Compressed Set Representations based on Set Difference

TL;DR

The paper introduces set-difference compressibility via $\Delta(\mathcal{S})$, enabling compact representations of a collection of sets $\mathcal{S}$ from a universe $\mathcal{U}$ by encoding symmetric differences. It proves this space bound equals the weight of a minimum spanning tree on a complete graph with edge weights $|S \triangle S'|$, yielding a two-tree representation rooted at $\emptyset$ and $\mathcal{U}$ that supports membership, access, rank, and predecessor/successor queries in $O(\log u)$ time with $O(\Delta(\mathcal{S}))$ space. The authors also present an improved MST-based construction, including suffix-tree techniques for fast symmetric-difference computations and a Prim-like incremental scheme, plus an enhanced insertion-compressibility framework and a hierarchy to support positive/negative access patterns. Together, these contributions enable efficient operations on compressed sets and show practical applicability to matrix/graph representations where row-wise similarity can be exploited. The work opens directions toward even more compact succinct representations and broader query capabilities on compressed combinatorial objects.

Abstract

We introduce a compressed representation of sets of sets that exploits how much they differ from each other. Our representation supports access, membership, predecessor and successor queries on the sets within logarithmic time. In addition, we give a new MST-based construction algorithm for the representation that outperforms standard ones.

Figures (4)

• Figure 1: An example of tree extraction, in which the original tree $T$ is shown on the left, $X=\{A, B, D, E, F, H, I, J, K, O\}$ is the set of shaded nodes, and the extracted tree $T_X$ is shown on the right.
• Figure 2: On the left, an insertion graph for sets $\mathcal{S} = \{ S_1,\ldots,S_9\}$, with the weights written as slanted values on the edges. It holds $\mathcal{I}(\mathcal{S})=20$. On the right, a corresponding insertion tree with 21 nodes. We write $S_i$ besides its corresponding node $v(S_i)$.
• Figure 3: On the left, a symdiff graph of minimum weight between the same sets of Figure \ref{['fig:ientropy']}, yielding $\Delta(\mathcal{S}) = 13$. To build it, it is sufficient to consider the full graph edges with weights up to $\ell=2$. Among these edges, those not belonging to the MST (i.e., the symdiff graph of minimum weight) are dashed. On the right, the indel trees, rooted at $\mathcal{U}$ (upside down) and at $\emptyset$, that represent $\mathcal{S}$ in space $O(\Delta(\mathcal{S}))$. We write $+x$ and $-x$ to indicate elements $x$ to add or to delete, respectively, from the parent node.
• Figure 4: The hierarchical extraction process for $\mathcal{T}^- = \mathcal{T}^-_{0,7}$, starting from the (upside down) tree $\mathcal{T}$ rooted at $\mathcal{U}$ of Figure \ref{['fig:dentropy']}. The rightward bold arrows lead from $\mathcal{T}^-_{a,b}$ to $\mathcal{T}^-_{a,m}$ (with label $0$) and to $\mathcal{T}^-_{m+1,b}$ (with label 1). We show the labels of the trees $\mathcal{T}^-_{a,b}$ inside the nodes, and the original symbols in small font near the boxes.

Theorems & Definitions (16)

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