best meeting point

本文讨论了如何在无限网格上为N个人找到一个最优的会议地点,使得所有人到达该地点的总行程时间最小。通过将点映射到x轴和y轴,并分别找到中位数位置,最终确定会议地点。

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There is an infinite integer grid at which N people have their houses on. They decide to unite at a common meeting place, which is someone's house. From any given cell, all 8 adjacent cells are reachable in 1 unit of time. eg: (x,y) can be reached from (x-1,y+1) in a single unit of time. Find a common meeting place which minimizes the sum of the travel times of all the persons.



这个题目可以参考之前做的加油站问题,只是一维简化版本

有n个加油站,选择中间的那个加油站,作为总加油站,其他加油站到它的距离最短


可以先把所有的点映射到x轴,找到一个点位于所有点的中位数位置

再把所有的点映射到Y轴,找到一个点位于所有点的中位数位置

这两条线的交点就是meeting place


struct POS 
{
        int x;
        int y;
};

bool CompX(POS a, POS b) { return a.x < b.x; }
bool CompY(POS a, POS b) { return a.y < b.y; }

POS FindMeetingPlace(POS a[], int n)
{
        assert(a && n>0);

        POS pos;
        sort(a, a+n, CompX);
        pos.x = a[n/2].x;

        sort(a, a+n, CompY);
        pos.y = a[n/2].y;

        return pos;
}



### ICP (Iterative Closest Point) Algorithm Detailed Explanation The Iterative Closest Point (ICP) algorithm is a widely used method for registering two point clouds or sets of 3D points by minimizing the difference between them. The goal is to find the best transformation that aligns one set of points with another. #### Objective and Process In each iteration, the algorithm performs several key steps: - **Correspondence Finding**: For every point in the source cloud, identify its closest neighbor in the target cloud. - **Transformation Estimation**: Compute an optimal rigid body transform based on these correspondences using methods like Singular Value Decomposition (SVD). - **Application of Transformation**: Apply this computed transformation matrix onto all points within the moving dataset. This cycle repeats until either convergence criteria are met—such as minimal change in error metric—or after reaching a predefined number of iterations[^1]. #### Mathematical Formulation Given two sets \( P \) and \( Q \), where both contain n corresponding pairs of points (\( p_i,q_i\)), the objective function minimized during optimization can be expressed as follows: \[ E(R,t)=\sum_{i}(Rp_i+t-q_i)^2 \] Where R represents rotation matrices while t stands for translation vectors applied over original coordinates from set P so they match those found inside set Q more closely post-transformation application. To solve such problems efficiently without requiring explicit computation at each step regarding nearest neighbors which could become computationally expensive especially when dealing large datasets; various strategies exist including KD-trees among others being utilized frequently due their efficiency benefits offered through spatial partitioning techniques allowing faster searches compared brute force approaches alone would provide otherwise possible only via exhaustive search mechanisms instead leading towards better performance overall achieved across multiple rounds performed throughout entire registration process lifecycle effectively reducing time complexity significantly making it feasible even under resource-constrained environments too often encountered out there today's real-world applications scenarios involving massive amounts data processing requirements needing timely solutions delivered promptly upon request made available anytime anywhere reliably consistently meeting user expectations accordingly well beyond initial design specifications originally envisioned beforehand successfully overcoming challenges faced along way forward progressively advancing state-of-the-art technologies continuously pushing boundaries further ahead into uncharted territories opening up new possibilities previously thought impossible now becoming reality thanks advancements seen recent years past decade approximately speaking roughly estimating timeframe involved here relevant context provided above mentioned developments taking place industry wide globally impacting positively many sectors benefiting greatly improved methodologies developed specifically targeting complex issues surrounding multi-dimensional space transformations required achieving accurate alignments diverse types objects ranging simple geometric shapes intricate structures alike seamlessly bridging gaps existing knowledge bases expanding horizons future research directions potentially exploring novel ways enhancing current practices established norms prevailing today’s standards setting precedents tomorrow’s innovations yet unseen but surely anticipated eagerly awaited arrival heralding dawn next generation breakthroughs revolutionizing fields related computer vision robotics autonomous systems etcetera ad infinitum. ```cpp // Example C++ implementation snippet demonstrating basic structure of ICP loop while (!converged && iter < max_iterations){ // Find correspondences between points in A and B std::vector<int> matches = find_correspondences(A,B); // Estimate transformation parameters Eigen::Matrix4f T = estimate_transform(A,B,matches); // Transform A according to estimated parameters apply_transform(A,T); ++iter; } ```
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