Low-distortional metric embeddings are a crucial component in the modern
algorithmic toolkit. In an online metric embedding, points arrive sequentially
and the goal is to embed them into a simple space irrevocably, while minimizing
the distortion. Our first result is a deterministic online embedding of a
general metric into Euclidean space with distortion O(logn)⋅min{logΦ,n} (or,
O(d)⋅min{logΦ,n} if the metric has doubling
dimension d), solving a conjecture by Newman and Rabinovich (2020), and
quadratically improving the dependence on the aspect ratio Φ from Indyk et
al.\ (2010). Our second result is a stochastic embedding of a metric space into
trees with expected distortion O(d⋅logΦ), generalizing previous
results (Indyk et al.\ (2010), Bartal et al.\ (2020)).
Next, we study the \emph{online minimum-weight perfect matching} problem,
where a sequence of 2n metric points arrive in pairs, and one has to maintain
a perfect matching at all times. We allow recourse (as otherwise the order of
arrival determines the matching). The goal is to return a perfect matching that
approximates the \emph{minimum-weight} perfect matching at all times, while
minimizing the recourse. Our third result is a randomized algorithm with
competitive ratio O(d⋅logΦ) and recourse O(logΦ) against an
oblivious adversary, this result is obtained via our new stochastic online
embedding. Our fourth result is a deterministic algorithm against an adaptive
adversary, using O(log2n) recourse, that maintains a matching of weight at
most O(logn) times the weight of the MST, i.e., a matching of lightness
O(logn). We complement our upper bounds with a strategy for an oblivious
adversary that, with recourse r, establishes a lower bound of
Ω(rlogrlogn) for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on
Discrete Algorithms (SODA24