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usaco-guide/content/6_Plat/Slope.md
Benjamin Qi af1719ad14 order, rename
2020-06-15 21:06:49 -04:00

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id title author prerequisites
slope Slope Trick Benjamin Qi
some familiarity with at least one of the two tutorials mentioned in this module

Slope trick refers to manipulating piecewise linear convex functions. Includes a simple solution to Landscaping.

Tutorials

From the latter link (modified):

Slope trick is a way to represent a function that satisfies the following conditions:

  • It can be divided into multiple sections, where each section is a linear function (usually) with an integer slope.
  • It is a convex/concave function. In other words, the slope of each section is non-decreasing or non-increasing when scanning the function from left to right.

It's generally applicable as a DP optimization. Usually you can come up with a slower DP (ex. O(N^2)) first and then optimize it to O(N\log N) with slope trick.

Buy Low Sell High

Slow Solution: Let dp[i][j] denote the maximum amount of money you can have on day i if you have exactly j shares of stock on that day. The final answer will be dp[N][0]. This easily leads to an O(N^2) DP.

Of course, we never used the fact that the DP is concave down! Specifically, let dif[i][j]=dp[i][j]-dp[i][j+1]\ge 0. Then dif[i][j]\le dif[i][j+1] for all j\ge 0 (ignoring the case when we get dp values of -\infty).

We'll process the shares in order. Suppose that on the current day shares are worth p. We can replace (buy or sell a share) in the statement with (buy, then sell between 0 and 2 shares).

  • If we currently have j shares and overall balance b, then after buying, j increases by one and b decreases by p. The differences between every two consecutive elements do not change.
  • If we choose to buy a share, this is equivalent to setting dp[i][j]=\max(dp[i][j],dp[i][j+1]+p) for all j. By the concavity condition, dp[i][j]=dp[i][j+1]+p will hold for all j less than a certain threshold while dp[i][j+1] will hold for all others. So this is equivalent to inserting p into the list of differences while maintaining the condition that the differences are in sorted order.
  • So we add p to the list of differences two times. After that, we should pop the smallest difference in the list because we can't end up with a negative amount of shares.

(insert diagram)

(insert example)

The implementation is quite simple; maintain a priority queue that allows you to pop the minimum element.

My Solution 
#include <bits/stdc++.h>
using namespace std;
 
int main() {
	int N; cin >> N;
	priority_queue<int,vector<int>,greater<int>> pq;
	long long ans = 0;
	for (int i = 0; i < N; ++i) {
		int p; cin >> p; ans -= p;
		pq.push(p); pq.push(p); pq.pop();
	}
	for (int i = 0; i < N; ++i) {
		ans += pq.top();
		pq.pop();
	}
	cout << ans << "\n";
}

Potatoes

Let dif_i=a_i-b_i. Defining d_j=\sum_{i=1}^jdif_i, our goal is to move around the potatoes such that d_0,d_1,\ldots,d_N is a non-decreasing sequence. Moving a potato is equivalent to changing exactly one of the d_i (aside from d_0,d_N) by one.

Slow Solution: Let dp[i][j] be the minimum cost to determine d_0,d_1,\ldots,d_i such that d_i\le j for each 0\le j\le d_N. This gives a O(N\cdot d_N) solution.

As before, this DP is concave up for a fixed i! Given a piecewise linear function DP_x, we need to support the following operations.

  • Add |x-k| to the function for some k
  • Set DP_x=\min(DP_x,DP_{x-1}) for all x

Again, these can be done with a priority queue in O(N\log N) time!

My Solution 
#include <bits/stdc++.h>
using namespace std;

typedef long long ll;

int N;
ll fst = 0; // value of DP function at 0
priority_queue<ll> points; // points where DP function changes slope

int main() {
	cin >> N;
	vector<ll> dif(N+1);
	for (int i = 1; i <= N; ++i) {
		int a,b; cin >> a >> b;
		dif[i] = a-b+dif[i-1];
	}
	assert(dif[N] >= 0);
	for (int i = 1; i < N; ++i) {
		if (dif[i] < 0) fst -= dif[i], dif[i] = 0;
		fst += dif[i];
		points.push(dif[i]); points.push(dif[i]);
		points.pop();
	}
	while (points.size()) {
		ll a = points.top(); points.pop();
		fst -= min(a,dif[N]);
	}
	cout << fst << "\n";
}

Landscaping

This is quite similar to the previous task, so it's easy to guess that slope trick is applicable.

Again, let's first come up with a slow DP. Let dp[i][j] equal the number of ways to move dirt around the first i flowerbeds such that the first i-1 flowerbeds all have the correct amount of dirt while the $i$-th flowerbed has j extra units of dirt (or lacks -j units of dirt if j is negative). The answer will be dp[N][0].

This DP is concave up for any fixed i. To get dp[i+1] from dp[i] we must be able to support the following operations.

  • Shift the DP curve A_i units to the right.
  • Shift the DP curve B_i units to the left.
  • Add Z\cdot |j| to DP[j] for all j.
  • Set DP[j] = \min(DP[j],DP[j-1]+X) and DP[j] = \min(DP[j],DP[j+1]+Y) for all j.

As before, it helps to look at the differences dif[j]=DP[j+1]-dif[j] instead. Then the last operation is equivalent to the following:

  • For all j\ge 0, we set dif[j] = \min(dif[j]+Z,X)
  • For all j<0, we set dif[j] = \max(dif[j]-Z,-Y).

If we maintain separate deques for dif depending on whether j\ge 0 or j<0 and update all of the differences in the deques "lazily" then we can do this in O(\sum A_i+\sum B_i) time.

Bonus: Solve this problem when \sum A_i+\sum B_i is not so small.

My Solution 
#include <bits/stdc++.h>

using namespace std;

int N,X,Y,Z;
int difl, difr;
deque<int> L, R;
long long ans;

void rig() { // shift right A
	if (L.size() == 0) L.push_back(-Y-difl);
	int t = L.back()+difl; L.pop_back();
	t = max(t,-Y); ans -= t;
	R.push_front(t-difr);
}

void lef() { // shift left B
	if (R.size() == 0) R.push_front(X-difr);
	int t = R.front()+difr; R.pop_front();
	t = min(t,X); ans += t;
	L.push_back(t-difl);
}

int main() {
	freopen("landscape.in","r",stdin);
	freopen("landscape.out","w",stdout);
	cin >> N >> X >> Y >> Z; 
	for (int i = 0; i < N; ++i) {
		int A,B; cin >> A >> B; 
		for (int j = 0; j < A; ++j) rig(); // or we can just do |A-B| shifts in one direction
		for (int j = 0; j < B; ++j) lef(); 
		difl -= Z, difr += Z; // adjust slopes differently for left and right of j=0
	}
	cout << ans << "\n";
}

Problems