## Cave

This is a nice problem following a recent IOI trend of having problems where the limiting factor is not execution time, but some abstract measure of efficiency. This is slightly easier than some of the previous problems in each vein.
To achieve 100%, we can use up to 14 queries per door. Since 214 is slightly more than the number of doors, this strongly suggests some form of binary search on each door. It will be easiest if we attack the doors in order. Once we know which switch and position opens a door, we can lock that switch in place so that we will always be able to see the next door, and thereafter pretend the switch does not even exist.
To solve for a door, we can start with all switches down, which will immediately tell us which switch position opens the door. At this point every switch is a candidate. We can then test with half the candidates down and half up, which will eliminate half the candidates. This process is then repeated until only one candidate remains.

## Robots

As is commonly the case for problems that ask how long some agents need to achieve a goal, the answer can be found by a binary search on the answer. So we need to decide whether the robots can clear the floor in S seconds.
We can simplify the problem slightly by noting that for each toy, we only need to know how many weak and how many small robots are able to move it, which can be found by binary searching the two lists (after sorting them). Of course, if a toy cannot be moved by any robot, return ­ -1.
Let's first decide the actions of the weak robots, starting from the weakest. There will be some set of toys it can handle. Since they effectively differ only in size, the weakest robot should work from the largest downwards, so as to make the job of the small robots easier. Also, there is never any reason for it to move fewer than S toys, unless it runs out. Now consider the 2nd-weakest robot. There will be extra toys it can handle, plus any light toys that the weakest robot didn't have time for. Since the weakest robot is finished, the difference in weights are irrelevant, and the 2nd-weakest robot should again work in decreasing order of size amongst the toys it can handle. The same process can be continued for the remaining weak robots.
Now consider the small robots, from largest to smallest. These can again take up to S toys, starting from the largest remaining one. If a robot is unable to handle the largest remaining toy, then S was too small.
Implementation can be done using a priority queue, implemented as a binary heap, representing toys that are light enough to be handled by the current robot and ordered with the largest at the head. The toys are initially sorted by weight. Each time a new weak robot is considered, new elements are added from the list of toys to the priority queue, and the robot then removes items starting from the head of the queue.
Assuming that T is larger than A, B, the running time will be O(T (log T)2): one log T for the binary search, the other for the priority queue operations. I think that it may be possible to reduce this to something like O(T·log T·log(max(A, B))) using the right sort of data structure for the priority queue (to allow S items to be removed in log time): something like an interval tree for the number of toys of each size.

## Game

I disliked this problem because it has a nice solution that takes just a bit too much memory. I only managed to get 80% for it in the time I spent on it, and I didn't feel inspired to modify my solution to pass fully.
In 1D, this can be solved by a fairly straightforward use of a segment tree: each node stores the GCD of its two children. Since R can be quite big, this needs to be a sparse segment tree; another alternative would be a balanced binary tree.
In 2D, it is tempting to use a quadtree, but in fact that doesn't guarantee poly-logarithmic time. A 1xC query will force refinement down to the individual non-zero cell entries. Instead, we can use a range tree, which is a tree of trees: an outer tree is built over the columns, and for each column span corresponding to a node in this tree, we have an inner tree over the rows. Each node in this inner tree corresponds to a rectangle of the grid, and stores the GCD of elements from this rectangle. A query now uses the columns to select O(log C) nodes from the outer tree i.e., O(log C) inner trees, and applies a query to each of them. Queries thus take O(log R·log C) time when the implementation uses segment trees for the inner and outer trees. With balanced binary trees, it would only be O((log Nu)2).
Unfortunately, using segment trees also requires O(log R·log C) memory per non-zero element, which just exceeds the available memory. Using balanced binary trees instead should fit within memory, but is a lot more painful to implement. I think it might also be possible to make it work by sticking with segment trees, but compressing the representation by compacting chains of nodes with one child.

## Wednesday, July 17, 2013

### IOI 2013 day 1 analysis

I finally got around to actually solving the IOI day 1 tasks in the contest system - I'd had ideas, but no time to test them out. So here are my solutions (I haven't looked around for any other writeups, although no doubt they exist). I'm hoping to have time to tackle day 2 before the contest system is shut down.

## Art class

This is a heuristic problem that can probably be solved in many ways. Since it was a hammer I have, I decided to hit it with a very simple wavelet-based spectral analysis. To find the highest-frequency components, downsample the image, upsample it again, and subtract it from the original. Then take the sum of squared values in the residual. Now start with the downsampled image and repeat recursively to get progressively lower frequencies. For the downsample and upsample I took a simple box filter. I kept 6 frequencies, since 26 is less than the minimum image size.
For each style, I computed the mean and standard deviation of each of the 6 power values from the examples. To classify a picture, I sum up the squared differences from the means, with the differences scaled using the standard deviations. It's not 100% accurate, but good enough to get a perfect score.

## Dreaming

This is a combination of a few common tree processing tasks. Firstly, the longest path might just be within one of the original trees, i.e., a tree diameter. This can be computed recursively on a tree by determining, for each node, the two longest paths downwards via different children (one or both can be zero if there are fewer than 2 children). The diameter path will have a highest node, and so the diameter will be the sum of these two lengths.
When add an edge to a tree, we must decide where to make the connection. The longest path from the connection point to anywhere in the tree ought to be as short as possible, and so for each point in the tree we need to know the distance to the furthest point. This is slightly more complex than before, since we also have to consider paths that start upwards. However, a second recursive walk (this time computing top-down instead of bottom-up) allows the shortest such paths to be found. For a given tree, let the radius be the distance from the optimal connection point to the furthest point in the tree.
Finally, we must decide how to connect the trees together. Sort the trees by decreasing radius r1 > r2 > .... Clearly, there will be a path of at least length r1 + r2 + L. If there at least three trees, they can't all be connected to each other, so there must also be a path of at least length r2 + r3 + 2L. Conversely, by connecting the first tree to every other tree (always using the optimal connection points), it is not hard to see that there are no other paths that can be longer than the worst of these.

## Wombats

I found this to be the most difficult of the tasks. Apart from being conceptually difficult, it also required a reasonably amount of tuning, and my solution still takes over 10s in many cases.
The basis of the solution is to note that C is relatively small, and so it is feasible to precompute the costs to get from any point on row X to any point on row Y, for some X and Y. Let's write such a table as {X, Y}. What's less obvious is that it's possible to combine {X, Y} and {Y, Z} to produce {X, Z} in O(C2) time. The trick is to use the fact that optimal paths won't cross over each other. Thus, if i < j, (X, i) to (Z, j-1) goes via (Y, p), and (X, i+1) to (Z, j) goes via (Y, q), then the optimal path from (X, i) to (Z, j) will go via (Y, r) where p ≤ r ≤ q. By iterating in order of increasing j - i, it is possible to compute {X, Z} in quadratic time.
We can combine this observation with a segment tree: for each appropriate i and a, we maintain {a·2i, (a+1)·2i}, computing it either directly (i = 0) or by combining two smaller intervals as above (where the upper bound exceeds R - 1, it is clamped). Each change invalidates O(log R) of these, so the time to update after a change is O(C^2 log R). Queries can be answered in O(1) time using the root of the segment tree.
The catch with this approach is that it requires too much memory: we can't even afford R different {X, Y} tables. Instead of keeping {X, X+1} at the finest level of the segment tree, we can instead store, say, {10X, 10X+10} for each X, and use 1/10th the memory. The cost is that updating the base level of the segment tree will now take 10 times as long.