  In the online metric bipartite matching problem, we are given a set $S$ of
server locations in a metric space. Requests arrive one at a time, and on its
arrival, we need to immediately and irrevocably match it to a server at a cost
which is equal to the distance between these locations. A $\alpha$-competitive
algorithm will assign requests to servers so that the total cost is at most
$\alpha$ times the cost of $M_{OPT}$ where $M_{OPT}$ is the minimum cost
matching between $S$ and $R$.
  We consider this problem in the adversarial model for the case where $S$ and
$R$ are points on a line and $|S|=|R|=n$. We improve the analysis of the
deterministic Robust Matching Algorithm (RM-Algorithm, Nayyar and Raghvendra
FOCS'17) from $O(\log^2 n)$ to an optimal $\Theta(\log n)$. Previously, only a
randomized algorithm under a weaker oblivious adversary achieved a competitive
ratio of $O(\log n)$ (Gupta and Lewi, ICALP'12). The well-known Work Function
Algorithm (WFA) has a competitive ratio of $O(n)$ and $\Omega(\log n)$ for this
problem. Therefore, WFA cannot achieve an asymptotically better competitive
ratio than the RM-Algorithm.
