Algoritmi za određivanje najbližeg para

Abstract

U ovom radu smo proučili šest algoritama za rješavanje problema najbližeg para u $\mathbb{R}^{2}$. Pokazali smo kako se problem može riješiti determinističkim algoritmom složenosti $O(n\log n)$, a nedeterminističkim algoritmima složenost možemo spustiti do $O(n)$. Ipak, testiranjem smo utvrdili da je na praktičnim veličinama ulaznih podataka od egzaktnih algoritama najefikasniji onaj koji je predstavljen u odjeljku 3.1 pod imenom Divide_and_Conquer, čija složenost je $O(n\log^{2}n)$. Na kraju smo predstavili i jedan parametrizirani heuristički algoritam složenosti $O(\frac{n}{\log n})$ koji pronalazi približno rješenje, čija točnost se može povećati nauštrb vremena izvođenja.In this paper we have examined six different algorithms for solving the closest pair of points problem in $\mathbb{R}^{2}$. We have shown that the problem can be solved with deterministic algorithms in $O(n\log n)$ time and in $O(n)$ time with non-deterministic algorithms. However, testing these algorithms has revealed that, for practical input sizes, the most efficient exact algorithm is the one we have labelled Divide_and_Conquer which was presented in section 3.1 and which runs in $O(n\log^{2}n)$time. In the final chapter we have presented a parameterised simulated annealing algorithm which finds an approximate solution in $O(\frac{n}{\log n})$ time, accuracy of which can be increased at the expense of speed