Locality in Open Addressing Hash Tables

Abstract: Open-addressed hash tables without reordering, such as linear probing and uniform probing, are among the simplest and most widely used data structures. Their performance is traditionally measured by probe count. We study a complementary parameter: locality, defined as the geometric distance from the first probed location to the farthest cell inspected or used. At load factor $1-\varepsilon$, uniform probing achieves the optimal $Θ(1/\varepsilon)$ probe count among greedy schemes, but has essentially no locality, whereas linear probing is highly local but performs $Θ(1/\varepsilon<sup>2)$ probes. We show that this quadratic locality scale is fundamental: no open-addressing algorithm without reordering can achieve locality $o(1/\varepsilon<sup>2)$ simultaneously at every load $1-\varepsilon$. We also prove an amortized expected-locality lower bound of $Ω(1/\varepsilon)$ over any sequence of $(1-\varepsilon)n$ insertions, even when the final load is known in advance. Our lower bound further implies that page size $B=Ω(1/\varepsilon<sup>2)$ is necessary for $1+o(1)$ expected page span in immutable open addressing. We complement these lower bounds with two upper bounds. When the target load is known in advance, every insertion and every successful or unsuccessful search can be given expected probe count and locality $\widetilde O(1/\varepsilon)$, essentially deamortizing the amortized lower bound. We also give a load-oblivious greedy scheme with optimal expected probe count $Θ(1/\varepsilon)$ whose $i$-th probe is at distance $O(i<sup>2)$ from the first probe. Its analysis gives a general variance bound for occupied-cell densities in symmetric probing schemes, implying an $O(\log n/\varepsilon<sup>2)$ expected probe bound for every fixed-shift probing sequence and every load $1-\varepsilon$.

• The paper proves that no-reordering open-addressing schemes incur expected locality of Ω(1/ε²) at some loads, establishing linear probing’s quadratic locality as an inherent all-load barrier and showing that page sizes of Ω(1/ε²) are necessary for near-one-transfer operations.
• The paper develops a distribution-free optimal-transport lower bound and a variance theorem for symmetric probing schemes, then uses them to design load-oblivious and known-load constructions with near-optimal probe counts and locality up to polylogarithmic factors.
• The results show that probe count and locality are distinct design objectives: linear probing is optimal for all-load locality, while proposed schemes approach O(1/ε) expected probes, leaving open whether matching locality guarantees and stronger tail bounds are possible.

This paper, "Locality in Open Addressing Hash Tables" (2607.16390) by Or Zamir, initiates the formal study of locality as a performance measure for open-addressed hash tables, complementing the classical probe-count metric. The central finding is that the quadratic locality scale of linear probing is not an artifact of that particular scheme but an inherent barrier: in open addressing without reordering, expected locality $\Omega(1/\varepsilon^2)$ is unavoidable at all loads $1-\varepsilon$. This provides a theoretical explanation for the persistent practical dominance of linear probing, whose probes scan contiguous memory and interact well with caches and prefetching, despite being suboptimal in probe count.

Model and cost measure

The table is identified with the cyclic group $\mathbb{Z}_n$. For each inserted key, the paper distinguishes the home location $H_i$ (first probed cell) from the placement $Y_i$ (final storage cell). The locality radius $R_i$ of an insertion is the maximum cyclic distance from $H_i$ to any cell inspected during the insertion, including the final placement cell. The setting is deliberately permissive: the algorithm may be adaptive, randomized, table-aware, and need not be greedy or follow a fixed probe sequence. The only structural restriction is no reordering — once a key is placed, it is never moved. This immutable setting is common in practice and has been the focus of recent work on open addressing beyond greedy schemes (Farach-Colton et al., 4 Jan 2025).

The contrast motivating the paper is well known: at load $1-\varepsilon$, uniform probing achieves the optimal greedy probe count $\Theta(1/\varepsilon)$ [Yao 1985] but probes essentially random locations, while linear probing incurs $\Theta(1/\varepsilon^2)$ probes due to primary clustering yet is maximally local.

Lower bounds via optimal transport

The technical core of the lower bound is a distribution-free transport statement on the cycle. Given $1-\varepsilon$0 i.i.d. home locations $1-\varepsilon$1 drawn from an arbitrary distribution $1-\varepsilon$2 on $1-\varepsilon$3, the expected optimal transport cost (Earth Mover's Distance) to a configuration of distinct cells is always $1-\varepsilon$4. The proof proceeds in two steps. First, for the uniform distribution, an interval-overload argument shows that intervals of length $1-\varepsilon$5 frequently contain more home locations than they can hold: the excess, $1-\varepsilon$6 for $1-\varepsilon$7, is $1-\varepsilon$8 by a Berry–Esseen normal-approximation argument, and each unit of excess forces at least unit transport distance via a combinatorial covering argument. Second, a convexity lemma — $1-\varepsilon$9 is convex in $\mathbb{Z}_n$0, since $\mathbb{Z}_n$1 — combined with Jensen's inequality shows that the uniform distribution minimizes the expected interval overload, so the bound holds for every $\mathbb{Z}_n$2.

The reduction to hashing is direct. Inserting uniformly random elements from a universe of size $\mathbb{Z}_n$3 makes the home locations i.i.d. samples from some induced distribution $\mathbb{Z}_n$4, and since every insertion must inspect its final placement cell, $\mathbb{Z}_n$5. This yields:

• Amortized lower bound. Any no-reordering algorithm has $\mathbb{Z}_n$6, i.e., amortized expected locality $\mathbb{Z}_n$7, even when $\mathbb{Z}_n$8, $\mathbb{Z}_n$9, and $H_i$0 are known in advance. The assumptions are minimal — open addressing, stateless first probes, and a large universe — and each is shown necessary for the statement to be meaningful.
• All-load lower bound. No such algorithm achieves expected insertion locality $H_i$1 at every load $H_i$2. This follows by integrating the amortized bound over the load profile: a hypothetical $H_i$3 guarantee at all loads contradicts the $H_i$4 total. The all-load quantifier matters because search costs depend on insertions performed at earlier, possibly different, loads.

Both bounds are matched exactly by linear probing, which has expected locality $H_i$5 at slack $H_i$6 and $H_i$7 total over the insertion sequence. The lower bounds therefore establish that linear probing is optimal with respect to all-load locality, even though its probe count is far from optimal.

External-memory consequence

The locality measure directly controls cache and disk behavior. In the external-memory model with pages of $H_i$8 consecutive cells, an operation of locality $H_i$9 spans $Y_i$0 pages. Consequently, if a no-reordering scheme achieves $Y_i$1 expected page span at load $Y_i$2, then page size $Y_i$3 is necessary. This supplies the first lower bound explaining the long-standing $Y_i$4 barrier for $Y_i$5 block transfers in immutable open addressing, previously achieved constructively by Jensen and Pagh but never shown necessary. The result is complementary to graveyard hashing (Bender et al., 2021) and zombie hashing, which do beat the $Y_i$6 threshold by rebuilding and reordering — confirming that the barrier is specific to the no-reordering regime.

Upper bound: load-oblivious greedy scheme with optimal probes

The paper's main technical contribution on the upper-bound side is a variance bound for symmetric probing schemes: in any greedy, translation-invariant scheme (covering linear probing, double hashing, uniform probing, and the new construction), for every fixed set $Y_i$7 the number of occupied cells satisfies $Y_i$8. The proof uses the Efron–Stein inequality together with an elegant coupling argument: resampling one key's random data changes the final occupied set by at most a single swap of two cells, and by translation invariance each swapped cell lands in $Y_i$9 with probability at most $R_i$0. This shows that the occupied-set density in any fixed test set has binomial-like variance, despite the insertion process being highly adaptive — a robust pseudorandomness property that may be of independent interest.

Two consequences follow. First, for any fixed-offset probe sequence (even with dependent, randomized shifts), expected probe count at load $R_i$1 is $R_i$2 at every load. This is weaker than the constant-load result of Kuszmaul and Xi for quadratic probing but is, to the author's knowledge, the first all-load guarantee for arbitrary fixed-shift schemes; in regimes $R_i$3 it gives $R_i$4, precluding uniformly worse polynomial behavior.

Second, the paper analyzes an expanding-window sampler: it guesses the slack $R_i$5, and for each guess samples $R_i$6 cells uniformly from an interval of length $R_i$7 around the home location. A Chebyshev argument on the variance lemma shows such an interval contains an $R_i$8 fraction of empty cells with constant probability, giving expected probe count $R_i$9 — optimal among greedy schemes — with the $H_i$0-th probe at distance $H_i$1 from the home location, so $H_i$2 deterministically. This unconditionally delivers the locality profile one would obtain from quadratic probing if its conjectured $H_i$3 high-load probe count held.

A notable caveat: the combination $H_i$4 and $H_i$5 yields locality $H_i$6 only with high constant probability, not in expectation, since a first-moment bound on $H_i$7 does not control $H_i$8. Closing this gap requires stronger tail bounds on occupied-cell densities, which the variance lemma does not provide.

Upper bound: deamortized known-load construction

When the target slack $H_i$9 is known in advance, the amortized $1-\varepsilon$0 lower bound can essentially be deamortized. The construction replaces temporal cost discrepancy with per-insertion probabilistic discrepancy: a hierarchical block structure in which each key carries reproducible flexibility bits at each level; with probability $1-\varepsilon$1 the key is proactively sent to a least-loaded child block, and otherwise it stays in its home block. Most keys pay a small cost, while a small fraction intentionally pay more, leaving local space for the rest.

The analysis rests on two lemmas: a biased minimum-allocation lemma proved via an exponential potential function, showing that interleaving uniform allocations with least-loaded choices bounds the maximum bin imbalance by $1-\varepsilon$2; and a top-block overload lemma using Chernoff bounds and a no-crossing monotonicity argument (valid because there are no deletions, so a currently nonfull block was never full). A two-level version achieves $1-\varepsilon$3, and a $1-\varepsilon$4-level hierarchy with $1-\varepsilon$5 levels yields:

$1-\varepsilon$6

for every insertion and every successful or unsuccessful search. Thus the amortized lower bound is tight up to polylogarithmic factors, and no individual insertion pays the quadratic cost. The same guarantee covers successful searches for fixed earlier keys, via a tagged version of the overload lemma.

Limitations and open problems

The paper is explicit about the remaining gaps. The known-load upper bound carries a $1-\varepsilon$7 factor; removing it, to match the amortized lower bound exactly with $1-\varepsilon$8 per operation, is open, as is any separation between the amortized bound and the worst per-operation cost. The load-oblivious sampler lacks an expected-locality guarantee — only high-probability locality $1-\varepsilon$9 is known — and the missing ingredient is a tail bound beyond the variance lemma. More generally, the paper asks what moment or tail bounds hold uniformly for $\Theta(1/\varepsilon)$0 across all symmetric probing mechanisms. Finally, the lower bounds apply only to no-reordering schemes; recent non-greedy constructions beat the greedy probe-count barrier, and whether such schemes can retain optimal $\Theta(1/\varepsilon)$1 all-load locality while achieving $\Theta(1/\varepsilon)$2 probes remains open, as does extending the locality framework to schemes that move or rebuild keys.

Conclusion

This paper establishes locality as a rigorous second axis for evaluating open-addressed hash tables and proves that, without reordering, linear probing's quadratic locality is optimal: no immutable scheme can achieve $\Theta(1/\varepsilon)$3 expected locality at all loads, and page size $\Theta(1/\varepsilon)$4 is necessary for $\Theta(1/\varepsilon)$5 expected page span in external memory. The lower bounds follow from a distribution-free optimal-transport theorem on the cycle, and the upper bounds from a general variance bound for symmetric probing schemes together with a hierarchical deamortization construction achieving $\Theta(1/\varepsilon)$6 expected probe count and locality per operation. The results cleanly delineate what recent probe-count improvements can and cannot deliver: probe count can be pushed below the greedy optimum, but the all-load locality scale cannot.