randomPickWithWeight.java

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 <= w.length <= 10000
1 <= w[i] <= 10^5
pickIndex will be called at most 10000 times.  (KEY!!!)

BASICALLY, we have an array with weighted inputs

// This means binary search will be used alot, but the
// intial O(n) to make the prearray will only take O(N) one time!!


BRUTE FORCE:
// Generate an array with x occurences of each index, where x is the weight
// Won't be optimal because we will have to build a HUGE array



Optimized:

1. Create an array with accumlted frequency, last element arr[length-1] will contain total weight

2. Generate random number from (1, arr[length-1]) 

3. Binary search for random number

//TC: O(n) to build to initial array, O(logn) for each pick index call!
//SC: O(n) to store the array


class Solution {

    Random random;
    int[] wSums;
    
    public Solution(int[] w) {
        this.random = new Random();
        for(int i=1; i<w.length; ++i)
            w[i] += w[i-1];
        this.wSums = w;
    }
    
    public int pickIndex() {
        int len = wSums.length;
        int idx = random.nextInt(wSums[len-1]) + 1;
        int left = 0, right = len - 1;
        // search position 
        while(left < right){
            int mid = left + (right-left)/2;
            if(wSums[mid] == idx)
                return mid;
            else if(wSums[mid] < idx)
                left = mid + 1;
            else
                right = mid;
        }
        return left;
    }
}