牛客 最长递增子序列

题目链接：https://www.nowcoder.com/practice/30fb9b3cab9742ecae9acda1c75bf927?tpId=101&tqId=33093&tPage=1&rp=1&ru=/ta/programmer-code-interview-guide&qru=/ta/programmer-code-interview-guide/question-ranking

题目大意

　　略。

分析

　　经典 DP，具体怎么 DP 的推荐看左神的书，这里解释一下为什么从后往前不断找第一个 dp[i] 减少并且位于 LIS 上的元素就可以找到字典序最小的 LIS。

　　下面以样例 2,1,5,3,6,4,8,9,7,5,7,2,3 来演示 smallEnds 数组的更新。

　　

　　从图中可以直观地看出来，后更新的，并且位于 LIS 路径上的点，一定能取代被它更新的，并且位于 LIS 路径上的点。

代码如下

  1 #include <bits/stdc++.h>
  2 using namespace std;
  3  
  4 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
  5 #define Rep(i,n) for (int i = 0; i < (int)(n); ++i)
  6 #define For(i,s,t) for (int i = (int)(s); i <= (int)(t); ++i)
  7 #define rFor(i,t,s) for (int i = (int)(t); i >= (int)(s); --i)
  8 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
  9 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
 10 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
 11 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
 12  
 13 #define pr(x) cout << #x << " = " << x << "  "
 14 #define prln(x) cout << #x << " = " << x << endl
 15  
 16 #define LOWBIT(x) ((x)&(-x))
 17  
 18 #define ALL(x) x.begin(),x.end()
 19 #define INS(x) inserter(x,x.begin())
 20 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
 21 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // 删去 x 中所有 c 
 22 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
 23 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
 24  
 25 #define ms0(a) memset(a,0,sizeof(a))
 26 #define msI(a) memset(a,0x3f,sizeof(a))
 27 #define msM(a) memset(a,-1,sizeof(a))
 28 
 29 #define MP make_pair
 30 #define PB push_back
 31 #define ft first
 32 #define sd second
 33  
 34 template<typename T1, typename T2>
 35 istream &operator>>(istream &in, pair<T1, T2> &p) {
 36     in >> p.first >> p.second;
 37     return in;
 38 }
 39  
 40 template<typename T>
 41 istream &operator>>(istream &in, vector<T> &v) {
 42     for (auto &x: v)
 43         in >> x;
 44     return in;
 45 }
 46 
 47 template<typename T>
 48 ostream &operator<<(ostream &out, vector<T> &v) {
 49     Rep(i, v.size()) out << v[i] << " \n"[i == v.size() - 1];
 50     return out;
 51 }
 52  
 53 template<typename T1, typename T2>
 54 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
 55     out << "[" << p.first << ", " << p.second << "]" << "\n";
 56     return out;
 57 }
 58 
 59 inline int gc(){
 60     static const int BUF = 1e7;
 61     static char buf[BUF], *bg = buf + BUF, *ed = bg;
 62     
 63     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
 64     return *bg++;
 65 } 
 66 
 67 inline int ri(){
 68     int x = 0, f = 1, c = gc();
 69     for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
 70     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
 71     return x*f;
 72 }
 73 
 74 template<class T>
 75 inline string toString(T x) {
 76     ostringstream sout;
 77     sout << x;
 78     return sout.str();
 79 }
 80 
 81 inline int toInt(string s) {
 82     int v;
 83     istringstream sin(s);
 84     sin >> v;
 85     return v;
 86 }
 87 
 88 //min <= aim <= max
 89 template<typename T>
 90 inline bool BETWEEN(const T aim, const T min, const T max) {
 91     return min <= aim && aim <= max;
 92 }
 93 
 94 typedef unsigned int uI;
 95 typedef long long LL;
 96 typedef unsigned long long uLL;
 97 typedef vector< int > VI;
 98 typedef vector< bool > VB;
 99 typedef vector< char > VC;
100 typedef vector< double > VD;
101 typedef vector< string > VS;
102 typedef vector< LL > VL;
103 typedef vector< VI > VVI;
104 typedef vector< VB > VVB;
105 typedef vector< VS > VVS;
106 typedef vector< VL > VVL;
107 typedef vector< VVI > VVVI;
108 typedef vector< VVL > VVVL;
109 typedef pair< int, int > PII;
110 typedef pair< LL, LL > PLL;
111 typedef pair< int, string > PIS;
112 typedef pair< string, int > PSI;
113 typedef pair< string, string > PSS;
114 typedef pair< double, double > PDD;
115 typedef vector< PII > VPII;
116 typedef vector< PLL > VPLL;
117 typedef vector< VPII > VVPII;
118 typedef vector< VPLL > VVPLL;
119 typedef vector< VS > VVS;
120 typedef map< int, int > MII;
121 typedef unordered_map< int, int > uMII;
122 typedef map< LL, LL > MLL;
123 typedef map< string, int > MSI;
124 typedef map< int, string > MIS;
125 typedef set< int > SI;
126 typedef stack< int > SKI;
127 typedef deque< int > DQI;
128 typedef queue< int > QI;
129 typedef priority_queue< int > PQIMax;
130 typedef priority_queue< int, VI, greater< int > > PQIMin;
131 const double EPS = 1e-8;
132 const LL inf = 0x7fffffff;
133 const LL infLL = 0x7fffffffffffffffLL;
134 const LL mod = 1e9 + 7;
135 const int maxN = 1e5 + 7;
136 const LL ONE = 1;
137 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
138 const LL oddBits = 0x5555555555555555;
139 
140 int N;
141 int arr[maxN], maxLen; 
142 int ans[maxN];
143 // dp[i] 表示以 arr[i] 结尾的最长递增子序列 
144 int dp[maxN];
145 // smallEnds[i] 表示LIS长度为 i + 1 的最小结尾 
146 VI smallEnds;
147 
148 int main() {
149     //freopen("MyOutput.txt","w",stdout);
150     //freopen("input.txt","r",stdin);
151     //INIT();
152     scanf("%d", &N);
153     For(i, 1, N) scanf("%d", &arr[i]);
154     
155     For(i, 1, N) {
156         auto it = lower_bound(ALL(smallEnds), arr[i]);
157         dp[i] = it - smallEnds.begin() + 1;
158         
159         if(it != smallEnds.end()) *it = arr[i];
160         else smallEnds.PB(arr[i]);
161     }
162     maxLen = smallEnds.size();
163     
164     int len = maxLen, nextLen = maxLen + 1;
165     rFor(i, N, 1) {
166         if(dp[i] == len && len == nextLen - 1) {
167             ans[len] = arr[i];
168             --len;
169             --nextLen;
170         }
171     }
172     
173     For(i, 1, maxLen) printf("%d%c", ans[i], " \n"[i == N]);
174     return 0;
175 }

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