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3407. Find All Possible Stable Binary Arrays II

Hard
Dynamic Programming
Prefix Sum
Solve on LeetCode View Solutions

Description

You are given 3 positive integers zero, one, and limit.

A binary array arr is called stable if:

  • The number of occurrences of 0 in arr is exactly zero.
  • The number of occurrences of 1 in arr is exactly one.
  • Each subarray of arr with a size greater than limit must contain both 0 and 1.

Return the total number of stable binary arrays.

Since the answer may be very large, return it modulo 109 + 7.

 

Example 1:

Input: zero = 1, one = 1, limit = 2

Output: 2

Explanation:

The two possible stable binary arrays are [1,0] and [0,1].

Example 2:

Input: zero = 1, one = 2, limit = 1

Output: 1

Explanation:

The only possible stable binary array is [1,0,1].

Example 3:

Input: zero = 3, one = 3, limit = 2

Output: 14

Explanation:

All the possible stable binary arrays are [0,0,1,0,1,1], [0,0,1,1,0,1], [0,1,0,0,1,1], [0,1,0,1,0,1], [0,1,0,1,1,0], [0,1,1,0,0,1], [0,1,1,0,1,0], [1,0,0,1,0,1], [1,0,0,1,1,0], [1,0,1,0,0,1], [1,0,1,0,1,0], [1,0,1,1,0,0], [1,1,0,0,1,0], and [1,1,0,1,0,0].

 

Constraints:

  • 1 <= zero, one, limit <= 1000

Hints

Hint 1
Let <code>dp[x][y][z = 0/1]</code> be the number of stable arrays with exactly <code>x</code> zeros, <code>y</code> ones, and the last element is <code>z</code>. (0 or 1). <code>dp[x][y][0] + dp[x][y][1]</code> is the answer for given <code>(x, y)</code>.
Hint 2
If we have already placed <code>x</code> 1 and <code>y</code> 0, if we place a group of <code>k</code> 0, the number of ways is <code>dp[x-k][y][1]</code>. We can place a group with size <code>i</code>, where <code>i</code> varies from 1 to <code>min(limit, zero - x)</code>. Similarly, we can solve by placing a group of ones.
Hint 3
Speed up the calculation using prefix arrays to store the sum of <code>dp</code> states.

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