Homework 1 - Problem 1: Driving in Manhattan

In Manhattan, the avenues run north-south and the streets run east-west. For simplicity, each point can be thought of as a (street, avenue) pair, where streets are numbered 1 to $n$ and avenues are numbered from 1 to $m$. Suppose you want to get from 14th St. & 8th Ave. to anywhere north of 96th St. in the shortest time. For simplicity, assume all streets are two-way; going east-west from one avenue to an adjacent one takes 3 units of time and going north-south from one street to another takes 1 unit of time.

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(a) [2 points] Construct a deterministic state space model for this situation, that is, what are the states, actions, start state and goal test? Be precise.

(b) [2 points] Suppose you also need to stop at a grocery store (choose any of $k$ different locations $p_1, \dots, p_k$). Redefine the states, actions, start state, and goal test so that the new constraint is accommodated. Again, be precise.

(c) [2 points] Your friend calls and asks that you need to actually stop at all the grocery stores along the way, but you can do it in any order. What are the states now? You don't need to specify the actions, etc. How many states are there as a function of $m$ and $n$ and $k$?

(d) [2 points] What if you need to stop at all the grocery stores but in a particular known order $p_1, \dots, p_k$. What is the new state space and how may states are there?

(e) [1 point] From these examples, reflect on whether adding constraints to a problem leads to a smaller or larger number of states.