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Can you solve this problem in o(n^5)?
Автор not_amir, 2 месяца назад, По-английски

For a contest I was one of the main contributors to, I came up with the following problem:

Problem

There are $$$n$$$ points on a circle numbered 0 to $$$n-1$$$. You are given $$$k$$$ pairs $$$(x_i, y_i)$$$ where $$$x_i\neq y_i$$$. For the $$$i$$$-th pair, you need to choose $$$S_i$$$ to be $$$[x_i, y_i)$$$ or $$$[y_i, x_i)$$$. This is a cyclic range, meaning if $$$a \gt b$$$, then $$$[a, b) = {a \dots n-1, 0 \dots b-1}$$$.

We say an unordered pair $$$(i, j)$$$ (possibly $$$i=j$$$) is connected if there exists some $$$l$$$ such that $$$i \in S_l$$$ and $$$j \in S_l$$$. Note that although I call it connected, I don't mean it to be a transitive property.

Let $$$S$$$ be the set of connected unordered pairs. For each unordered pair $$$(i, j)$$$, you are given a nonnegative cost $$$C[i][j]$$$. The cost of $$$S$$$ is defined to be the sum of the costs of all unordered pairs in $$$S$$$. The task is to find the minimal possible cost of $$$S$$$, given the cost matrix $$$C$$$ and the $$$k$$$ pairs.

I found the problem quite difficult, although the final structure feels deceptively simple in hindsight. Below is a description of the solution my friends and I came up with, which has a time complexity of $$$\mathcal{O}(n^5)$$$. I encourage you to try to solve the problem yourselves first!

Solution

Let's start by solving a specific case of the problem: $$$C[i][j] = 1$$$, and every unordered pair $$${x_i,y_i}$$$ appears exactly once in the input, thus $$$k = \binom{n}{2}$$$. The answer should depend only on $$$n$$$.

We can think about the problem in the following way: We have a bit matrix $$$A$$$ of size $$$n \times n$$$ initially filled with 0s, and for every given pair $$$(x, y)$$$, we need to color either $$$[x, y)^2$$$ or $$$[y, x)^2$$$. We want to maximize the number of 0s at the end of the process (counting only up to the main diagonal).

A brute-force search for small values of $$$n$$$ suggests that the maximal number of 0s is exactly $$$n$$$, and every optimal solution will have a very specific structure. The set of 0s is always a right-up path starting at $$$(n - 1 - a, 0)$$$ and ending at $$$(n - 1, b)$$$ of length $$$n$$$ (containing $$$n$$$ points, i.e. $$$a + b + 1 = n$$$).

Optimal Solution Structure

We can see why it has to be a right-up path: there can't be two 0s on the same $$$x+y$$$ diagonal because between them there will be a "barrier". Whichever interval we choose, all 0s must remain on the same side of that barrier:

Barrier on the diagonal

In this picture, we need to choose to color either the blue area or the red area. This puts a barrier (the green barrier) between the two yellow points. For any 2 points with the same $$$x+y$$$ value there exists such a barrier.

But why must it be of length $$$n$$$? For that, we need to expand our definition of a "diagonal".

Let's try to understand the topology we are working with. $$$(n, x)$$$ is identified with $$$(0, x)$$$, and $$$(x, n)$$$ is identified with $$$(x, 0)$$$ (in the sense that after point $$$n-1$$$ on the circle comes point 0). This quotient is famously known to be a torus. But we also have another condition: $$$(x, y)$$$ is identified with $$$(y, x)$$$ because the pairs are unordered. This turns our topology into a Möbius band. Here is a fundamental polygon calculation that shows this is the case:

Polygon Calculation

Arrows of the same color are identified in the indicated direction; the sequence of cuts and gluings converts the quotient into the standard fundamental polygon of a Möbius band.

The "columns" of this Möbius band are exactly the diagonals we looked for! We have $$$n$$$ diagonals, and each of them is the set of points $$$(x, y)$$$ with the same $$$(x + y) \pmod n$$$.

Diagonals

Every one of the $$$k$$$ pairs lands on one of the $$$n$$$ diagonals and forces us to choose a side. Because in this case we are given all $$$\binom{n}{2}$$$ pairs, there can be at most one 0 on each of the $$$n$$$ diagonals, so the number of 0s is at most $$$n$$$. (The maximum number of 0s, as it turns out, is exactly $$$n$$$, the matching construction will follow from the claims below).

Let's try to generalize this to a full solution. How does our Möbius band transformation look? Every point is adjacent to up two points in every neighboring diagonal, giving us a hex-like grid which looks a bit different depending on the parity of $$$n$$$. Every one of the given $$$k$$$ pairs serves as a barrier on one of the diagonals.

For example, for the following input:

$$$ \begin{aligned} n=5, \qquad C_{uv}=1, \qquad k=5, \\[5pt] \{(x_i, y_i)\} = \{(2,4), (0,3), (1,3), (0,4), (2,3)\}. \end{aligned} $$$

We will get the following structure:

Grid Structure

We call a set $$$\mathcal{L}$$$ of points a valid loop if:

  1. It forms a contiguous segment in every column of the Möbius band.
  2. Every two neighboring columns connect nicely (see picture). Concretely, the respective endpoints of the neighboring columns should be adjacent in the hex grid.
  3. It contains no barriers inside its area.

We define the weight of $$$\mathcal{L}$$$ as the sum of the costs of all points in it. I argue that the answer is the sum of the costs of all pairs minus the maximum weight of a valid loop. So for the example input above, the maximum valid loop will be:

Maximum Valid Loop

Here the points outside the black region form the valid loop. This has weight 8, so the answer is $$$\binom{6}{2} - 8 = 7$$$.

Proof

Direction 1: Every valid loop $$$\mathcal{L}$$$ is contained in the disconnected set of some assignment

Proof: For each barrier, choose the side that avoids $$$\mathcal{L}$$$ in the barrier's column. We just need to show that this also avoids $$$\mathcal{L}$$$ in every other column. Property 2 guarantees this: as we move between neighboring columns, the endpoints of $$$\mathcal{L}$$$ can move only along adjacent grid edges, which is exactly the rate at which either side of a barrier expands.

Direction 2: Every set of disconnected pairs of some assignment can be extended to a valid loop

Proof: If we look at a range in one column, we can extend the set in a way such that all points in its "shrinking" are also contained in the set.

Shrinking Range

Explanation: Assume the range in the left column is contained in the set of disconnect pairs. We show that we can obtain all the non black points in the picture. Assume for the sake of contradiction that there is a barrier (the green barrier) contained in the white area. Then one of its sides, the blue side or the red side, contains a point in the original range, contradicting the assumption that the original range consists of disconnected pairs under some assignment.

This shows that, once every column contains at least one disconnected pair, the disconnected set can be extended to satisfy the conditions of a valid loop.

So let's assume there are two diagonals (I am talking now about diagonals in the original matrix representation) with some nonempty range of disconnected pairs on them, and every diagonal between them has no disconnected pairs on it. Then we can extend the set of disconnected pairs in the following way:

Extending Set

Explanation: In case 1, we shrink the red range on the left to a single point, and do the same thing for the blue range on the right but in the opposite direction. We connect these two points with some right-up path. In case 2, we do a similar thing except that we fill the area between the two points with a rectangle. Any barrier in the orange region will have one side contain a point from the original red range, and the other side will contain a point from the original blue range, contradicting our assumption. There are a couple more cases that you can check if you want but it's all really the same thing. The point is that if there are two disconnected pairs on the same row (in the original matrix representation) then everything between them also has to be disconnected.

Note that because all costs are nonnegative, extending the set of disconnected pairs can only increase its total weight.

Algorithm

We now do the most naive algorithm possible for calculating the maximum weight of a valid loop. Define dp[s,e][l,r][i] as the maximum weight of a valid walk starting in column 0 with the range $$$[s, e]$$$ that ends in column $$$i$$$ with the range $$$[l, r]$$$. We then take the maximum over all walks that connect at the ends (this roughly corresponds to dp[s,e][n/2-e, n/2-s][n-1]). Each state considers only $$$\mathcal{O}(1)$$$ previous states, so the time complexity is $$$\mathcal{O}(n^5)$$$. For exact details on the DP transitions, see the implementation.

Implementation

You can also do DP directly on the original table representation (a valid loop is a right-up band going around the matrix with $$$(n, x)\sim(x, 0)$$$), but I think this approach is more elegant.

Can you solve this problem in $$$o(n^5)$$$?

The reason I am writing this blog is because I am frustrated that I wasn't able to reduce the time complexity down from $$$\mathcal{O}(n^5)$$$. It feels very disappointing that we made all those claims about the geometry of the problem, and then once we understood it, we did the most naive thing we could have done.

I have spent a lot of time trying to reduce the time complexity, but everything I tried failed. For example, attempts on supermodularity properties (which this DP does have, but precisely in a way that doesn't help us) didn't work out.

It would be amazing if someone could optimize the algorithm. It doesn't feel quite right that this is the best we can do.

More Thoughts

We can try to extend the problem in a couple of different directions:

  1. Negative costs: What if the costs could have been negative? I think there is a solution in $$$\mathcal{O}(n^7)$$$ or $$$\mathcal{O}(n^8)$$$ but this doesn't sound too interesting.
  2. Higher dimensions: What if we increased the dimension of the problem? Instead of counting pairs, we would count $$$d$$$-tuples (or a weighted combination of them). The topology of this is quite interesting. For odd $$$d$$$, we get a $$$d$$$-dimensional analogue of the solid torus. For even $$$d$$$ we get the twisted, non-orientable version of this. Concretely, for odd $$$d$$$ we have $$$SP^d(S^1) \cong S^1 \times \Delta^{d-1}$$$ and for even $$$d$$$ we have $$$SP^d(S^1) \cong S^1 \tilde{\times} \Delta^{d-1}$$$ where $$$\tilde{\times}$$$ denotes the non-trivial (twisted) product over $$$S^1$$$. For example, for $$$d=3$$$ we get the solid torus. I think I can use this to make an algorithm with time complexity somewhere between $$$\mathcal{O}(n^{11})$$$ and $$$\mathcal{O}(n^{15})$$$ for the $$$d=3$$$ case, although I am not quite sure about that. It makes a lot of sense to conjecture that there is an algorithm of time complexity $$$\mathcal{O}(n^{f(d)})$$$ for every $$$d$$$. My intuition says that as we get to higher dimensions, the barriers get sparser, so the problem may become easier. Does there exist a polynomial algorithm with $$$d$$$ as part of the input (with the cost function defined somehow implicitly, for example an interesting case is a positive linear combination)?
  3. Infinite dimensions: What if we take this to the limit? A generalization worth considering is counting connected subsets, again with the cost function defined implicitly. I didn't give much thought to this. Topologically, I am not quite sure what we have. I have some ideas, but I don't want to be imprecise.
  4. Different choices: What if instead of choosing between $$$[x, y)$$$ we are given a sequence $$$a_0 \lt \dots \lt a_{l-1}$$$ and we need to choose some $$$[a_i, a_{(i+1)\bmod l})$$$? I think we are stepping into NP-hard territory, but I am ready to be surprised.

Special thanks to DnlPrs809 for thinking with me about the problem and reviewing the blog!

Thank you for reading! I hope you found the question(s) interesting. Hope you liked my Microsoft Paint skills :)

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Maintaining mst with online edge insertions — no LCT needed
Автор not_amir, 2 года назад, По-английски

This blog is about maintaining mst with online edge insertions (or msf — minimum spanning forest to be exact) by using a data structure I came up with in $$$O(\log n)$$$ for edge insertion. although this problem can solved quite easily for anyone who knows LCT (link-cut tree), coding LCT is not really practical and has really bad constants. The data structure I present has very good constants because as you'll see all it does is manipulation on arrays. For obvious reasons that will be shown later I have called this data structure "weighted DSU".

Warmup problem

The problem is stated as follows: you are given an undirected graph where each edge has a time $$$t_i$$$. you are asked to answer queries of the form "what is the smallest time t such that you can get between u and v with all edges in time $$$ \lt = t$$$". This problem is analogous to calculating the msf of the graph where the weights are the times and being able to answer what is the maximum on the simple path between u and v. This problem can be solved a lot of different ways, I will present the dsu solution:
first we want to sort the edges by time/weight, than we will add the edges in a dsu like way but without path compression. Additionally we we will store on each edge in the dsu the weight of the edges it represents. To be specific we will store 3 arrays: parent, weight and size. when connecting two representatives u to v we update $$$parent[u] = v, weight[u] = w, size[v] += size[u]$$$. now we can find the representative of v at time w. One way to answer the queries will be to binary search on the answer, which will be $$$O(\log^2 n)$$$. A faster way is to essentially find the LCA of u and v in the dsu tree. this can be done by moving up with the vertex with current smallest out-edge weight. This will be $$$O(\log n)$$$.

The data structure

We will use the same idea from the warmup problem and try to turn it to a dynamic one. When adding an edge we want to think what would have happened if we had added the edges in increasing order, this will result in the following almost complete algorithm:
when adding an edge with weight w between u and v we can get the root of u at time w, get the root of v at time w and connect the one with smaller subtree size to the bigger. assume we connect u to v, this means we discarded the out-edge of u. To correct this we will recursively add the edge between u and parent[u] with weight weight[u] (adding the edge we deleted).

This algorithm has 3 problems:
1. The underlying increasing nature of the weight of the edges going up was what our original DS was relying on, now we may destroy it.
2. the array size is dynamic so we cant rely on it like we did in the warmup problem.
3. what if the vertices are already connected?

The first problem has a straightforward solution — when trying to access the parent of a vertex, jump over all the edges that have weight less than the edge to the parent and connect the vertex to the new parent. The get root function will now look like this:

int getRoot(int v, int w = INF - 1) {
	while (weight[v] <= w) {
		while (weight[parent[v]] <= weight[v])
			parent[v] = parent[parent[v]];
		v = parent[v];
	}
	return v;
}

The second problem can be solved in an elegant and simple way — instead of connecting by size connect by a random index! To be specific we create an array index that is initialized with random values. We connect the vertex with smaller index to the one with the bigger index. (this union technique is known to be with same expected complexity as union by size which is optimal complexity)
To solve the third problem we can use mst properties. when adding an edge {w, u, v} we find the "main" edge along the path between u and v — the edge with maximum weight (or an edge if there are multiple). if it is weight is bigger than w, than we delete it and add the edge with the algorithm described above. Putting all of this together we get the following DS:

code

Time complexity

While I do not have a formal proof for the time complexity, I can provide a rough sketch/intuition for it:
what our DS does is it tries to replicate the DS from the warmup problem. It does that pretty good but has minor defects as described in problem 2. My claim is that this defects constitute at most the depth of the tree in the warmup problem, which will still result in max depth of $$$O(\log n)$$$. Testing that I had done highly support this claim. If anyone has a formal proof please share it!
another advantage of this DS are the constants — while in theory LCT has $$$O(\log n)$$$ complexity, it has really bad constants and this data structure easily defeats it often time being 3-6 times faster.

Additional operations

Deleting maximal edge:
Assume that all weights in the graph are distinct. We will notice that the maximal edge in each connected component is necessary — it is a bottleneck edge. Therefore we can delete the maximal edge and it will break the component into two components. Deleting the edge wont create a problem because no other edge addition used it to "jump" up. The easiest way to deal with the problem if the weights are not distinct is to just make them distinct.

Saving connected component size:
For some problems we may want to maintain the size of each connected component in the msf, even after edge deletions. We can do this buy maintaining the subtree size of each node in the tree that is saved in our DS. that is, we maintain an array size such that for each node v size[v] equals the number of nodes are there such that v is a parent of them. The easiest and most elegant way I found doing it was to change the size array as if the the edges in the parent path from u and v were disconnected and when going through an edge add the size back.

code

Problems

Offline Dynamic Connectivity

solution

codeforces 1423 — problem H

solution

BOI 2020 Joker

solution

codeforces 76 — problem A

solution

codeforces 603 — problem E

solution

Comment if you can think of more uses of this DS

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