You are given a 0-indexed integer array nums. You are allowed to permute nums into a new array perm of your choosing.

We define the greatness of nums be the number of indices 0 <= i < nums.length for which perm[i] > nums[i].

Return the maximum possible greatness you can achieve after permuting nums.

Example 1:

Input: nums = [1,3,5,2,1,3,1]
Output: 4
Explanation: One of the optimal rearrangements is perm = [2,5,1,3,3,1,1].
At indices = 0, 1, 3, and 4, perm[i] > nums[i]. Hence, we return 4.

Example 2:

Input: nums = [1,2,3,4]
Output: 3
Explanation: We can prove the optimal perm is [2,3,4,1].
At indices = 0, 1, and 2, perm[i] > nums[i]. Hence, we return 3.

Constraints:

• 1 <= nums.length <= 105
• 0 <= nums[i] <= 109

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Problem 1: Maximum Valid Connections in a Directed Tree

You are given a directed tree with n nodes, where each node is labeled from 0 to n-1. The tree structure is defined by three arrays:

• from_nodes and to_nodes, where from_nodes[i] represents the parent node of to_nodes[i], forming a directed edge (from_nodes[i] → to_nodes[i]).
• cost, where cost[i] represents the cost associated with the edge (from_nodes[i] → to_nodes[i]).
• Additionally, you are given an array value of length n, where value[i] represents the value associated with node i.

Valid Connection Criteria:

A connection (u → v) is valid if and only if:

1. u is an ancestor of v in the tree (i.e., there is a directed path from u to v).
2. The cost of the connection cost[i] is strictly less than value[v].

Output:

Return the maximum number of valid connections in the tree.

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Problem 2: House Coloring with Constraints

There are n houses along a street, and we need to color each house using one of three colors while following two constraints:

1. Adjacent houses must have different colors.
2. Symmetric houses must have different colors. Specifically:
   • The 1st and nth house must have different colors.
   • The 2nd and (n-1)th house must have different colors.
   • The 3rd and (n-2)th house must have different colors, and so on.

Input:

• N (even integer) representing the number of houses.

Output:

• Return the total number of valid colorings modulo 10⁷.

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