A.最长上升子序列
Key:动态规划—线性DP
AC代码
未优化版
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 5010, MAX = 1e9, INF = -1e9;
int n;
int a[N];
int dp[N];
int ans=0;
void solve(){
cin>>n;
for(int i=1;i<=n;i++){
cin>>a[i];
dp[i]=1;
}
for(int i=1;i<=n;i++){
for(int j=1;j<i;j++){
if(a[j]<a[i])dp[i]=max(dp[i],dp[j]+1);
ans=max(ans,dp[i]);
}
}
cout<<ans<<endl;
return ;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
二分优化
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 1e5+10, MAX = 1e9, INF = -1e9;
int n;
int a[N];
int dp[N];
vector<int> v;
void solve(){
cin>>n;
for(int i=1;i<=n;i++){
cin>>a[i];
}
v.pb(a[1]);
for(int i=2;i<=n;i++){
if(a[i]>v[v.size()-1])v.pb(a[i]);
else{
int p=lower_bound(v.begin(),v.end(),a[i])-v.begin();
v[p]=a[i];
}
}
cout<<v.size()<<endl;
return ;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
B.编辑距离
Key:动态规划—线性DP
AC代码
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 2010, MAX = 1e9, INF = -1e9;
string a,b;
int dp[N][N];
void solve(){
cin>>a>>b;
int n=a.size();int m=b.size();
a=" "+a;b=" "+b;
for(int i=1;i<N;i++)dp[i][0]=dp[0][i]=i;
for(int i=1;i<=n;i++){
for(int j=1;j<=m;j++){
dp[i][j] = min(dp[i - 1][j] + 1, dp[i][j - 1] + 1);
dp[i][j] = min(dp[i][j], dp[i - 1][j - 1] + (a[i] != b[j]));
}
}
cout<<dp[n][m]<<endl;
return ;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
C.采药
Key:动态规划—背包问题—01背包
AC代码
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 105, MAX = 1e9, INF = -1e12, M = 1005;
int n,V;
int v[N];
int w[N];
int dp[N][M];
void solve(){
cin>>V>>n;
for(int i=1;i<=n;i++){
cin>>v[i]>>w[i];
}
memset(dp,0,sizeof dp);
for(int i=1;i<=n;i++){
for(int j=1;j<=V;j++){
dp[i][j]=dp[i-1][j];
if(j-v[i]>=0)dp[i][j]=max(dp[i][j],dp[i-1][j-v[i]]+w[i]);
}
}
cout<<dp[n][V]<<endl;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
D.第37次CSP认证第二题：机器人饲养指南
Key:动态规划—背包问题—完全背包
AC代码
未优化版
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 105, MAX = 1e9, INF = -1e12, M = 10005;
int n,m;
int w[N];
int dp[N][M];
void solve(){
cin>>n>>m;
for(int i=1;i<=m;i++){
cin>>w[i];
}
memset(dp,0,sizeof dp);
for(int i=1;i<=m;i++){
for(int j=0;j<=n;j++){
for(int k=0;k*i<=j;k++){
dp[i][j]=max(dp[i][j],dp[i-1][j-k*i]+k*w[i]);
}
}
}
cout<<dp[m][n]<<endl;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
优化版
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 105, MAX = 1e9, INF = -1e12, M = 10005;
int n,m;
int w[N];
int dp[N][M];
void solve(){
cin>>n>>m;
for(int i=1;i<=m;i++){
cin>>w[i];
}
memset(dp,0,sizeof dp);
for(int i=1;i<=m;i++){
for(int j=0;j<=n;j++){
dp[i][j]=dp[i-1][j];
if(j>=i)dp[i][j]=max(dp[i][j],dp[i][j-i]+w[i]);
}
}
cout<<dp[m][n]<<endl;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
E.石子合并（弱化版）
Key:动态规划—区间DP
AC代码
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 305, MAX = 1e9, INF = -1e9;
int n;
int a[N];
int dp[N][N];
int s[N];
void solve(){
cin>>n;
memset(dp,0x3f,sizeof dp);
for(int i=1;i<=n;i++){
cin>>a[i];
dp[i][i]=0;
s[i]=s[i-1]+a[i];
}
for(int len=1;len<=n;len++){
for(int i=1;i<=n-len;i++){
int l=i;int r=i+len;
for(int k=l;k<=r-1;k++){
dp[l][r]=min(dp[l][k]+dp[k+1][r]+s[r]-s[l-1],dp[l][r]);
}
}
}
cout<<dp[1][n]<<endl;
return ;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
另解：记忆化搜索
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 305, MAX = 1e9, INF = -1e9;
int n;
int a[N];
int s[N];
int dp[N][N];
int dfs(int l,int r){
if(r==l){
return 0;
}
int &ans=dp[l][r];
if(ans!=-1)return ans;
ans=MAX;
for(int k=l;k<r;k++){
ans=min(ans,dfs(l,k)+dfs(k+1,r)+a[r]-a[l-1]);
}
return ans;
}
void solve(){
cin>>n;
memset(dp,-1,sizeof dp);
for(int i=1;i<=n;i++){
cin>>a[i];
a[i]+=a[i-1];
}
cout<<dfs(1,n)<<endl;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
F.没有上司的舞会
Key:动态规划—树形DP
AC代码
#include<bits/stdc++.h>
#define endl '\n'
#define int int long long
#define pb push_back
#define bs bitset
#define val(a) (a<'a' ? (a-'A'+'a') : a)
#define fast() ios::sync_with_stdio(false),cin.tie(nullptr),cout.tie(nullptr)
using namespace std;
typedef pair<char,int> PCI;
typedef pair<int,int> PII;
typedef pair<long long, long long> PLL;
typedef priority_queue<int> PQ;
typedef priority_queue<int, vector<int>, greater<int>> Q;
const int N = 2e5+10, MAX = 1e9, INF = -1e9;
int n;
int e[N];
bool st[N];
vector<vector<int>> v(N);
int root;
int dp[N][2];
void dfs(int p){
dp[p][1]+=e[p];
for(auto i:v[p]){
dfs(i);
dp[p][1]+=dp[i][0];
dp[p][0]+=max(dp[i][0],dp[i][1]);
}
}
void solve(){
cin>>n;
for(int i=1;i<=n;i++)cin>>e[i];
for(int i=1;i<=n-1;i++){
int l,r;
cin>>l>>r;
st[l]=true;
v[r].pb(l);
}
for(int i=1;i<=n;i++){
if(!st[i]){
root=i;
break;
}
}
dfs(root);
cout<<max(dp[root][0],dp[root][1])<<endl;
return ;
}
signed main()
{
fast();
int t=1;
//cin>>t;
while(t--){
solve();
}
return 0;
}
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