Analiza complexității unui algoritm


Este evident că durata execuţiei unui program depinde de dimensiunea şi de particularităţile datelor de intrare. De exemplu, putem spune cu certitudine că sortarea unui şir care conţine o mie de numere va dura mai mult decât sortarea unui şir care conţine o sută de numere.

În general , timpul de execuţie al unui algoritm creşte pe măsură ce creşte dimensiunea datelor de intrare. Din acest motiv, este natural să descriem timpul de execuţie al unui program ca o funcţie care depinde de această dimensiune.


Măsurarea timpului

În principiu, timpul de execuţie al unui program este durata ( exprimată în secunde, milisecunde, nanosecunde ) necesară pentru că programul să preia datele de intrare, să efectueze anumite operaţii asupra lor şi să furnizeze datele de ieşire. Evident acest timp depinde întotdeauna de calculatorul pe care este executat programul.

Chiar şi pentru calculatoarele de ultima generaţie vom avea timpi de execuţie diferiţi care vor depinde, în primul rând, de arhitectură şi frecvenţa procesorului. Considerând din nou exemplul programului care sortează un milion de numere, vom avea timpi de execuţie diferiţi pe calculatoare cu procesoare din familii diferite.


O măsură mult mai utilă a timpului de execuţie este dată de numărul paşilor elementari necesari execuţiei unui algoritm. Noţiunea de pas trebuie definită astfel încât să nu depindă de tipul calculatorului pe care este executat programul.

Cu o oarecare aproximare, putem considera că o linie descrisă în pseudocod se execută întotdeauna în acelaşi interval de timp. Evident pentru linii diferite vom avea intervale diferite, dar o aceeaşi linie se va execută întotdeauna în acelaşi timp. Există câteva argumente împotriva unei astfel de alegeri. De exemplu, este posibil ca anumite linii să nu se execute în timp constant. Dacă avem o linie în care se precizează că se sortează un şir, atunci timpul de execuţie al acestei linii va depinde de dimensiunea şirului. Totuşi, o astfel de linie “ascunde” de fapt o mulţime de paşi care nu au mai fost prezentaţi pentru a simplifică descrierea. De aceea, va trebui să luăm în considerare astfel de cazuri în momentul în care analizăm timpii de execuţie ai algoritmilor.

Dacă dimensiunea intrării este n, putem exprimă numărul de paşi sub formă unei funcţii T(n). Dacă algoritmul constă într-un-un număr de paşi egal cu pătratul valorii lui n vom avea T(n)=n^2. Dacă din diverse motive mai avem încă n paşi vom avea T(n)=n^2 + n


Cazul cel mai defavorabil şi cazul mediu

Timpul de execuţie al unui algoritm nu depinde doar de dimensiunea datelor de intrare, ci şi de configuratia acestora. De exemplu, pentru a sorta un şir de numere “aproape sortat” vom avea nevoie , în principiu, de mai puţin timp decât pentru un şir “sortat în ordine inversă”.

Din acest motiv au fost introduce noţiunea de caz mediu şi caz  defavorabil. Timpul de execuţie în cel mai defavorabil caz reprezintă cel mai mare dintre timpii de execuţie corespunzători configuraţiilor posibile ale unei intrări de dimensiunea dată.

De obicei, în analiză algoritmilor se utilizează cazul cel mai defavorabil.


Ordinul de complexitate

Deoarece funcţiile obţinute pentru timpii de execuţie pot fi uneori foarte complicate, se vor elimina toţi termenii care nu sunt importanţi.

De exemplu, dacă avem T(n)=2*( n^2) + 5*n +3 , ordinul de complexitate este O(n^2).

Dacă ordinul de complexitate al operaţiei de citire a datelor de intrare este O(n), ordinul de complexitate al operatilor de prelucrare este O(n^2) şi ordinul de complexitate al operaţiei de scriere a datelor de ieşire este O(n). În acest caz, ordinul de complexitate al algoritmului va fi O(n)+O(n^2)+O(n)=O(n^2);

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