By Benoit A., Robert Y., Vivien F.
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Additional resources for A guide to algorithm design paradigms, methods, and complexity analysis
3. Recall briefly the usual merging algorithm and give its complexity. 4. Prove that for n = m, we cannot do better than the usual algorithm. Therefore, the lower bound of Question 2 cannot be matched. 8: The toolbox (solution p. 26) In a toolbox, there are n nuts, all of different sizes, and n corresponding bolts. However, everything is mixed up, and you wish to associate each nut with the corresponding bolt. The size differences are so small that it is not possible to decide if a nut (or a bolt) is larger than another one just by looking at them.
Therefore, the lower bound of Question 2 cannot be matched. 8: The toolbox (solution p. 26) In a toolbox, there are n nuts, all of different sizes, and n corresponding bolts. However, everything is mixed up, and you wish to associate each nut with the corresponding bolt. The size differences are so small that it is not possible to decide if a nut (or a bolt) is larger than another one just by looking at them. The only way to proceed consists of trying one nut with one bolt, and each operation can lead to three possible answers: (i) the nut is strictly larger than the bolt, (ii) the bolt is strictly larger than the nut, and (iii) they correspond to each other.
Therefore, to be able to determine the maximum, the graph must contain a single connected component, and at least n 1 comparisons must have been performed. 3. Average complexity in the number of comparisons. Whatever the instance, the algorithm always performs n 1 comparisons, and the average complexity is equal to the best-case and worst-case complexity. Average complexity in the number of assignments. The average complexity of an algorithm A on data of size n is defined as: avgA (n) = p(d) costA (d), d data of size n where p(d) is the probability that d is an entry of algorithm A and costA (d) is the complexity reached by algorithm A on entry d.