arXiv subjects

John Kuszmaul

Publications and source records attributed to John Kuszmaul.

8 recordsLinked to original sources

arXiv2026

Dynamic and Optimal Function Inversion in the Small-Time Regime

The classic function-inversion problem considers the task of constructing a data structure which, given access to a constant-time oracle for a function $f : [N] \rightarrow [N]$, supports efficient inverse-queries on $f$. This problem has been studied extensively in the small-space/large-time regime, where one wishes to use space $S$, say, $N^{1 - Ω(1)}$ bits, and where the query time is intended to be a small polynomial of $N$. Much less attention has been given to the \emph{small-time/large-space} regime, where $S = (N \log N) / t$ for some relatively small $t$, and where the goal is to achieve a good space bound as a function of $t$. In this paper, we give an optimal solution in the small-time regime, achieving space $S = O(N \log N / t)$ and time $O(t)$ for any $t \le O(\log N / \log \log N)$. This matches a lower bound by Yao (and is the first parameter regime where the lower bound has been matched for general functions). Additionally, we extend our solution to support point-updates to $f$, also in $O(t)$ time. Our techniques for supporting point updates also extend to the classic function-inversion solution of Fiat and Naor. As a sample application of our results, we show how to construct dynamic unordered graphs that use space $(1 + ε)$-close to information-theoretically optimal while offering adjacency queries, neighborhood queries, and edge insertions/deletions in amortized time $O(ε^{-1})$.

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arXiv2026

Strengths and Limitations of Greedy in Cup Games

In the cup game, an adversary distributes 1 unit of water among $n$ cups every time step. The player then selects a single cup from which to remove 1 unit of water. In the bamboo trimming problem, the adversary must choose fixed rates for the cups, and the player is additionally allowed to empty the chosen cup entirely. Past work has shown that the optimal backlog in these two settings is $Θ(\log n)$ and 2 respectively. The greedy algorithm has been shown in previous work to be exactly optimal in the general cup game and asymptotically optimal in the bamboo setting. The greedy algorithm has been conjectured [16] to achieve the exactly optimal backlog of 2 in the bamboo setting as well. In this paper, we prove a lower bound of $2.076$ for the backlog of the greedy algorithm, disproving the conjecture of [16]. We also introduce a new algorithm, a hybrid greedy/Deadline-Driven, which achieves backlog $O(\log n)$ in the general cup game, and remains exactly optimal for the bamboo trimming problem and the fixed-rate cup game -- this constitutes the first algorithm that achieves asymptotically optimal performance across all three settings. Additionally, we introduce a new model, the semi-oblivious cup game, in which the player is uncertain of the exact heights of each cup. We analyze the performance of the greedy algorithm in this setting, which can be viewed as selecting an arbitrary cup within a constant multiplicative factor of the fullest cup. We prove matching upper and lower bounds showing that the greedy algorithm achieves a backlog of $Θ(n^{\frac{c-1}{c}})$ in the semi-oblivious cup game. We also establish matching upper and lower bounds of $2^{Θ(\sqrt{\log n})}$ in the semi-oblivious cup flushing game. Finally, we show that in an additive error setting, greedy is actually able to achieve backlog $Θ(\log n)$, via matching upper and lower bounds.

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arXiv2026

Preprocessed 3SUM for Unknown Universes with Subquadratic Space

We consider the classic 3SUM problem: given sets of integers $A, B, C $, determine whether there is a tuple $(a, b, c) \in A \times B \times C$ satisfying $a + b + c = 0$. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets $A, B$ of size $n$, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form $A', B', C'$, where $A' \subseteq A$ and $B' \subseteq B$. A series of results have achieved $\tilde{O}(n^2)$ preprocessing time, $\tilde{O}(n^2)$ space, and query time improving progressively from $\tilde{O}(n^{1.9})$ [CL15] to $\tilde{O}(n^{11/6})$ [CVX23] to $\tilde{O}(n^{1.5})$ [KPS25]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes? We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving $\tilde{O}(n^{1.5 + \varepsilon})$ query time and $\tilde{O}(n^{2 - 2\varepsilon/3})$ space complexity. Furthermore, our algorithm has $\tilde{O}(n^2)$ preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least $n^{2-o(1)}$ under the 3SUM Hypothesis.

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arXiv2025

Fully Energy-Efficient Randomized Backoff: Slow Feedback Loops Yield Fast Contention Resolution

Contention resolution addresses the problem of coordinating access to a shared channel. Time proceeds in slots, and a packet transmission can be made in any slot. A packet is successfully sent if no other packet is also transmitted during that slot. If two or more packets are sent in the same slot, then none of these transmissions succeed. Listening during a slot gives ternary feedback, indicating if that slot had (0) silence, (1) a successful transmission, or (2+) noise. No other feedback is available. Packets are (adversarially) injected into the system over time. A packet departs the system once it is successful. The goal is to send all packets while optimizing throughput, which is roughly the fraction of successful slots. Most prior algorithms with constant throughput require a short feedback loop, in the sense that a packet's sending probability in slot t+1 is fully determined by its internal state at slot t and the channel feedback at slot t. An open question is whether these short feedback loops are necessary; that is, how often must listening and updating occur in order to achieve constant throughput? This question addresses energy efficiency, since both listening and sending consume significant energy. The channel can also suffer adversarial noise ("jamming"), which causes any listener to hear noise, even when no packets are sent. How does jamming affect our goal of long feedback loops/energy efficiency? Connecting these questions, we ask: what does a contention-resolution algorithm have to sacrifice to reduce channel accesses? Must we give up on constant throughput or robustness to noise? Here, we show that we need not concede anything. Suppose there are N packets and J jammed slots, where the input is determined by an adaptive adversary. We give an algorithm that, with high probability in N+J, has constant throughput and polylog(N+J) channel accesses per packet.

cs.DC↗

arXiv2022

A Nearly Tight Lower Bound for the $d$-Dimensional Cow-Path Problem

In the $d$-dimensional cow-path problem, a cow living in $\mathbb{R}^d$ must locate a $(d - 1)$-dimensional hyperplane $H$ whose location is unknown. The only way that the cow can find $H$ is to roam $\mathbb{R}^d$ until it intersects $\mathcal{H}$. If the cow travels a total distance $s$ to locate a hyperplane $H$ whose distance from the origin was $r \ge 1$, then the cow is said to achieve competitive ratio $s / r$. It is a classic result that, in $\mathbb{R}^2$, the optimal (deterministic) competitive ratio is $9$. In $\mathbb{R}^3$, the optimal competitive ratio is known to be at most $\approx 13.811$. But in higher dimensions, the asymptotic relationship between $d$ and the optimal competitive ratio remains an open question. The best upper and lower bounds, due to Antoniadis et al., are $O(d^{3/2})$ and $Ω(d)$, leaving a gap of roughly $\sqrt{d}$. In this note, we achieve a stronger lower bound of $\tildeΩ(d^{3/2})$.

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arXiv2022

Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too

The bamboo trimming problem considers $n$ bamboo with growth rates $h_1, h_2, \ldots, h_n$ satisfying $\sum_i h_i = 1$. During a given unit of time, each bamboo grows by $h_i$, and then the bamboo-trimming algorithm gets to trim one of the bamboo back down to height zero. The goal is to minimize the height of the tallest bamboo, also known as the backlog. The bamboo trimming problem is closely related to many scheduling problems, and can be viewed as a variation of the widely-studied fixed-rate cup game, but with constant-factor resource augmentation. Past work has given sophisticated pinwheel algorithms that achieve the optimal backlog of 2 in the bamboo trimming problem. It remained an open question, however, whether there exists a simple algorithm with the same guarantee -- recent work has devoted considerable theoretical and experimental effort to answering this question. Two algorithms, in particular, have appeared as natural candidates: the Reduce-Max algorithm (which always cuts the tallest bamboo) and the Reduce-Fastest$(x)$ algorithm (which cuts the fastest-growing bamboo out of those that have height at least $x$). Both algorithms are conjectured to achieve backlog 2. This paper improves the bounds for both Reduce-Fastest and Reduce-Max. Among other results, we show that the exact optimal backlog for Reduce-Fastest$(x)$ is $x + 1$ for all $x \ge 2$ (proving a conjecture of D'Emidio, Di Stefano, and Navarra in the case of $x = 2$), and we show that Reduce-Fastest$(1)$ does not achieve backlog 2 (disproving a conjecture of D'Emidio, Di Stefano, and Navarra). Finally, we show that there is a different algorithm, which we call the Deadline-Driven Strategy, that is both very simple and achieves the optimal backlog of 2. This resolves the question as to whether there exists a simple worst-case optimal algorithm for the bamboo trimming problem.

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arXiv2022

Contention Resolution for Coded Radio Networks

Randomized backoff protocols, such as exponential backoff, are a powerful tool for managing access to a shared resource, often a wireless communication channel (e.g., [1]). For a wireless device to transmit successfully, it uses a backoff protocol to ensure exclusive access to the channel. Modern radios, however, do not need exclusive access to the channel to communicate; in particular, they have the ability to receive useful information even when more than one device transmits at the same time. These capabilities have now been exploited for many years by systems that rely on interference cancellation, physical layer network coding and analog network coding to improve efficiency. For example, Zigzag decoding [56] demonstrated how a base station can decode messages sent by multiple devices simultaneously. In this paper, we address the following question: Can we design a backoff protocol that is better than exponential backoff when exclusive channel access is not required. We define the Coded Radio Network Model, which generalizes traditional radio network models (e.g., [30]). We then introduce the Decodable Backoff Algorithm, a randomized backoff protocol that achieves an optimal throughput of $1-o(1)$. (Throughput $1$ is optimal, as simultaneous reception does not increase the channel capacity.) The algorithm breaks the constant throughput lower bound for traditional radio networks [47-49], showing the power of these new hardware capabilities.

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arXiv2021

On the Optimal Time/Space Tradeoff for Hash Tables

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 Theta(log n) bits each, then it is possible to achieve constant-time insertions/deletions/queries while wasting only O(loglog n) bits of space per key when compared to the information-theoretic optimum. Even prior to this bound being achieved, the target of O(loglog 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(loglog n) wasted bits per key is not the end of the line for hashing. In fact, for any k \in [log* n], it is possible to achieve O(k)-time insertions/deletions, O(1)-time queries, and O(\log^{(k)} n) wasted bits per key (all with high probability in n). This means that, each time we increase insertion/deletion time by an \emph{additive constant}, we reduce the wasted bits per key \emph{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.

cs.DS↗