato|
| $(colorant"grey98")       | nodo default |
"""

# ╔═╡ 56e74cdc-fbe0-4ae9-847d-fe0fcdd8c2d6
md"""
##### Complessità
---
L'algoritmo esegue $|V|$ iterazioni del ciclo *while*, una per ogni nodo $u$, dove :

| funzione    |       costo        | 
|:-----------:|:------------------:|
|*decreaseKey*| $log{\|V\|}⋅deg(u)$| 
| *deleteMin* |    $\log{\|V\|}$   |   

costo di un ciclo:
$θ(\log{|V|}⋅deg(u) + \log|V|)$
   

abbiamo quindi che il costo totale dell'algoritmo è dato dalla sommatoria :

$\sum_{u \in V} \ θ(deg(u)⋅\log{|V|}) = \log{|V|} ⋅ \sum_{u \in V} \ deg(u) = \log{2|E|} = θ(|E|⋅\log{|V|})$


Le considerazioni su questi costi suppongono la coda con priorità implementata con  heap binomiali o d-heap.
Se si implementa la coda con heap di fibonacci si avrà un costo ridotto a causa 
della funzione decreaseKey che costa θ(1).

Costo utilizzando *heap di fibonacci* :
$O(|E|+|V|\log{|V|})$
	
"""

# ╔═╡ 64168c91-09a4-4dbc-b719-ffe6a2a692fc
# per tenere la cronologia dei passi
begin
	mutable struct PrimStruct
		G::AbstractSimpleWeightedGraph
		T::AbstractSimpleWeightedGraph
		nodi_visitati::Vector # ordinati
		nodi_visti::Vector{Vector}
		archi_scelti::Vector  # ordinati
		PrimStruct(G::AbstractSimpleWeightedGraph) = new(G, SimpleWeightedGraph(nv(G)), Int[], Vector[], Edge[])
	end

	function add_step_prim!(s::PrimStruct, nodo_corrente::Int, nodi_visti::Vector)
		
		# salva gli archi scelti
		if !isempty(s.nodi_visitati)
			arco_scelto = Edge(-1,-1)
			peso_min = Inf
			for node in s.nodi_visitati
				if has_edge(s.G, node, nodo_corrente)
					if s.G.weights[node,nodo_corrente] < peso_min
						arco_scelto = Edge(node, nodo_corrente)
						peso_min = s.G.weights[node,nodo_corrente]
					end
				end
			end
			push!(s.archi_scelti, arco_scelto)
		end

		#salva i nodi
		push!(s.nodi_visitati, nodo_corrente)

		# salva i nodi aggiornati
		push!(s.nodi_visti, nodi_visti)
		setdiff!(s.nodi_visti[end], s.nodi_visitati)
		
	end
	
	PrimStruct
end

# ╔═╡ 712f751e-3d36-4e15-8ae9-c8bd365d833c
"""
Input:
- `G::AbstractSimpleWeightedGraph` il grafo

Output:
- `simulazione::PrimStruct` il risultato della visita di Prim
"""
function Prim(G::AbstractSimpleWeightedGraph)::PrimStruct

	simulazione = PrimStruct(G)
	
	T = SimpleGraph(nv(G))
	Q = PriorityQueue(1:nv(G) .=> Inf)
	S = Set()

	is_visto = Dict(vertices(G) .=> false)
	nodi_visti = []
	
	while !isempty(Q)
		u = dequeue!(Q)		#deleteMin
		push!(S, u)
		is_visto[u] = true
		
		for v ∈ neighbors(G, u)
			
			if !is_visto[v]
				is_visto[v] = true
				push!(nodi_visti, v)
			end
			
			if !in(v,S) && G.weights[u,v] < Q[v]
				Q[v] = G.weights[u,v]  					#decreaseKey
				# rendi u nuovo padre di v in T 
				for w ∈ neighbors(T, v) 				# solo 1 vicino (il padre)
					rem_edge!(T,v,w)
				end
				add_edge!(T, Edge(u,v))
			end
		end
		add_step_prim!(simulazione, u, copy(nodi_visti))
	end
	

	simulazione.T = SimpleWeightedGraph(T)
	
	#imposta i pesi
	for e ∈ edges(simulazione.T)
		simulazione.T.weights[e.src,e.dst] = simulazione.T.weights[e.dst, e.src] = G.weights[e.src, e.dst]
	end
	
	return simulazione
end

# ╔═╡ adeba58b-0872-4662-9052-b97bcb7ffcff
md"""
### Algoritmo di Kruskal
---
Dato un grafo $G$ ordina gli archi in ordine crescente di costo e successivamente li analizzia singolarmente, inserisce l'arco nella soluzione $T ⊆ E$ se *non forma cicli* con gli archi precedentemente selezionati. Notiamo che ad ogni passo, se abbiamo più archi con lo stesso costo, è indifferente quale viene scelto. 
"""

# ╔═╡ d8cb3ce5-1295-4b5b-b32f-e2686c78d1e9
md"""
Più in dettaglio:
- Ordina gli archi in ordine non decrescente di peso → *sorted_edges*
- mantiene un vettore di insiemi, i cui insiemi sono le componenti connesse. Inizialmente ogni nodo appartiene ad una componente connessa diversa.
- per ogni arco $(u,v)$ in *sorted_edges*:
   - se $u$ e $v$ non appartengono alla stessa componente, allora $(u,v)$ fa parte della soluzione, inserisce l'arco in $T$.
      - viene quindi, fatta l'**unione** delle componenti connesse che contengono i nodi $u$ e $v$.
   - Altrimenti l'arco $(u,v)$ sta creando un ciclo, quindi non fa parte della soluzione.

	Un'unione diminuisce il numero di componenti connesse di 1 e inizialmente abbiamo 
    |V| componenti connesse. Al più quindi, si possono fare |V| - 1 unioni. Infatti 
    Kruskal esegue |V| - 1 unioni così che alla fine rimane un'unica componente 
    connessa.
"""

# ╔═╡ f9bad8d1-7b6d-497f-92ca-080b45274c8a
md"""
!!! note
	Osserviamo che ad un grafo possono corrispondere più MST con lo stesso costo minimo
"""

# ╔═╡ 5d99e996-129f-4eb5-87a5-d1d47088b202
md"""
Confronto tra l'MST calcolato con *Kruskal* e con *Prim* : 
"""

# ╔═╡ 93821fc3-02f9-418f-8188-c5527ff926ab
md"""
Se sono uguali possiamo provare ad impostare tutti i pesi degli archi uguali ad uno, in questo modo avremo buona probabilità che gli MST saranno diversi, ovviamente anche aumentando il numero di archi e nodi del grafo la probabilità aumenta.

- Imposta pesi a uno : $@bind pesi_a_uno CheckBox(default=false)

"""

# ╔═╡ 1e357bc3-bc31-4bef-a7f9-40a5fcd66e19
md"""
##### Simulazione passo per passo
---
"""

# ╔═╡ 752b6429-2dcf-4114-9c6d-0654119008c9
md"""
- L'arco più "spesso" è quello che Kruskal sta esaminando.
- Gli archi colorati sono gli archi dell'MST (quelli scelti da Kruskal)
- Ogni colore identifica in modo univoco una componente connessa. 
"""

# ╔═╡ 45ddcc19-0249-4c99-85a2-3330faf2cdfb
# per tenere la cronologia dei passi
begin
	mutable struct KruskalStruct
		G::AbstractSimpleWeightedGraph
		T::AbstractSimpleWeightedGraph
		archi_ord::Vector
		comp_connesse::Vector{Vector}
		archi_scelti::Vector
		KruskalStruct(G::AbstractSimpleWeightedGraph) = new(G, SimpleWeightedGraph(nv(G)),
		sort(collect(edges(G)), by=weight),Vector[], Edge[])
		
	end

	function add_step_kruskal!(s::KruskalStruct, comp_connesse::Vector, arco_scelto::Edge)
		
		# salva le componenti connesse
		push!(s.comp_connesse, comp_connesse)

		push!(s.archi_scelti, arco_scelto)
		
	end

	# solo per printare
	function makeSet()
	end
	function find()
	end

	KruskalStruct
end

# ╔═╡ 13728ce1-f7e6-4008-bea5-a3d79ccab07d
"""
Input:
- `G::AbstractSimpleWeightedGraph` il grafo

Output:
- `simulazione::KruskalStruct` il risultato dell'esecuzione di Kruskal
"""
function Kruskal(G::AbstractSimpleWeightedGraph)::KruskalStruct
	
	simulazione = KruskalStruct(G)
	T = SimpleGraph(nv(G))
	
	archi_ord = sort(collect(edges(G)), by=weight, alg=MergeSort)

	comp_conn = Set.(1:nv(G))	
	arco_scelto = Edge(-1,-1)
	
	for e in archi_ord
		set_1 = Set()
		set_2 = Set()

		
		add_step_kruskal!(simulazione, [copy.(comp_conn)], arco_scelto)
		
		for set in comp_conn
			if length(intersect(set, Set([e.src, e.dst]))) == 1
				if isempty(set_1)
					set_1 = set
				else
					set_2 = copy(set)
					#cancella set_2 da comp_conn
					filter!(e->e≠set,comp_conn)
				end
			end
		end

		if !isempty(set_1) && !isempty(set_2)
			union!(set_1,set_2)	
			add_edge!(T, e.src, e.dst)
			arco_scelto = Edge(e.src, e.dst)
		end

		if size(comp_conn)[1] == 1
			add_step_kruskal!(simulazione, [copy.(comp_conn)], arco_scelto)
		end
		
	end
	
	simulazione.T = SimpleWeightedGraph(T)
	
	#imposta i pesi
	for e ∈ edges(simulazione.T)
		simulazione.T.weights[e.src,e.dst] = simulazione.T.weights[e.dst, e.src] = G.weights[e.src, e.dst]
	end
	
	return simulazione
end

# ╔═╡ c07294c0-251e-4fe7-a06f-a1971009bd38
md"""
# k-Clustering
Dato un insieme $U$ di $n$ oggetti $p₁,...pₙ$,si vuole classificare questi oggetti in gruppi coerenti. Definiamo la *funzione distanza* come il valore numerico che specifica la "vicinanza" di 2 oggetti : 
	
$d(pᵢ, pⱼ)$ 

dove pᵢ e pⱼ sono due oggetti. 

**Problema** : Dividi gli oggetti in cluster (gruppi) in base alla distanza degli oggetti, in modo tale che punti in differenti cluster sono più lontani possibile.

Ad esempio possiamo dire che, più due oggetti sono simili e meno è la loro distanza, così che vengano raggruppati oggetti tra loro simili.

Piu formalmente, assumiamo che la funzione rispetti le proprietà naturali, ossia :

|                           |
|:-------------------------:|
|$d(pᵢ, pⱼ) = 0 ⟺ pᵢ = pⱼ$  |
|$d(pᵢ, pⱼ) ≥ 0$            |
|$d(pᵢ, pⱼ) = d(pⱼ, pᵢ)$    |

Definiamo lo **spacing** come la minima distanza tra una coppia di punti in differenti cluster.

###### Obbiettivo : 
Dato un intero **k** vogliamo trovare un **k-clustering**, cioè **k** partizioni disgiunte di $U$, di *spacing massimale*.
"""

# ╔═╡ 4b5b0a42-5105-45b7-ac4b-22aa04b04056
md"""
Esempio di spacing tra due cluster 

|   |
|:-:|
|$(load(spacing_image)[1:219, 215:450])|

"""

# ╔═╡ ba7b2fcc-127f-4853-ac3a-b5fd983270ce
md"""
### Modello del problema
---

**Idea principale**: usare un grafo $G(V,E, d)$, dove $d:E→\mathbb{R}^+$,  per rappresentare l'istanza, dove :
-  $V = U = \{p₁,..., pₙ\}$
-  $E = \{(u,v) : u, v ∈ V ⩓ u ≠ v \}$
-  $peso(pᵢ,pⱼ) = d(pᵢ,pⱼ) , ∀ (pᵢ,pⱼ) ∈ E$

La soluzione cercata sarà una *k-partizione* di $V$.
"""

# ╔═╡ 0523463f-d64d-4a2d-a4ea-54558b8c42f3
md"""
### Relazione con Kruskal
---
Algoritmo per il *k-clustering*, dato $U$ l'insieme degli oggetti e $k$ un intero:
1. Forma il grafo descritto nel modello del problema, e ad ogni nodo corrisponde un cluster differente
2. Trova la coppia più vicina di oggetti $(p,p')$ tale che $p$ e $p'$ non sono nello stesso cluster.
3. Unisce $p$ e $p'$ nello stesso cluster
4. Riparte dal punto *2* 

Questo viene rifatto per $|V|-k$ volte

	Perchè |V|-k volte? 
	Si parte da |V| componenti, e si vuole arrivare ad avere k componenti. Ogni 
    iterazione unisce 2 componenti e quindi diminuisce il numero di componenti di 1. 
    Quindi abbiamo |V| - num_iterazioni = k → num_iterazioni = |V| - k.


Ci avete fatto caso? Quest'algoritmo è uguale all'algoritmo di **Kruskal** solo che si ferma quando si hanno k componenti !

!!! note
	Un altro metodo per calcolare un k-clustering è quello di ricavare l'MST, dal grafo costruito nel punto 1, e cancellare i k-1 archi di peso maggiore.
"""

# ╔═╡ 08093235-d9da-4faa-b2ca-27284a25f1fb
md"""
### k-Clustering punti in un piano
Adesso vediamo una semplice applicazione di *k-Clustering* per un insieme di punti posti su un piano. Dove la funzione distanza associa ad ogni coppia di punti, la loro distanza Euclidea.
"""

# ╔═╡ 70d2917e-a0be-4bd9-b3cf-0515a06ec550
md"""
---

Se si vuole modificare il grafo : 

|   |   | 
|:-:|:-:|
|numero nodi |$@bind num_nodi Slider(2:20, show_value=true, default=10)|
 
"""

# ╔═╡ 20a8694c-718c-4eda-976d-21e4a84e52dc
begin 
	g_c = SimpleWeightedGraph(num_nodi)
	locs_x_c, locs_y_c = random_layout(g_c)
	nothing
end

# ╔═╡ 1bb898b1-eab8-4e5f-bec9-f235a216a0f9
md"""
posizione nodo $(@bind nodo_sel Select(1:num_nodi))
"""

# ╔═╡ 55b55dd5-8b48-4c0d-94bd-5ad97eeeb723
md"""
x : $@bind pos_x Slider(0:0.01:1, show_value=true, default=locs_x_c[nodo_sel])
"""

# ╔═╡ 149fc894-9b8b-4bb0-b6ef-f9664bb2ced9
md"""
y : $@bind pos_y Slider(0:0.01:1, show_value=true, default=locs_y_c[nodo_sel])
"""

# ╔═╡ cf19ce0b-fcf9-4def-893f-02ff6c8b203e
begin
	locs_x_c[nodo_sel] = pos_x
	locs_y_c[nodo_sel] = pos_y

	
	# impostiamo le distanze
	for u ∈ 1:nv(g_c)
		for v ∈ 1:nv(g_c)
			if u != v
				distanza = sqrt((locs_x_c[u]-locs_x_c[v])^2+(locs_y_c[u]-locs_y_c[v])^2)

				add_edge!(g_c, u, v, trunc(distanza*10,digits=2))
			end
		end
	end

	#pesi_c = weight.(collect(edges(g_c)))

	# precalcolo iterazioni il numero di componenti cambia
	cluster_s = Kruskal(g_c)
	iter_comp = [1]
	local n_comp = nv(g_c)
	for iter in 1:ne(g_c)
		temp = size(cluster_s.comp_connesse[iter][])[1]
		if n_comp > temp
			push!(iter_comp, iter)
			n_comp = temp
		end
	end
	
	@bind k Slider(iter_comp)
end

# ╔═╡ dc56ea28-9d6f-4e50-9f4d-484448a6b2a1
md"""
# Funzioni ausiliarie
"""

# ╔═╡ 9df7b24f-abc2-4e34-891c-6fcedd00db8e
function createWeightedGraph()
	g = smallgraph(tipo_grafo)
	locs_x, locs_y = spring_layout(g)

	#assegna pesi
	wg = SimpleWeightedGraph(g)
	for e ∈ edges(g)
		if !pesi_a_uno
			wg.weights[e.src, e.dst] = wg.weights[e.dst, e.src] = round(rand(1:1:10))
		else
			wg.weights[e.src, e.dst] = wg.weights[e.dst, e.src] = 1
		end
	end

	# array pesi
	weights = weight.(collect(edges(wg)))
	
	return wg, weights, locs_x, locs_y
end

# ╔═╡ a588cd4f-be81-4744-946d-2a6c2c631fbc
function showGraph(G::SimpleWeightedGraph, weights::Vector, locs_x=nothing, locs_y=nothing)

	if locs_x == nothing && locs_y == nothing
		locs_x, locs_y = spring_layout(G)
	end
	gplot(G, locs_x, locs_y, nodelabel=1:nv(G), edgelabel=weights, nodefillc="white", nodestrokec="grey", nodestrokelw=true, EDGELABELSIZE=6, NODELABELSIZE=5)
end

# ╔═╡ 965facde-6da3-41ff-a5b8-2a85a5d9bca2
begin
	wg, pesi, locs_x, locs_y = createWeightedGraph()
	showGraph(wg,Int.(pesi),locs_x,locs_y)
end

# ╔═╡ c8502f00-5f25-40c1-b5b4-8ae7e55ccaea
"""
	plot_Prim
Funzione che mostra lo stato della `visita di Prim` al tempo `t`.
"""
function plot_Prim(
	simulazione::PrimStruct,
	t::Int,
	locs_x=locs_x,
	locs_y=locs_y,
	nodelabel=nothing,
	edgelabel= nothing,
	)

	grafo = simulazione.G

	# colora i nodi visitati
	nodi_visitati = simulazione.nodi_visitati[begin:t]
	nodi_visti = simulazione.nodi_visti[t]
	
	colorenodo = [colorant"palegreen3", colorant"grey98", colorant"lightskyblue1"]
	
	colori_nodi = fill(2, nv(grafo))
	colori_nodi[nodi_visitati] .= 1
	colori_nodi[nodi_visti] .= 3
	nodefillc = colorenodo[colori_nodi]
	
	if !isempty(nodi_visitati)
		nodefillc[nodi_visitati[t]] = colorant"plum3" # colora il nodo corrente
	end
	
	#colora gli archi scelti
	archi_scelti = simulazione.archi_scelti[begin:t-1]

	colorearco = [colorant"lightgrey", colorant"brown3"]
	colori_archi = fill(1, ne(grafo))

	for edge in archi_scelti
		index = 1
		for e in edges(grafo)
			if Edge(edge.src, edge.dst) == e || 
			   Edge(edge.dst, edge.src) == e
				
				colori_archi[index] = 2
			end
			index = index + 1
		end
	end
	
	edgestrokec = colorearco[colori_archi]

	
	gplot(
		grafo, locs_x, locs_y,
		nodefillc=nodefillc,
		nodestrokec="black",
		nodelabel=1:nv(grafo),
		nodestrokelw=true,
		edgelabel=Int.(pesi),
		edgestrokec = edgestrokec,
		EDGELABELSIZE = 6,
		NODELABELSIZE=5
	)
	
end

# ╔═╡ 28115885-5bdc-4d9c-9b79-01ee4e0cd6e6
begin
	md"""
	Simulazione Prim : $t$ $(@bind t₁ Slider(1:nv(wg), show_value=true))
	"""
end

# ╔═╡ 594d8851-ccea-496f-b431-3200f092a172
plot_Prim(Prim(wg), t₁)

# ╔═╡ c0992ba6-cff2-4900-ade1-c12a5ec236e8
"""
	plot_Kruskal
Funzione che mostra lo stato dell'esecuzione di Kruskal al tempo `t`.
"""
function plot_Kruskal(
	simulazione::KruskalStruct,
	t::Int,
	locs_x=locs_x,
	locs_y=locs_y,
	edgestrokec=nothing, # imposta white solo per il clustering
	edgelabelc=colorant"black"   # solo per il clustering
	)

	grafo = simulazione.G

	# colora le componenti
	nodefillc = distinguishable_colors(nv(grafo), [RGB(0.9,0.9,0.9), RGB(0.1,0.1,0.1)], dropseed=true)

	componenti = simulazione.comp_connesse[t]
	
	for sets in componenti
		for set in sets
			color_set = 0
			for nodo in set
				if color_set == 0
					color_set = nodefillc[nodo]
				end
				nodefillc[nodo] = color_set
			end
		end
	end

	if edgestrokec == nothing
		#colora gli archi scelti nelle componenti
		archi_scelti = simulazione.archi_scelti[begin:t]
		edgestrokec = fill(colorant"lightgrey", ne(grafo))
	
		for edge in archi_scelti[2:end]
			index = 1
			for e in edges(grafo)
				if Edge(edge.src, edge.dst) == e || 
				   Edge(edge.dst, edge.src) == e
			
					edgestrokec[index] = nodefillc[edge.src]
				end
				index = index + 1
			end
		end
	end

	#segna l'arco corrente
	curr_edge_t = copy(t) #indice per l'arco corrente

	# bound superiore
	if curr_edge_t > ne(grafo)
		curr_edge_t = curr_edge_t - (curr_edge_t - ne(grafo))
	end 
	
	current_edge = simulazione.archi_ord[curr_edge_t]
	edgelinewidth = fill(1.0, ne(grafo))
	

	index = 1
	for e in edges(grafo)
		if Edge(current_edge.src, current_edge.dst) == e || 
			Edge(current_edge.dst, current_edge.src) == e
		
			edgelinewidth[index] = 3.0
		end
		index = index + 1
	end

	
	pesi = weight.(collect(edges(grafo)))
	
	gplot(
		grafo, locs_x, locs_y,
		nodefillc=nodefillc,
		nodestrokec="black",
		nodelabel=1:nv(grafo),
		nodestrokelw=true,
		edgelabel= pesi,
		edgestrokec = edgestrokec,
		edgelinewidth = edgelinewidth,
		EDGELABELSIZE = 6,
		NODELABELSIZE=5,
		edgelabelc=edgelabelc
	)
end

# ╔═╡ ae0aecdf-f8fd-4276-9f04-14c2814e4dfc
begin
	kappa = size(cluster_s.comp_connesse[k][])[1]
	plot_Kruskal(cluster_s, k, locs_x_c, locs_y_c, colorant"white", colorant"white")
end

# ╔═╡ 2e5789b6-03e6-477b-95ef-729bc19aeff8
md"""
Puoi modificare il numero di cluster :
#### k : $kappa
"""

# ╔═╡ d9a20ee6-09fa-481a-a394-5d64dc5c2e0b
md"""
In questo modo abbiamo raggruppato i punti in $kappa cluster (Ogni cluster ha un suo colore associato).
"""

# ╔═╡ 0fea9759-343a-4d01-a6c0-b7f27417bd4d
begin
	md"""
	Simulazione Kruskal : $t$ $(@bind t₂ Slider(1:ne(wg)+1, show_value=true))
	"""
end

# ╔═╡ b8e91235-fca7-44d6-8cbf-6c753b3e7a60
plot_Kruskal(Kruskal(wg), t₂)

# ╔═╡ d39858aa-932e-47fd-aa7e-443798336f77
begin
	prim_struct = Prim(wg)
	
	pesiT_prim = Int.(weight.(collect(edges(prim_struct.T))))
	
	if !locs
		showGraph(prim_struct.T, pesiT_prim)
	else
		showGraph(prim_struct.T, pesiT_prim, locs_x, locs_y)	
	end
end 

# ╔═╡ a7488240-4fa2-44e9-879e-af5497c1d1f4
md"""
###### Costo soluzione : $(sum(pesiT_prim)) 
"""

# ╔═╡ b945af30-8729-4c8f-9a74-c9c0e07c4a31
begin
	kruskalStruct = Kruskal(wg)
	pesiT_kruskal = Int.(weight.(collect(edges(kruskalStruct.T))))

	["Kruskal" => showGraph(kruskalStruct.T, pesiT_kruskal, locs_x, locs_y),"Prim"=> showGraph(prim_struct.T, pesiT_prim, locs_x, locs_y)]	
end 

# ╔═╡ 05c76e3f-b7e1-4ed3-a1cb-766cc56bea35
md"""
###### Costo soluzione Kruskal : $(sum(pesiT_kruskal)) 
"""

# ╔═╡ 2e17fe64-4dbc-4668-b3cd-3bae27e8b7c7
begin
	uguali = "non uguali"
	if (kruskalStruct.T == prim_struct.T)
		uguali = "uguali"
	end

	nothing
end

# ╔═╡ 0f099b10-7bb0-441b-ace7-8f496097ee04
md"""
In questo caso gli MST calcolati risulano **$uguali**
"""

# ╔═╡ bf7f25bc-126c-4d9d-9c2c-d4eb4c471851
begin 
	costo_sort="O(|E|log|V|)"
	costo_makeSet ="O(1)"
	costo_union = "O(|V|log|V|)"
	costo_find = "O(1)"

	nothing
end

# ╔═╡ ff5609e6-7a6e-4755-9d5c-545ca39c5177
md"""
##### Complessità
---
Per quanto riguarda la complessità, utilizzando la struttura dati **UnionFind**,  implementata con gli alberi **QuickFind** con *union by size*, per gestire le componenti connesse, abbiamo che l'algoritmo esegue queste funzioni con il rispettivo costo e numero di esecuzioni:
	
| funzione    | costo        | numero esecuzioni |
|:-----------:|:------------:|:-----------------:|
|   *Sort*    | $costo_sort  |      $1$          |
|  *makeSet*  |$costo_makeSet|      $\|V\|$      |
|   *union*   | $costo_union |      $*$          |
|   *find*    | $costo_find  |      $2\|E\|$     |


	Il costo della union (by size) è dato da un'analisi ammortizzata (più precisa) sull'intera sequenza di |V|-1 union e quindi non di una singola operazione.
 
Il costo totale trascurando l'ordinamento è : 

$O(|V|+ 2|E| + |V|\log{|V|}) = O(|E|+|V|\log{|V|})$

Questo costo è asintoticamente più piccolo del costo di ordinamento, quindi il costo finale dell'algoritmo di Kruskal è :

$O(|E|\log{|V|})$
"""

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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "78736dab31ae7a53540a6b752efc61f77b304c5b"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.8.6+1"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─7532e879-36ae-4a3c-bdac-3b3524f802d6
# ╟─2278b12a-62b0-45cc-abe2-50312fcfcea7
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# ╟─712f751e-3d36-4e15-8ae9-c8bd365d833c
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# ╟─c8502f00-5f25-40c1-b5b4-8ae7e55ccaea
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# ╟─28115885-5bdc-4d9c-9b79-01ee4e0cd6e6
# ╟─594d8851-ccea-496f-b431-3200f092a172
# ╟─56e74cdc-fbe0-4ae9-847d-fe0fcdd8c2d6
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# ╟─adeba58b-0872-4662-9052-b97bcb7ffcff
# ╟─d8cb3ce5-1295-4b5b-b32f-e2686c78d1e9
# ╟─13728ce1-f7e6-4008-bea5-a3d79ccab07d
# ╟─f9bad8d1-7b6d-497f-92ca-080b45274c8a
# ╟─5d99e996-129f-4eb5-87a5-d1d47088b202
# ╟─b945af30-8729-4c8f-9a74-c9c0e07c4a31
# ╟─05c76e3f-b7e1-4ed3-a1cb-766cc56bea35
# ╟─0f099b10-7bb0-441b-ace7-8f496097ee04
# ╟─93821fc3-02f9-418f-8188-c5527ff926ab
# ╟─2e17fe64-4dbc-4668-b3cd-3bae27e8b7c7
# ╟─1e357bc3-bc31-4bef-a7f9-40a5fcd66e19
# ╟─c0992ba6-cff2-4900-ade1-c12a5ec236e8
# ╟─0fea9759-343a-4d01-a6c0-b7f27417bd4d
# ╟─b8e91235-fca7-44d6-8cbf-6c753b3e7a60
# ╟─752b6429-2dcf-4114-9c6d-0654119008c9
# ╟─ff5609e6-7a6e-4755-9d5c-545ca39c5177
# ╟─45ddcc19-0249-4c99-85a2-3330faf2cdfb
# ╟─c07294c0-251e-4fe7-a06f-a1971009bd38
# ╟─4b5b0a42-5105-45b7-ac4b-22aa04b04056
# ╟─c639f755-3d8d-4a25-a6f5-057016e79656
# ╟─ba7b2fcc-127f-4853-ac3a-b5fd983270ce
# ╟─0523463f-d64d-4a2d-a4ea-54558b8c42f3
# ╟─08093235-d9da-4faa-b2ca-27284a25f1fb
# ╟─20a8694c-718c-4eda-976d-21e4a84e52dc
# ╟─2e5789b6-03e6-477b-95ef-729bc19aeff8
# ╟─cf19ce0b-fcf9-4def-893f-02ff6c8b203e
# ╟─ae0aecdf-f8fd-4276-9f04-14c2814e4dfc
# ╟─d9a20ee6-09fa-481a-a394-5d64dc5c2e0b
# ╟─70d2917e-a0be-4bd9-b3cf-0515a06ec550
# ╟─1bb898b1-eab8-4e5f-bec9-f235a216a0f9
# ╟─55b55dd5-8b48-4c0d-94bd-5ad97eeeb723
# ╟─149fc894-9b8b-4bb0-b6ef-f9664bb2ced9
# ╟─dc56ea28-9d6f-4e50-9f4d-484448a6b2a1
# ╟─9df7b24f-abc2-4e34-891c-6fcedd00db8e
# ╟─a588cd4f-be81-4744-946d-2a6c2c631fbc
# ╟─bf7f25bc-126c-4d9d-9c2c-d4eb4c471851
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002