USACO JAN10 Gold

hayturn

动态规划， \(F_m\)表示当前要做选择的奶牛在可以选择\(w_{m\ldots n-1}\)时可以获得的最大值。 \(S_m\)表示当前要做选择的奶牛做完\(w_{m\ldots n-1}\)的最优决策后，下一个奶牛可以取得的最大值。

#include <stdio.h>

#ifdef WIN32
#define LLD "%I64d"
#else
#define LLD "%lld"
#endif

int w[700000];

int main()
{
    int n;
    scanf("%d", &n);
    for (int i = 0; i < n; ++i)
        scanf("%d", &w[i]);
    long long a = 0, b = 0;
    for (int i = n; --i >= 0; )
        if (b+w[i] >= a)
        {
            long long t = a;
            a = b+w[i];
            b = t;
        }
    printf(LLD" "LLD"\n", a, b);
}

island

根据USACL Analysis（后附），根据一个格子周围格子的布局，先把一些点转换为A，然后绕着A走一圈。

#include <cstdio>
using namespace std;

const int H = 1000, W = 1000;
const int dr[] = {0,1,0,-1}, dc[] = {1,0,-1,0};
char a[H][W+1];
int tmp[4], h, w;

void DFS(int r, int c)
{
    if (a[r][c] != '.' || !r || r == h-1 || !c || c == w-1) return;
    int cntA = 0;
    for (int d = 0; d < 4; ++d)
        if (a[r+dr[d]][c+dc[d]] == 'A')
            tmp[cntA++] = d;
    switch (cntA)
    {
        case 2:
            if ((tmp[0]+tmp[1])%2 == 0 || a[r-dr[tmp[0]]-dr[tmp[1]]][c-dc[tmp[0]]-dc[tmp[1]]] != '.')
                break;
            //fall through
        case 3:
        case 4:
            a[r][c] = 'A';
            for (int d = 0; d < 4; ++d)
                DFS(r+dr[d], c+dc[d]);
    }
}

inline bool check(int r, int c)
{
    return 0 <= r && r < h && 0 <= c && c < w && a[r][c] == '.';
}

int main()
{
    scanf("%d%d", &h, &w);
    for (int r = 0; r < h; ++r)
        scanf("%s", a[r]);
    for (int r = 1; r < h-1; ++r)
        for (int c = 1; c < w-1; ++c)
            DFS(r, c);

    int r, c, rr, cc, d = 0, len = 0;
    for (r = 0; r < h; ++r)
        for (c = 0; c < w; ++c)
            if (a[r][c] == 'A')
            {
                --r;
                goto L1;
            }
L1:
    rr = r;
    cc = c;
    do
    {
        int rrr = r+dr[d+1&3], ccc = c+dc[d+1&3];
        if (check(rrr, ccc))
        {
            r = rrr;
            c = ccc;
            d = d+1 & 3;
        }
        else if (rrr = r+dr[d], ccc = c+dc[d], check(rrr, ccc))
        {
            r = rrr;
            c = ccc;
        }
        else if (rrr = r+dr[d+3&3], ccc = c+dc[d+3&3], check(rrr, ccc))
        {
            r = rrr;
            c = ccc;
            d = d+3 & 3;
        }
        ++len;
    }while (r != rr || c != cc);
    printf("%d\n", len);
}

telephone

先把题目中给出的树有根化，对于一个顶点u，如果它有不超过K/2个孩子还未被分配， 可以把它们中最多2*K个在u处连接起来。如果有孤立孩子并且还未配对完K对孩子， 那么只能和u子树外的顶点配对，这相当于u是其parent的未分配顶点。

#include <cstdio>
#include <vector>
using namespace std;

const int N = 100000;
int k, res = 0;
vector<int> e[N];

int compute(int u, int p)
{
    int t = e[u].size() == 1;
    for (vector<int>::iterator it = e[u].begin(); it != e[u].end(); ++it)
        if (*it != p)
            t += compute(*it, u);
    if (t <= k*2)
        return res += t/2, t%2;
    res += k;
    return 0;
}

int main()
{
    int n;
    scanf("%d%d", &n, &k);
    for (int i = n; --i; )
    {
        int u, v;
        scanf("%d%d", &u, &v);
        --u; --v;
        e[u].push_back(v);
        e[v].push_back(u);
    }
    compute(0, -1);
    printf("%d\n", res);
}

USACO JAN10 Problem 'island' Analysis

by John Pardon

This problem requires nothing more than a modified flood-fill algorithm. Our strategy is too add squares to the main island until the optimal path is just to follow the boundary of the main island. Note that this would not work without the assumption that the main island is connected.

Consider the following four configurations:

?.?    ?..    ?.x    ?.A
A.A    A..    A..    A..
?A?    ?A?    ?A?    ?A?
(I)   (II)   (III)  (IV)
Let's focus on the center square of each of these configurations.

In (I), clearly there is no reason why FJ's ship would ever want to visit the center square; he would just have to leave again. Thus the problem remains unchanged if we change the center square to 'A'.

In (II), the situation is similar. A path like:

 V
?|.
A+->
?A?
can be changed to:

 V
?++
A.+>
?A?
with no increase in length. Thus we may change the center to square to 'A'.

In (III), we may not change the center square to an 'A', because then it wouldn't be possible to go around the main island without also going around the 'x' in the upper-right corner.

In (IV), we may also not change the center square to an 'A', at least not unless the top center square or the right center square is changed to an 'A' first.



After making the modifications (I) and (II) as many times as possible, the optimal solution is to follow the boundary of the main island, and this can just be simulated in order to find its length. Doing (I) and (II) as many times as possible is a modified flood-fill and can be done with a DFS. A solution follows: