CareerCup Divide n cakes to k different people

本文探讨如何在派对中将不同口味的蛋糕平均分配给K名成员,确保每位成员获得相同体积的单一口味蛋糕,并尽量减少浪费。通过二分搜索算法实现最优分配策略。

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In a party there are n different-flavored cakes of volume V1, V2, V3 ... Vn each. Need to divide them into K people present in the 
party such that 
- Each member of party gets equal volume of cake (say V, which is the solution we are looking for) 
- A given member should get a cake of single flavour only i.e. You cannot distribute parts of different flavored cakes to same 
member. 
- Minimum volume of cake gets wasted after distribution so that means a maximum distribution policy


-------------------------------------------------------------------------------------


Binary Search: We assume the volumes are int. If not, we should define an epi.


class Solution {
 public:
  bool isDivided(vector<int>& vi, int K, int v) {
     
  }
  int getVol(vector<int>& vi, int K) {
    int sum = 0, l = 0, r = 0, maxVol = 0, mid;
    sort(vi.begin(),vi.end(),greater<int>);
    for (int i = 0; i < vi.size(); ++i)
      sum += vi[i];
    l = vi[0] / K;
    r = sum / K;
    while (l <= r) {
      mid = (l + r) / 2;
      if (isDivided(vi, K, mid)) {
        if (mid > maxVol) {
          maxVol = mid;
          l = mid + 1;
        }
        else 
          r = mid - 1;
      }
      else
        r = mid - 1;
    }
    return maxVol;
  }
}


### K-Means聚类算法中的分治法实现与优化 #### 分治策略概述 分治法是一种通过将一个问题分解成若干个小规模子问题来求解的技术。对于K-Means聚类算法而言,可以采用分治策略减少计算量并提高效率。 #### 数据分割 数据集可以根据空间分布特性被划分为多个互不相交的小区域[^1]。这种划分可以通过预先设定边界框或其他几何形状完成。每个子区域内独立运行标准的K-Means过程: ```python import numpy as np from sklearn.cluster import MiniBatchKMeans def divide_data(X, num_partitions=4): """ 将输入的数据X按照指定数量分区 """ n_samples = X.shape[0] indices = np.random.permutation(n_samples) partition_size = int(np.ceil(n_samples / num_partitions)) partitions = [] for i in range(num_partitions): start_idx = i * partition_size end_idx = min((i + 1) * partition_size, n_samples) partitions.append(X[indices[start_idx:end_idx]]) return partitions partitions = divide_data(data_matrix, num_partitions=8) local_clusters = {} for idx, part in enumerate(partitions): mbkmeans = MiniBatchKMeans(n_clusters=k).fit(part) local_clusters[idx] = (mbkmeans.cluster_centers_, mbkmeans.labels_) ``` #### 局部中心点聚合 各个子区间的局部质心会被收集起来作为全局范围内的初始簇中心候选集合。接着利用这些新的种子再次执行完整的K-Means迭代直到收敛: ```python global_centroids = np.vstack([centers for centers,_ in local_clusters.values()]) final_kmeans = MiniBatchKMeans(n_clusters=k).fit(global_centroids) final_labels = final_kmeans.predict(data_matrix) ``` 这种方法不仅能够加速训练速度而且有助于克服大规模数据集中存在的内存瓶颈问题。
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