Stony Brook Algorithm Repository

Generating Subsets


Input Description: An integer \(n\).
Problem: Generate (1) all, or (2) a random, or (3) the next subset of the integers \(1\) to \(n\).

Excerpt from The Algorithm Design Manual: A subset describes a selection of objects, where the order among them does not matter. Many of the algorithmic problems in this catalog seek the best subset of a group of things: vertex cover seeks the smallest subset of vertices to touch each edge in a graph; knapsack seeks the most profitable subset of items of bounded total size; and set packing seeks the smallest subset of subsets that together cover each item exactly once.

There are \(2n\) distinct subsets of an \(n\)-element set, including the empty set as well as the set itself. This grows exponentially, but at a considerably smaller rate than the \(n!\) permutations of \(n\) items. Indeed, since \(220 =\) 1,048,576, a brute-force search through all subsets of 20 elements is easily manageable, although by \(n=30\), \(230 =\) 1,073,741,824, so you will certainly be pushing things.


CAGES (rating 9)
cppitertools (rating 8)
Frank Ruskey's Combinatorial Generation Resources (rating 8)
Nijenhuis and Wilf (rating 8)
FFT (rating 8)
combinatorics.js (rating 7)
Combinatorica (rating 7)
Netlib (rating 3)

Recommended Books

The Art of Computer Programming, Volume 4 Fascicle 3: Generating All Combinations and Partitions by D. E. Knuth Combinatorial Algorithms : Generation, Enumeration, and Search by Donald L. Kreher and Douglas R. Stinson Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica by S. Pemmaraju and S. Skiena
Combinatorial Algorithms: an update by H. Wilf Combinatorial Algorithms for Computers and Calculators by A. Nijenhuis and H. Wilf

Related Problems

Generating Partitions

Generating Permutations

Random Number Generation

Set Data Structures

Go To Main Page