In modern data warehousing, data skipping is essential for high query performance. While index structures such as B-trees or hash tables allow for precise pruning, their large storage requirements make them impractical for indexing secondary columns. Therefore, many systems rely on approximate indexes such as min/max sketches (ZoneMaps) or Bloom filters for cost-effective data pruning. For example, Google PowerDrill skips more than 90% of data on average using such indexes. In this paper, we introduce Cuckoo Index (CI), an approximate secondary index structure that represents the many-to-many relationship between keys and data partitions in a highly space-efficient way. At its core, CI associates variable-sized fingerprints in a Cuckoo filter with compressed bitmaps indicating qualifying partitions. With our approach, we target equality predicates in a read-only (immutable) setting and optimize for space efficiency under the premise of practical build and lookup performance. In contrast to per-partition (Bloom) filters, CI produces correct results for lookups with keys that occur in the data. CI allows to control the ratio of false positive partitions for lookups with non-occurring keys. Our experiments with real-world and synthetic data show that CI consumes significantly less space than per-partition filters for the same pruning power for low-to-medium cardinality columns. For high cardinality columns, CI is on par with its baselines.

Cardinality estimation has a wide range of applications and is of particular importance in database systems. Various algorithms have been proposed in the past, and the HyperLogLog algorithm is one of them. In this paper, we present a series of improvements to this algorithm that reduce its memory requirements and significantly increase its accuracy for an important range of cardinalities. We have implemented our proposed algorithm for a system at Google and evaluated it empirically, comparing it to the original HyperLogLog algorithm. Like HyperLogLog, our improved algorithm parallelizes perfectly and computes the cardinality estimate in a single pass.

Column-oriented database systems have been a real game changer for the industry in recent years. Highly tuned and performant systems have evolved that provide users with the possibility of answering ad hoc queries over large datasets in an interactive manner. In this paper we present the column-oriented datastore developed as one of the central components of PowerDrill. It combines the advantages of columnar data layout with other known techniques (such as using composite range partitions) and extensive algorithmic engineering on key data structures. The main goal of the latter being to reduce the main memory footprint and to increase the efficiency in processing typical user queries. In this combination we achieve large speed-ups. These enable a highly interactive Web UI where it is common that a single mouse click leads to processing a trillion values in the underlying dataset.

For a given number L , an L -length-bounded edge-cut (node-cut, respectively) in a graph G with source s and sink t is a set C of edges (nodes, respectively) such that no s - t -path of length at most L remains in the graph after removing the edges (nodes, respectively) in C . An L -length-bounded flow is a flow that can be decomposed into flow paths of length at most L . In contrast to classical flow theory, we describe instances for which the minimum L -length-bounded edge-cut (node-cut, respectively) is Θ( n 2/3 )-times (Θ(√ n )-times, respectively) larger than the maximum L -length-bounded flow, where n denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is NP -hard to approximate within a factor of 1.1377 for L ≥ 5 in the case of node-cuts and for L ≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio O (min{ L , n/L }) ⊆ O √ n in the node case and O (min { L , n 2 / L 2 ,√ m } ⊆ O 2/3 in the edge case, where m denotes the number of edges. Concerning L -length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is NP -complete to decide whether there is a fractional length-bounded flow of a given value. We analyze the structure of optimal solutions and present further complexity results.

An efficient technique for mapping application tasks to heterogeneous processing elements (PEs) on a network-on-chip (NoC) platform, operating at multiple voltage levels, is presented in this paper. The goal of the mapping is to minimize energy consumption subject to the performance constraints. Such a mapping involves solving several subproblems. Most of the research effort in this area often address these subproblems in a sequential fashion or a subset of them. We take a unified approach to the problem without compromising the solution time and provide techniques for optimal and heuristic solutions. We prove that the voltage assignment component of the problem itself is NP-hard and is in approximable within any constant factor. Our optimal solution utilizes a mixed integer linear program (MILP) formulation of the problem. The heuristic utilizes MILP relaxation and randomized rounding. Experimental results based on E3S benchmark applications and a few real applications show that our heuristic produces near-optimal solution in a fraction of time needed to find the optimal.

Consider the problem of discovering (or verifying) the edges and non-edges of a network, modeled as a connected undirected graph, using a minimum number of queries. A query at a vertex v discovers (or verifies) all edges and non-edges whose endpoints have different distance from v. In the network discovery problem, the edges and non-edges are initially unknown, and the algorithm must select the next query based only on the results of previous queries. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio $O(\sqrt{nlogn})$ for graphs with n vertices. We also show that no deterministic algorithm can have competitive ratio better than 3. In the network verification problem, the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and non-edges. This problem has previously been studied as the problem of placing landmarks in a graph or determining the metric dimension of a graph. We show that there is no approximation algorithm for this problem with ratio o(log n) unless $\mathcal{P} = \mathcal{NP}$ .