Submission #494910

#	Time	Username	Problem	Language	Result	Execution time	Memory
494910		Protostar	Harbingers (CEOI09_harbingers)	C++17	100 / 100	214 ms	25916 KiB

harbingers

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

using ll = long long;
using ld = long double;
using pll = pair<ll, ll>;

const int MAXN = 1e5 + 1;

/** A simple line class for linear functions */ 
struct Line {
	ll m, b;

	Line(ll m = 0, ll b = 0) : m(m), b(b) {}

	/** Evaluates the linear function at position x */ 
	ll operator()(ll x) { return m * x + b; }

	/** Returns the x-coordinate of the intersection of two lines */ 
	friend ld intersect(Line l1, Line l2) { 
		return (ld)(l2.b - l1.b) / (ld)(l1.m - l2.m); 
	}
};

int N;
ll S[MAXN];  // Start time for harbinger at node
ll V[MAXN];  // Velocity for harbinger at node
ll min_time[MAXN];  // Minimum time to reach the capital from node

vector<pll> T[MAXN];  // Stores tree edges

Line stk[MAXN];  // convex hull of lines
int stk_max = 0;  // end of convex hull stack

/**
 * Given the convex hull of points in stk[],
 * finds the minimum y value for the given x value
 * out of all lines in the hull
 * @param x The x position
 * @return The minimum y value in the convex hull
 */ 
ll line_min(ll x) {
	// binary search for min value
	int l = 0, r = stk_max - 1;
	while (l < r) {
		int m = (l + r) / 2;
		if (intersect(stk[m], stk[m + 1]) < x) {
			l = m + 1;
		} else {
			r = m;
		}
	}
	return stk[l](x);
}

/**
 * Gives the position such that if this line was to be added 
 * into the convex hull, it would occupy that position. The 
 * convex hull is stored in stk. If no lines are to be 
 * removed to make room for the new line, then the size of the
 * convex hull is returned. Note that the convex hull 
 * is NOT modified in this function. 
 * @param line The line to be queried
 * @return An index between 1 and stk_max (inclusive) specifying 
 *		 the location the given line would occupy should it 
 *		 be added to the hull
 */ 
ll remove_pos(Line line) {
	// check if hull is trivial or if line belongs at the end
	if (stk_max < 2 || 
		intersect(line, stk[stk_max - 1]) >= intersect(stk[stk_max - 1], stk[stk_max - 2])) {
		return stk_max;
	}

	// begin at l = 1 because we need to check intersection between k and k - 1
	int l = 1, r = stk_max - 1;
	while (l < r) {
		int m = (l + r) / 2;
		if (intersect(line, stk[m]) < intersect(stk[m], stk[m - 1])) {
			r = m;
		} else {
			l = m + 1;
		}
	}
	return l;
}

/**
 * Finds the smallest dp value for all nodes in the 
 * current subtree
 * @param u Current node
 * @param p Parent of current node
 * @param d Distance from root
 */ 
void dfs(int u = 1, int p = 0, ll d = 0) {
	int prev_max, prev_pos;
	Line prev_line;
	if (u == 1) {
		min_time[u] = 0;
		stk[stk_max++] = {0, 0}; 
	} else {
		min_time[u] = line_min(V[u]) + d * V[u] + S[u];  // get dp value by querying convex hull
		Line l(-d, min_time[u]);  // construct new line that might be added into hull
		prev_max = stk_max;  // store previous hull size
		prev_pos = stk_max = remove_pos(l);  // update hull size to include new line
		prev_line = stk[stk_max];  // store previous line at replacement position
		stk[stk_max++] = l;  // update replacement position to new line
	}

	// recurse to children
	for (auto [v, w] : T[u]) {
		if (v != p) {
			dfs(v, u, d + w);
		}
	}

	// reset any changes made at this step
	if (u != 1) {
		stk_max = prev_max;  // reset convex hull size 
		stk[prev_pos] = prev_line;  // reset replacement position
	}
}

int main() {
	cin >> N;
	for (int i = 1; i < N; i++) {
		int u, v, w;
		cin >> u >> v >> w;
		T[u].emplace_back(v, w);
		T[v].emplace_back(u, w);
	}
	for (int i = 2; i <= N; i++) {
		cin >> S[i] >> V[i];
	}

	dfs();

	for (int i = 2; i <= N; i++) {
		cout << min_time[i] << ' ';
	}
	cout << '\n';
}

• Subtask 1: 100 / 100