Engineering Predecessor Data Structures for Dynamic Integer Sets

We present highly optimized data structures for the dynamic predecessor problem, where the task is to maintain a set S of w-bit numbers under insertions, deletions, and predecessor queries (return the largest element in S no larger than a given key). The problem of finding predecessors can be viewed as a generalized form of the membership problem, or as a simple version of the nearest neighbour problem. It lies at the core of various real-world problems such as internet routing. In this work, we engineer (1) a simple implementation of the idea of universe reduction, similar to van-Emde-Boas trees (2) variants of y-fast tries [Willard, IPL\u2783], and (3) B-trees with different strategies for organizing the keys contained in the nodes, including an implementation of dynamic fusion nodes [P?tra?cu and Thorup, FOCS\u2714]. We implement our data structures for w = 32,40,64, which covers most typical scenarios. Our data structures finish workloads faster than previous approaches while being significantly more space-efficient, e.g., they clearly outperform standard implementations of the STL by finishing up to four times as fast using less than a third of the memory. Our tests also provide more general insights on data structure design, such as how small sets should be stored and handled and if and when new CPU instructions such as advanced vector extensions pay off

Weighted ancestors in suffix trees revisited

The weighted ancestor problem is a well-known generalization of the predecessor problem to trees. It is known to require O(log log n) time for queries provided O(n polylog n) space is available and weights are from [0..n], where n is the number of tree nodes. However, when applied to suffix trees, the problem, surprisingly, admits an O(n)-space solution with constant query time, as was shown by Gawrychowski, Lewenstein, and Nicholson (Proc. ESA 2014). This variant of the problem can be reformulated as follows: given the suffix tree of a string s, we need a data structure that can locate in the tree any substring s[p..q] of s in O(1) time (as if one descended from the root reading s[p..q] along the way). Unfortunately, the data structure of Gawrychowski et al. has no efficient construction algorithm, limiting its wider usage as an algorithmic tool. In this paper we resolve this issue, describing a data structure for weighted ancestors in suffix trees with constant query time and a linear construction algorithm. Our solution is based on a novel approach using so-called irreducible LCP values.Peer reviewe

Greedy routing and virtual coordinates for future networks

At the core of the Internet, routers are continuously struggling with ever-growing routing and forwarding tables. Although hardware advances do accommodate such a growth, we anticipate new requirements e.g. in data-oriented networking where each content piece has to be referenced instead of hosts, such that current approaches relying on global information will not be viable anymore, no matter the hardware progress. In this thesis, we investigate greedy routing methods that can achieve similar routing performance as today but use much less resources and which rely on local information only. To this end, we add specially crafted name spaces to the network in which virtual coordinates represent the addressable entities. Our scheme enables participating routers to make forwarding decisions using only neighbourhood information, as the overarching pseudo-geometric name space structure already organizes and incorporates "vicinity" at a global level. A first challenge to the application of greedy routing on virtual coordinates to future networks is that of "routing dead-ends" that are local minima due to the difficulty of consistent coordinates attribution. In this context, we propose a routing recovery scheme based on a multi-resolution embedding of the network in low-dimensional Euclidean spaces. The recovery is performed by routing greedily on a blurrier view of the network. The different network detail-levels are obtained though the embedding of clustering-levels of the graph. When compared with higher-dimensional embeddings of a given network, our method shows a significant diminution of routing failures for similar header and control-state sizes. A second challenge to the application of virtual coordinates and greedy routing to future networks is the support of "customer-provider" as well as "peering" relationships between participants, resulting in a differentiated services environment. Although an application of greedy routing within such a setting would combine two very common fields of today's networking literature, such a scenario has, surprisingly, not been studied so far. In this context we propose two approaches to address this scenario. In a first approach we implement a path-vector protocol similar to that of BGP on top of a greedy embedding of the network. This allows each node to build a spatial map associated with each of its neighbours indicating the accessible regions. Routing is then performed through the use of a decision-tree classifier taking the destination coordinates as input. When applied on a real-world dataset (the CAIDA 2004 AS graph) we demonstrate an up to 40% compression ratio of the routing control information at the network's core as well as a computationally efficient decision process comparable to methods such as binary trees and tries. In a second approach, we take inspiration from consensus-finding in social sciences and transform the three-dimensional distance data structure (where the third dimension encodes the service differentiation) into a two-dimensional matrix on which classical embedding tools can be used. This transformation is achieved by agreeing on a set of constraints on the inter-node distances guaranteeing an administratively-correct greedy routing. The computed distances are also enhanced to encode multipath support. We demonstrate a good greedy routing performance as well as an above 90% satisfaction of multipath constraints when relying on the non-embedded obtained distances on synthetic datasets. As various embeddings of the consensus distances do not fully exploit their multipath potential, the use of compression techniques such as transform coding to approximate the obtained distance allows for better routing performances