Silo Switchboard

# Silo Switchboard ## Problem The Cow State Fair has N grain silos that need to be configured for a special synchronized display. Each silo can be set to one of three states: **0** (idle), **1** (processing), or **2** (active). You have a control system that can set each silo to any state, but each configuration change has a cost. Additionally, due to structural concerns about resonance, **no two adjacent silos may end in the same state**. Given: - The number of silos N - A target state pattern (for reference) - The cost matrix where `cost[i][j]` represents the cost to set silo `i` to state `j` Find the **minimum total cost** to configure all silos such that no two adjacent silos have the same final state. ## Input Format The first line contains a single integer **N** (1 ≤ N ≤ 1000), the number of silos. The second line contains N space-separated integers representing the target state pattern (for reference only; you must find the minimum cost valid configuration, which may not match the target exactly). The next N lines contain three space-separated integers each: - Line i+2 contains `c0 c1 c2`, where `cj` is the cost to set silo `i` to state `j`. All costs are non-negative integers ≤ 10^6. ## Output Format Output a single integer: the minimum cost to configure all silos such that no two adjacent silos have the same final state. ## Constraints - 1 ≤ N ≤ 1000 - 0 ≤ each cost ≤ 10^6 - All silos must be assigned a state from {0, 1, 2} - For all adjacent pairs (i, i+1), the final states must differ ## Sample Input ``` 3 1 2 0 3 0 2 2 1 5 0 2 1 ``` ## Sample Output ``` 3 ``` ## Sample Explanation With 3 silos and costs: - Silo 0: costs [3, 0, 2] to set to states [0, 1, 2] - Silo 1: costs [2, 1, 5] to set to states [0, 1, 2] - Silo 2: costs [0, 2, 1] to set to states [0, 1, 2] One optimal configuration is [2, 1, 0] (no adjacent silos are the same): - Set silo 0 to state 2: cost 2 - Set silo 1 to state 1: cost 1 (state 1 ≠ 2, satisfies adjacency) - Set silo 2 to state 0: cost 0 (state 0 ≠ 1, satisfies adjacency) - **Total cost: 2 + 1 + 0 = 3** Another optimal configuration is [1, 0, 2]: - Set silo 0 to state 1: cost 0 - Set silo 1 to state 0: cost 2 (state 0 ≠ 1, satisfies adjacency) - Set silo 2 to state 2: cost 1 (state 2 ≠ 0, satisfies adjacency) - **Total cost: 0 + 2 + 1 = 3** Both achieve the minimum possible cost while respecting the adjacency constraint.