An O(n\log n)-Time Algorithm for the k-Center Problem in Trees
نویسندگان
چکیده
We consider a classical k-center problem in trees. Let T be a tree of n vertices and every vertex has a nonnegative weight. The problem is to find k centers on the edges of T such that the maximum weighted distance from all vertices to their closest centers is minimized. Megiddo and Tamir (SIAM J. Comput., 1983) gave an algorithm that can solve the problem in O(n log n) time by using Cole’s parametric search. Since then it has been open for over three decades whether the problem can be solved in O(n log n) time. In this paper, we present an O(n log n) time algorithm for the problem and thus settle the open problem affirmatively.
منابع مشابه
Efficient Algorithms for Two Generalized 2-Median Problems on Trees
The p-median problem on a tree T is to find a set S of p vertices on T that minimize the sum of distances from T’s vertices to S. For this problem, Tamir [14] had an O(pn)-time algorithm, while Gavish and Sridhar [6] had an O(nlog n)-time algorithm for the case of p=2. In this paper, we study two generalizations of the 2-median problem, which are obtained by imposing constraints on a 2-median: ...
متن کاملParallel Generation of t-ary Trees
A parallel algorithm for generating t-ary tree sequences in reverse B-order is presented. The algorithm generates t-ary trees by 0-1 sequences, and each 0-1 sequences is generated in constant average time O(1). The algorithm is executed on a CREW SM SIMD model, and is adaptive and cost-optimal. Prior to the discussion of the parallel algorithm a new sequential generation with O(1) average time ...
متن کاملAn efficient algorithm for finding the semi-obnoxious $(k,l)$-core of a tree
In this paper we study finding the $(k,l)$-core problem on a tree which the vertices have positive or negative weights. Let $T=(V,E)$ be a tree. The $(k,l)$-core of $T$ is a subtree with at most $k$ leaves and with a diameter of at most $l$ which the sum of the weighted distances from all vertices to this subtree is minimized. We show that, when the sum of the weights of vertices is negative, t...
متن کاملFr{'e}chet and Hausdorff Queries on $x$-Monotone Trajectories
vspace{0.2cm}In this paper, we design a data structure for the following problem. Let $pi$ be an $x$-monotone trajectory with $n$ vertices in the plane and $epsilon >0$. We show how to preprocess $pi$ and $epsilon$ into a data structure such that for any horizontal query segment $Q$ in the plane, one can quickly determine the minimal continuous fraction of $pi$ whose Fr{'e}chet and Hausdo...
متن کاملA novel local search method for microaggregation
In this paper, we propose an effective microaggregation algorithm to produce a more useful protected data for publishing. Microaggregation is mapped to a clustering problem with known minimum and maximum group size constraints. In this scheme, the goal is to cluster n records into groups of at least k and at most 2k_1 records, such that the sum of the within-group squ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1705.02752 شماره
صفحات -
تاریخ انتشار 2017