# Set Data Structures

 Input Output

Input Description: A universe of objects $$U = \{ u_1,...,u_n\}$$, and a collection of subsets $$S_1,...,S_m$$, $$S_i \subset U$$.
Problem: Represent each subset so as to efficiently (1) test whether $$u_i \in S_j$$, (2) find the union or intersection of $$S_i$$ and $$S_j$$, (3) insert or delete members of $$S_i$$.

Excerpt from The Algorithm Design Manual: We distinguish sets from two other kinds of objects: strings and dictionaries. If there is no fixed-size universal set, a collection of objects is best thought of as a dictionary. If the order does matter in a subset, i.e. if {A,B,C} is not the same as {B,C,A}, then your structure is more profitably thought of as a string.

When each subset has cardinality exactly two, they form edges in a graph whose vertices are the universal set. A system of subsets with no restrictions on the cardinality of its members is called a hypergraph. It often can be profitable to consider whether your problem has a graph-theoretical analogy, like connected components or shortest path in a hypergraph.

Your primary alternatives for representing arbitrary systems of subsets are:

• Bit Vectors If your universal set U contains n items, an n-bit vector or array can represent any subset S subset of U. Bit i will be 1 if i is contained in S, otherwise bit i is 0. Since only one bit is used per element, bit vectors can be very space efficient for surprisingly large values of |U|. Element insertion and deletion simply flips the appropriate bit. Intersection and union are done by and-ing'' or or-ing'' the bits together. The only real drawback of a bit vector is that for sparse subsets, it takes O(n) time to explicitly identify all members of S.
• Containers or dictionaries A subset can also be represented using a linked list, array, binary tree, or dictionary containing exactly the elements in the subset. No notion of a fixed universal set is needed for such a data structure. For sparse subsets, dictionaries can be more space and time efficient than bit vectors and easier to work with and program. For efficient union and intersection operations, it pays to keep the elements in each subset sorted, so a linear-time traversal through both subsets identifies all duplicates.

### Related Problems

 Generating Partitions Generating Subsets Graph Data Structures Minimum Spanning Tree Set Cover

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