照亮体育馆 Barisal Stadium
题目为逆时针顺序编号，这里直接将数组开两倍来处理环。（然而不知为啥开到1000也能过）
定义：
Corners[i]Corners[i]Corners[i]为体育馆点的坐标。
Lights[i]Lights[i]Lights[i]为灯的坐标及费用。
IsShineOnCur[i]IsShineOn_{Cur}[i]IsShineOnCur[i]表示第CurCurCur盏灯是否能照到iii，用于计算范围。
RangeCurRange_{Cur}RangeCur为第CurCurCur盏灯的照射范围，从LeftLeftLeft到RightRightRight，用于初始化dpdpdp
dp[Left][Right]dp[Left][Right]dp[Left][Right]为从第LeftLeftLeft盏灯照到第RightRightRight盏灯的最少花费。
初始化IsShineOnCur[i]\pmb{IsShineOn_{Cur}[i]}IsShineOnCur[i]IsShineOnCur[i]IsShineOnCur[i]：
令A、BA、BA、B两点为体育馆的两点，CCC为体育馆的一盏灯，那么CCC要想照亮边ABABAB，那么必须有0∘<∠BAC<180∘0^{\circ}< ∠BAC< 180^{\circ}0∘<∠BAC<180∘（手动转一转就知道了）
因为有：
sin(∠BAC)=AB⃗×AC⃗sin(∠BAC)=\vec{AB}×\vec{AC}sin(∠BAC)=AB×AC并且当sinθ>0sin\theta>0sinθ>0时，θ∈(0∘,180∘)\theta\in(0^{\circ},180^{\circ})θ∈(0∘,180∘)。
因此只要叉乘大于0即可。注意是逆时针的顺序。
//Cur为灯编号
void InitIsShineOn(const int& Cur) {
memset(IsShineOn, false, sizeof(IsShineOn));
for (int i = 1; i <= N; ++i) {
//i以及i的下一个点位体育馆的点
int&& Position = i % N + 1;
//如果叉乘大于0
IsShineOn[i] =
(Lights[Cur].x - Corners[i].x) * (Corners[Position].y - Corners[i].y)
>
(Corners[Position].x - Corners[i].x) * (Lights[Cur].y - Corners[i].y);
}
return;
}初始化RangeCur\pmb{Range_{Cur}}RangeCurRangeCurRangeCur：
扫一遍即可
void InitRange(const int& Cur) {
if (IsShineOn[1] && IsShineOn[N]) {
Range.Left = N;
Range.Right = 1;
while (IsShineOn[Range.Left]) {
--Range.Left;
}
while (IsShineOn[Range.Right]) {
++Range.Right;
}
++Range.Left;
--Range.Right;
Range.Right += N;
}
else {
Range.Left = 1;
Range.Right = N;
while (!IsShineOn[Range.Left]) {
++Range.Left;
}
while (!IsShineOn[Range.Right]) {
--Range.Right;
}
}
return;
}dp初始化：
dp=infif(第Cur盏灯能照亮Left到Right)dp[Left][Right]=min(dp[Left][Right],Lights[i].Cost])dp=inf\\if(第Cur盏灯能照亮Left到Right)\\dp[Left][Right]=min(dp[Left][Right],Lights[i].Cost])dp=infif(第Cur盏灯能照亮Left到Right)dp[Left][Right]=min(dp[Left][Right],Lights[i].Cost])
//判断第Cur盏灯能否照亮(Left,Right)的范围
bool Judgt(const int& Left, const int& Right) {
return
Range.Left - N <= Left && Range.Right - N >= Right
||
Range.Left <= Left && Range.Right >= Right
||
Range.Left + N <= Left && Range.Right + N >= Right;
}
void InitDP(const int& Cur) {
for (int Length = 0; Length < N; ++Length) {
for (int Left = 1; Left <= 2 * N - 1; ++Left) {
int&& Right = Left + Length;
if (Right > 2 * N - 1) {
break;
}
if (Judgt(Left, Right)) {
dp[Left][Right] = min(dp[Left][Right], Lights[Cur].Cost);
}
}
}
return;
}转移方程：
dp[Left][Right]=min(dp[Left][k]+dp[k+1][Right])Left≤k<Rightdp[Left][Right]=min(dp[Left][k]+dp[k+1][Right])\\Left\leq k<Rightdp[Left][Right]=min(dp[Left][k]+dp[k+1][Right])Left≤k<Right经典的区间dpdpdp
最后答案为min(dp[i][i+N−1]),(1≤1≤N)min(dp[i][i+N-1]),(1\leq 1\leq N)min(dp[i][i+N−1]),(1≤1≤N)（双倍数组处理环的答案）
AC代码：
#include<iostream>
#include<string>
#include<cstring>
#include<algorithm>
#include<vector>
#include<cmath>
#include<map>
using namespace std;
constexpr static long long inf = 0x3f3f3f3f3f3f3f3f;
struct Corner {
double x, y;
}Corners[31];
struct Light {
double x, y;
long long Cost;
}Lights[1001];
struct {
int Left, Right;
}Range;
int N, M;
bool IsShineOn[1001]{ false };
long long dp[1001][1001];
bool Input() {
cin >> N;
if (!N) {
return false;
}
for (int i = 1; i <= N; ++i) {
cin >> Corners[i].x >> Corners[i].y;
}
cin >> M;
for (int i = 1; i <= M; ++i) {
cin >> Lights[i].x >> Lights[i].y >> Lights[i].Cost;
}
return true;
}
void InitIsShineOn(const int& Cur) {
memset(IsShineOn, false, sizeof(IsShineOn));
for (int i = 1; i <= N; ++i) {
int&& Position = i % N + 1;
IsShineOn[i] =
(Lights[Cur].x - Corners[i].x) * (Corners[Position].y - Corners[i].y)
>
(Corners[Position].x - Corners[i].x) * (Lights[Cur].y - Corners[i].y);
}
return;
}
void InitRange(const int& Cur) {
if (IsShineOn[1] && IsShineOn[N]) {
Range.Left = N;
Range.Right = 1;
while (IsShineOn[Range.Left]) {
--Range.Left;
}
while (IsShineOn[Range.Right]) {
++Range.Right;
}
++Range.Left;
--Range.Right;
Range.Right += N;
}
else {
Range.Left = 1;
Range.Right = N;
while (!IsShineOn[Range.Left]) {
++Range.Left;
}
while (!IsShineOn[Range.Right]) {
--Range.Right;
}
}
return;
}
bool Judgt(const int& Left, const int& Right) {
return
Range.Left - N <= Left && Range.Right - N >= Right
||
Range.Left <= Left && Range.Right >= Right
||
Range.Left + N <= Left && Range.Right + N >= Right;
}
void InitDP(const int& Cur) {
for (int Length = 0; Length < N; ++Length) {
for (int Left = 1; Left <= 2 * N - 1; ++Left) {
int&& Right = Left + Length;
if (Right > 2 * N - 1) {
break;
}
if (Judgt(Left, Right)) {
dp[Left][Right] = min(dp[Left][Right], Lights[Cur].Cost);
}
}
}
return;
}
void Compute() {
for (int Length = 0; Length < N; ++Length) {
for (int Left = 1; Left <= 2 * N - 1; ++Left) {
int&& Right = Left + Length;
if (Right > 2 * N - 1) {
break;
}
for (int k = Left; k < Right; ++k) {
if (dp[Left][k] == inf && dp[k + 1][Right] == inf) {
continue;
}
dp[Left][Right] = min(dp[Left][Right], dp[Left][k] + dp[k + 1][Right]);
}
}
}
}
long long DP() {
memset(dp, 0x3f, sizeof(dp));
for (int i = 1; i <= M; ++i) {
InitIsShineOn(i);
InitRange(i);
InitDP(i);
}
Compute();
long long Ans = inf;
for (int i = 1; i <= N; ++i) {
Ans = min(Ans, dp[i][i + N - 1]);
}
return Ans;
}
int main() {
while (Input()) {
long long&& Ans = DP();
if (Ans == inf) {
puts("Impossible.");
}
else {
cout << Ans << endl;
}
}
return 0;
}手机扫一扫
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