Karp et al. (1988) described Deferred Data Structures for Multisets as "lazy" data structures which partially sort data to support online rank and select queries, with the minimum amount of work in the worst case over instances of size n and number of queries q fixed. Barbay et al. (2016) refined this approach to take advantage of the gaps between the positions hit by the queries (i.e., the structure in the queries). We develop new techniques in order to further refine this approach and take advantage all at once of the structure (i.e., the multiplicities of the elements), some notions of local order (i.e., the number and sizes of runs) and global order (i.e., the number and positions of existing pivots) in the input; and of the structure and order in the sequence of queries. Our main result is a synergistic deferred data structure which outperforms all solutions in the comparison model that take advantage of only a subset of these features. As intermediate results, we describe two new synergistic sorting algorithms, which take advantage of some notions of structure and order (local and global) in the input, improving upon previous results which take advantage only of the structure (Munro and Spira 1979) or of the local order (Takaoka 1997) in the input; and one new multiselection algorithm which takes advantage of not only the order and structure in the input, but also of the structure in the queries.

We prove the existence of an algorithm A for computing 2D or 3D convex hulls that is optimal for every point set in the following sense: for every sequence σ of n points and for every algorithm A ′ in a certain class A , the running time of A on input σ is at most a constant factor times the running time of A ′ on the worst possible permutation of σ for A ′. In fact, we can establish a stronger property: for every sequence σ of points and every algorithm A ′, the running time of A on σ is at most a constant factor times the average running time of A ′ over all permutations of σ. We call algorithms satisfying these properties instance optimal in the order-oblivious and random-order setting. Such instance-optimal algorithms simultaneously subsume output-sensitive algorithms and distribution-dependent average-case algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input are given in a random order. The class A under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instance-specific lower bound, we deviate from traditional Ben-Or-style proofs and adopt a new adversary argument. For 2D convex hulls, we prove that a version of the well-known algorithm by Kirkpatrick and Seidel [1986] or Chan, Snoeyink, and Yap [1995] already attains this lower bound. For 3D convex hulls, we propose a new algorithm. We further obtain instance-optimal results for a few other standard problems in computational geometry, such as maxima in 2D and 3D, orthogonal line segment intersection in 2D, finding bichromatic L ∞ -close pairs in 2D, offline orthogonal range searching in 2D, offline dominance reporting in 2D and 3D, offline half-space range reporting in 2D and 3D, and offline point location in 2D. Our framework also reveals a connection to distribution-sensitive data structures and yields new results as a byproduct, for example, on online orthogonal range searching in 2D and online half-space range reporting in 2D and 3D.

The action of teaching reinforces one's learning but requires some external quality control when done by nonprofessionals (e.g., a professor supervising teaching assistants). This quality control is costly and has limited the adoption of peer teaching in schools. Our solution to this problem is to ask students to create pedagogical material that will then be evaluated by the students themselves, where they evaluate a mix of new and already rated materials. One advantage of this technique is that it allows students to develop critical thinking skills, since they must judge if the presented material adequately covers a specific topic. We describe the “Teaching is Learning” project, that implements these ideas. We first discuss two pilot studies: 1) since 2009, engineering students from the University of Chile have been creating and validating pedagogical 3D animations; and 2) during 2015, seventh graders from the Blest Gana secondary school created and evaluated pedagogical videos using their cell phones, editing the videos during their Technology class. We then give an overview of existing work on measuring student learning, and discuss how we can evaluate the effectiveness of our technique. We conclude by describing our plans for implementing the “Teaching is Learning” project at a larger scale during 2016, both in Chile and in Brazil.

The Swap-Insert Correction distance from a string S of length n to another string L of length $$m\ge n$$ on the alphabet [1..d] is the minimum number of insertions, and swaps of pairs of adjacent symbols, converting S into L. Contrarily to other correction distances, computing it is NP-Hard in the size d of the alphabet. We describe an algorithm computing this distance in time within $$O(d^2 nm g^{d-1})$$ , where there are $$n_\alpha $$ occurrences of $$\alpha $$ in S, $$m_\alpha $$ occurrences of $$\alpha $$ in L, and where $$g=\max _{\alpha \in [1..d]} \min \{n_\alpha ,m_\alpha -n_\alpha \}$$ measures the difficulty of the instance. The difficulty g is bounded by above by various terms, such as the length of the shortest string S, and by the maximum number of occurrences of a single character in S. The latter bound yields a running time within $$O(d(n+m)+(d/(d-1)^{d-2})\cdot n^{d}(m-n))$$ in the worst case over instances of fixed lengths n and m for S and L, which further simplifies to within $$O(n^d(m-n)+m)$$ when d is fixed, the state of the art for this problem. This illustrates how, in many cases, the correction distance between two strings can be easier to compute than in the worst case scenario.

We define and design succinct indexes for several abstract data types (ADTs). The concept is to design auxiliary data structures that ideally occupy asymptotically less space than the information-theoretic lower bound on the space required to encode the given data, and support an extended set of operations using the basic operators defined in the ADT. The main advantage of succinct indexes as opposed to succinct (integrated data/index) encodings is that we make assumptions only on the ADT through which the main data is accessed, rather than the way in which the data is encoded. This allows more freedom in the encoding of the main data. In this article, we present succinct indexes for various data types, namely strings, binary relations and multilabeled trees. Given the support for the interface of the ADTs of these data types, we can support various useful operations efficiently by constructing succinct indexes for them. When the operators in the ADTs are supported in constant time, our results are comparable to previous results, while allowing more flexibility in the encoding of the given data. Using our techniques, we design a succinct encoding that represents a string of length n over an alphabet of size σ using n H k ( S ) + lg σ · o ( n ) + O ( n lg σ/lg lg lg σ) bits to support access/rank/select operations in o ((lg lg σ) 1+ϵ ) time, for any fixed constant ϵ > 0. We also design a succinct text index using n H 0 ( S ) + O ( n lg σ/lg lg σ) bits that supports finding all the occ occurrences of a given pattern of length m in O ( m lg lg σ + occ lg n /lg ϵ σ) time, for any fixed constant 0 < ϵ < 1. Previous results on these two problems either have a lg σ factor instead of lg lg σ in the running time, or are not compressed. Finally, we present succinct encodings of binary relations and multi-labeled trees that are more compact than previous structures.

We prove the existence of an algorithm A for computing 2-d or 3-dconvex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A' in a certain class A, the running time of A on the worst permutation of S for A is at most a constant factor times the running time of A' on the worst permutation of S for A'. In fact, we can establish a stronger property: for every S and A', the running time of A on S is at most a constant factor times the average running time of A' over all permutations of S. We call algorithms satisfying these properties instance-optimal in the order-oblivious and random-order setting. Such instance-optimal algorithms simultaneously subsume output-sensitive algorithms and distribution-dependent average-case algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input is given in a random order. The class A under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instance-specific lower bound, we deviate from traditional Ben-Or-style proofs and adopt an interesting adversary argument. For 2-d convex hulls, we prove that a version of the well known algorithm by Kirkpatrick and Seidel (1986) or Chan, Snoeyink, and Yap(1995) already attains this lower bound. For 3-d convex hulls, we propose a new algorithm. We further obtain instance-optimal results for a few other standard problems in computational geometry, such as maxima in 2-d and 3-d, orthogonal line segment intersection in 2-d, offline orthogonal range searching in 2-d, off-line halfspace range reporting in 2-d and 3-d, and off-line point location in 2-d. The theory we develop also neatly reveals connections to entropy-dependent data structures, and yields as a byproduct new expected case results, e.g., for on-line orthogonal range counting in 2-d.

We explore various techniques to compress a permutation $\pi$ over $n$ integers, taking advantage of ordered subsequences in $\pi$, while supporting its application $\pi(i)$ and the application of its inverse $\pi^{-1}(i)$ in small time. Our compression schemes yield several interesting byproducts, in many cases matching, improving or extending the best existing results on applications such as the encoding of a permutation in order to support iterated applications $\pi^{k}(i)$ of it, of integer functions, and of inverted lists and suffix arrays.

The intersection of sorted arrays problem has applications in search engines such as Google. Previous work has proposed and compared deterministic algorithms for this problem, in an adaptive analysis based on the encoding size of a certificate of the result (cost analysis). We define the alternation analysis , based on the nondeterministic complexity of an instance. In this analysis we prove that there is a deterministic algorithm asymptotically performing as well as any randomized algorithm in the comparison model. We define the redundancy analysis , based on a measure of the internal redundancy of the instance. In this analysis we prove that any algorithm optimal in the redundancy analysis is optimal in the alternation analysis, but that there is a randomized algorithm which performs strictly better than any deterministic algorithm in the comparison model. Finally, we describe how these results can be extended beyond the comparison model.