• Corpus ID: 12640119

# OPTIMAL PREPROCESSING FOR S ANSWERING ON-LINE PRODUCT QUERIE

@inproceedings{Alon1987OPTIMALPF,
title={OPTIMAL PREPROCESSING FOR S ANSWERING ON-LINE PRODUCT QUERIE},
author={Noga Alon and Baruch Schieber},
year={1987}
}
• Published 1987
• Mathematics, Computer Science
e q We examine the amount of preprocessing needed for answering certain on-lin ueries as fast as possible. We start with the following basic problem. Suppose we are e given a semigroup (S ,°). Let s , . . . , s be elements of S . We want to answer on-lin 1 n i i +1 j −1 j n 1 queries of the form, "What is the product s °s ° . . . °s °s ?" for any give ≤ i ≤ j ≤ n . We show that a preprocessing of Θ(n λ(k ,n )) time and space is both . T necessary and sufficient to answer each such query in at…
19 Citations

### Path Queries in Weighted Trees

• Computer Science
ISAAC
• 2011
A linear space data structure is presented that answers path reporting queries in $O(\lg \sigma + occ \lg\lgg\sigma)$ time, which are the first data structures that answer path reporting queries under the word RAM model.

### Optimal Parallel Shortest Paths in Small Treewidth Digraphs

• Computer Science, Mathematics
ESA
• 1995
These are the first parallel algorithms which achieve bounds for any class of graphs except trees, when the treewidth is a constant: computing a shortest path tree, or finding a negative cycle in O(log2n) time using O(n) work.

### Simple Parallel Algorithms for Dynamic Range Products

• C. Zaroliagis
• Computer Science
Algorithms, Probability, Networks, and Games
• 2015
We consider here the problem of answering range product queries on an n-node unrooted tree labelled with elements of a semigroup provided with an associative operator only. We present simple parallel

### Shortest Paths in Digraphs of Small Treewidth . Part II : Optimal Parallel Algorithms

• Computer Science, Mathematics
• 1995
We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be eeciently answered. We

### Succinct Indices for Path Minimum, with Applications

• Computer Science
Algorithmica
• 2016
Novel succinct indices for path minimum queries are designed under the indexing model, for which weights of nodes are read-only and can be accessed with ranks of nodes in the preorder traversal sequence of the input tree.

### Shortest Path Queries in Digraphs of Small Treewidth

• Computer Science, Mathematics
ICALP
• 1995
We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered.

### In-Memory Storage for Labeled Tree-Structured Data

This thesis designs in-memory data structures for labeled and weights trees, so that various types of path queries or operations can be supported with efficient query time and presents the first succinct representations for dynamic labeled ordinal trees that support several labelbased operations including finding the level ancestor with a given label.

### A Faster Algorithms for Dynamic Algebraic Queries in Basic RSMs with Constant

• Computer Science
• 2019
This work considers possible multiple queries as required in many applications such as in alias analysis, and considers a general framework with RSMs where the transitions are labeled from a semiring, and path properties are algebraic with semiring operations.

### The Euler Path to Static Level-Ancestors

The purpose of this note is to expose another simple algorithm that lies under a complicated PRAM algorithm by Berkman and Vishkin (1990,1994).

## References

SHOWING 1-7 OF 7 REFERENCES

### Fast Algorithms for Finding Nearest Common Ancestors

• Computer Science, Mathematics
SIAM J. Comput.
• 1984
An algorithm for a random access machine with uniform cost measure (and a bound of $\Omega (\log n)$ on the number of bits per word) that requires time per query and preprocessing time is presented, assuming that the collection of trees is static.

### Applications of Path Compression on Balanced Trees

A method for computing functions defined on paths in trees based on tree manipulation techniques first used for efficiently representing equivalence relations, which has an almost-linear running time and is useful for solving certain kinds of pathfinding problems on reducible graphs.

### Efficiency of a Good But Not Linear Set Union Algorithm

It is shown that, if t(m, n) is seen as the maximum time reqmred by a sequence of m > n FINDs and n -- 1 intermixed UNIONs, then kima(m), n is shown to be related to a functional inverse of Ackermann's functmn and as very slow-growing.

### Parallelism in Comparison Problems

The worst-case time complexity of algorithms for multiprocessor computers with binary comparisons as the basic operations is investigated and the algorithm for finding the maximum is shown to be optimal for all values of k and n.

### A Separator Theorem for Planar Graphs

• Mathematics
• 1977
Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more

### Data Structures for On-Line Updating of Minimum Spanning Trees, with Applications

Data structures are presented for the problem of maintaining a minimum spanning tree on-line under the operation of updating the cost of some edge in the graph. For the case of a general graph, mai...

### Zum Hilbertschen Aufbauder reellen Zahlen "

• SIAM J . Comput .