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Register a new userMultidimensional segment trees can do range updates in poly-logarithmic time
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Added by
Nabil Ibtehaz
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Updating and querying on a range is a classical algorithmic problem with a multitude of applications. The Segment Tree data structure is particularly notable in handling the range query and update operations. A Segment Tree divides the range into disjoint segments and merges them together to perform range queries and range updates elegantly. Although this data structure is remarkably potent for 1-dimensional problems, it falls short in higher dimensions. Lazy Propagation enables the operations to be computed in $O(logn)$ time in a single dimension. However, the concept of lazy propagation could not be translated to higher-dimensional cases, which imposes a time complexity of $O(n^{k-1} ; logn)$ for operations on $k$-dimensional data. In this work, we have made an attempt to emulate the idea of lazy propagation differently so that it can be applied for 2-dimensional cases. Moreover, the proposed modification should be capable of performing most general aggregate functions similar to the original Segment Tree, and can also be extended to even higher dimensions. Our proposed algorithm manages to perform range sum queries and updates in $O(log^2 n)$ time for a 2-dimensional problem, which becomes $O(log^d n)$ for a $d$-dimensional situation.
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The vertex connectivity of an $m$-edge $n$-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in $tilde O(m^{alpha})$ time for any $alpha ge 1$, if there is a $m^{alpha}$-time maxflow algorithm. Using the current best maxflow algorithm that runs in $m^{4/3+o(1)}$ time (Kathuria, Liu and Sidford, FOCS 2020), this yields a $m^{4/3+o(1)}$-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an $tilde O(mn)$-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an $o(mn)$ running time was known before our work, even if we assume an $tilde O(m)$-time maxflow algorithm. Our new technique is robust enough to also improve the best $tilde O(mn)$-time bound for directed vertex connectivity to $mn^{1-1/12+o(1)}$ time
We give an $n^{O(loglog n)}$-time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over ${pm 1}^n$. Even in the realizable setting, the previous fastest runtime was $n^{O(log n)}$, a consequence of a classic algorithm of Ehrenfeucht and Haussler. Our algorithm shares similarities with practical heuristics for learning decision trees, which we augment with additional ideas to circumvent known lower bounds against these heuristics. To analyze our algorithm, we prove a new structural result for decision trees that strengthens a theorem of ODonnell, Saks, Schramm, and Servedio. While the OSSS theorem says that every decision tree has an influential variable, we show how every decision tree can be pruned so that every variable in the resulting tree is influential.
In this paper we present novel algorithmic techniques with a O(H(N)+N/H(N)) time complexity for performing several types of queries and updates on general rooted trees, binary search trees and lists of size N. For rooted trees we introduce a new compressed super-node tree representation which can be used for efficiently addressing a wide range of applications. For binary search trees we discuss the idea of globally rebuilding the entire tree in a fully balanced manner whenever the height of the tree exceeds the value of a conveniently chosen function of the number of tree nodes. In the end of the paper we introduce the H-list data structure which supports concatenation, split and several types of queries. Note that when choosing H(N)=sqrt(N) we obtain O(H(N)+N/H(N))=O(sqrt(N)).
Let $G$ be a DAG with $n$ vertices and $m$ edges. Two vertices $u,v$ are incomparable if $u$ doesnt reach $v$ and vice versa. We denote by emph{width} of a DAG $G$, $w_G$, the maximum size of a set of incomparable vertices of $G$. In this paper we present an algorithm that computes a dominance drawing of a DAG G in $k$ dimensions, where $w_G le k le frac{n}{2}$. The time required by the algorithm is $O(kn)$, with a precomputation time of $O(km)$, needed to compute a emph{compressed transitive closure} of $G$, and extra $O(n^2w_G)$ or $O(n^3)$ time, if we want $k=w_G$. Our algorithm gives a tighter bound to the dominance dimension of a DAG. As corollaries, a new family of graphs having a 2-dimensional dominance drawing and a new upper bound to the dimension of a partial order are obtained. We also introduce the concept of transitive module and dimensional neck, $w_N$, of a DAG $G$ and we show how to improve the results given previously using these concepts.
We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online $O(log log n)$-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.
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