Codeforces 437E The Child and Polygon (interval DP)

Link: Codeforces 437E The Child and Polygon

Question: Give a polygon and ask how many kinds of segmentation methods are available. Divide the Polygon into multiple triangles.

Solution: first, we need to understand the nature of the Cross Product of the vector. At the beginning, we need to convert the given point to clockwise, and then use the interval dp for calculation. Dp [I] [j] indicates that a dp [I] [j] cutting method can be provided from point I to Point j. Then, determine whether I and j can be used as the cutting line. If k is selected in I and j, the point k must be in the clockwise direction of I and j.

#include 
  
   #include 
   
    #include using namespace std;typedef long long ll;const int N = 205;const ll MOD = 1e9+7;struct point {    ll x, y;    ll operator * (const point& c) const {        return x * c.y - y * c.x;    }    point operator - (const point& c) const {        point u;        u.x = x - c.x;        u.y = y - c.y;        return u;    }}p[N];int n;ll dp[N][N];void init () {    scanf("%d", &n);    memset(dp, -1, sizeof(dp));    for (int i = 0; i < n; i++)        scanf("%lld %lld", &p[i].x, &p[i].y);    ll tmp = 0;    p[n] = p[0];    for (int i = 0; i < n; i++)        tmp += p[i] * p[i+1];    if (tmp < 0)        reverse(p, p + n);}ll solve (int l, int r) {    if (dp[l][r] != -1)        return dp[l][r];    ll& ans = dp[l][r];    ans = 0;    if (r - l == 1)        return ans = 1;    for (int i = l + 1; i < r; i++) {        if ((p[l] - p[r]) * (p[i] - p[r]) > 0)            ans = (ans + solve(l, i) * solve(i, r)) % MOD;    }    return ans;}int main () {    init();    printf("%lld\n", solve(0, n-1));    return 0;}