We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,…,m − 1} for some integer m while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i), which returns the ith smallest element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n, m) p o(n) p O(lg lg m) bits to store a set of size n, where B(n, m) e ⌊lg (m/n)⌋ is the minimum number of bits required to store any n-element subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen [1995] and Pagh [2001]. We present extensions and applications of our indexable dictionary data structure, including: —an information-theoretically optimal representation of a k-ary cardinal tree that supports standard operations in constant time; —a representation of a multiset of size n from {0,…,m − 1} in B(n, m p n) p o(n) bits that supports (appropriate generalizations of) rank and select operations in constant time; and p O(lg lg m) —a representation of a sequence of n nonnegative integers summing up to m in B(n, m p n) p o(n) bits that supports prefix sum queries in constant time.

/pdf/succinct-indexable-dictionaries-with-applications-to-2oo5tkxgm3.pdf

Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets

/pdf/the-disjoint-paths-problem-in-quadratic-time-5ckbmxaxd7.pdf

The disjoint paths problem in quadratic time

This paper considers space-efficient data structures for storing an approximation S' to a set S such that S ⊆ S' and any element not in S belongs to S' with probability at most ∈. The Bloom filter data structure, solving this problem, has found widespread use. Our main result is a new RAM data structure that improves Bloom filters in several ways:• The time for looking up an element in S' is O(1), independent of ∈.• The space usage is within a lower order term of the lower bound.• The data structure uses explicit hash function families.• The data structure supports insertions and deletions on S in amortized expected constant time.The main technical ingredient is a succinct representation of dynamic multisets. We also consider three recent generalizations of Bloom filters.

/pdf/an-optimal-bloom-filter-replacement-j0ikuvardf.pdf

An optimal Bloom filter replacement

Preprocessing, or data reduction, is a standard technique for simplifying and speeding up computation. Written by a team of experts in the field, this book introduces a rapidly developing area of preprocessing analysis known as kernelization. The authors provide an overview of basic methods and important results, with accessible explanations of the most recent advances in the area, such as meta-kernelization, representative sets, polynomial lower bounds, and lossy kernelization. The text is divided into four parts, which cover the different theoretical aspects of the area: upper bounds, meta-theorems, lower bounds, and beyond kernelization. The methods are demonstrated through extensive examples using a single data set. Written to be self-contained, the book only requires a basic background in algorithmics and will be of use to professionals, researchers and graduate students in theoretical computer science, optimization, combinatorics, and related fields.

Kernelization: Theory of Parameterized Preprocessing

Sur le probléme des courbes gauches en Topologie

We consider the following problem: Given a subset S of size n of a universe {0,..., u-1}, construct a minimal perfect hash function for S, i.e., a bijection h from S to {0,..., n - 1}. The parameters of interest are the space needed to store h, its evaluation time, and the time required to compute h from S. The number of bits needed for the representation of h, ignoring the other parameters, has been thoroughly studied and is known to be n log e + loglog u ± O(log n), where "log" denotes the binary logarithm. A construction by Schmidt and Siegel uses O(n + loglogu) bits and offers constant evaluation time, but the time to find h is not discussed. We present a simple randomized scheme that uses n log e+log log u+o(n+log log u) bits and has constant evaluation time and O(n + log log u) expected construction time.

Efficient Minimal Perfect Hashing in Nearly Minimal Space

We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.

Approximation Algorithms for Intersection Graphs

Given four distinct vertices s1,s2,t1, and t2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p1 from s1 to t1 and p2 from s2 to t2, if such paths exist. Disjoint can mean vertex- or edge-disjoint. Both, the edge- and the vertex-disjoint version of the problem, are NP-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified to solve the 2-vertex-disjoint paths problem in only O(n + mα(m,n)) time, where m is the number of edges in G, n is the number of vertices in G, and where α denotes the inverse of the Ackermann function. Our result also improves the running time for the 2-edge-disjoint paths problem on undirected graphs as well as the running times for the 2-vertex- and the 2-edge-disjoint paths problem on dags.

Solving the 2-Disjoint Paths Problem in Nearly Linear Time

Given four distinct vertices s 1 , s 2 , t 1 , and t 2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p 1 from s 1 to t 1 and p 2 from s 2 to t 2 , if such paths exist. Disjoint can mean vertex- or edge-disjoint. Both, the edge- and the vertex-disjoint version of the problem, are NP-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified so as to solve the 2-(vertex-)disjoint paths problem in O(n + mα(m, n)) time, where m is the number of edges in G, n is the number of vertices in G, and α denotes the inverse of the Ackermann function. Our result also improves the running time for the 2-edge-disjoint paths problem on undirected graphs as well as the running times for the decision versions of the 2-vertex- and the 2-edge-disjoint paths problem on dags.

Solving the 2-disjoint paths problem in nearly linear time

Many algorithms have been developed for NP-hard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NP-hard problems is the computation of a tree decomposition of width O(k). In particular, by the bidimensional theory, there are many linear extended monadic second order problems that can be solved on n-vertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width O(k) can be found in such a time.We present the first algorithm that, on n-vertex planar graphs with treewidth k, finds a tree decomposition of width O(k) in such a time. In more detail, our algorithm has a running time of O(nk3 log k). The previous best algorithm with a running time subexponential in k was the algorithm of Gu and Tamaki [12] with a running time of O(n1+e log n) and an approximation ratio 1.5 + 1/e for any e > 0. The running time of our algorithm is also better than the running time of O(f(k) · n log n) of Reed's algorithm [18] for general graphs, where f is a function exponential in k.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973099.57

Torsten Tholey

Papers

Efficient Minimal Perfect Hashing in Nearly Minimal Space

Approximation Algorithms for Intersection Graphs

Solving the 2-Disjoint Paths Problem in Nearly Linear Time

Solving the 2-disjoint paths problem in nearly linear time

Approximate tree decompositions of planar graphs in linear time