Random Pick Index

Given an array of integers with possible duplicates, randomly output the index of a given target number. You can assume that the given target number must exist in the array.

Note:

The array size can be very large. Solution that uses too much extra space will not pass the judge.

Example:

int[] nums = new int[] {1,2,3,3,3};
Solution solution = new Solution(nums);

// pick(3) should return either index 2, 3, or 4 randomly. Each index should have equal probability of returning.
solution.pick(3);

// pick(1) should return 0. Since in the array only nums[0] is equal to 1.
solution.pick(1);

Tips:

Randomly select an int from 0 to the nums of target. If x equals 0, set the res as the current index. The probability is always 1/nums for the latest appeared number. For example, 1 for 1st num, 1/2 for 2nd num, 1/3 for 3nd num (1/2 * 2/3 for each of the first 2 nums).

O(1) momery, O(n) time

Code:

public class Solution {
int[] nums;
Random rand;
public Solution(int[] nums) {
this.nums = nums;
this.rand = new Random();
}

public int pick(int target) {
int result = -1;
int count = 0;
for (int i = 0; i < nums.length; i++) {
if (nums[i] != target)
continue;
if (rand.nextInt(++count) == 0)
result = i;
}
return result;
}
}

Reservoir Sampling

PROBLEM:

Choose k entries from n numbers. Make sure each number is selected with the probability of k/n BASIC IDEA:

Choose 1, 2, 3, ..., k first and put them into the reservoir.

For k+1, pick it with a probability of k/(k+1), and randomly replace a number in the reservoir.
For k+i, pick it with a probability of k/(k+i), and randomly replace a number in the reservoir.
Repeat until k+i reaches n

PROOF:

For k+i, the probability that it is selected and will replace a number in the reservoir is k/(k+i)
For a number in the reservoir before (let's say X), the probability that it keeps staying in the reservoir is
P(X was in the reservoir last time) × P(X is not replaced by k+i)
= P(X was in the reservoir last time) × (1 - P(k+i is selected and replaces X))
= k/(k+i-1) × （1 - k/(k+i) × 1/k）
= k/(k+i)
When k+i reaches n, the probability of each number staying in the reservoir is k/n

EXAMPLE

Choose 3 numbers from [111, 222, 333, 444]. Make sure each number is selected with a probability of 3/4

First, choose [111, 222, 333] as the initial reservior
Then choose 444 with a probability of 3/4
For 111, it stays with a probability of
P(444 is not selected) + P(444 is selected but it replaces 222 or 333)
= 1/4 + 3/4*2/3
= 3/4
The same case with 222 and 333
Now all the numbers have the probability of 3/4 to be picked