Ting-An Wu | Searcharxiv

arXiv2026

A Low-Diameter Nash-Williams--Tutte Theorem

The Nash-Williams--Tutte theorem characterizes the existence of $k$ edge-disjoint spanning trees by a condition on every vertex partition. We prove that, for simple graphs, the same condition guarantees $k$ edge-disjoint spanning trees, each of diameter $O(n/δ)$ and with $O((n/δ)\log(δ+1))$ internal vertices, where $n$ is the number of vertices and $δ$ is the minimum degree. Both bounds hold simultaneously for every tree, with absolute implicit constants independent of $k$, and each is asymptotically optimal in general. Our proof is constructive and yields a randomized algorithm with expected running time $\tilde{O}(n^5)$. We derive applications to short edge-disjoint routes and weak linkages, small Steiner trees, and survivable networks with short paths. The joint bounds also yield low-diameter spanning Eulerian subgraphs and short tours for postman and general routing problems.

math.CO↗

arXiv2026

Efficient Streaming Algorithms for Two-Dimensional Congruence Testing and Congruence Hashing

Geometric congruence asks whether two point multisets are identical up to translation and rotation, while congruence hashing seeks compact encodings that support efficient congruence queries. We study these problems in the streaming model under finite-precision rational inputs, where each coordinate is $p/q$ with $|p|,|q|\le U$. Our main results are two randomized polylogarithmic-space algorithms for 2D congruence identification (CI), which additionally requires outputting a valid transformation when congruent. With probability at least $1-δ$, our 3-pass product-anchor algorithm uses $O((\log n+\log U+\log \frac{1}δ)\log\log n\log \frac{1}δ)$ space in the turnstile model, while our 3-pass complex-moment algorithm uses $O(\log n(\log n+\log U+\log \frac{1}δ))$ space in the insertion-only model. Using CI as a building block, we obtain a 4-pass insertion-only congruence hashing algorithm over $m$ query sets using $O(m(\log n+\log U+\log m+\log \frac{1}δ))$ space and producing signatures of length $O(\log \frac{1}δ+\log U+\log m)$. Both algorithms presample primes for modular hashing to handle precision. The former algorithm uses finite-field embeddings and number-theoretic guarantees to recover rotations, and the latter hinges on a new non-vanishing complex moment criterion, thus avoiding additional number-theoretic conditions and the classical vanishing-moment obstacle. For the hardness results, we prove that any $p$-pass randomized streaming algorithm for 2D CI with error at most $δ$ requires $Ω(\frac{1}{p}(\log n+\log U+\log \frac{1}δ))$ space, matching the turnstile upper bound up to a $\log\log n$ factor. Moreover, approximate CI requires $\text{poly}(n)$ space even with $\text{poly}(n)$ passes. This sharply contrasts with the RAM model, where both exact and approximate versions are solvable in polynomial time.

cs.DS↗