Optimal Static Dictionary with Worst-Case Constant Query Time

TL;DR

The paper tackles building a succinct static dictionary with worst-case constant query time, achieving space of OPT plus a subpolynomial redundancy $n^{\varepsilon}$, where OPT $= \log\binom{U}{n}+n\log\sigma$. It introduces augmented retrieval and spillover representations to store variable-length components and per-bucket data, enabling near-OPT space while ensuring constant-time queries in the cell-probe model and extending to Word RAM. A key technical contribution is a sparsification technique via a hierarchical matrix construction and base-conversion-enabled retrieval, which allows concatenation of many small structures without incurring large redundancy. The work improves prior results by combining worst-case optimal space with constant-time queries and offers a framework (augmented redundancy) that may influence future static data-structure designs and open problems related to redundancy budgeting and construction complexity.

Abstract

In this paper, we design a new succinct static dictionary with worst-case constant query time. A dictionary data structure stores a set of key-value pairs with distinct keys in $[U]$ and values in $[σ]$, such that given a query $x\in [U]$, it quickly returns if $x$ is one of the input keys, and if so, also returns its associated value. The textbook solution to dictionaries is hash tables. On the other hand, the (information-theoretical) optimal space to encode such a set of key-value pairs is only $\text{OPT} := \log\binom{U}{n}+n\log σ$. We construct a dictionary that uses $\text{OPT} + n^ε$ bits of space, and answers queries in constant time in worst case. Previously, constant-time dictionaries are only known with $\text{OPT} + n/\text{poly}\log n$ space [Pǎtraşcu 2008], or with $\text{OPT}+n^ε$ space but expected constant query time [Yu 2020]. We emphasize that most of the extra $n^ε$ bits are used to store a lookup table that does not depend on the input, and random bits for hash functions. The "main" data structure only occupies $\text{OPT}+\text{poly}\log n$ bits.

Figures (4)

• Figure 1: Sparsifying a row
• Figure 2: Storing all spills of type $s$. (1) First, we convert part of $m_{\textup{fix}}$ from base $2^w$ to base $P^{(s)}$ using \ref{['thm:changing_base']}, getting $m_{\text{conv}}^{(s)}$; we also "round up" the universe of spills from $[K^{(s)}]$ to $[P^{(s)}]$. (2) Next, we build the augmented retrieval by \ref{['thm:augmented_retrieval']} using $m_{\text{conv}}^{(s)}$ as augmented elements, getting the representation of this data structure $m_{\text{retr}}^{(s)}$. (3) Finally, we convert it back to binary words using \ref{['thm:changing_base']}.
• Figure 3: Tree of blocks with $h = 4$ levels. Every rectangle represents a block on the tree, in which the hatched area represents the supplementary columns of a block, and the gray area represents the locations of non-zero entries in the matrix.
• Figure 4: Gaussian elimination on matrix $M$. (a) The matrix is partitioned into $3 \times 3$ blocks, where they correspond to row sets $Q_0, Q_1, Q_2$ from top to bottom, and column sets $C_1, C_2, C_0$ from left to right. The third column set $C_0$ consists of the $2\Delta_i$ supplementary columns of block $u$. Gray area of the matrix represents unknown non-zero entries. (b) Perform elimination within each child's submatrix, obtaining two identity submatrices. (c) Use two children's submatrices to eliminate entries on $Q_0$ above the identity parts. The non-zero entries in the hatched area changed during this step. (d) Permute the columns. We show that $M_{Q_0, T}$ has full rank.

Theorems & Definitions (31)

• Theorem 1.1
• Lemma 3.1
• Corollary 3.2
• Lemma 3.2
• Theorem 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4: NextPrime
• Theorem 4.5: Implicit in dodis2010changing
• proof : Proof of \ref{['thm:cell_probe_main']}