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3699. Number of ZigZag Arrays I

HardDynamic Programming Preparing judge… LeetCode

You are given three integers n, l, and r.

A ZigZag array of length n is defined as follows:

  • Each element lies in the range [l, r].
  • No two adjacent elements are equal.
  • No three consecutive elements form a strictly increasing or strictly decreasing sequence.

Return the total number of valid ZigZag arrays.

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

A sequence is said to be strictly increasing if each element is strictly greater than its previous one (if exists).

A sequence is said to be strictly decreasing if each element is strictly smaller than its previous one (if exists).

 

Example 1:

Input: n = 3, l = 4, r = 5

Output: 2

Explanation:

There are only 2 valid ZigZag arrays of length n = 3 using values in the range [4, 5]:

  • [4, 5, 4]
  • [5, 4, 5]​​​​​​​

Example 2:

Input: n = 3, l = 1, r = 3

Output: 10

Explanation:

There are 10 valid ZigZag arrays of length n = 3 using values in the range [1, 3]:

  • [1, 2, 1], [1, 3, 1], [1, 3, 2]
  • [2, 1, 2], [2, 1, 3], [2, 3, 1], [2, 3, 2]
  • [3, 1, 2], [3, 1, 3], [3, 2, 3]

All arrays meet the ZigZag conditions.

 

Constraints:

  • 3 <= n <= 2000
  • 1 <= l < r <= 2000
Dynamic ProgrammingPrefix Sum

Hints

Hint 1
Use dynamic programming: let dp[i][dir][x] be the count of length-i sequences ending at value x where dir is the required next comparison (0 = down, 1 = up).
Hint 2
If the required move is up (dir=1) do dp[i+1][0][y] += sum(dp[i][1][x]) for x < y; if the required move is down (dir=0) do dp[i+1][1][y] += sum(dp[i][0][x]) for x > y.
Hint 3
Speed up with prefix/suffix sums so each layer updates in O(m) instead of O(m2); take values mod 109+7.