USACO 4.3.1 Buy Low, Buy Lower

线性DP, 笨得WA了10次才AC，而且高精度加法还是借用的别人的，自己的压位高精度大数算的好奇怪。。。
本题的重点在于第二问的判重。可以证明，对于相同price的i, j(i<j)，应该选择j，因为j包含i的所有情况。
那么怎么判断某个price是待计算的i前面的最后一个那个j呢，这里我实在太挫没有思路。
后来参考了别人的做法，我们可以给每一个j构造一个next指针指向下一个相同数字的下标，如果这个下标>i那么就表明j是最后一个。
然后就很简单了，把所有的最后一个加起来就行了。不过本题要使用高精度算法，总时间复杂度为O(N2)。
Compile: OK

Executing...
   Test 1: TEST OK [0.011 secs, 5896 KB]
   Test 2: TEST OK [0.011 secs, 5896 KB]
   Test 3: TEST OK [0.011 secs, 5896 KB]
   Test 4: TEST OK [0.011 secs, 5896 KB]
   Test 5: TEST OK [0.011 secs, 5896 KB]
   Test 6: TEST OK [0.022 secs, 5896 KB]
   Test 7: TEST OK [0.032 secs, 5896 KB]
   Test 8: TEST OK [0.011 secs, 5896 KB]
   Test 9: TEST OK [0.065 secs, 5896 KB]
   Test 10: TEST OK [0.248 secs, 5896 KB]

All tests OK.

Your program ('buylow') produced all correct answers! This is your submission #11 for this problem. Congratulations!

 1 #include <iostream>
 2 #include <iomanip>
 3 #include <cstring>
 4 #include <cstdio>
 5 using namespace std;
 6 
 7 int N, price[5001];
 8 int eqn[5001];
 9 
10 int dp[5001];
11 
12 struct bign
13 {
14     char n[500];
15     int len;
16     bign():len(1)
17     {
18         memset(n,0,sizeof(n));
19     }
20     void operator = (const int v)
21     {
22         len = 1;
23         n[0] = v;
24     }
25 
26     void operator += (const bign &rhs)
27     {
28         if(rhs.len>len)
29             len=rhs.len;
30         for(int i=0;i<len;++i)
31         {
32             n[i]+=rhs.n[i];
33             if(n[i]>9)
34             {
35                 ++n[i+1];
36                 n[i]-=10;
37             }
38         }
39         if(n[len])
40             ++len;
41     }
42     void print()
43     {
44         for(int i=len-1;i>=0;--i)
45             cout << (int)n[i];
46     }
47 }dp2[5001];
48 
49 int main(void)
50 {
51     freopen("buylow.in", "r", stdin);
52     freopen("buylow.out", "w", stdout);
53     int i, m;
54 
55     cin >> N;
56     for(i=0; i<N; ++i) cin >> price[i];
57     N++;    // 增加一个超级节点, price[supernode] = 0;
58 
59     for(i=0; i<N-1; ++i) {
60         int j;
61         for(j=i+1; j<N; ++j)
62             if (price[j] == price[i]) {
63                 eqn[i] = j;
64                 break;
65             }
66         if (j >= N) eqn[i] = N;
67     }
68     eqn[N-1] = N;
69 
70     for(i=0; i<N; ++i) {
71         dp[i] = 1;
72         dp2[i] = 1;
73         int j;
74         for(j=0; j<i; ++j) {
75             if (price[i] < price[j]) {
76                 if (eqn[j] > i) {
77                     const int k = dp[j] + 1;
78                     if (k > dp[i]) {
79                         dp[i] = k;
80                         dp2[i] = dp2[j];
81                     } else if (k == dp[i])
82                         dp2[i] += dp2[j];
83                 }
84             }
85         }
86     }
87     cout << dp[N-1] - 1 << ' ';
88     dp2[N-1].print();
89     cout << endl;
90 
91     return 0;
92 }

2014/5/20更新：

终于用上自己的高精度了，效率貌似还可以。

Compiling...
Compile: OK

Executing...
   Test 1: TEST OK [0.008 secs, 5400 KB]
   Test 2: TEST OK [0.011 secs, 5400 KB]
   Test 3: TEST OK [0.008 secs, 5400 KB]
   Test 4: TEST OK [0.008 secs, 5400 KB]
   Test 5: TEST OK [0.014 secs, 5400 KB]
   Test 6: TEST OK [0.024 secs, 5400 KB]
   Test 7: TEST OK [0.022 secs, 5400 KB]
   Test 8: TEST OK [0.024 secs, 5400 KB]
   Test 9: TEST OK [0.054 secs, 5400 KB]
   Test 10: TEST OK [0.270 secs, 5400 KB]

All tests OK.
#include <iostream>
#include <iomanip>
#include <cstring>
#include <cstdio>
using namespace std;

int N, price[5001];
int eqn[5001];

int dp[5001];

#define min(a,b)    ((a)>(b)?(b):(a))
#define max(a,b)    ((a)>(b)?(a):(b))

#define BIGNUM_LL            100
#define BIGNUM_BL            "4"
#define BIGNUM_MOD            10000UL
typedef typeof(BIGNUM_MOD) bignum_block;

struct bignum {
    unsigned len;
    bignum_block blks[BIGNUM_LL];

    bignum() {blks[0]=0, len=1;}

    bignum(unsigned num) {
        len = 0;
        while(num) {
            blks[len++] = num % BIGNUM_MOD;
            num /= BIGNUM_MOD;
        }
    }

    void operator += (bignum &b) {
        int i;
        bignum_block carry;

        i = len; do blks[i++] = 0; while(i<=b.len);

        carry = i = 0;
        for(; i<b.len; ++i) {
            blks[i] += b.blks[i];
            blks[i+1] += blks[i] / BIGNUM_MOD;
            blks[i] %= BIGNUM_MOD;
        }
        for(; i<len; ++i) {
            blks[i+1] +=  blks[i] / BIGNUM_MOD;
            blks[i] %= BIGNUM_MOD;
        }
        len = i + (blks[i] > 0);
    }

    void print() {
        printf("%d", blks[len-1]);
        for(int i=len-1; i--; )
            printf("%0" BIGNUM_BL "d", blks[i]);
    }
}dp2[5001];

int main(void)
{
    freopen("buylow.in", "r", stdin);
    freopen("buylow.out", "w", stdout);
    int i, m;

    cin >> N;
    for(i=0; i<N; ++i) cin >> price[i];
    N++;    // 增加一个超级节点, price[supernode] = 0;

    for(i=0; i<N-1; ++i) {
        int j;
        for(j=i+1; j<N; ++j)
            if (price[j] == price[i]) {
                eqn[i] = j;
                break;
            }
        if (j >= N) eqn[i] = N;
    }
    eqn[N-1] = N;

    for(i=0; i<N; ++i) {
        dp[i] = 1;
        dp2[i] = 1;
        int j;
        for(j=0; j<i; ++j) {
            if (price[i] < price[j]) {
                if (eqn[j] > i) {
                    const int k = dp[j] + 1;
                    if (k > dp[i]) {
                        dp[i] = k;
                        dp2[i] = dp2[j];
                    } else if (k == dp[i])
                        dp2[i] += dp2[j];
                }
            }
        }
    }
    cout << dp[N-1] - 1 << ' ';
    dp2[N-1].print();
    cout << endl;

    return 0;
}
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原文地址：https://www.cnblogs.com/e0e1e/p/usaco_buylow.html