# On Computing the Subset Graph of a Collection of Sets

@article{Pritchard1999OnCT, title={On Computing the Subset Graph of a Collection of Sets}, author={Paul Pritchard}, journal={J. Algorithms}, year={1999}, volume={33}, pages={187-203} }

Abstract Let a given collection of sets have size N measured by the sum of the cardinalities. Yellin and Jutla presented an algorithm which constructed the partial order induced by the subset relation (a “subset graph”) in a claimed O ( N 2 /log N ) operations over a dictionary ADT, and exhibited a collection whose subset graph had Θ( N 2 /log 2 N ) edges. This paper first establishes a matching upper bound on the number of edges in a subset graph. It also presents a finer analysis of the…

## 25 Citations

### A Fast Bit-Parallel Algorithm for Computing the Subset Partial Order

- Computer Science, MathematicsAlgorithmica
- 1999

This paper gives implementations of a variant of a previously proposed algorithm which exploit bit-parallel operations on a RAM with Theta(log N) -bit words to have a worst-case complexity of O(N2 log log N / log2N) operations.

### Computing the subset partial order for dense families of sets

- Computer Science, MathematicsInf. Process. Lett.
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### The Subset Partial Order: Computing and Combinatorics

- Computer ScienceANALCO
- 2010

It is shown that the size of the partial order graph induced by the subset relation (called the subset graph) is [EQUATION], where B = log (N/ log2 N), which implies a simpler proof to the O(N2/log2 N) bound concluded in [2].

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It is shown that, if the Strong Exponential Time Hypothesis is true, both problems cannot be solved in O ( N ) time for α-acyclic hypergraphs and any constant ε > 0, even if the created graph is sparse.

### Fully dynamic algorithms for maintaining extremal sets in a family of sets

- Computer Science, MathematicsInt. J. Comput. Math.
- 1998

These algorithms are the first linear-time fully dynamic algorithms for maintaining extremal sets, which, requiring extra words in space within the same bound O(N 2), improve the time complexity of the existing result by a factor of O( N).

### Strongly connected components of directed hypergraphs

- Mathematics, Computer ScienceArXiv
- 2011

A superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs is established, showing that it is combinatorially more complex than in directed graphs.

### Fast Algorithms for Finding Extremal Sets

- Computer ScienceSDM
- 2011

Two new algorithms for the extremal set finding task are described and it is shown that both these algorithms provide excellent performance in practice, and can identify all extremal sets from multi-gigabyte itemset data using only a single processor core.

### Into the Square: On the Complexity of Some Quadratic-time Solvable Problems

- Mathematics, Computer ScienceElectron. Notes Theor. Comput. Sci.
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### Computing the Subset Partial Order: Progress and Open Problems Extended Abstract

- Computer Science
- 2008

The published work on the problem of eeciently computing this partial order is reviewed, recent progress is described, and the outstanding questions are described.

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