I've encountered a Dynamic Programming problem which is a variation of the thief one.

Say you are a thief and you are given a number of houses in a row you should rob :

$$House_1,House_2 \dots House_N$$

with each house having the following values : $$(x_i \geq y_i \geq z_i \gt0)$$

You profit X if you rob a house but none of the adjacent houses.

You profit Y if you rob a house and at least one of the adjacent houses.

You profit Z if you rob a house and both of the adjacent houses.

Obviously you can't utilize the Z value for the first and the last house and each set of values is random.

Now the question is : Which houses should you rob to get the maximum profit?

My main issue is that i can't establish a base case for this problem.

At first i thought of creating a N*M array with M being the maximum amount of houses i can rob from 0-N when every house is not robbed and think like : Rob it - Don't rob it but came up with nothing.

Any tips or directions would be appreciated.

Thanks in advance.

I've encountered a Dynamic Programming problem which is a variation of the thief one.

Say you are a thief and you are given a number of houses in a row you should rob :

$$House_1,House_2 \dots House_N$$

with each house having the following values : $$(x_i \geq y_i \geq z_i \gt0)$$

You profit X if you rob a house but none of the adjacent houses.

You profit Y if you rob a house and exactly one of the adjacent houses.

You profit Z if you rob a house and both of the adjacent houses.

Cases with houses A-B-C would be :

$$Profit(001)=0+0+C_x$$ $$Profit(101)=A_x+0+C_x$$ $$Profit(110)=A_y+B_y+0$$ $$Profit(111)=A_y+B_z+C_y$$

Where 1 stands for robbing the house and 0 for not robbing the house

Obviously you can't utilize the Z value for the first and the last house and each set of values is random.

Now the question is : Which houses should you rob to get the maximum profit?

My main issue is that i can't establish a base case for this problem.

At first i thought of creating a N*M array with M being the maximum amount of houses i can rob from 0-N when every house is not robbed and think like : Rob it - Don't rob it but came up with nothing.

Any tips or directions would be appreciated.

Thanks in advance.