Informed State-Space Search

The second fundamental strategy, informed state-space search, is an application of the A* algorithm. Here we modeled the task of placing $n$ cookies as a sequence of transitions between nodes in the search tree, where each transition consists of placing a cookie on the dough in a particular way. When we expand a node, we create a set of $b$ candidate nodes. The worst case running time for this algorithm is $O(b^n)$.

For each node $n$, we decomposed our cost function, $f(n) = g(n)+h(n)$, as follows:

• the depth factor $g(n)$ for a node in the search tree is equal to the rightmost boundary of the cookies placed so far.
• the heuristic factor $h(n)$ for each node, is equal to the sum of the areas of the cookies yet to be placed, multiplied by $\alpha$ where $\alpha$ is a real number greater than 1. Since the height of the dough is defined to be one, there is no difference between measuring area (h) and measuring distance (g).

Running the A* algorithm starting with the empty node (no cookies) until a goal node is reached (all cookies placed) would produce an approximately optimal placement. We note that this approach is highly sensitive to the parameters b (the number of nodes expanded from each node) and $\alpha$, our overestimation factor. To avoid the exponentional running time $O(b^n)$ and to force the algorithm to run in a reasonable amount of time, we need to ensure that $\alpha$ is high enough and $b$ is not too large. As $\alpha$ increases and $b$ decreases, the algorithm runs faster because it inspects fewer nodes in the tree, but the result becomes less optimal.