POJ_2566_Bound_Found_(尺取法+前缀和)
描述
http://poj.org/problem?id=2566
给出一个整数序列,并给出非负整数t,求数列中连续区间和的绝对值最接近k的区间左右端点以及这个区间和的绝对值.
Bound Found
Time Limit: 5000MS | Memory Limit: 65536K | |||
Total Submissions: 2592 | Accepted: 789 | Special Judge |
Description
Signals of most probably extra-terrestrial origin have been received and digitalized by The Aeronautic and Space Administration (that must be going through a defiant phase: "But I want to use feet, not meters!"). Each signal seems to come in two parts: a sequence of n integer values and a non-negative integer t. We'll not go into details, but researchers found out that a signal encodes two integer values. These can be found as the lower and upper bound of a subrange of the sequence whose absolute value of its sum is closest to t.You are given the sequence of n integers and the non-negative target t. You are to find a non-empty range of the sequence (i.e. a continuous subsequence) and output its lower index l and its upper index u. The absolute value of the sum of the values of the sequence from the l-th to the u-th element (inclusive) must be at least as close to t as the absolute value of the sum of any other non-empty range.
Input
The input file contains several test cases. Each test case starts with two numbers n and k. Input is terminated by n=k=0. Otherwise, 1<=n<=100000 and there follow n integers with absolute values <=10000 which constitute the sequence. Then follow k queries for this sequence. Each query is a target t with 0<=t<=1000000000.Output
For each query output 3 numbers on a line: some closest absolute sum and the lower and upper indices of some range where this absolute sum is achieved. Possible indices start with 1 and go up to n.Sample Input
5 1 -10 -5 0 5 10 3 10 2 -9 8 -7 6 -5 4 -3 2 -1 0 5 11 15 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 15 100 0 0
Sample Output
5 4 4 5 2 8 9 1 1 15 1 15 15 1 15
Source
Ulm Local 2001分析
尺取法.
一眼看过去就是用尺取法找和t绝对值相差最小的区间和,但是这道题里的序列并不是非负的,这意味着固定左端点,移动右端点时,区间和不是单调递增的.尺取法的模板题中区间和是单调的,所以找到大于某一值的区间右端点就可以固定.而这道题中因为存在负值,所以区间的变化不是单调的,不能按照普通的方法解决.我们考虑一段区间和,不仅可以认为是a[l]+a[l+1]+...+a[r],还可以看作是sum[r]-sum[l-1],而这道题中要求的区间和的绝对值,就可以看作是max(sum[r],sum[l-1])-min(sum[r],sum[l-1]).这样求一个前缀和,进行排序,双指针确定区间左右端点,这样sum[r]-sum[l](排序后的编号)就代表一个区间和的绝对值,这样的区间就是单调的.如此一来,确定左端点,移动右端点,sum[r]-sum[l]就是单增的,找到sum[r]-sum[l]>t的位置即可停止,然后l++,sum[r-1]-sum[l]比之前的sum[r-1]-sum[l]更小(减数增大),本来就比t小,现在小得更多了,对于任意r'<=r-1都是如此,就不必考虑,从sum[r]-sum[l]开始继续即可.
注意:
1.sum[r]-sum[l-1],因为1<=l<=r,所以l-1>=0,注意sum[0].s=0,sum[0].num=0,要参与排序,并且每一次都要重新赋值,因为上一组数据排序后sum[0].s不一定是0,这将影响到语句"sum[i]=point(sum[i-1].s,i)" .
2.对于区间右端点表示的状态,最好是用来表示当前右端点的位置,然后每一次更新状态时都要判断,与ans比较,看是否更新最优解.
3.由于r>=l所以r>l-1,也就是说两个区间端点不能是同一个点,当l追上r时,r++.
4.其实尺取法的写法可以优化,现在的写法是两层循环,l改变后进行一次操作,然后对于同一个l改变r,每次进行一次操作,这样在一个外层循环内部要写两遍操作.其实对于l和r的改变是一样,都是区间状态的改变,所以只要一层循环即可.
即
while (l<=n&&r<=n&&ans!=0)
#include<cstdio> #include<cstdlib> #include<algorithm> using std :: sort; using std :: min; using std :: max; const int maxn=100005,INF=0x7fffffff; int a[maxn],s[maxn]; int n,q,t; struct point { int s,num; point() {} point(int a,int b) : s(a) , num(b) {} }sum[maxn]; bool comp(point x,point y) { return x.s<y.s; } int value(int l,int r) { return abs(sum[r].s-sum[l].s); } int L(int l,int r) { return min(sum[l].num,sum[r].num)+1; } int R(int l,int r) { return max(sum[l].num,sum[r].num); } void solve(int t) { int ans=INF,res,idxl,idxr; int r=1; for(int l=0;l<=n;l++) { if(l==r) r++; if(r>n) break; int now=value(l,r); int d=abs(now-t); if(d<=ans) { ans=d; res=now; idxl=L(l,r); idxr=R(l,r); } while(r<n&&now<t) { r++; now=value(l,r); int d=abs(now-t); if(d<=ans) { ans=d; res=now; idxl=L(l,r); idxr=R(l,r); } } if(now<t) break; if(now==t) { break; } } printf("%d %d %d\n",res,idxl,idxr); } void init() { while(scanf("%d%d",&n,&q)&&(n!=0||q!=0)) { sum[0].s=0; sum[0].num=0; for(int i=1;i<=n;i++) { scanf("%d",&a[i]); sum[i]=point(sum[i-1].s+a[i],i); } sort(sum,sum+n+1,comp); for(int i=1;i<=q;i++) { scanf("%d",&t); solve(t); } } } int main() { freopen("bound.in","r",stdin); freopen("bound.out","w",stdout); init(); fclose(stdin); fclose(stdout); return 0; }View Code
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