arXiv subjects

Richard Hladík

Publications and source records attributed to Richard Hladík.

8 recordsLinked to original sources

arXiv2026

Bidirectional Dijkstra's Algorithm is Instance-Optimal

Although Dijkstra's algorithm has near-optimal time complexity for the problem of finding a shortest path from a given vertex $s$ to a given vertex $t$, in practice other algorithms are often superior on huge graphs. A prominent example is bidirectional search, which concurrently executes Dijkstra's algorithm forward from $s$ and backward from $t$, and stops when these executions meet. In this paper, we give a strong theoretical justification for the use of bidirectional search to find a shortest $st$-path. We prove that for weighted multigraphs, both directed and undirected, a careful implementation of bidirectional search is instance-optimal with respect to the number of edges it examines. That is, we prove that no correct algorithm can outperform our implementation of bidirectional search on any single instance by more than a constant factor. For unweighted graphs, we show that bidirectional breadth-first search is instance-optimal up to a factor of $O(Δ)$ where $Δ$ is the maximum degree of the graph. We also show that this is best possible.

cs.DS↗

arXiv2026

Heaps and Their Working Sets

We construct a heap with strong beyond-worst-case performance guarantees and explore the analysis of such heaps. First, we unify existing notions of the working-set bound for heaps by proving that essentially all of them are equivalent - with the notable exception of the so-called stack-like bound, which is strictly stronger. This equivalence simplifies the theoretical landscape and extends the range of applications of heaps with working-set bounds. Second, we present the first heap implementation that has the amortized stack-like bound and supports $\mathcal O(1)$-time decrease-key and $o(\log^*n)$-time insert.

cs.DS↗

arXiv2026

Splay trees are almost dynamically optimal

Sleator and Tarjan [JACM, 1985] conjectured that splay trees are dynamically optimal -- that on every access sequence, they perform within a constant factor of the optimal offline dynamic binary search tree. Despite four decades of work, no $o(\log n)$ competitive ratio was known. We prove that splay trees are $O(\log\log n \cdot \log^2\log\log n)=\tilde{O}(\log\log n)$-competitive.

cs.DS↗

arXiv2026

Fast and Simple Sorting Using Partial Information

We consider the problem of sorting $n$ items, given the outcomes of $m$ pre-existing comparisons. We present a simple and natural deterministic algorithm that runs in $O(m + \log T)$ time and does $O(\log T)$ comparisons, where $T$ is the number of total orders consistent with the pre-existing comparisons. Our running time and comparison bounds are best possible up to constant factors, thus resolving a problem that has been studied intensely since 1976 (Fredman, Theoretical Computer Science). The best previous algorithm with a bound of $O(\log T)$ on the number of comparisons has a time bound of $O(n^{2.5})$ and is more complicated. Our algorithm combines three classic algorithms: topological sort, heapsort with the right kind of heap, and efficient search in a sorted list. It outputs the items in sorted order one by one. It can be modified to stop early, thereby solving the important and more general top-$k$ sorting problem: Given $k$ and the outcomes of some pre-existing comparisons, output the smallest $k$ items in sorted order. The modified algorithm solves the top-$k$ sorting problem in minimum time and comparisons, to within constant factors.

cs.DS↗

arXiv2026

Stronger Directed Low-Diameter Decompositions with Sub-Logarithmic Diameter and Separation

This paper significantly strengthens directed low-diameter decompositions in several ways. We define and give the first results for separated low-diameter decompositions in directed graphs, tighten and generalize probabilistic guarantees, and prove new independence results between (far away) edges. Our results are the first to give meaningful guarantees for decompositions with small diameters $D = Ω(\log\log n)$ in contrast to the state of the art that only applies to super-logarithmic diameters $D = ω(\log n)$. These results transfer several important and widely used aspects of undirected low-diameter decompositions to the directed setting. All our results are algorithmic -- small modifications to two existing directed low-diameter decompositions [BFHL25; Li25] can be used to sample decompositions with our new guarantees in near-linear time $\tilde{O}(m)$.

cs.DS↗

arXiv2025

Universal Optimality of Dijkstra via Beyond-Worst-Case Heaps

In this paper we prove that Dijkstra's shortest-path algorithm, if implemented with a sufficiently efficient heap, is universally optimal in its running time, and with suitable small additions is also universally optimal in its number of comparisons. Universal optimality is a powerful beyond-worst-case performance guarantee for graph algorithms that informally states that a single algorithm on a problem involving graphs with arc and/or vertex weights performs as well as possible on every graph, assuming a worst-case choice of weights. We give the first application of this notion to any sequential algorithm. We design a new heap data structure with a working-set bound, which guarantees that the heap takes advantage of a certain kind of locality in the heap operations. Our heap has the optimal (amortized) bounds of Fibonacci heaps but also has the beyond-worst-case guarantee that the cost of deleting the minimum item is logarithmic in the number of items inserted after it but before it is deleted, instead of logarithmic in the size of the heap when the item is deleted. That is, deletion of recently inserted items is especially efficient. We prove that our working-set bound guarantees universal optimality for the problem of ordering vertices by their distance from the source vertex, which we call the distance order problem. Our result relies on the observation that the sequence of heap operations generated by any run of Dijkstra's algorithm on a fixed graph possesses enough locality that one can couple the number of comparisons performed by any heap with our working-set bound to the minimum number of comparisons required to solve the distance order problem on this graph for a worst-case choice of arc lengths.

cs.DS↗

arXiv2024

Smooth Sensitivity Revisited: Towards Optimality

Smooth sensitivity is one of the most commonly used techniques for designing practical differentially private mechanisms. In this approach, one computes the smooth sensitivity of a given query $q$ on the given input $D$ and releases $q(D)$ with noise added proportional to this smooth sensitivity. One question remains: what distribution should we pick the noise from? In this paper, we give a new class of distributions suitable for the use with smooth sensitivity, which we name the PolyPlace distribution. This distribution improves upon the state-of-the-art Student's T distribution in terms of standard deviation by arbitrarily large factors, depending on a "smoothness parameter" $γ$, which one has to set in the smooth sensitivity framework. Moreover, our distribution is defined for a wider range of parameter $γ$, which can lead to significantly better performance. Moreover, we prove that the PolyPlace distribution converges for $γ\rightarrow 0$ to the Laplace distribution and so does its variance. This means that the Laplace mechanism is a limit special case of the PolyPlace mechanism. This implies that out mechanism is in a certain sense optimal for $γ\to 0$.

cs.CR↗

arXiv2024

Near-Universally-Optimal Differentially Private Minimum Spanning Trees

Devising mechanisms with good beyond-worst-case input-dependent performance has been an important focus of differential privacy, with techniques such as smooth sensitivity, propose-test-release, or inverse sensitivity mechanism being developed to achieve this goal. This makes it very natural to use the notion of universal optimality in differential privacy. Universal optimality is a strong instance-specific optimality guarantee for problems on weighted graphs, which roughly states that for any fixed underlying (unweighted) graph, the algorithm is optimal in the worst-case sense, with respect to the possible setting of the edge weights. In this paper, we give the first such result in differential privacy. Namely, we prove that a simple differentially private mechanism for approximately releasing the minimum spanning tree is near-optimal in the sense of universal optimality for the $\ell_1$ neighbor relation. Previously, it was only known that this mechanism is nearly optimal in the worst case. We then focus on the $\ell_\infty$ neighbor relation, for which the described mechanism is not optimal. We show that one may implement the exponential mechanism for MST in polynomial time, and that this results in universal near-optimality for both the $\ell_1$ and the $\ell_\infty$ neighbor relations.

cs.CR↗