POJ2374 Fence Obstacle Course

Posted 2021-11-04 autoint

题意

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Fence Obstacle Course

Time Limit: 3000MS		Memory Limit: 65536K
Total Submissions: 2900		Accepted: 1042

Description

Farmer John has constructed an obstacle course for the cows‘ enjoyment. The course consists of a sequence of N fences (1 <= N <= 50,000) of varying lengths, each parallel to the x axis. Fence i‘s y coordinate is i.

The door to FJ‘s barn is at the origin (marked ‘*‘ below). The starting point of the course lies at coordinate (S,N).

   +-S-+-+-+        (fence #N)

 +-+-+-+            (fence #N-1)

     ...               ...

   +-+-+-+          (fence #2)

     +-+-+-+        (fence #1)

=|=|=|=*=|=|=|      (barn)

-3-2-1 0 1 2 3    

FJ‘s original intention was for the cows to jump over the fences, but cows are much more comfortable keeping all four hooves on the ground. Thus, they will walk along the fence and, when the fence ends, they will turn towards the x axis and continue walking in a straight line until they hit another fence segment or the side of the barn. Then they decide to go left or right until they reach the end of the fence segment, and so on, until they finally reach the side of the barn and then, potentially after a short walk, the ending point.

Naturally, the cows want to walk as little as possible. Find the minimum distance the cows have to travel back and forth to get from the starting point to the door of the barn.

Input

* Line 1: Two space-separated integers: N and S (-100,000 <= S <= 100,000)

* Lines 2..N+1: Each line contains two space-separated integers: A_i and B_i (-100,000 <= A_i < B_i <= 100,000), the starting and ending x-coordinates of fence segment i. Line 2 describes fence #1; line 3 describes fence #2; and so on. The starting position will satisfy A_N <= S <= B_N. Note that the fences will be traversed in reverse order of the input sequence.

Output

* Line 1: The minimum distance back and forth in the x direction required to get from the starting point to the ending point by walking around the fences. The distance in the y direction is not counted, since it is always precisely N.

Sample Input

4 0 
-2 1
-1 2
-3 0
-2 1

Sample Output

4

Hint

This problem has huge input data,use scanf() instead of cin to read data to avoid time limit exceed.

INPUT DETAILS:

Four segments like this:

   +-+-S-+             Fence 4

 +-+-+-+               Fence 3

     +-+-+-+           Fence 2

   +-+-+-+             Fence 1

 |=|=|=*=|=|=|         Barn

-3-2-1 0 1 2 3      

OUTPUT DETAILS:

Walk positive one unit (to 1,4), then head toward the barn, trivially going around fence 3. Walk positive one more unit (to 2,2), then walk to the side of the barn. Walk two more units toward the origin for a total of 4 units of back-and-forth walking.

Source

USACO 2004 December Gold

分析

参照逐梦起航-带梦飞翔的题解，线段树优化DP

设f[i][0/1]表示在通过第i条栅栏后，处于栅栏左边/右边的最小路径长。

因为奶牛是直线下来的，所以最优方案当然是从上一个栅栏的这个位置下来。由于有栅栏的影响，奶牛们不能顺利的下来，此时到达这个位置的最优策略要么是从前面那个栅栏的左端点过来，要么从右端点过来。所以有
\[ f[i][0]=\min\{f[j][0]+|l_i-l_j|,f[j][1]+|l_i-r_j|\} \f[i][1]=\min\{f[j][0]+|r_i-l_j|,f[j][1]+|r_i-r_j|\} \]
其中的j就是上一个挡住了这个位置的栅栏。我们可以用线段树来维护这个栅栏的编号。当栅栏(l[i],r[i])，出现后，我们把线段树上(l[i],r[i])这段区间改成i，表示这个位置是栅栏i阻挡了。对于后面的栅栏，修改时直接覆盖前面的信息即可。我们只要实现一个改段求点的线段树即可。

特别的，线段树初始值为0。一个位置如果得到的j=0，那么说明它前面没有栅栏，它可以直接从s过来，路径=abs(s-p)。

时间复杂度\(O(n\log s)\)，也可以用线段树连边跑最短路，但这题用DP来做常数小。

代码

#include<iostream>
#include<cmath>
#define rg register
#define il inline
#define co const
template<class T>il T read(){
    rg T data=0,w=1;rg char ch=getchar();
    while(!isdigit(ch)) {if(ch=='-') w=-1;ch=getchar();}
    while(isdigit(ch)) data=data*10+ch-'0',ch=getchar();
    return data*w;
}
template<class T>il T read(rg T&x) {return x=read<T>();}
typedef long long ll;
using namespace std;

co int N=5e4+1,S=2e5+1,X=1e5;
int n,s,l[N],r[N],f[N][2];
struct T {int l,r,x;}t[S*4];
void build(int p,int l,int r){
    t[p].l=l,t[p].r=r,t[p].x=l==r?0:-1;
    if(l==r) return;
    int mid=l+r>>1;
    build(p<<1,l,mid),build(p<<1|1,mid+1,r);
}
void change(int p,int l,int r,int x){
    if(l<=t[p].l&&t[p].r<=r) return t[p].x=x,void();
    if(t[p].x!=-1) t[p<<1].x=t[p<<1|1].x=t[p].x,t[p].x=-1;
    int mid=t[p].l+t[p].r>>1;
    if(l<=mid) change(p<<1,l,r,x);
    if(r>mid) change(p<<1|1,l,r,x);
}
int ask(int p,int x){
    if(t[p].l==t[p].r) return t[p].x;
    if(t[p].x!=-1) t[p<<1].x=t[p<<1|1].x=t[p].x,t[p].x=-1;
    int mid=t[p].l+t[p].r>>1;
    return ask(x<=mid?p<<1:p<<1|1,x);
}
int main(){
    read(n),read(s);
    build(1,0,X*2);
    s+=X,l[0]=r[0]=X;
    for(int i=1,w;i<=n;++i){
        l[i]=read<int>()+X,r[i]=read<int>()+X;
        w=ask(1,l[i]);
        f[i][0]=min(f[w][0]+abs(l[i]-l[w]),f[w][1]+abs(l[i]-r[w]));
        w=ask(1,r[i]);
        f[i][1]=min(f[w][0]+abs(r[i]-l[w]),f[w][1]+abs(r[i]-r[w]));
        change(1,l[i],r[i],i);
    }
    printf("%d\n",min(f[n][0]+s-l[n],f[n][1]+r[n]-s));
    return 0;
}