UVa 10748 - Knights Roaming

Summary
Knights Roaming is a problem involving a Breadth-First Search and some guesswork heuristics to get an acceptable speed.

Explanation
We are given the location of $$n <= 30$$ knights, each having a corresponding starting location and number of moves $$(x, y, k)$$, where $$k <= 50$$. We need to determine how many different board locations the knights can cover.

The first thing to notice is that the board locations covered by different knights at a fixed distance have the same shape, they are just translated from one another. The next thing to notice is that the board locations reachable in k or less steps steps are a subset of reachable in k + 1 or less steps. Thus with a single 50 step Breadth-First Search we can determine the essential shape that every knight covers. From a single precomputed breadth first search, we can index board locations by the minimum distance it takes to reach them. So, for example, we will know that the points that are two moves away are (4, 2), (4, -2), (2, 4)...

However, the range of starting locations for the knights is quite large, in the range of $$-10^9$$ to $$10^9$$, so some sparse or compressed representation of the covered area is neccesary. A set of x, y pairs is a suitable representation.

One final note is that if you try insert every possible reachable location to a set of reachable points, you will timeout. One way to speed it up is to realize that in a single move, a knight moves exactly 3 manhattan steps away from it's start. Thus, the coverage from two knights which have manhattan distance 100 from each other, and one having 15 moves and the other having 10 moves cannot possibly interfere with one another, since $$100 > (15 + 10) \cdot 3 $$. We can extend this idea to make our solution much faster. For each knight, let $$k_i$$ be the number of moves for knight i, and $$d(i, j)$$ be the manhattan distance from knight i to knight j. Compute $$s_i = min_{j=1}^n(d(i, j) - 3 \cdot (k_i + k_j))$$ for $$1 \leq i \leq n$$. Then, by the above reasoning, the points less than or equal to $$s_i$$ moves away can be covered by only knight i, so the corresponding number of moves can simply be added into the number of covered cells without inserting them into the covered set.

Optimizations
See UsingHashMap for details on using an stl hash set as the sparse board representation.

Input
5 1 1 0 2 2 0 3 3 0 4 4 0 5 5 0

5 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1

4 -1 -1 2 2 2 1 3 3 3 4 4 3

30 0 0 50 1 1000 50 2 2000 50 3 3000 50 4 4000 50 5 5000 50 6 6000 50 7 7000 50 8 8000 50 9 9000 50 10 10000 50 11 11000 50 12 12000 50 13 13000 50 14 14000 50 15 15000 50 16 16000 50 17 17000 50 18 18000 50 19 19000 50 20 20000 50 21 21000 50 22 22000 50 23 23000 50 24 24000 50 25 25000 50 26 26000 50 27 27000 50 28 28000 50 29 29000 50

Output
5 33 155 1041150