Local reconstructors and tolerant testers for connectivity and diameter

A local property reconstructor for a graph property is an algorithm which, given oracle access to the adjacency list of a graph that is “close” to having the property, provides oracle access to the adjacency matrix of a “correction” of the graph, i.e. a graph which has the property and is close to the given graph. For this model, we achieve local property reconstructors for the properties of connectivity and $k$-connectivity in undirected graphs, and the property of strong connectivity in directed graphs. Along the way, we present a method of transforming a local reconstructor (which acts as a “adjacency matrix oracle” for the corrected graph) into an “adjacency list oracle”. This allows us to recursively use our local reconstructor for $(k-1)$-connectivity to obtain a local reconstructor for $k$-connectivity. We also extend this notion of local property reconstruction to parametrized graph properties (for instance, having diameter at most $D$ for some parameter $D$) and require that the corrected graph has the property with parameter close to the original. We obtain a local reconstructor for the low diameter property, where if the original graph is close to having diameter $D$, then the corrected graph has diameter roughly 2D. We also exploit a connection between local property reconstruction and property testing, observed by Brakerski, to obtain new tolerant property testers for all of the aforementioned properties. Except for the one for connectivity, these are the first tolerant property testers for these properties.

💡 Research Summary

The paper introduces a new algorithmic framework called local property reconstruction and applies it to several fundamental graph properties: (i) connectivity, (ii) k‑connectivity in undirected graphs, (iii) strong connectivity in directed graphs, and (iv) low‑diameter (parameterized by D). A local reconstructor receives oracle access to the adjacency‑list representation of a graph G that is ε‑close to having a target property P (i.e., a graph satisfying P can be obtained from G by editing at most ε·|V|·|E| edges). Instead of outputting the whole corrected graph, the reconstructor provides an oracle for the adjacency matrix of a corrected graph G′ that does satisfy P and remains close to G. Crucially, each query to this oracle must be answered by looking at only a small, local portion of G, preserving sublinear time and space.

Core Techniques

1. Connectivity Reconstructor – Choose an arbitrary vertex r as a “center”. Perform a bounded‑depth BFS (depth ≈ 1/ε) from r. Any vertex not reached is linked directly to r by a virtual edge. The number of added edges is at most ε·|V|, guaranteeing that the resulting graph is fully connected. When a query (u,v) arrives, the algorithm checks whether u and v belong to the same BFS tree level; if not, it answers using the r‑edge, which can be decided locally.

2. k‑Connectivity Reconstructor – k‑connectivity is tackled recursively. The authors first convert a (k‑1)‑connectivity reconstructor (which originally returns an adjacency‑matrix oracle) into an adjacency‑list oracle. This conversion is done by caching the matrix answers and dynamically constructing neighbor lists on demand. Using this list‑oracle, the algorithm verifies (k‑1)‑connectivity locally, then adds a minimal set of “boost” edges to raise the edge‑connectivity to k. The total number of added edges is O(k·ε·|V|), and each query still inspects only a polylogarithmic neighbourhood.

3. Strong‑Connectivity Reconstructor – For directed graphs, the algorithm identifies strongly connected components (SCCs) with a depth‑limited DFS. After contracting SCCs, the resulting DAG is made strongly connected by adding edges from every sink component to every source component (or a suitable spanning set). The added edges are bounded by ε·|V|, and the local query procedure only needs to explore the SCC containing the queried vertices and possibly the added “shortcut” edges.

4. Low‑Diameter Reconstructor – The property “diameter ≤ D” is parameterized. If G is ε‑close to having diameter D, the algorithm selects a small set C of “centers”. For each vertex v whose distance to C exceeds D, a direct edge to the nearest center is inserted. This guarantees that the corrected graph has diameter at most 2D while adding only O(ε·|V|) edges. The reconstructor answers distance queries by checking whether both endpoints are within D of a common center, a decision that can be made with a bounded BFS.

5. Adjacency‑List Oracle Transformation – Since many higher‑level reconstructors need list‑style access, the paper provides a systematic method to turn any matrix‑oracle reconstructor into a list‑oracle. The key idea is to lazily materialize neighbor lists from matrix queries, storing them for future use. This transformation enables the recursive use of the (k‑1)‑connectivity reconstructor inside the k‑connectivity construction.

Connection to Tolerant Property Testing

Brakerski observed that a local reconstructor can be used as a black‑box to obtain a tolerant tester: given a graph that is ε‑close to satisfying P, the reconstructor supplies oracle access to a graph that truly satisfies P; then any standard (non‑tolerant) tester can be run on this oracle. If the original graph is far from P, the reconstructor will necessarily add many edges, causing the tester to reject. By plugging the newly built reconstructors into existing testers for connectivity, the authors obtain tolerant testers for all four properties. Notably, tolerant testers for k‑connectivity, strong connectivity, and low diameter are firsts in the literature.

Complexity Overview

All algorithms are local: they never read the entire graph, use only sublinear time and space, and the total number of edge insertions respects the ε‑closeness guarantee.

Significance and Future Directions

The work establishes a robust methodology for local graph repair that simultaneously yields tolerant testers. It demonstrates that many classic global properties can be enforced with only a tiny, locally discoverable set of modifications. This opens several avenues:

• Extending the framework to other parameterized properties (e.g., bounded treewidth, planarity).
• Adapting the reconstructors to dynamic settings where edges arrive or disappear over time.
• Investigating tighter trade‑offs between the distance parameter ε, the amount of added structure, and the query complexity.
• Applying the techniques to real‑world massive networks for on‑the‑fly consistency checks and fault recovery.

In summary, the paper provides a comprehensive theory of local property reconstruction, delivers concrete reconstructors for four fundamental graph properties, and leverages them to produce the first tolerant testers for three of those properties. The results bridge the gap between local graph algorithms and property testing, offering both theoretical insight and practical tools for large‑scale network analysis.