Optimal in-place sorting of vectors and records
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
The complexity of implicit and space efficient priority queues
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
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Sorting is commonly meant as the task of arranging keys in increasing or decreasing order (or small variations of this order). Given n keys underlying a total order, the best organization in an array is maintaining them in sorted order. Searching requires 驴(log n) comparisons in the worst case, which is optimal. We demonstrate that this basic fact in data structures does not hold for the general case of multi-dimensional keys, whose comparison cost is proportional to their length. In two papers [1, 3] and the full version in [2], Andersson, Hagerup, Håstad and Petersson study the complexity of searching a sorted array of n keys, each of length k, arranged in lexicographic (or alphabetic) order for an arbitrary, possibly unbounded, ordered alphabet. They give sophisticated arguments for proving a tight bound in the worst case for this basic data organization, up to a constant factor, obtaining \Theta (\frac{{k\log \log n}}{{\log \log (4 + \frac{{k\log \log n}}{{\log n}})}} + k + \log n) character comparisons (or probes). Note that the bound is 驴(log n) when k = 1, which is the case that is well known in algorithmics. We describe a novel permutation of the n keys that is different from the sorted order, and sorting is just the starting point for describing our preprocessing. When keys are stored according to this "unsorted" order in the array, the complexity of searching drops to 驴(k + log n) character comparisons (or probes) in the worst case, which is optimal among all possible permutations of the n keys in the array, up to a constant factor. Again, the bound is 驴(log n) when k = 1. Jointly with the aforementioned result of Andersson et al., our finding provably shows that keeping k-dimensional keys sorted in an array is not the best data organization for searching. This fact was not observable before by just considering k = O(1) as sorting is an optimal organization in this case. More implications of our result are commented in the introduction.