CodeForces - 798D Mike and distribution 想法题，数学证明

题意:给你两个数列a,b,你要输出k个下标，使得这些下标对应的a的和大于整个a数列的和的1/2.同时这些下标对应的b

//题解：首先将条件换一种说法，就是要取floor(n/2)+1个数使得这些数大于剩下的数。然后想到两种取法，一种是排好序选前半段。令一种取法是两个两个分组，每组取大的那个。
//然后就可以（很困难地）想到一种巧妙的取法，先用一个结构体保存a,b,idx,按照a从大到小排序，先取第一个，再对剩下的每两个一组取其中b更大的。这样保证了被选出
//的b一定大于剩下的b，也就满足了条件。然后考虑a，按照前面的分组，每组的a并不一定选到了大的。但是我们（并没有）观察到第一个选的a （也就是a的最大值）一定大于第一组没选到的那个a、第一组选的a一定大于第二组没选出的a···；
//同理这样选出的a也一定大于剩下的a。所以算法就是读取，排序，一个循环 。（codeforces上python只有三行。。。）

ac代码：

#define _CRT_SECURE_NO_WARNINGS
#include<iostream>
#include<stdio.h>
#include<algorithm>
using namespace std;
const int maxn = 1e5 + ;
struct node { int a, b, idx; }s[maxn];
bool cmp(node x,node y) {
    return x.a > y.a;
}
int main() {
    int n;
    cin >> n;
    for (int i = ; i <= n; i++) {
        scanf("%d", &s[i].a);
        s[i].idx = i;
    }
    for (int i = ; i <= n; i++) {
        scanf("%d", &s[i].b);
    }

    sort(s + , s +  + n,cmp);
    cout << n /  +<<endl;
    cout << s[].idx;
    for (int i = ; i <= n; i += ) {
        int ans;
        s[i].b > s[i + ].b ? ans = s[i].idx : ans=s[i + ].idx;
        printf(" %d", ans);
    }
    //cin >> n;

}

附 ：codeforces上看到的神级头文件，

#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>
#include <vector>
#include <set>
#include <map>
#include <queue>
#include <cstdlib>
#include <string>
#include <complex>
#include <bitset>
#include <cmath>

using namespace std;

#define mst(x,v) memset(x,v,sizeof(x))
#define rp(k,x,v) for(int k = x;k<v;++k)
#define rps(s) for(int i = 0;s[i];++i)
#define RP(k,x,v) for(int k = x;k<=v;++k)
#define lp(k,x,v) for(int k = x;k>v;--k)
#define LP(k,x,v) for(int k = x;k>=v;--k)
#define ll long long
#define ull unsigned long long
#define fr(file) freopen(file,"r",stdin);
#define fw(file) freopen(file,"w",stdout);
#define lowbit(x) (x&-x)
#define adde(u,v) edg[++ecnt]=Edge(u,v,g[u]),g[u]=ecnt
#define addew(u,v,w) edg[++ecnt]=Edge(u,v,g[u],w),g[u]=ecnt
#define addedg(u,v,w,cap) edg[++ecnt]=Edge(u,v,g[u],w,cap),g[u]=ecnt,edg[++ecnt]=Edge(v,u,g[v],-w,0),g[v]=ecnt
#define fson for(int i = g[u];~i;i=edg[i].next)
#define LB lower_bound
#define UB upper_bound
#define inp(n,a) RP(i,1,n) ru(a[i]);
#define pii pair<int,int>
#define pll pair<ll,ll>
#define pil pair<int,ll>
#define pdd pair<double,double>
#define piii pair<pii,int>
#define plll pair<pll,ll>
#define mpt int mid = l+r>>1;
#define ls (k<<1)
#define rs (k<<1|1)
#define INF (1<<28)
#define LINF (1LL << 60)
#define initi(n,a) RP(i,1,n) a[i]=INF;
#define initl(n,a) RP(i,1,n) a[i]=LINF;
#define dsui(n,p) RP(i,1,n) p[i]=i;
#define all(a) a.begin(),a.end()
#define inr (ql<=l && r<=qr)
#define cpx complex<double>
#define spf(s,a) RP(i,1,n) s[i]=s[i-1]+a[i]
#define ssf(s,a) lp(i,n,0) g[i]=g[i+1]+a[i]
#define upmax(a,b) a=max(a,b)
#define upmin(a,b) a=min(a,b)
#define getmax(maxd,a,n) RP(i,1,n) upmax(maxd,a[i])
#define getmin(mind,a,n) RP(i,1,n) upmin(mind,a[i])

char READ_DATA;
int SIGNAL_INPUT;
template <typename Type>
inline Type ru(Type &v)
{
    SIGNAL_INPUT = ;
    while ((READ_DATA = getchar()) < '' || READ_DATA > '')
        if (READ_DATA == '-')
            SIGNAL_INPUT = -;
        else if (READ_DATA == EOF)
            return EOF;
    v = READ_DATA - '';
    while ((READ_DATA = getchar()) >= '' && READ_DATA <= '')
        v = v *  + READ_DATA - '';
    v *= SIGNAL_INPUT;
    return v;
}
template<typename A, typename B > inline char ru(A&x, B&y) { if (ru(x) == EOF) return EOF; ru(y); return ; }
template<typename A, typename B, typename C>inline char ru(A&x, B&y, C&z) { if (ru(x) == EOF) return EOF; ru(y); ru(z); return ; }
template<typename A, typename B, typename C, typename D>inline char ru(A&x, B&y, C&z, D&w) { if (ru(x) == EOF) return EOF; ru(y); ru(z); ru(w); return ; }
int TEMP;
inline void swap(int &a, int &b)
{
    TEMP = a;
    a = b;
    b = TEMP;
}
struct Vector
{
    double x, y;
    Vector(double _x = 0.0, double _y = 0.0)
    {
        x = _x;
        y = _y;
    }
    int operator<(const Vector &cmp) const
    {
        return x < cmp.x || (x == cmp.x && y < cmp.y);
    }
};
typedef Vector Point;
inline double dis(Point &a, Point &b)
{
    return sqrt(pow(a.x - b.x, ) + pow(a.y - b.y, ));
}

struct Edge
{
    int  u, v, next;
    int w;
    int cap, flow;
    Edge(int _u = , int _v = , int nxt = -, int _w = , int _cap = )
    {
        u = _u;
        v = _v;
        w = _w;
        cap = _cap;
        flow = ;
        next = nxt;
    }

    int operator<(const Edge &b) const
    {
        return w < b.w;
    }
};

namespace BalancedTreeNode
{
    struct Node
    {
        Node *ch[], *fa;
        int v, s, rev, setv, sum, lsum, rsum, mxsum;

        Node(int val = )
        {
            v = sum = val;
            s = rev = ;
            lsum = rsum = val;
            mxsum = val;
            setv = INF;
        }

        int cmpv(int val)
        {
            return val == v ? - : val>v;
        }

        int cmprk(int rk)
        {
            return rk == ch[]->s +  ? - : rk > ch[]->s + ;
        }

        void pushdown()
        {
            if (rev)
            {
                swap(ch[], ch[]);
                swap(lsum, rsum);
                if (ch[]->v != -INF) ch[]->rev ^= ;
                if (ch[]->v != -INF) ch[]->rev ^= ;
                rev = ;
            }

            if (setv != INF)
            {
                v = setv;
                lsum = rsum = mxsum = max(v, sum = v*s);
                if (ch[]->v != -INF) ch[]->setv = setv;
                if (ch[]->v != -INF) ch[]->setv = setv;
                setv = INF;
            }
        }

        void maintain()
        {
            pushdown();
            ch[]->pushdown(); ch[]->pushdown();
            s = ch[]->s + ch[]->s + ;
            sum = ch[]->sum + ch[]->sum + v;
            lsum = rsum = -INF;
            mxsum = -INF;
            upmax(lsum, max(ch[]->lsum, ch[]->sum + v + max(, ch[]->lsum)));
            upmax(rsum, max(ch[]->rsum, ch[]->sum + v + max(, ch[]->rsum)));
            upmax(mxsum, max(, ch[]->rsum) + v + max(, ch[]->lsum));
            upmax(mxsum, max(ch[]->mxsum, ch[]->mxsum));
        }

    };
    Node* null = new Node(-INF);

    Node* newnode(int val)
    {
        Node* k = new Node(val);
        k->ch[] = k->ch[] = k->fa = null;
        k->maintain();
        return k;
    }
    inline void rotate(Node *x, const int d)
    {
        Node* y = x->fa;
        y->ch[d ^ ] = x->ch[d];
        if (x->ch[d] != null) x->ch[d]->fa = y;
        x->fa = y->fa;
        if (y->fa->ch[] == y) x->fa->ch[] = x;
        else if (y->fa->ch[] == y) x->fa->ch[] = x;
        y->fa = x;
        x->ch[d] = y;
        y->maintain();
        x->maintain();
    }

    Node* insert(Node* &k, int val)
    {
        if (k == null)
            return k = newnode(val);
        else
        {
            int d = k->cmpv(val);
            if (d == -) return k;
            int reach = k->ch[d] == null;
            Node* p = insert(k->ch[d], val);
            k->maintain();
            if (reach) k->ch[d]->fa = k;
            return p;
        }
    }

    void Remove_Tree(Node* k)
    {
        if (k->ch[] != null) Remove_Tree(k->ch[]);
        if (k->ch[] != null) Remove_Tree(k->ch[]);
        delete k;
    }

    int Rank(Node *k, int val)
    {
        int rk = , d;
        while (k != null)
        {
            k->pushdown();
            d = k->cmpv(val);
            if (d) rk += k->ch[]->s + ;
            if (d == -) break;
            k = k->ch[d];
        }
        return rk;
    }

    Node* Kth(Node* k, int rk)
    {
        int d;
        while (k != null)
        {
            k->pushdown();
            d = k->cmprk(rk);
            if (d == -)
                return k;
            if (d) rk -= k->ch[]->s + ;
            k = k->ch[d];
        }
        return null;
    }
}
namespace Splay
{
    using namespace BalancedTreeNode;

    inline int isfa(Node *x, Node* &y)
    {
        return (y = x->fa) != null && (y->ch[] == x || y->ch[] == x);
    }
    inline void splay(Node* x)
    {
        x->pushdown();
        Node *y, *z;
        while (isfa(x, y))
        {
            if (isfa(y, z))
            {
                z->pushdown(); y->pushdown(); x->pushdown();
                int d = y == z->ch[];
                if (x == y->ch[d]) rotate(x, d ^ ), rotate(x, d);
                else rotate(y, d), rotate(x, d);
            }
            else
            {
                y->pushdown(); x->pushdown();
                rotate(x, x == y->ch[]);
                break;
            }
        }
    }

    inline void Insert(Node* &k, int val)
    {
        splay(k = insert(k, val));
    }

    Node* build(int *a, int l, int r)
    {
        if (l > r) return null;
        mpt;
        Node* k = newnode(a[mid]);
        k->ch[] = build(a, l, mid - );
        k->ch[] = build(a, mid + , r);
        if (k->ch[] != null) k->ch[]->fa = k;
        if (k->ch[] != null) k->ch[]->fa = k;
        if (mid == )
            k->sum = ;
        k->maintain();

        return k;
    }

    Node* merge(Node *left, Node *right)
    {
        splay(left = Kth(left, left->s));
        left->ch[] = right;
        right->fa = left;
        left->maintain();
        return left;
    }

    void split(Node *k, int rk, Node* &left, Node* &right)
    {
        splay(k = Kth(k, rk));
        left = k;
        right = left->ch[];
        left->ch[] = null;
        right->fa = null;
        left->maintain();
    }
};
namespace LCT
{
    using namespace Splay;

    Node* access(Node* k)
    {
        Node* p = null;
        while (k != null)
        {
            splay(k);
            k->ch[] = p;
            p = k;

            k->maintain();
            k = k->fa;
        }
        return p;
    }

    void makeroot(Node* k)
    {
        access(k)->rev ^= ;
        splay(k);
    }

    Node* getroot(Node* k)
    {
        k = access(k);
        k->pushdown();
        while (k->ch[] != null)
        {
            k = k->ch[];
            k->pushdown();
        }
        return k;
    }

    void link(Node* x, Node* y)
    {
        makeroot(x);
        x->fa = y;
        access(x);
    }

    void cut(Node* x, Node* y)
    {
        makeroot(x);
        access(y); splay(y);
        y->ch[]->fa = null;
        y->ch[] = null;
        y->maintain();
    }
}
ll gcd(ll a, ll b)
{
    while (b)
    {
        a %= b;
        swap(a, b);
    }
    return a;
}
void exgcd(ll a, ll b, ll &d, ll &x, ll &y)
{
    if (!b) d = a, x = , y = ;
    else exgcd(b, a%b, d, y, x), y -= x*(a / b);
}
ll inv(ll a, ll mod)
{
    ll d, x, y;
    exgcd(a, mod, d, x, y);
    return d ==  ? (x + mod) % mod : -;
}
int bitcnt(ll x)
{
    int cnt = ;
    while (x)
    {
        cnt += (x & );
        x >>= ;
    }
    return cnt;
}
ll key = ;
ll mod = 1e9 + ;
ll qkmul(ll a, ll b)
{
    ll ans = , x = a;
    while (b)
    {
        if (b & ) (ans += x) %= mod;
        (x += x) %= mod;
        b >>= ;
    }
    return ans;
}
ll qkpow(ll a, ll b)
{
    ll ans = , x = a;
    while (b)
    {
        if (b & ) (ans *= x) %= mod;
        (x *= x) %= mod;
        b >>= ;
    }
    return ans;
}
const int maxn = 1e5 + , maxp = 1e6;
const double pi = acos(-1.0), alpha = 0.65, eps = 1e-;
inline int dcmp(double a, double b) { return abs(a - b) < eps; }
struct DSU
{
    int p[maxn];

    void init(int n)
    {
        RP(i, , n) p[i] = i;
    }

    int find(int u)
    {
        return u == p[u] ? u : p[u] = find(p[u]);
    }

    void merge(int u, int v)
    {
        u = find(u); v = find(v);
        p[u] = (u^v) ? v : p[u];
    }
};
struct BIT
{
    ll *t, n;

    void init(int sz)
    {
        t = new ll[sz + ];
        memset(t, , sizeof(ll)*(sz + ));
        n = sz;
    }

    void add(int x, ll d)
    {
        while (x <= n)
        {
            t[x] += d;
            x += lowbit(x);
        }
    }

    ll qry(int x)
    {
        ll ans = ;
        while (x)
        {
            ans += t[x];
            x -= lowbit(x);
        }
        return ans;
    }

    int qry(int l, int r)
    {
        return qry(r) - qry(l - );
    }
};
int N;
struct SegTree
{
    struct Node
    {
        int lc, rc;
        ll sum, add;
    }t[maxn * ];

    void maintain(int k,int l,int r)
    {
        int lc = t[k].lc, rc = t[k].rc;
        (t[k].sum = t[lc].sum + t[rc].sum + t[k].add*(r - l + ) % mod) %= mod;
    }

    void pushdown(int k,int l,int r)
    {
        if (t[k].add)
        {
            if (l == r)
            {
                t[k].sum += t[k].add;
                t[k].add = ;
                return;
            }
            mpt;
            int lc = t[k].lc, rc = t[k].rc;
            (t[lc].add += t[k].add)%=mod; maintain(lc, l, mid);
            (t[rc].add += t[k].add)%=mod; maintain(rc, mid + , r);
            t[k].add = ;
        }
    }

    int ql, qr, tot;
    ll qv;
    void init(int _N)
    {
        N = _N;
        tot = ;
    }

    void build(int k = , int l = , int r = N)
    {
        if (l == r)
            return;
        else
        {
            mpt;
            build(t[k].lc = ++tot, l, mid);
            build(t[k].rc = ++tot, mid + , r);
        }
    }

    void add(int k = , int l = , int r = N)
    {
        if (inr)
        {
            (t[k].add += qv) %= mod;
        }
        else
        {
            mpt;
            if (ql <= mid)
                add(t[k].lc, l, mid);
            if (qr > mid)
                add(t[k].rc, mid + , r);
        }
        maintain(k, l, r);
    }

    ll qry(int k = , int l = , int r = N,ll add=)
    {
        if (inr)
        {
            return (t[k].sum + add*(r - l + ) % mod) % mod;
        }
        else
        {
            mpt;
            ll ans = ;
            add += t[k].add;
            if (ql <= mid)
                (ans += qry(t[k].lc, l, mid, add)) %= mod;
            if (qr > mid)
                (ans += qry(t[k].rc, mid + , r, add)) %= mod;
            return ans;
        }
    }

    void Add(int l, int r, ll v)
    {
        ql = l; qr = r; qv = v;
        add();
    }

    ll Qry(int l, int r)
    {
        ql = l; qr = r;
        return qry();
    }
};
struct MCMF
{
    int n, m, s, t;
    Edge edg[maxn];
    int ecnt, g[maxn], d[maxn], a[maxn], inq[maxn], pre[maxn];

    void init(int N)
    {
        n = N;
        mst(g, ecnt = -);
    }

    void addedge(int u, int v, int w, int cap)
    {
        addedg(u, v, w, cap);
    }

    queue<int> q;
    int bfs()
    {
        RP(i, , n) d[i] = INF;
        q.push(s);
        int u, v;
        d[s] = ;
        inq[s] = ;
        a[s] = INF;
        while (!q.empty())
        {
            u = q.front(); q.pop();
            fson
            {
                v = edg[i].v;
            if (edg[i].cap>edg[i].flow && d[v] > d[u] + edg[i].w)
            {
                d[v] = d[u] + edg[i].w;
                a[v] = min(a[u], edg[i].cap - edg[i].flow);
                pre[v] = i;
                if (!inq[v])
                {
                    q.push(v);
                    inq[v] = ;
                }
            }
            }
            inq[u] = ;
        }

        return d[t] != INF;
    }

    void augment(int &cost)
    {
        cost += a[t] * d[t];
        int u = t;
        while (u != s)
        {
            edg[pre[u]].flow += a[t];
            edg[pre[u] ^ ].flow -= a[t];
            u = edg[pre[u]].u;
        }
    }

    int mcmf(int S, int T)
    {
        s = S; t = T;
        int cost = ;
        while (bfs())
        {
            augment(cost);
        }
        return cost;
    }
};
struct Dinic
{
    int n, m, s, t;
    Edge edg[maxn];
    int ecnt, g[maxn], d[maxn], vis[maxn], cur[maxn];

    void init(int N)
    {
        n = N;
        mst(g, ecnt = -);
    }

    void addedge(int u, int v, int cap)
    {
        addedg(u, v, , cap);
    }

    queue<int> q;
    int bfs()
    {
        mst(vis, );
        q.push(s);
        int u, v;
        vis[s] = ;
        while (!q.empty())
        {
            u = q.front(); q.pop();
            fson
            {
                v = edg[i].v;
            if (edg[i].cap>edg[i].flow && !vis[v])
            {
                d[v] = d[u] + ;
                vis[v] = ;
                q.push(v);
            }
            }
        }

        return vis[t];
    }

    int dfs(int u, int a)
    {
        if (u == t || !a)
            return a;

        int flow = , aug, v;
        for (int &i = cur[u]; ~i; i = edg[i].next)
        {
            v = edg[i].v;
            if (d[v] == d[u] +  && (aug = dfs(v, min(a, edg[i].cap - edg[i].flow))))
            {
                flow += aug;
                a -= aug;
                edg[i].flow += aug;
                edg[i ^ ].flow -= aug;
                if (!a) break;
            }
        }
        return flow;
    }

    int maxflow(int S, int T)
    {
        s = S, t = T;
        int flow = ;
        while (bfs())
        {
            RP(i, , n) cur[i] = g[i];
            flow += dfs(s, INF);
        }
        return flow;
    }
};
struct Mat
{
    int n, m;
    ll a[][];

    Mat(int r = , int c = )
    {
        n = r; m = c;
        mst(a, );
    }

    static Mat idmat(int n)
    {
        Mat ans(n, n);
        rp(i, , n) ans[i][i] = ;
        return ans;
    }

    ll* operator[](int i)
    {
        return a[i];
    }

    Mat operator*(Mat b)
    {
        Mat ans(n, b.m);
        rp(i, , n)
            rp(j, , b.m)
                rp(k, , m)
                    (ans[i][j] += a[i][k] * b[k][j] % mod) %= mod;
        return ans;
    }
};
Mat qkpow(Mat &A, ll b)
{
    Mat ans = Mat::idmat(A.n), x = A;
    while (b)
    {
        if (b & ) ans = ans * x;
        x = x*x;
        b >>= ;
    }
    return ans;
}

int n, a[maxn], b[maxn], c[maxn];

int cmp(const int &i, const int &j)
{
    return a[i] > a[j];
}

int main()
{
    ru(n); inp(n, a); inp(n, b);
    RP(i, , n) c[i] = i;
    sort(c + , c + n + , cmp);
    printf("%d\n%d", n /  + , c[]);
    for (int i = ; i <= n; i += )
    {
        if (b[c[i]] > b[c[i + ]])
            printf(" %d", c[i]);
        else
            printf(" %d", c[i + ]);
    }
}

的和也大于整个b数列的和的1/2.k<= .(这个条件很关键)

//题解：首先将条件换一种说法，就是要取floor(n/2)+1个数使得这些数大于剩下的数。然后想到两种取法，一种是排好序选前半段。令一种取法是两个两个分组，每组取大的那个。
//然后就可以（很困难地）想到一种巧妙的取法，先用一个结构体保存a,b,idx,按照a从大到小排序，先取第一个，再对剩下的每两个一组取其中b更大的。这样保证了被选出
//的b一定大于剩下的b，也就满足了条件。然后考虑a，按照前面的分组，每组的a并不一定选到了大的。但是我们（并没有）观察到第一个选的a （也就是a的最大值）一定大于第一组没选到的那个a、第一组选的a一定大于第二组没选出的a···；
//同理这样选出的a也一定大于剩下的a。所以算法就是读取，排序，一个循环 。（codeforces上python只有三行。。。）

ac