  We present two improved algorithms for weighted discrete $p$-center problem
for tree networks with $n$ vertices. One of our proposed algorithms runs in
$O(n \log n + p \log^2 n \log(n/p))$ time. For all values of $p$, our algorithm
thus runs as fast as or faster than the most efficient $O(n\log^2 n)$ time
algorithm obtained by applying Cole's speed-up technique [cole1987] to the
algorithm due to Megiddo and Tamir [megiddo1983], which has remained
unchallenged for nearly 30 years. Our other algorithm, which is more practical,
runs in $O(n \log n + p^2 \log^2(n/p))$ time, and when $p=O(\sqrt{n})$ it is
faster than Megiddo and Tamir's $O(n \log^2n \log\log n)$ time algorithm
[megiddo1983].
