Given an array w of positive integers, where w[i] describes the weight of index i, write a function pickIndex which randomly picks an index in proportion to its weight.

Note:

1. 1 <= w.length <= 10000
2. 1 <= w[i] <= 10^5
3. pickIndex will be called at most 10000 times.

Example 1:

Input: 
["Solution","pickIndex"]
[[[1]],[]]
Output: [null,0]

Input: 
["Solution","pickIndex","pickIndex","pickIndex","pickIndex","pickIndex"]
[[[1,3]],[],[],[],[],[]]
Output: [null,0,1,1,1,0]

Explanation of Input Syntax:

The input is two lists: the subroutines called and their arguments. Solution‘s constructor has one argument, the array w. pickIndex has no arguments. Arguments are always wrapped with a list, even if there aren’t any.

Solution: Binary Search

1. Convert PDF to CDF
2. Uniformly sample a value s in [1, sum(weights)].
3. Use binary search to find first index such that PDF[index] >= s.

Time complexity: Init O(n), query O(logn)

Space complexity: O(n)

C++

// Author: Huahua
// Running time: 184 ms
class Solution {
public:
  Solution(vector<int> w) { 
    sums_ = std::move(w);
    for (int i = 1; i < sums_.size(); ++i)
      sums_[i] += sums_[i - 1];
  }

  int pickIndex() {
    int s = rand() / static_cast<double>(RAND_MAX) * sums_.back() + 1;
    return lower_bound(sums_.begin(), sums_.end(), s) - sums_.begin();
    // or
    // int l = 0;
    // int r = sums_.size();
    // while (l < r) {
    //   int m = (r - l) / 2 + l;
    //   if (sums_[m] < s)
    //     l = m + 1;
    //   else
    //     r = m;
    // }
    // return l;
  }
private:  
  vector<int> sums_;
};

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