Utilità del prodotto cartesiano

Definizione

Il prodotto cartesiano non è altro che l’insieme delle coppie ordinate di due insiemi. Ad esempio se ho due insiemi:
A = {a,b,c}
B = {d,e,f}
AxB = {(a,d),(b,d),(c,d),(c,e),(c,f)

Per tre, invece:
A = {a,b}
B = {c,d}
C = {e,f}
AxBxC = {(a,c,e),(a,d,e),(a,c,f),(a,d,f),(b,c,e),(b,d,e),(b,c,f),(b,d,f)}

Per i database il prodotto cartesiano dei dati è il modo naturale per unire due set differenti.

cartesianoLa caratteristica del prodotto cartesiano risiede nel modo di determinare la griglia: è sufficiente conoscerne gli elementi degli assi. L’immagine seguente, realizzata in Python, mostra proprio il prodotto cartesiano fra i punti di 3 assi, appunto, cartesiani.

Il prodotto cartesiano non voluto

Un esempio di prodotto cartesiano non desiderato è facilmente riscontrabile nell’esempio del negozio, riporto un mock semplificato di due tabelle:

Account:
+-----------------+--------+
+ NOME		  + ACCOID +
+-----------------+--------+
+ Cristiano 	  + 1      +
+ Mario		  + 2      +
++++++++++++++++++++++++++++
Ordini:
+-----------------+---------+
+ ORDINI	  + ACCOID  +
+-----------------+---------+
+ 100€		  + 1	    +
+ 50€		  + 1 	    +
+ 70€		  + 1	    +
+ 50€		  + 1 	    +
+ 30€		  + 2	    +
+ 10€		  + 2 	    +
+++++++++++++++++++++++++++++

Eseguendo una query SQL del tipo

SELECT * FROM Account, Ordini

si ottiene proprio il prodotto cartesiano fra le due, che non è probabilmente il risultato che si vuole ottenere! (Per intenderci otterremo 12 righe).

+-----------------+-------------+
+ ORDINI	  + NOME	+
+-----------------+-------------+
+ 100€		  + Cristiano   +
+ 50€		  + Cristiano   +
+ 70€		  + Cristiano   +
+ 50€		  + Cristiano   +
+ 30€		  + Cristiano   +
+ 10€		  + Cristiano   +
+ 100€		  + Mario	+
+ 50€		  + Mario	+
+ 70€		  + Mario	+
+ 50€		  + Mario	+
+ 30€		  + Mario	+
+ 10€		  + Mario	+
+++++++++++++++++++++++++++++++++

Il motivo per cui la query non produce il risultato che vogliamo, è che viene valutato di default il prodotto cartesiano fra le tabelle interrogate (operazione di CROSS JOIN): l’output è proprio l’insieme di tutte le possibili combinazioni di account e ordini, e non un insieme con il nome dell’account all’intero del db. Si dice che il JOIN cerca il percorso più lungo (o globale) per unire le due tabelle.

SELECT * FROM Account NATURAL JOIN Ordini

In questo modo si otterrà invece il risultato desiderato: i parametri con lo stesso nome e valore verranno utilizzati come fonte per l’unione. La colonna ACCOID verrà riprodotta una sola volta.

+-----------------+-----------------+
+ ORDINI	  + ACCOID	    +
+-----------------+-----------------+
+ 100€		  + Cristiano	    +
+ 50€		  + Cristiano	    +
+ 70€		  + Cristiano	    +
+ 50€		  + Cristiano 	    +
+ 30€		  + Mario	    +
+ 10€		  + Mario	    +
+++++++++++++++++++++++++++++++++++++

Lo scopo di questo articolo non è presentare le JOIN, motivo per cui non mi soffermerò ulteriormente su altri esempi di questo tipo o su come modificare la query nel caso in cui ci siano colonne da unire con nomi diversi.

Quando è utile

In realtà le applicazioni del prodotto cartesiano sono moltissime. L’esempio più chiaro è nella creazione di grafici e report, ovvero ogni qual volta la combinazione di due valori deve dare un unica coppia. Ad esempio:

SELECT Negozi.Negozio, Prodotti.Prodotto FROM Negozi CROSS JOIN Prodotti

In questo modo avremo per ogni negozio i prodotti disponibili in una matrice.

Per fare un altro esempio, la prossima volta che dovete creare tutte le combinazioni di data e ora, potreste fare il prodotto cartesiano fra ore e minuti invece che innestare un ciclo dentro un altro.

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