arXiv subjects

William Kuszmaul

Publications and source records attributed to William Kuszmaul.

At least 37 records · Page 2Linked to original sources

arXiv2026

History-Independent Load Balancing

We give a (strongly) history-independent two-choice balls-and-bins algorithm on $n$ bins that supports both insertions and deletions on a set of up to $m$ balls, while guaranteeing a maximum load of $m / n + O(1)$ with high probability, and achieving an expected recourse of $O(\log \log (m/n))$ per operation. To the best of our knowledge, this is the first history-independent solution to achieve nontrivial guarantees of any sort for $m/n \ge ω(1)$ and is the first fully dynamic solution (history independent or not) to achieve $O(1)$ overload with $o(m/n)$ expected recourse.

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arXiv2026

A Simple and Combinatorial Approach to Proving Chernoff Bounds and Their Generalizations

The Chernoff bound is one of the most widely used tools in theoretical computer science. It's rare to find a randomized algorithm that doesn't employ a Chernoff bound in its analysis. The standard proofs of Chernoff bounds are beautiful but in some ways not very intuitive. In this paper, I'll show you a different proof that has four features: (1) the proof offers a strong intuition for why Chernoff bounds look the way that they do; (2) the proof is user-friendly and (almost) algebra-free; (3) the proof comes with matching lower bounds, up to constant factors in the exponent; and (4) the proof extends to establish generalizations of Chernoff bounds in other settings. The ultimate goal is that, once you know this proof (and with a bit of practice), you should be able to confidently reason about Chernoff-style bounds in your head, extending them to other settings, and convincing yourself that the bounds you're obtaining are tight (up to constant factors in the exponent).

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arXiv2025

Succinct Dynamic Rank/Select: Bypassing the Tree-Structure Bottleneck

We show how to construct a dynamic ordered dictionary, supporting insert/delete/rank/select on a set of $n$ elements from a universe of size $U$, that achieves the optimal amortized expected time complexity of $O(1 + \log n / \log \log U)$, while achieving a nearly optimal space consumption of $\log \binom{U}{n} + n / 2^{(\log n)^{Ω(1)}} + \text{polylog}\, U$ bits in the regime where $U = \text{poly}(n)$. This resolves an open question by Pibiri and Venturini as to whether a redundancy (a.k.a. space overhead) of $o(n)$ bits is possible, and is the first dynamic solution to bypass the so-called tree-structure bottleneck, in which the bits needed to encode some dynamic tree structure are themselves enough to force a redundancy of $\widetildeΩ(n)$ bits. Our main technical building block is a dynamic balanced binary search tree, which we call the compressed tabulation-weighted treap, that itself achieves a surprising time/space tradeoff. The tree supports $\text{polylog}\, n$-time operations and requires a static lookup table of size $\text{poly}(n) + \text{polylog}\, U$ -- but, in exchange for these, the tree is able to achieve a remarkable space guarantee. Its total space redundancy is $O(\log U)$ bits. In fact, if the tree is given $n$ and $U$ for free, then the redundancy further drops to $O(1)$ bits.

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arXiv2025

Static Retrieval Revisited: To Optimality and Beyond

In the static retrieval problem, a data structure must answer retrieval queries mapping a set of $n$ keys in a universe $[U]$ to $v$-bit values. Information-theoretically, retrieval data structures can use as little as $nv$ bits of space. For small value sizes $v$, it is possible to achieve $O(1)$ query time while using space $nv + o(n)$ bits -- whether or not such a result is possible for larger values of $v$ (e.g., $v = Θ(\log n)$) has remained open. In this paper, we obtain a tight lower bound (as well as matching upper bounds) for the static retrieval problem. In the case where values are large, we show that there is actually a significant tension between time and space. It is not possible, for example, to get $O(1)$ query time using $nv + o(n)$ bits of space, when $v = Θ(\log n)$ (and assuming the word RAM model with $O(\log n)$-bit words). At first glance, our lower bound would seem to render retrieval unusable in many settings that aim to achieve very low redundancy. However, our second result offers a way around this: We show that, whenever a retrieval data structure $D_1$ is stored along with another data structure $D_2$ (whose size is similar to or larger than the size of $D_1$), it is possible to implement the combined data structure $D_1 \cup D_2$ so that queries to $D_1$ take $O(1)$ time, operations on $D_2$ take the same asymptotic time as if $D_2$ were stored on its own, and the total space is $nv + \mathrm{Space}(D_2) + n^{0.67}$ bits.

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arXiv2025

Fingerprint Filters Are Optimal

Dynamic filters are data structures supporting approximate membership queries to a dynamic set $S$ of $n$ keys, allowing a small false-positive error rate $\varepsilon$, under insertions and deletions to the set $S$. Essentially all known constructions for dynamic filters use a technique known as fingerprinting. This technique, which was first introduced by Carter et al. in 1978, inherently requires $$\log \binom{n \varepsilon^{-1}}{n} = n \log \varepsilon^{-1} + n \log e - o(n)$$ bits of space when $\varepsilon = o(1)$. Whether or not this bound is optimal for all dynamic filters (rather than just for fingerprint filters) has remained for decades as one of the central open questions in the area. We resolve this question by proving a sharp lower bound of $n \log \varepsilon^{-1} + n \log e - o(n)$ bits for $\varepsilon = o(1)$, regardless of operation time.

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arXiv2025

Optimal Non-Oblivious Open Addressing

A hash table is said to be open-addressed (or non-obliviously open-addressed) if it stores elements (and free slots) in an array with no additional metadata. Intuitively, open-addressed hash tables must incur a space-time tradeoff: The higher the load factor at which the hash table operates, the longer insertions/deletions/queries should take. In this paper, we show that no such tradeoff exists: It is possible to construct an open-addressed hash table that supports constant-time operations even when the hash table is entirely full. In fact, it is even possible to construct a version of this data structure that: (1) is dynamically resized so that the number of slots in memory that it uses, at any given moment, is the same as the number of elements it contains; (2) supports $O(1)$-time operations, not just in expectation, but with high probability; and (3) requires external access to just $O(1)$ hash functions that are each just $O(1)$-wise independent. Our results complement a recent lower bound by Bender, Kuszmaul, and Zhou showing that oblivious open-addressed hash tables must incur $Ω(\log \log \varepsilon^{-1})$-time operations. The hash tables in this paper are non-oblivious, which is why they are able to bypass the previous lower bound.

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arXiv2025

Optimal Bounds for Open Addressing Without Reordering

In this paper, we revisit one of the simplest problems in data structures: the task of inserting elements into an open-addressed hash table so that elements can later be retrieved with as few probes as possible. We show that, even without reordering elements over time, it is possible to construct a hash table that achieves far better expected search complexities (both amortized and worst-case) than were previously thought possible. Along the way, we disprove the central conjecture left by Yao in his seminal paper ``Uniform Hashing is Optimal''. All of our results come with matching lower bounds.

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arXiv2025

Tight Analyses of Ordered and Unordered Linear Probing

Linear-probing hash tables have been classically believed to support insertions in time $Θ(x^2)$, where $1 - 1/x$ is the load factor of the hash table. Recent work by Bender, Kuszmaul, and Kuszmaul (FOCS'21), however, has added a new twist to this story: in some versions of linear probing, if the \emph{maximum} load factor is at most $1 - 1/x$, then the \emph{amortized} expected time per insertion will never exceed $x \log^{O(1)} x$ (even in workloads that operate continuously at a load factor of $1 - 1/x$). Determining the exact asymptotic value for the amortized insertion time remains open. In this paper, we settle the amortized complexity with matching upper and lower bounds of $Θ(x \log^{1.5} x)$. Along the way, we also obtain tight bounds for the so-called path surplus problem, a problem in combinatorial geometry that has been shown to be closely related to linear probing. We also show how to extend Bender et al.'s bounds to say something not just about ordered linear probing (the version they study) but also about classical linear probing, in the form that is most widely implemented in practice.

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arXiv2025

The Multiplicative Version of Azuma's Inequality, with an Application to Contention Analysis

Azuma's inequality is a tool for proving concentration bounds on random variables. The inequality can be thought of as a natural generalization of additive Chernoff bounds. On the other hand, the analogous generalization of multiplicative Chernoff bounds does not appear to be widely known. We formulate a multiplicative-error version of Azuma's inequality. We then show how to apply this new inequality in order to greatly simplify (and correct) the analysis of contention delays in multithreaded systems managed by randomized work stealing.

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arXiv2025

Efficient $d$-ary Cuckoo Hashing at High Load Factors by Bubbling Up

A $d$-ary cuckoo hash table is an open-addressed hash table that stores each key $x$ in one of $d$ random positions $h_1(x), h_2(x), \ldots, h_d(x)$. In the offline setting, where all items are given and keys need only be matched to locations, it is possible to support a load factor of $1 - ε$ while using $d = \lceil \ln ε^{-1} + o(1) \rceil$ hashes. The online setting, where keys are moved as new keys arrive sequentially, has the additional challenge of the time to insert new keys, and it has not been known whether one can use $d = O(\ln ε^{-1})$ hashes to support $\poly(ε^{-1})$ expected-time insertions. In this paper, we introduce bubble-up cuckoo hashing, an implementation of $d$-ary cuckoo hashing that achieves all of the following properties simultaneously: (1) uses $d = \lceil \ln ε^{-1} + α\rceil$ hash locations per item for an arbitrarily small positive constant $α$. (2) achieves expected insertion time $O(δ^{-1})$ for any insertion taking place at load factor $1 - δ\le 1 - ε$. (3) achieves expected positive query time $O(1)$, independent of $d$ and $ε$. The first two properties give an essentially optimal value of $d$ without compromising insertion time. The third property is interesting even in the offline setting: it says that, even though \emph{negative} queries must take time $d$, positive queries can actually be implemented in $O(1)$ expected time, even when $d$ is large.

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arXiv2024

Tight Bounds and Phase Transitions for Incremental and Dynamic Retrieval

Retrieval data structures are data structures that answer key-value queries without paying the space overhead of explicitly storing keys. The problem can be formulated in four settings (static, value-dynamic, incremental, or dynamic), each of which offers different levels of dynamism to the user. In this paper, we establish optimal bounds for the final two settings (incremental and dynamic) in the case of a polynomial universe. Our results complete a line of work that has spanned more than two decades, and also come with a surprise: the incremental setting, which has long been viewed as essentially equivalent to the dynamic one, actually has a phase transition, in which, as the value size $v$ approaches $\log n$, the optimal space redundancy actually begins to shrink, going from roughly $n \log \log n$ (which has long been thought to be optimal) all the way down to $Θ(n)$ (which is the optimal bound even for the seemingly much-easier value-dynamic setting).

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arXiv2024

Tight Bounds for Classical Open Addressing

We introduce a classical open-addressed hash table, called rainbow hashing, that supports a load factor of up to $1 - \varepsilon$, while also supporting $O(1)$ expected-time queries, and $O(\log \log \varepsilon^{-1})$ expected-time insertions and deletions. We further prove that this tradeoff curve is optimal: any classical open-addressed hash table that supports load factor $1 - \varepsilon$ must incur $Ω(\log \log \varepsilon^{-1})$ expected time per operation. Finally, we extend rainbow hashing to the setting where the hash table is dynamically resized over time. Surprisingly, the addition of dynamic resizing does not come at any time cost -- even while maintaining a load factor of $\ge 1 - \varepsilon$ at all times, we can support $O(1)$ queries and $O(\log \log \varepsilon^{-1})$ updates. Prior to our work, achieving any time bounds of the form $o(\varepsilon^{-1})$ for all of insertions, deletions, and queries simultaneously remained an open question.

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arXiv2024

Scheduling Jobs with Work-Inefficient Parallel Solutions

This paper introduces the \emph{serial-parallel decision problem}. Consider an online scheduler that receives a series of tasks, where each task has both a parallel and a serial implementation. The parallel implementation has the advantage that it can make progress concurrently on multiple processors, but the disadvantage that it is (potentially) work-inefficient. As tasks arrive, the scheduler must decide for each task which implementation to use. We begin by studying \emph{total awake time}. We give a simple \emph{decide-on-arrival} scheduler that achieves a competitive ratio of $3$ for total awake time -- this scheduler makes serial/parallel decisions immediately when jobs arrive. Our second result is an \emph{parallel-work-oblivious} scheduler that achieves a competitive ratio of $6$ for total awake time -- this scheduler makes all of its decisions based only on the size of each serial job and without needing to know anything about the parallel implementations. Finally, we prove a lower bound showing that, if a scheduler wishes to achieve a competitive ratio of $O(1)$, it can have at most one of the two aforementioned properties (decide-on-arrival or parallel-work-oblivious). We also prove lower bounds of the form $1 + Ω(1)$ on the optimal competitive ratio for any scheduler. Next, we turn our attention to optimizing \emph{mean response time}. Here, we show that it is possible to achieve an $O(1)$ competitive ratio with $O(1)$ speed augmentation. This is the most technically involved of our results. We also prove that, in this setting, it is not possible for a parallel-work-oblivious scheduler to do well. In addition to these results, we present tight bounds on the optimal competitive ratio if we allow for arrival dependencies between tasks (e.g., tasks are components of a single parallel program), and we give an in-depth discussion of the remaining open questions.

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arXiv2024

A Nearly Quadratic Improvement for Memory Reallocation

In the Memory Reallocation Problem a set of items of various sizes must be dynamically assigned to non-overlapping contiguous chunks of memory. It is guaranteed that the sum of the sizes of all items present at any time is at most a $(1-\varepsilon)$-fraction of the total size of memory (i.e., the load-factor is at most $1-\varepsilon$). The allocator receives insert and delete requests online, and can re-arrange existing items to handle the requests, but at a reallocation cost defined to be the sum of the sizes of items moved divided by the size of the item being inserted/deleted. The folklore algorithm for Memory Reallocation achieves a cost of $O(\varepsilon^{-1})$ per update. In recent work at FOCS'23, Kuszmaul showed that, in the special case where each item is promised to be smaller than an $\varepsilon^4$-fraction of memory, it is possible to achieve expected update cost $O(\log\varepsilon^{-1})$. Kuszmaul conjectures, however, that for larger items the folklore algorithm is optimal. In this work we disprove Kuszmaul's conjecture, giving an allocator that achieves expected update cost $O(\varepsilon^{-1/2} \operatorname*{polylog} \varepsilon^{-1})$ on any input sequence. We also give the first non-trivial lower bound for the Memory Reallocation Problem: we demonstrate an input sequence on which any resizable allocator (even offline) must incur amortized update cost at least $Ω(\log\varepsilon^{-1})$. Finally, we analyze the Memory Reallocation Problem on a stochastic sequence of inserts and deletes, with random sizes in $[δ, 2 δ]$ for some $δ$. We show that, in this simplified setting, it is possible to achieve $O(\log\varepsilon^{-1})$ expected update cost, even in the ``large item'' parameter regime ($δ> \varepsilon^4$).

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arXiv2024

Nearly Optimal List Labeling

The list-labeling problem captures the basic task of storing a dynamically changing set of up to $n$ elements in sorted order in an array of size $m = (1 + Θ(1))n$. The goal is to support insertions and deletions while moving around elements within the array as little as possible. Until recently, the best known upper bound stood at $O(\log^2 n)$ amortized cost. This bound, which was first established in 1981, was finally improved two years ago, when a randomized $O(\log^{3/2} n)$ expected-cost algorithm was discovered. The best randomized lower bound for this problem remains $Ω(\log n)$, and closing this gap is considered to be a major open problem in data structures. In this paper, we present the See-Saw Algorithm, a randomized list-labeling solution that achieves a nearly optimal bound of $O(\log n \operatorname{polyloglog} n)$ amortized expected cost. This bound is achieved despite at least three lower bounds showing that this type of result is impossible for large classes of solutions.

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arXiv2024

Layered List Labeling

The list-labeling problem is one of the most basic and well-studied algorithmic primitives in data structures, with an extensive literature spanning upper bounds, lower bounds, and data management applications. The classical algorithm for this problem, dating back to 1981, has amortized cost $O(\log^2 n)$. Subsequent work has led to improvements in three directions: \emph{low-latency} (worst-case) bounds; \emph{high-throughput} (expected) bounds; and (adaptive) bounds for \emph{important workloads}. Perhaps surprisingly, these three directions of research have remained almost entirely disjoint -- this is because, so far, the techniques that allow for progress in one direction have forced worsening bounds in the others. Thus there would appear to be a tension between worst-case, adaptive, and expected bounds. List labeling has been proposed for use in databases at least as early as PODS'99, but a database needs good throughput, response time, and needs to adapt to common workloads (e.g., bulk loads), and no current list-labeling algorithm achieve good bounds for all three. We show that this tension is not fundamental. In fact, with the help of new data-structural techniques, one can actually \emph{combine} any three list-labeling solutions in order to cherry-pick the best worst-case, adaptive, and expected bounds from each of them.

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arXiv2023

Iceberg Hashing: Optimizing Many Hash-Table Criteria at Once

Despite being one of the oldest data structures in computer science, hash tables continue to be the focus of a great deal of both theoretical and empirical research. A central reason for this is that many of the fundamental properties that one desires from a hash table are difficult to achieve simultaneously; thus many variants offering different trade-offs have been proposed. This paper introduces Iceberg hashing, a hash table that simultaneously offers the strongest known guarantees on a large number of core properties. Iceberg hashing supports constant-time operations while improving on the state of the art for space efficiency, cache efficiency, and low failure probability. Iceberg hashing is also the first hash table to support a load factor of up to $1 - o(1)$ while being stable, meaning that the position where an element is stored only ever changes when resizes occur. In fact, in the setting where keys are $Θ(\log n)$ bits, the space guarantees that Iceberg hashing offers, namely that it uses at most $\log \binom{|U|}{n} + O(n \log \log n)$ bits to store $n$ items from a universe $U$, matches a lower bound by Demaine et al. that applies to any stable hash table. Iceberg hashing introduces new general-purpose techniques for some of the most basic aspects of hash-table design. Notably, our indirection-free technique for dynamic resizing, which we call waterfall addressing, and our techniques for achieving stability and very-high probability guarantees, can be applied to any hash table that makes use of the front-yard/backyard paradigm for hash table design.

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arXiv2022

IcebergHT: High Performance PMEM Hash Tables Through Stability and Low Associativity

Modern hash table designs strive to minimize space while maximizing speed. The most important factor in speed is the number of cache lines accessed during updates and queries. This is especially important on PMEM, which is slower than DRAM and in which writes are more expensive than reads. This paper proposes two stronger design objectives: stability and low-associativity. A stable hash table doesn't move items around, and a hash table has low associativity if there are only a few locations where an item can be stored. Low associativity ensures that queries need to examine only a few memory locations, and stability ensures that insertions write to very few cache lines. Stability also simplifies scaling and crash safety. We present IcebergHT, a fast, crash-safe, concurrent, and space-efficient hash table for PMEM based on the design principles of stability and low associativity. IcebergHT combines in-memory metadata with a new hashing technique, iceberg hashing, that is (1) space efficient, (2) stable, and (3) supports low associativity. In contrast, existing hash-tables either modify numerous cache lines during insertions (e.g. cuckoo hashing), access numerous cache lines during queries (e.g. linear probing), or waste space (e.g. chaining). Moreover, the combination of (1)-(3) yields several emergent benefits: IcebergHT scales better than other hash tables, supports crash-safety, and has excellent performance on PMEM (where writes are particularly expensive).

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