19 research outputs found
Searching, sorting and randomised algorithms for central elements and ideal counting in posets
By the Central Element Theorem of Linial and Saks, it follows that for the problem of (generalised) searching in posets, the information--theoretic lower bound of comparisons (where is the number of order--ideals in the poset) is tight asymptotically. We observe that this implies that the problem of (generalised) sorting in posets has complexity (where is the number of elements in the poset). We present schemes for (efficiently) transforming a randomised generation procedure for central elements (which often exists for some classes of posets) into randomised procedures for approximately counting ideals in the poset and for testing if an arbitrary element is central
On the Complexity of Mining Itemsets from the Crowd Using Taxonomies
We study the problem of frequent itemset mining in domains where data is not
recorded in a conventional database but only exists in human knowledge. We
provide examples of such scenarios, and present a crowdsourcing model for them.
The model uses the crowd as an oracle to find out whether an itemset is
frequent or not, and relies on a known taxonomy of the item domain to guide the
search for frequent itemsets. In the spirit of data mining with oracles, we
analyze the complexity of this problem in terms of (i) crowd complexity, that
measures the number of crowd questions required to identify the frequent
itemsets; and (ii) computational complexity, that measures the computational
effort required to choose the questions. We provide lower and upper complexity
bounds in terms of the size and structure of the input taxonomy, as well as the
size of a concise description of the output itemsets. We also provide
constructive algorithms that achieve the upper bounds, and consider more
efficient variants for practical situations.Comment: 18 pages, 2 figures. To be published to ICDT'13. Added missing
acknowledgemen
Universal Sorting: Finding a DAG using Priced Comparisons
We resolve two open problems in sorting with priced information, introduced
by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC
2000]. In this setting, different comparisons have different (potentially
infinite) costs. The goal is to sort with small competitive ratio (algorithmic
cost divided by cheapest proof).
1) When all costs are in , we give an algorithm that has
competitive ratio. Our algorithm generalizes the
algorithms for generalized sorting (all costs are either or ), a
version initiated by [Huang, Kannan, Khanna, FOCS 2011] and addressed recently
by [Kuszmaul, Narayanan, FOCS 2021].
2) We answer the problem of bichromatic sorting posed by [CFGKRS]: The input
is split into and , and and comparisons are more expensive
than an comparisons. We give a randomized algorithm with a O(polylog n)
competitive ratio.
These results are obtained by introducing the universal sorting problem,
which generalizes the existing framework in two important ways. We remove the
promise of a directed Hamiltonian path in the DAG of all comparisons. Instead,
we require that an algorithm outputs the transitive reduction of the DAG. For
bichromatic sorting, when and comparisons cost , this
generalizes the well-known nuts and bolts problem. We initiate an
instance-based study of the universal sorting problem. Our definition of
instance-optimality is inherently more algorithmic than that of the competitive
ratio in that we compare the cost of a candidate algorithm to the cost of the
optimal instance-aware algorithm. This unifies existing lower bounds, and opens
up the possibility of an -instance optimal algorithm for the bichromatic
version.Comment: 40 pages, 5 figure
The Quicksort algorithm and related topics
Sorting algorithms have attracted a great deal of attention and study, as they have numerous applications to Mathematics, Computer Science and related fields. In this thesis, we first deal with the mathematical analysis of the Quicksort algorithm and its variants. Specifically, we study the time complexity of the algorithm and we provide a complete demonstration of the variance of the number of comparisons required, a known result but one whose detailed proof is not easy to read out of the literature. We also examine variants of Quicksort, where multiple pivots are chosen for the partitioning of the array. The rest of this work is dedicated to the analysis of finding the true order by further pairwise comparisons when a partial order compatible with the true order is given in advance. We discuss a number of cases where the partially ordered sets arise at random. To this end, we employ results from Graph and Information Theory. Finally, we obtain an alternative bound on the number of linear extensions when the partially ordered set arises from a random graph, and discuss the possible application of Shellsort in merging chains
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum