HDU 1025 高效最长上升子序列（二分查）

用到了LIS（Longest Increasing Subsequence）最长上升（不下降）子序列，它有两种算法复杂度为O(n*logn)和O(n^2)。前者使用二分查找，后者朴素查找。它的算法大致思路：设置两个a,b数组，a数组存储数据，b数组存储最长不降序序列。此算法关键在于设计二分查找。

刚开始用到了朴素dp算法，果断超时，超时代码为：

#include <iostream>
#include <algorithm>
using namespace std;
const int Max = 500010;
int dp[Max];
typedef struct
{
int x, y;
} Node;
Node node[Max];

int cmp(const void * a, const void * b)
{
return (*(Node*)a).x - (*(Node*)b).x;
}
int main()
{
int n;
int t = 1;
while(cin >> n)
{
memset(dp, 0, sizeof(dp));
for(int i = 0; i < n; i++)
scanf("%d%d", &node[i].x, &node[i].y);
//qsort(node, n, sizeof(Node), cmp);
/*for(int i = 0;  i < n; i++)
printf("%d %d\n", node[i].x, node[i].y);*/
for(int i = 0; i < n; i++)
{
dp[i] = 1;
for(int j = i - 1; j >= 0; j--)
{
if(dp[j] + 1 > dp[i] && node[i].y > node[j].y)
dp[i] = dp[j] + 1;
}
}
int max = 0;
for(int i = 0; i < n; i++)
if(dp[i] > max)
max = dp[i];
printf("Case %d:\n", t++);
printf("My king, at most %d road can be built.\n\n", max);
}
return 0;
}

改进后，用到二分查找，一直莫名奇妙Wa，还没找到原因，数据上有问题？
Wa代码：

#include <iostream>
#include <algorithm>
using namespace std;
const int Max = 500010;
int dp[Max];
int B[Max];
typedef struct
{
int x, y;
} Node;
Node node[Max];

int cmp(const void * a, const void * b)
{
return (*(Node*)a).x - (*(Node*)b).x;
}
int main()
{
freopen("1025.txt", "r", stdin);
int n;
int t = 1;
while(cin >> n)
{
memset(dp, 0, sizeof(dp));
memset(B, 0, sizeof(B));
for(int i = 0; i < n; i++)
scanf("%d%d", &node[i].x, &node[i].y);
qsort(node, n, sizeof(Node), cmp);
B[0] = -1000;
B[1] = node[0].y;
int len = 1;
int p, r, m;
for(int i = 1; i < n; i++)
{
p = 1;
r = len;
while(p <= r)
{
m = (p+r) / 2;
if(B[m] < node[i].y)
p = m + 1;
else
r = m - 1;
}
B[p] = node[i].y;
if(p > len) len = p;
}
printf("Case %d:\n", t++);
printf("My king, at most %d road can be built.\n\n", len);
}
return 0;
}

最后是AC代码，源于/article/6994645.html

#include<stdio.h>
int dp[50005],a[50005];

int Lis(int n)
{
int len=1,i,low,high,mid;
dp[1]=a[1];
for(i=2;i<=n;i++)
{
low=1;
high=len;
while(low<=high)
{
mid=(low+high)/2;
if(a[i]>dp[mid])
low=mid+1;
else
high=mid-1;
}
dp[low]=a[i];
if(low>len)
len=low;
}
return len;
}
int main()
{
int n,x,y,i,ans,k=1;
while(scanf("%d",&n)!=EOF)
{
for(i=0;i<n;i++)
{
scanf("%d%d",&x,&y);
a[x]=y;
}
ans=Lis(n);
printf("Case %d:\n",k++);
if(ans==1)
printf("My king, at most 1 road can be built.\n");
else
printf("My king, at most %d roads can be built.\n",ans);
printf("\n");
}
return 0;
}

其实显然这个数组开小了，不过能ac。
改天继续查找Wa原因