arXiv subjects

Dariusz Kowalski

Publications and source records attributed to Dariusz Kowalski.

3 recordsLinked to original sources

arXiv2026

Navigating Small-World Networks with Distance Predictions

The small-world phenomenon was given an algorithmic foundation by Kleinberg, who showed that in an augmented $k$-dimensional lattice a decentralized greedy algorithm delivers a message in $O(\log^2 n)$ expected steps. We study predicted-greedy routing, in which a mobile agent forwarding the message moves at each step to the neighbor minimizing a noisy $(\varepsilon,δ)$-prediction of its distance to the target, redrawn at every step from an oracle conditioned on the full routing history. Two cases arise from what this agent can observe. An agent with the coordinate awareness can still compute lattice distance exactly, but not graph distance in the shortcut-augmented network, since that depends on the shortcuts of nodes it has not yet visited; given an $(\varepsilon,δ)$-prediction of graph distance, information the classical model never supplies, it achieves expected delivery time $O(\log n/(1-4k\varepsilonδ))$, an asymptotic improvement over $Θ(\log^2 n)$. An agent with no coordinate awareness at all, the natural model for a privacy-preserving network whose nodes never disclose their coordinates, cannot compute even lattice distance; given an $(\varepsilon,δ)$-prediction of lattice distance instead, it still reaches the target in $O(n/(1-4k\varepsilonδ))$ expected steps. Together these results show that a modest amount of predicted information, of the right kind, is enough to accelerate decentralized routing well below Kleinberg's classical bound, and that even when nodes reveal no coordinates at all, reliable delivery remains achievable.

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arXiv2026

Short Graph Sketches Suffice for Error-resilient Leader Verification in CONGEST

Locally Checkable Proofs (LCPs) enable the verification of global graph properties using locally checkable certificates assigned by a prover. Recently, this framework was extended to Locally Checkable Proofs-with-Errors (LCPE), where an adversary may corrupt some certificates. Existing LCPE algorithms, however, are designed for the LOCAL model, whose unbounded communication makes them unsuitable for direct implementation in the bandwidth-restricted CONGEST model. We initiate the study of efficient CONGEST implementations of LCPE through the \textsc{unique-leader} verification problem on trees. The main challenge is that tolerating $\varepsilon$ certificate errors requires each node to reason about its $(2\varepsilon+1)$-hop neighborhood, whose exact topology may require up to $O(Δ^{2\varepsilon+1}\log n)$ bits to communicate. To overcome this bottleneck, we introduce \emph{local graph sketches}, together with the notions of \emph{imagined trees} and \emph{imagined certifications}, which encode precisely the information needed for verification using only $O(\varepsilon^2\log n)$ bits per node. Based on these sketches, we design an LCPE algorithm that tolerates up to $\varepsilon$ adversarial certificate errors and constructs the required sketches in $O(\varepsilon^2)$ communication rounds in the CONGEST model. We complement our algorithm with a matching impossibility result: even in the strictly more powerful LOCAL model, and even with unbounded certificate size, no verification scheme with view distance at most $\varepsilon$ can tolerate $\varepsilon$ adversarial certificate errors. Since every CONGEST algorithm can be simulated in LOCAL, this lower bound immediately applies to CONGEST, showing that a view distance exceeding $\varepsilon$ is unavoidable.

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arXiv2026

Distributed Local Verification using Proofs with(out) Errors

We study local verification of graph properties in distributed networks under the framework of \emph{locally checkable proofs} (LCPs). In an LCP, a prover assigns proof labels to nodes, and a distributed verifier must make all nodes accept if the graph satisfies the property, while at least one node rejects otherwise. Each node bases its decision on a local neighborhood, called its \emph{view distance}. Our focus is twofold. First, we study cycle existence, i.e., whether a graph contains a cycle (as opposed to cycle-freeness). We show that cycle existence admits verification with only $3$ proof labels and view distance $1$, and establish a matching lower bound. More importantly, inspired by direction-encoding techniques based on BFS distances, we introduce a novel gadget that encodes direction using only $2$ labels and view distance $3$ through repeated occurrences of the string $001101$. Although developed for cycle existence, this gadget may be useful for other verification tasks. Second, we introduce an \emph{erroneous proof} model in which an adversary may corrupt proof labels of at most $i$ nodes within the $(2i+1)$-hop neighborhood of each node. We present an algorithmic framework, called \textbf{\texttt{refix}}, that transforms an error-free verifier into one that tolerates such errors at the cost of a view distance of $2i+1$. We demonstrate the framework on cycle existence, cycle-freeness, and bipartiteness, and establish lower bounds relating the number of errors to the required view distance. Finally, we show that our $2$-label, view-distance-$3$ verifier for cycle existence admits a $3$-round implementation in the \textsc{CONGEST} model, providing a first step toward implementing LCPs under communication constraints.

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