## Monday, June 12, 2017

### Flagpoles

My solution during the contest was essentially the same as the official analysis. Afterwards I realised a potential slight simplification: if one starts by computing the second-order differences (i.e., the differences of the differences), then one is looking for the longest run of zeros, rather than the longest run of the same value. That removes the need to communicate the value used in the runs at the start and end of each section.

### Number Bases

I missed the trick of being able to uniquely determine the base from the first point at which X[i] + Y[i] ≠ Z[i]. Instead, at every point where X[i] + Y[i] ≠ Z[i], I determine two candidate bases (depending on whether there is a carry of not). Then I collect the candidates and test each of them. If more than three candidates are found, then the test case is impossible, since there must be two disjoint candidate pairs.

### Broken Memory

My approach was slightly different. Each node binary searches for its broken value, using two other nodes to help (and simultaneously helping two other nodes). Let's say we know the broken value is in a particular interval. Split that interval in half, and compute hashes for each half on the node (h1 and h2) and on two other nodes (p1 and p2, q1 and q2). If h1 equals p1 or q1, then the broken value must be in interval 2, or vice versa. If neither applies, then nodes p and q both have broken values, in the opposite interval to that of the current node. We can tell which by checking whether p1 = q1 or p2 = q2.

This does rely on not having collisions in the hash function. In the contest I relied on the contest organisers not breaking my exact choice of hash function, but it is actually possible to write a solution that works on all test data. Let P be a prime greater than $$10^{18}$$. To hash an interval, compute the sums $$\sum m_i$$ and $$\sum i m_i$$, both mod P, giving a 128-bit hash. Suppose two sequences p and q collide, but differ in at most two positions. The sums are the same, so they must differ in exactly two positions j and k, with $$p_j - q_j = q_k - p_k$$ (all mod P). But then the second sums will differ by
$$jp_j + kp_k - jq_j - kq_k = (j - k)(p_j - q_j)$$, and since P is prime and each factor is less than P, this will be non-zero.

### Alternative Code Jam 2017 R3 solution

This last weekend was round 3 of Code Jam 2017 and round 2 of Distributed Code Jam 2017. I'm not going to describe how to solve all the problems since there are official analyses (here and here), but just mention some alternatives. For this post I'll just talk about one problem from Code Jam; some commentary on DCJ will follow later.

### Slate Modern (Code Jam)

The idea I had for this seems simpler (in my opinion, without having tried implementing both) than the official solution, but unfortunately I had a bug that I couldn't find until about 10 minutes after the contest finished.

As noted in the official analysis, one can first check whether a solution is possible by comparing each pair of fixed cells: if the difference is value is greater than D times the Manhattan distance, then it is impossible; if no such pair exists, it is possible. The best solution is then found by setting each cell to the smallest lower bound imposed by any of the fixed cells.

Let's try to reduce the complexity by a factor of C, by computing the sum of a single row quickly. If we look at the upper bound imposed by one fixed cell, it has the shape $$b + |x - c| D$$, where $$x$$ is the column and b, c are constants. When combining the upper bounds, we take the lowest of them. Each function will be the smallest for some contiguous (possibly empty) interval. By sweeping through the fixed cells in order of c, we can identify those that contribute, similar to finding a Pareto front. Then, by comparing adjacent functions one can find the range in which each function is smallest, and a bit of algebra gives the sum over that range.

This reduces the complexity to O(RN + N²) (the N² is to check whether a solution is possible, but also hides an O(N log N) to sort the fixed cells). That's obviously still too slow. The next insight is that most of the time, moving from one row to the next changes very little: each of the functions increases or decreases by one (depending on whether the corresponding fixed cell is above or below the row), and the range in which each function is smallest grows or shrinks slightly. It thus seems highly likely that the row sum will be a low-degree polynomial in the row number. From experimentation with the small dataset, I found that it is actually a quadratic (conceptually it shouldn't be too hard to prove, but I didn't want to get bogged down in the details).

Note that I said "most of the time". This will only be true piecewise, and we need to find the "interesting" rows where the coefficients change. It is fairly obvious that rows containing fixed cells will be interesting. Additionally, when the range in which a function is smallest disappears or appears will be interesting. Consider just two fixed cells, and colour the whole grid according to which of the two fixed cells gives the lower upper bound. The boundary between the two colours can have diagonal, horizontal and vertical portions, and it is these horizontal portions that are (potentially) interesting. I took the conservative approach of adding all O(N²) such rows as interesting.

Now that we have partitioned the rows into homogeneous intervals, we need to compute the sum over each interval efficiently. Rather than determine the quadratic coefficients analytically, I just interfered them by taking the row sums of three consecutive rows (if there are fewer than three rows in the interval, just add up their row sums directly). A bit more algebra to find the sum of a quadratic series, and we're done! There are O(N²) intervals to sum and it requires O(N) to evaluate each of the row sums needed, giving a running time of O(N³). I suspect that this could probably be reduced to O(N² log N) by being smarter about how interesting rows are picked, but it is not necessary given the constraints on N.