kattis Grid MST (最小曼哈顿生成树)

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Grid MST

This is a very simple problem. You are given N points. Some points may be repeated. The weight (distance) between two points is given by the Manhattan distance between the two points. Find the weight of a Minimum Spanning Tree that spans these N points.

Input

The input consists of:

One line with one integer N (1≤N≤100000), the number of points,
N lines each with two integers x and y (0≤x,y<1000), the coordinates of each point.

Output

Output one line with a single integer: The weight of a Minimum Spanning Tree that spans these N points.

Sample Input 1Sample Output 1

40 00 11 01 1

Sample Input 2Sample Output 2

50 010 010 011 112 2

解题思路：最小曼哈顿生成树裸题，最小曼哈顿生成树难点在于建图，其实不必每个点都要与其他n - 1个点相连，每个点只需要与周围最近的八个点相连就行，这八个点分别对应于以这个点为中心的八个区域，然后我们可以通过坐标变换，变成只处理右上方45度的区域，然后处理四个区域，也就是变换四次就行（因为要求的是最小生成树，a ->b, b->a相同的效果），然后具体处理是用树状数组或者线段树按y - x离散，维护一个x + y的最小值就行。

线段树：

#include <bits/stdc++.h>using namespace std;typedef long long LL;const int maxn = 100000 + 10;LL inf = 1e18;int n;int tot;int father[maxn];struct point{    int x, y, id;    bool operator <(const point &res) const    {        if(x == res.x) return y < res.y;        else return x < res.x;    }} Point[maxn<<4];struct edge{    int u, v;    LL w;    bool operator <(const edge &res) const    {        return w < res.w;    }} Edge[maxn<<10];struct node{    int l, r, id;    LL Min;} Node[maxn<<6];void add(int u, int v, LL w){    Edge[++tot].u = u;    Edge[tot].v = v;    Edge[tot].w = w;}void Pushup(int i){    int lson = i<<1;    int rson = lson|1;    if(Node[lson].Min <= Node[rson].Min)    {        Node[i].Min = Node[lson].Min;        Node[i].id = Node[lson].id;    }    else    {        Node[i].Min = Node[rson].Min;        Node[i].id = Node[rson].id;    }}void build(int i, int l, int r){    Node[i].l = l;    Node[i].r = r;    Node[i].Min = inf;    Node[i].id = -1;    if(l == r) return;    int f = i;    int mid = (l + r)>>1;    i <<= 1;    build(i, l, mid);    build(i|1, mid + 1, r);    //Pushup(f);}void update(int i, int loc, LL value, int id){    if(Node[i].l == Node[i].r)    {        if(value >= Node[i].Min) return;        Node[i].Min = value;        Node[i].id = id;        return;    }    int f = i;    i <<= 1;    if(loc <= Node[i].r) update(i, loc, value, id);    else update(i|1, loc, value, id);    Pushup(f);}pair<LL, int> query(int i, int l, int r){    if(Node[i].l == l && Node[i].r == r)    {        return make_pair(Node[i].Min, Node[i].id);    }    i <<= 1;    if(r <= Node[i].r) return query(i, l, r);    else if(l >= Node[i|1].l) return query(i|1, l, r);    else    {        pair<LL, int> pp1 = query(i, l, Node[i].r);        pair<LL, int> pp2 = query(i|1, Node[i|1].l, r);        if(pp1.first <= pp2.first) return pp1;        else return pp2;    }}void Tran(int type){    if(type == 1)    {        sort(Point + 1, Point + n + 1);        return;    }    else if(type == 2 || type == 4)    {        for(int i = 1; i <= n; i++)        {            swap(Point[i].x, Point[i].y);        }        sort(Point + 1, Point + n + 1);        return;    }    else if(type == 3)    {        for(int i = 1; i <= n; i++)        {            Point[i].x = -Point[i].x;        }        sort(Point + 1, Point + n + 1);        return;    }}void getGraph(){    for(int i = 1; i <= 4; i++)    {        Tran(i);        build(1, 0, 4000);        for(int j = n; j >= 1; j--)        {            int loc = Point[j].y - Point[j].x;            loc += 2000;            pair<LL, int> pp = query(1, loc, 4000);            if(pp.first < inf)            {                int id = pp.second;                add(Point[j].id, id, pp.first - LL(Point[j].x + Point[j].y));            }            update(1, loc, (LL)(Point[j].x + Point[j].y), Point[j].id);        }    }}void init(){    tot = 0;    for(int i = 1; i <= n; i++) father[i] = i;}int Find(int x){    if(father[x] == x) return x;    else return father[x] = Find(father[x]);}LL kruscal(){    LL ans = 0;    int judge = 0;    sort(Edge + 1, Edge + tot + 1);    for(int i = 1; i <= tot; i++)    {        int u = Edge[i].v;        int v = Edge[i].u;        LL w = Edge[i].w;        int f1 = Find(u);        int f2 = Find(v);        if(f1 != f2)        {            judge++;            ans += w;            father[f1] = f2;        }        if(judge == n - 1) break;    }    return ans;}int main(){    freopen("C:\\Users\\creator\\Desktop\\in1.txt","r",stdin) ;    scanf("%d", &n);    for(int i = 1; i <= n; i++)    {        scanf("%d%d", &Point[i].x, &Point[i].y);        Point[i].id = i;    }    init();    getGraph();    printf("%lld\n", kruscal());    return 0;}

树状数组：

#include <bits/stdc++.h>using namespace std;typedef long long LL;inline bool scan_d(int &num){        char in;bool IsN=false;        in=getchar();        if(in==EOF) return false;        while(in!='-'&&(in<'0'||in>'9')) in=getchar();        if(in=='-'){ IsN=true;num=0;}        else num=in-'0';        while(in=getchar(),in>='0'&&in<='9'){                num*=10,num+=in-'0';        }        if(IsN) num=-num;        return true;}const int maxn = 100000 + 10;LL inf = 1e18;int n;int tot;int father[maxn];struct point{    int x, y, id;    bool operator <(const point &res) const    {        if(x == res.x) return y < res.y;        else return x < res.x;    }} Point[maxn<<4];struct edge{    int u, v;    LL w;    bool operator <(const edge &res) const    {        return w < res.w;    }} Edge[maxn<<10];struct BITnode{    LL w;    int id;}Tree[maxn<<1];LL H[maxn<<1];int num;int lowbit(int x){    return x&(-x);}int Hash(LL key){    return lower_bound(H, H + num, key) - H + 1;}void add_edge(int u, int v, LL w){    Edge[++tot].u = u;    Edge[tot].v = v;    Edge[tot].w = w;}void Tran(int type){    if(type == 1)    {        sort(Point + 1, Point + n + 1);        return;    }    else if(type == 2 || type == 4)    {        for(int i = 1; i <= n; i++)        {            swap(Point[i].x, Point[i].y);        }        sort(Point + 1, Point + n + 1);        return;    }    else if(type == 3)    {        for(int i = 1; i <= n; i++)        {            Point[i].x = -Point[i].x;        }        sort(Point + 1, Point + n + 1);        return;    }}void add(int loc, LL v, int id){    for(int i = loc; i >= 1; i -= lowbit(i))    {        if(Tree[i].w > v)        {            Tree[i].w = v;            Tree[i].id = id;        }    }}int query(int loc){    LL Min = inf;    int id = -1;    for(int i = loc; i <= num; i += lowbit(i))    {        if(Tree[i].w < Min)        {            Min = Tree[i].w;            id = i;        }    }    return id;}void getGraph(){    for(int i = 1; i <= 4; i++)    {        Tran(i);        num = 0;        for(int j = n; j >= 1; j--)        {            Tree[j].id = -1;            Tree[j].w = inf;            LL key = LL(Point[j].y - Point[j].x);            H[num++] = key;        }        sort(H, H + num);        num = unique(H, H + num) - H;        for(int j = n; j >= 1; j--)        {            LL key = LL(Point[j].y - Point[j].x);            LL v = LL(Point[j].x + Point[j].y);            int loc = Hash(key);            int id = query(loc);            if(id != -1)            {                add_edge(Point[j].id, Tree[id].id, Tree[id].w - v);            }            add(loc, v, Point[j].id);        }    }}void init(){    tot = 0;    for(int i = 1; i <= n; i++) father[i] = i;}int Find(int x){    if(father[x] == x) return x;    else return father[x] = Find(father[x]);}LL kruscal(){    LL ans = 0;    int judge = 0;    sort(Edge + 1, Edge + tot + 1);    for(int i = 1; i <= tot; i++)    {        int u = Edge[i].v;        int v = Edge[i].u;        LL w = Edge[i].w;        int f1 = Find(u);        int f2 = Find(v);        if(f1 != f2)        {            judge++;            ans += w;            father[f1] = f2;        }        if(judge == n - 1) break;    }    return ans;}int main(){   // freopen("C:\\Users\\creator\\Desktop\\in1.txt","r",stdin) ;    scanf("%d", &n);    for(int i = 1; i <= n; i++)    {        //scanf("%d%d", &Point[i].x, &Point[i].y);        scan_d(Point[i].x);        scan_d(Point[i].y);        Point[i].id = i;    }    init();    getGraph();    printf("%lld\n", kruscal());    return 0;}

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