1523. K-inversions
Time limit: 1.0 second
Memory limit: 64 MB
Memory limit: 64 MB
Consider a permutation a1, a2,
…, an (all ai are different integers in range from 1 to n). Let us call k-inversion a
sequence of numbers i1, i2, …, ik such
that 1 ≤ i1 < i2 < … < ik ≤ n andai1 > ai2 > … > aik.
Your task is to evaluate the number of different k-inversions in a given permutation.
…, an (all ai are different integers in range from 1 to n). Let us call k-inversion a
sequence of numbers i1, i2, …, ik such
that 1 ≤ i1 < i2 < … < ik ≤ n andai1 > ai2 > … > aik.
Your task is to evaluate the number of different k-inversions in a given permutation.
Input
The first line of the input contains two integers n and k (1 ≤ n ≤ 20000, 2 ≤ k ≤ 10). The second line is filled with n numbers ai.
Output
Output a single number — the number of k-inversions in a given permutation. The number must be taken modulo 109.
Samples
input | output |
---|---|
3 2 3 1 2 |
2 |
5 3 5 4 3 2 1 |
10 |
题意:求长度为k的不连续的严格递减子序列的总个数
题解:可以设dp【i】【j】表示以i结尾长度为j的子序列的个数,那么更新就是dp【i】【j】=∑dp【k】【j-1】,其中k<i,而且a【k】>a【i】。而要更新dp值,可以用树状数组维护,按顺序插入序列值,那么树状数组的值就可以表示比它小的长度为j-1的所有子序列的和,这样就可以在logn的时间更新dp值了,所以总复杂度是O(n*k*logn)
#include<stdio.h> #include<string.h> #define MAXN 20008 #define mod 1000000000 int a[MAXN],tree[MAXN],dp[MAXN][13]; int low_bit(int x){ return x&(-x); } void add(int x,int y,int n) { while(x<=n) { tree[x]=(tree[x]+y)%mod; x+=low_bit(x); } } int query(int x) { int res=0; while(x) { res=(res+tree[x])%mod; x-=low_bit(x); } return res; } int main() { int i,j,n,k,res; while(scanf("%d%d",&n,&k)>0) { for(i=1;i<=n;i++) { scanf("%d",a+i); dp[i][1]=1; } for(i=1,j=n;i<j;i++,j--) res=a[i],a[i]=a[j],a[j]=res; for(i=2;i<=k;i++) { memset(tree,0,sizeof(tree)); for(j=1;j<=n;j++) { dp[j][i]=query(a[j]); add(a[j],dp[j][i-1],n); } } for(res=0,i=1;i<=n;i++) res=(res+dp[i][k])%mod; printf("%d\n",res); } return 0; }