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1. Tiny Pointers

   Michael A. Bender; Alex Conway; Martín Farach-Colton; William Kuszmaul; Guido Tagliavini (January 2023, Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA))

   This paper introduces a new data-structural object that we call the tiny pointer. In many applications, traditional log n-bit pointers can be replaced with o(log n)-bit tiny pointers at the cost of only a constant-factor time overhead and a small probability of failure. We develop a comprehensive theory of tiny pointers, and give optimal constructions for both fixed-size tiny pointers (i.e., settings in which all of the tiny pointers must be the same size) and variable-size tiny pointers (i.e., settings in which the average tiny-pointer size must be small, but some tiny pointers can be larger). If a tiny pointer references an element in an array filled to load factor 1 — δ, then the optimal tiny-pointer size is Θ(log log log n + log δ-1) bits in the fixed-size case, and Θ(log δ-1) expected bits in the variable-size case. Our tiny-pointer constructions also require us to revisit several classic problems having to do with balls and bins; these results may be of independent interest. Using tiny pointers, we revisit five classic data-structure problems. We show that: • A data structure storing n v-bit values for n keys with constant-time modifications/queries can be implemented to take space nv + O(n log(r) n) bits, for any constant r > 0, as long as the user stores a tiny pointer of expected size O(1) with each key—here, log(r) n is the r-th iterated logarithm. • Any binary search tree can be made succinct with constant-factor time overhead, and can even be made to be within O(n) bits of optimal if we allow for O(log* n)-time modifications—this holds even for rotation-based trees such as the splay tree and the red-black tree. • Any fixed-capacity key-value dictionary can be made stable (i.e., items do not move once inserted) with constant-time overhead and 1 + o(1) space overhead. • Any key-value dictionary that requires uniform-size values can be made to support arbitrary-size values with constant-time overhead and with an additional space consumption of log(r) n + O(log j) bits per j-bit value for an arbitrary constant r > 0 of our choice. • Given an external-memory array A of size (1 + ε)n containing a dynamic set of up to n key-value pairs, it is possible to maintain an internal-memory stash of size O(n log ε-1) bits so that the location of any key-value pair in A can be computed in constant time (and with no IOs). These are all well studied and classic problems, and in each case tiny pointers allow for us to take a natural space-inefficient solution that uses pointers and make it space-efficient for free. 

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2. On the Optimal Time/Space Tradeoff for Hash Tables

   Bender, Michael; Farach-Colton, Martin; Kuszmaul, John; Kuszmaul, William; Liu, Mingmou (January 2022, STOC)

   For nearly six decades, the central open question in the study of hash tables has been to determine the optimal achievable tradeoff curve between time and space. State-of-the-art hash tables offer the following guarantee: If keys/values are Θ(logn) bits each, then it is possible to achieve constant-time insertions/deletions/queries while wasting only O(loglogn) bits of space per key when compared to the information-theoretic optimum. Even prior to this bound being achieved, the target of O(log log n) wasted bits per key was known to be a natural end goal, and was proven to be optimal for a number of closely related problems (e.g., stable hashing, dynamic retrieval, and dynamically-resized filters). This paper shows that O(log log n) wasted bits per key is not the end of the line for hashing. In fact, for any k ∈ [log∗ n], it is possible to achieve O(k)-time insertions/deletions, O(1)-time queries, and O(log(k) n) = Ologlog···logn 􏰟 􏰞􏰝 􏰠 k wasted bits per key (all with high probability in n). This means that, each time we increase inser- tion/deletion time by an additive constant, we reduce the wasted bits per key exponentially. We further show that this tradeoff curve is the best achievable by any of a large class of hash tables, including any hash table designed using the current framework for making constant-time hash tables succinct. Our results hold not just for fixed-capacity hash tables, but also for hash tables that are dynamically resized (this is a fundamental departure from what is possible for filters); and for hash tables that store very large keys/values, each of which can be up to no(1) bits (this breaks with the conventional wisdom that larger keys/values should lead to more wasted bits per key). For very small keys/values, we are able to tighten our bounds to o(1) wasted bits per key, even when k = O(1). Building on this, we obtain a constant-time dynamic filter that uses n􏰕logε−1􏰖+nloge+o(n) bits of space for a wide choice of 

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3. Approximate Nearest Neighbors in Limited Space

   https://doi.org/10.1137/1.9781611975031  

   Indyk, Piotr; Wagner, Tal (January 2018, Conference on Learning Theory)

   We consider the (1+ϵ)-approximate nearest neighbor search problem: given a set X of n points in a d-dimensional space, build a data structure that, given any query point y, finds a point x∈X whose distance to y is at most (1+ϵ)minx∈X ‖x−y‖ for an accuracy parameter ϵ∈(0,1). Our main result is a data structure that occupies only O(ϵ^−2 n log(n)log(1/ϵ)) bits of space, assuming all point coordinates are integers in the range {−n^O(1)…n^O(1)}, i.e., the coordinates have O(logn) bits of precision. This improves over the best previously known space bound of O(ϵ^−2 n log(n)^2), obtained via the randomized dimensionality reduction method of Johnson and Lindenstrauss (1984). We also consider the more general problem of estimating all distances from a collection of query points to all data points X, and provide almost tight upper and lower bounds for the space complexity of this problem. 

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4. Almost Optimal Exact Distance Oracles for Planar Graphs

   https://doi.org/10.1145/3580474  

   Charalampopoulos, Panagiotis; Gawrychowski, Paweł; Long, Yaowei; Mozes, Shay; Pettie, Seth; Weimann, Oren; Wulff-Nilsen, Christian (April 2023, Journal of the ACM)

   We consider the problem of preprocessing a weighted directed planar graph in order to quickly answer exact distance queries. The main tension in this problem is between space S and query time Q , and since the mid-1990s all results had polynomial time-space tradeoffs, e.g., Q = ~ Θ( n/√ S ) or Q = ~Θ( n 5/2 /S 3/2 ). In this article we show that there is no polynomial tradeoff between time and space and that it is possible to simultaneously achieve almost optimal space n 1+ o (1) and almost optimal query time n o (1) . More precisely, we achieve the following space-time tradeoffs: n 1+ o (1) space and log 2+ o (1) n query time, n log 2+ o (1) n space and n o (1) query time, n 4/3+ o (1) space and log 1+ o (1) n query time. We reduce a distance query to a variety of point location problems in additively weighted Voronoi diagrams and develop new algorithms for the point location problem itself using several partially persistent dynamic tree data structures. 

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5. Two-Dimensional Longest Common Extension Queries in Compact Space

   https://doi.org/10.4230/LIPICS.STACS.2025.38  

   Ganguly, Arnab; Gibney, Daniel; Shah, Rahul; Thankachan, Sharma V (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Beyersdorff, Olaf; Pilipczuk, Michał; Pimentel, Elaine; Thắng, Nguyễn Kim (Ed.)

   For a length n text over an alphabet of size σ, we can encode the suffix tree data structure in 𝒪(nlog σ) bits of space. It supports suffix array (SA), inverse suffix array (ISA), and longest common extension (LCE) queries in 𝒪(log^ε_σ n) time, which enables efficient pattern matching; here ε > 0 is an arbitrarily small constant. Further improvements are possible for LCE queries, where 𝒪(1) time queries can be achieved using an index of space 𝒪(nlog σ) bits. However, compactly indexing a two-dimensional text (i.e., an n× n matrix) has been a major open problem. We show progress in this direction by first presenting an 𝒪(n²log σ)-bit structure supporting LCE queries in near 𝒪((log_σ n)^{2/3}) time. We then present an 𝒪(n²log σ + n²log log n)-bit structure supporting ISA queries in near 𝒪(log n ⋅ (log_σ n)^{2/3}) time. Within a similar space, achieving SA queries in poly-logarithmic (even strongly sub-linear) time is a significant challenge. However, our 𝒪(n²log σ + n²log log n)-bit structure can support SA queries in 𝒪(n²/(σ log n)^c) time, where c is an arbitrarily large constant, which enables pattern matching in time faster than what is possible without preprocessing. We then design a repetition-aware data structure. The δ_2D compressibility measure for two-dimensional texts was recently introduced by Carfagna and Manzini [SPIRE 2023]. The measure ranges from 1 to n², with smaller δ_2D indicating a highly compressible two-dimensional text. The current data structure utilizing δ_2D allows only element access. We obtain the first structure based on δ_2D for LCE queries. It takes 𝒪^{~}(n^{5/3} + n^{8/5}δ_2D^{1/5}) space and answers queries in 𝒪(log n) time. 

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