Minimize L1 Distance with k Cluster Centers in Array
1D k-Center (Minimax L1) Clustering
Problem Description
You are given an array of n integers, representing points on a number line, and an integer k (where 1 <= k <= n). Your task is to place k cluster centers on the real line such that the maximum L1 (absolute) distance between any point and its nearest center is minimized.
- The distance between a point
xand a centercis defined as|x - c|. - Cluster centers can be placed anywhere on the real line; they are not restricted to the input point locations.
Goal
Return the minimum possible maximum distance, denoted as R*. This R* represents the optimal radius such that all given points can be covered by k intervals, each of length 2R*, centered around the chosen k cluster centers. No smaller radius should be sufficient to cover all points with k centers.
Notes
- Consider this problem as covering all sorted points with
kintervals, each of length2R. - A common approach involves sorting the input points, performing a binary search on the possible values of
R*, and using a greedy feasibility check for eachRvalue. - If the true optimal radius
R*is a half-integer (e.g., 2.5) and an integer output is required, you may output a floating-point value (e.g., with1e-6precision) or consider scaling all coordinates by 2 to maintain integer arithmetic throughout the process.
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Core dilemma
The central engineering conflict is efficiently finding the smallest possible maximum L1 distance (R*) that allows all 'n' points to be covered by exactly 'k' cluster centers. This involves a search over a continuous range of possible R* values, where each candidate R* requires a fast and provably optimal feasibility check.
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Minimize L1 Distance with k Cluster Centers in Array
1D k-Center (Minimax L1) Clustering
Problem Description
You are given an array of n integers, representing points on a number line, and an integer k (where 1 <= k <= n). Your task is to place k cluster centers on the real line such that the maximum L1 (absolute) distance between any point and its nearest center is minimized.
- The distance between a point
xand a centercis defined as|x - c|. - Cluster centers can be placed anywhere on the real line; they are not restricted to the input point locations.
Goal
Return the minimum possible maximum distance, denoted as R*. This R* represents the optimal radius such that all given points can be covered by k intervals, each of length 2R*, centered around the chosen k cluster centers. No smaller radius should be sufficient to cover all points with k centers.
Notes
- Consider this problem as covering all sorted points with
kintervals, each of length2R. - A common approach involves sorting the input points, performing a binary search on the possible values of
R*, and using a greedy feasibility check for eachRvalue. - If the true optimal radius
R*is a half-integer (e.g., 2.5) and an integer output is required, you may output a floating-point value (e.g., with1e-6precision) or consider scaling all coordinates by 2 to maintain integer arithmetic throughout the process.