Online Sorting via Searching and Selection

In this paper, we present a framework based on a simple data structure and parameterized algorithms for the problems of finding items in an unsorted list of linearly ordered items based on their rank (selection) or value (search). As a side-effect of answering these online selection and search queries, we progressively sort the list. Our algorithms are based on Hoare's Quickselect, and are parameterized based on the pivot selection method. For example, if we choose the pivot as the last item in a subinterval, our framework yields algorithms that will answer q<=n unique selection and/or search queries in a total of O(n log q) average time. After q=\Omega(n) queries the list is sorted. Each repeated selection query takes constant time, and each repeated search query takes O(log n) time. The two query types can be interleaved freely. By plugging different pivot selection methods into our framework, these results can, for example, become randomized expected time or deterministic worst-case time. Our methods are easy to implement, and we show they perform well in practice

Robust and Adaptive Search

Binary search finds a given element in a sorted array with an optimal number of log n queries. However, binary search fails even when the array is only slightly disordered or access to its elements is subject to errors. We study the worst-case query complexity of search algorithms that are robust to imprecise queries and that adapt to perturbations of the order of the elements. We give (almost) tight results for various parameters that quantify query errors and that measure array disorder. In particular, we exhibit settings where query complexities of log n + ck, (1+epsilon) log n + ck, and sqrt(cnk)+o(nk) are best-possible for parameter value k, any epsilon > 0, and constant c

On-the-Fly Array Initialization in Less Space

We show that for all given n,t,w in {1,2,...} with n<2^w, an array of n entries of w bits each can be represented on a word RAM with a word length of w bits in at most nw+ceil(n(t/(2 w))^t) bits of uninitialized memory to support constant-time initialization of the whole array and O(t)-time reading and writing of individual array entries. At one end of this tradeoff, we achieve initialization and access (i.e., reading and writing) in constant time with nw+ceil(n/w^t) bits for arbitrary fixed t, to be compared with nw+Theta(n) bits for the best previous solution, and at the opposite end, still with constant-time initialization, we support O(log n)-time access with just nw+1 bits, which is optimal for arbitrary access times if the initialization executes fewer than n steps

Dichotomic Selection on Words: A Probabilistic Analysis

The paper studies the behaviour of selection algorithms that are based on dichotomy principles. On the entry formed by an ordered list L and a searched element x not in L, they return the interval of the list L the element x belongs to. We focus here on the case of words, where dichotomy principles lead to a selection algorithm designed by Crochemore, Hancart and Lecroq, which appears to be "quasi-optimal". We perform a probabilistic analysis of this algorithm that exhibits its quasi-optimality on average

No sorting? Better searching!

Questions about order versus disorder in systems and models have been fascinating scientists over the years. In Computer Science, order is intimately related to sorting, commonly meant as the task of arranging keys in increasing or decreasing order with respect to an underlying total order relation. The sorted organization is amenable for searching a set of n keys, since each search requires Θ(log n) comparisons in the worst case, which is optimal if the cost of a single comparison can be considered a constant. Nevertheless, we prove that disorder implicitly provides more information than order does. For the general case of searching an array of multidimensional keys, whose comparison cost is proportional to their length (and hence cannot be considered a constant), we demonstrate that “suitable” disorder gives better bounds than those derivable by using the natural lexicographic order. We start out from previous work done by Andersson, Hagerup, H˚astad and Petersson [SIAM Journal on Computing, 30(2), 2001], who proved that k log log

No Sorting? Better Searching!

Questions about order versus disorder in systems and models have been fascinating scientists over the years. In computer science, order is intimately related to sorting, commonly meant as the task of arranging keys in increasing or decreasing order with respect to an underlying total order relation. The sorted organization is amenable for searching a set of n keys, since each search requires Θ(log n) comparisons in the worst case, which is optimal if the cost of a single comparison can be considered a constant. Nevertheless, we prove that disorder implicitly provides more information than order does. For the general case of searching an array of multidimensional keys whose comparison cost is proportional to their length (and hence which cannot be considered a constant), we demonstrate that “suitable” disorder gives better bounds than those derivable by using the natural lexicographic order. We start from previous work done by Andersson et al. [2001], who proved that Θ(k log log n/log log(4 + klog log n/log n) + k + log n) character comparisons (or probes) comprise the tight complexity for searching a plain sorted array of n keys, each of length k, arranged in lexicographic order. We describe a novel permutation of the n keys that is different from the sorted order. When keys are kept “unsorted” in the array according to this permutation, the complexity of searching drops to Θ(k + log n) character comparisons (or probes) in the worst case, which is optimal among all possible permutations, up to a constant factor. Consequently, disorder carries more information than does order; this fact was not observable before, since the latter two bounds are Θ(log n) when k = O(1). More implications are discussed in the article, including searching in the bit-probe model

No sorting? Better searching!

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 Theta(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, Hastad 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 [GRAPHICS] character comparisons (or probes). Note that the bound is Theta (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 front 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 Theta (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 Theta(log n) when k = 1