Thursday, July 28, 2016

BZOJ 1041 [HAOI2008] 圆上的整点 题解与分析 - 初学者 - 博客频道 - CSDN.NET


BZOJ 1041 [HAOI2008] 圆上的整点 题解与分析 - 初学者 - 博客频道 - CSDN.NET
平面上有一个圆, 圆心坐标为(0,0),半径为n. 问圆周上有多少个整点. 整点的定义即x,y坐标均为整数的点. 

input

输入文件只有一个正整数n.

Output

输出文件为一个正整数, 即圆周上有多少个整点.

Sample Input

4

Sample Output

4

HINT

20%的测试数据, n<=10000.
100%的测试数据, n<=2000 000 000, 保证答案在long long/int64 范围内.

样例说明:
4个整点分别为:(-4,0),(4,0),(0,4),(0,-4)

【分析】:           
          样例图示:
                                                                          
        首先,最暴力的算法显而易见:枚举x轴上的每个点,带入圆的方程,检查是否算出的值是否为整点,这样的枚举量为2*N,显然过不了全点。
        然后想数学方法。
       
        
         有了上面的推理,那么实现的方法为:
         枚举d∈[1,sqrt(2R)],然后根据上述推理可知:必先判d是否为2R的一约数。
         此时d为2R的约数有两种情况:d=d或d=2R/d。
         第一种情况:d=2R/d。枚举a∈[1,sqrt(2R/2d)] <由2*a*a < 2*R/d转变来>,算出对应的b=sqrt(2R/d-a^2),检查是否此时的A,B满足:A≠B且A,B互质 <根据上面的推理可知必需满足此条件>,若是就将答案加1
         第二种情况:d=d。枚举a∈[1,sqrt(d/2)] <由2*a*a < d转变来>,算出对应的b=sqrt(d-a^2),检查是否此时的A,B满足:A≠B且A,B互质 <根据上面的推理可知必需满足此条件>,若是就将答案加1
         因为这样只算出了第一象限的情况<上面枚举时均是从1开始枚举>,根据圆的对称性,其他象限的整点数与第一象限中的整点数相同,最后,在象限轴上的4个整点未算,加上即可,那么最后答案为ans=4*第一象限整点数+4

【时间复杂度分析】:
        枚举d:O(sqrt(2R)),然后两次枚举a:O(sqrt(d/2))+O(sqrt(R/d)),求最大公约数:O(logN)
http://hzwer.com/1457.html
long long R,ans;
long long gcd(long long x,long long y){return x%y==0? y:gcd(y,x%y);}
bool check(long long y,double x)
{
      if(x==floor(x))
      {
            long long x1=(long long)floor(x);
            if(gcd(x1*x1,y*y)==1&&x1*x1!=y*y)
                  return true;
      }
      return false;
}
int main()
{
      scanf("%lld",&R);
      for(long long d=1;d<=sqrt(2*R);d++)
      {
            if((2*R)%d==0)
            {
                  for(long long a=1;a<=(long long)sqrt(2*R/(2*d));a++)
                  {
                        double b=sqrt(((2*R)/d)-a*a);
                        if(check(a,b))ans++;
                  }
                  if(d!=(2*R)/d)
                  {
                        for(long long a=1;a<=(long long)sqrt(d/2);a++)
                        {
                              double b=sqrt(d-a*a);
                              if(check(a,b))
                                    ans++;
                        }
                  }
            }
      }    
      printf("%lld",ans*4+4);
      return 0;
}
http://edward-mj.com/archives/166
现在给定了一个方程a^2+b^2=c^2  (c已知)
这,就是勾股数。
然后勾股数有一个很霸气的定理。
1、a=m^2-n^2
2、b=2*m*n
3、c=m^2+n^2
4、gcd(n,m)=1。
5、gcd(a,b,c)=1。
同时满足这五条式的是一组勾股数,而且对于所有满足这五条式的(m,n)乘一个k(k>=1),即(km,kn)。就可以表示所有的勾股数,并且勾股数和三元对(m,n,k)一一对应。
换句话说,每个勾股数都只能表示为一个三元对(m,n,k)。
好了,现在回到这题的算法上。
我们枚举k^2(程序中是变量:div),但是k这里有一个小技巧,并不需要枚举1<=k^2<=r,
只要枚举1<=k^2<=sqrt(r)即可。因为我们可以保证拆成的两个乘数的大小顺序。
然后枚举了k^2以后,
问题转化成能不能找到一对二元对(m,n),满足:
1、gcd(m,n)=1
2、m^2+n^2==r/(k^2)
3、gcd(m^2-n^2,2*m*n,m^2+n^2)=1
然后这样暴力还是会很慢的,于是师傅就把第三个条件直接抽出来先判断。
我们可以从:1、3两式推出  r%4=1!
于是直接预先判断,里面那个复杂度为O(sqrt(r))的部分就很少会执行了!
然后算法甚至达到了接近O(sqrt(r))的复杂度。
【其它】
时间到了0MS。空间也没用什么。
膜拜AekdyCoin大神。。。
【CODE】
#include #include using namespace std;
int ans=0;
int gcd(int a,int b){return b?gcd(b,a%b):a;}
void solve(int r){
if (r==1 || r%4!=1) return;
int m,n;
for (n=1;n*n*2 m=(int)(sqrt(r-n*n)+1e-5);
if (m*m+n*n!=r) continue;
if (gcd(m,n)==1 && m>n) ans++;
}
}
int main(){
int r,div;
cin >> r;
ans=0;
for (div=1;div*div<=r;div++){
if (r%div) continue;
solve(r/div);
solve(div);
}
ans*=8;
ans+=4;
cout << ans << endl;
}
http://www.voidcn.com/blog/hbhcy98/article/p-5771408.html
Read full article from BZOJ 1041 [HAOI2008] 圆上的整点 题解与分析 - 初学者 - 博客频道 - CSDN.NET

No comments:

Post a Comment

Labels

GeeksforGeeks (1107) LeetCode (985) Algorithm (795) Review (759) to-do (631) LeetCode - Review (506) Classic Algorithm (324) Dynamic Programming (292) Classic Interview (288) Google Interview (242) Tree (145) POJ (139) Difficult Algorithm (132) LeetCode - Phone (127) EPI (125) Different Solutions (120) Bit Algorithms (118) Lintcode (113) Cracking Coding Interview (110) Smart Algorithm (109) Math (107) HackerRank (89) Binary Search (81) Binary Tree (80) Graph Algorithm (74) Greedy Algorithm (72) DFS (66) LeetCode - Extended (62) Interview Corner (61) Stack (60) List (58) Advanced Data Structure (56) Codility (54) BFS (53) ComProGuide (52) Algorithm Interview (50) Geometry Algorithm (48) Binary Search Tree (46) USACO (46) Trie (45) Mathematical Algorithm (42) ACM-ICPC (41) Interval (41) Data Structure (40) Knapsack (40) Space Optimization (40) Jobdu (39) Recursive Algorithm (39) LeetCode Hard (38) Matrix (38) String Algorithm (38) Backtracking (36) Codeforces (36) Introduction to Algorithms (36) Must Known (36) Beauty of Programming (35) Sort (35) Union-Find (34) Array (33) prismoskills (33) Segment Tree (32) Sliding Window (32) Data Structure Design (31) HDU (31) Google Code Jam (30) Permutation (30) Puzzles (30) Array O(N) (29) Company-Airbnb (29) Company-Zenefits (28) Microsoft 100 - July (28) Palindrome (28) to-do-must (28) Priority Queue (27) Random (27) Graph (26) Company - LinkedIn (25) GeeksQuiz (25) Logic Thinking (25) Pre-Sort (25) hihocoder (25) Queue (24) Company-Facebook (23) High Frequency (23) TopCoder (23) Algorithm Game (22) Hash (22) Post-Order Traverse (22) Binary Indexed Trees (21) Bisection Method (21) DFS + Review (21) Lintcode - Review (21) Brain Teaser (20) CareerCup (20) Company - Twitter (20) Merge Sort (20) Follow Up (19) O(N) (19) Time Complexity (19) Two Pointers (19) UVA (19) Ordered Stack (18) Probabilities (18) Company-Uber (17) Game Theory (17) Topological Sort (17) Codercareer (16) Heap (16) Shortest Path (16) String Search (16) Tree Traversal (16) itint5 (16) Difficult (15) Iterator (15) BST (14) Number (14) Number Theory (14) Amazon Interview (13) Basic Algorithm (13) Codechef (13) Euclidean GCD (13) KMP (13) Long Increasing Sequence(LIS) (13) Majority (13) mitbbs (13) Combination (12) Computational Geometry (12) LeetCode - Classic (12) Modify Tree (12) Reconstruct Tree (12) Reservoir Sampling (12) Reverse Thinking (12) 尺取法 (12) AOJ (11) DFS+Backtracking (11) Fast Power Algorithm (11) Graph DFS (11) LCA (11) LeetCode - DFS (11) Miscs (11) Princeton (11) Proof (11) Tree DP (11) X Sum (11) 挑战程序设计竞赛 (11) Bisection (10) Bucket Sort (10) Coin Change (10) Company - Microsoft (10) DFS+Cache (10) Facebook Hacker Cup (10) HackerRank Easy (10) O(1) Space (10) Rolling Hash (10) SPOJ (10) Theory (10) Tutorialhorizon (10) DP-Multiple Relation (9) DP-Space Optimization (9) Divide and Conquer (9) Kadane - Extended (9) Mathblog (9) Max-Min Flow (9) Prefix Sum (9) Quick Sort (9) Simulation (9) Stack Overflow (9) Stock (9) System Design (9) TreeMap (9) Use XOR (9) Book Notes (8) Bottom-Up (8) Company-Amazon (8) DFS+BFS (8) LeetCode - DP (8) Left and Right Array (8) Linked List (8) Longest Common Subsequence(LCS) (8) Prime (8) Suffix Tree (8) Tech-Queries (8) Traversal Once (8) 穷竭搜索 (8) Algorithm Problem List (7) Expression (7) Facebook Interview (7) Fibonacci Numbers (7) Game Nim (7) Graph BFS (7) HackerRank Difficult (7) Hackerearth (7) Interval Tree (7) Inversion (7) Kadane’s Algorithm (7) Level Order Traversal (7) Math-Divisible (7) Probability DP (7) Quick Select (7) Radix Sort (7) n00tc0d3r (7) 蓝桥杯 (7) Catalan Number (6) Classic Data Structure Impl (6) DFS+DP (6) DP - Tree (6) DP-Print Solution (6) Dijkstra (6) Dutch Flag (6) How To (6) Interviewstreet (6) Knapsack - MultiplePack (6) Manacher (6) Minimum Spanning Tree (6) Morris Traversal (6) Multiple Data Structures (6) One Pass (6) Programming Pearls (6) Pruning (6) Rabin-Karp (6) Randomized Algorithms (6) Sampling (6) Schedule (6) Stream (6) Suffix Array (6) Threaded (6) TreeSet (6) Xpost (6) reddit (6) AI (5) Algorithm - Brain Teaser (5) Art Of Programming-July (5) Big Data (5) Brute Force (5) Code Kata (5) Codility-lessons (5) Coding (5) Company - WMware (5) Convex Hull (5) Crazyforcode (5) Cycle (5) DP-Include vs Exclude (5) Fast Slow Pointers (5) Graph Cycle (5) Hash Strategy (5) Immutability (5) Java (5) Matrix Chain Multiplication (5) Maze (5) Microsoft Interview (5) Pre-Sum (5) Quadtrees (5) Quick Partition (5) Quora (5) SPFA(Shortest Path Faster Algorithm) (5) Subarray Sum (5) Sudoku (5) Sweep Line (5) Word Search (5) jiuzhang (5) 单调栈 (5) 树形DP (5) 1point3acres (4) Abbreviation (4) Anagram (4) Anagrams (4) Approximate Algorithm (4) Backtracking-Include vs Exclude (4) Brute Force - Enumeration (4) Chess Game (4) Consistent Hash (4) Distributed (4) Eulerian Cycle (4) Find Rule (4) Flood fill (4) Graph-Classic (4) HackerRank AI (4) Histogram (4) Kadane Max Sum (4) Knapsack - Mixed (4) Knapsack - Unbounded (4) LeetCode - Recursive (4) LeetCode - TODO (4) MST (4) MinMax (4) N Queens (4) Nerd Paradise (4) Parallel Algorithm (4) Practical Algorithm (4) Probability (4) Programcreek (4) Spell Checker (4) Stock Maximize (4) Subset Sum (4) Subsets (4) Symbol Table (4) Triangle (4) Water Jug (4) algnotes (4) fgdsb (4) to-do-2 (4) 最大化最小值 (4) A Star (3) Algorithm - How To (3) Algorithm Design (3) B Tree (3) Big Data Algorithm (3) Caterpillar Method (3) Coins (3) Company - Groupon (3) Company - Indeed (3) Cumulative Sum (3) DP-Fill by Length (3) DP-Two Variables (3) Dedup (3) Dequeue (3) Dropbox (3) Easy (3) Finite Automata (3) Github (3) GoLang (3) Graph - Bipartite (3) Include vs Exclude (3) Joseph (3) Jump Game (3) K (3) Knapsack-多重背包 (3) LeetCode - Bit (3) Linked List Merge Sort (3) LogN (3) Master Theorem (3) Min Cost Flow (3) Minesweeper (3) Missing Numbers (3) NP Hard (3) O(N) Hard (3) Online Algorithm (3) Pascal's Triangle (3) Pattern Match (3) Project Euler (3) Rectangle (3) Scala (3) SegmentFault (3) Shuffle (3) Sieve of Eratosthenes (3) Stack - Smart (3) State Machine (3) Subtree (3) Transform Tree (3) Trie + DFS (3) Two Pointers Window (3) Warshall Floyd (3) With Random Pointer (3) Word Ladder (3) bookkeeping (3) codebytes (3) Activity Selection Problem (2) Advanced Algorithm (2) AnAlgorithmADay (2) Application of Algorithm (2) Array Merge (2) BOJ (2) BT - Path Sum (2) Balanced Binary Search Tree (2) Bellman Ford (2) Binary Search - Smart (2) Binomial Coefficient (2) Bit Counting (2) Bit Mask (2) Bit-Difficult (2) Bloom Filter (2) Book Coding Interview (2) Branch and Bound Method (2) Clock (2) Codesays (2) Company - Baidu (2) Company-Snapchat (2) Complete Binary Tree (2) DFS+BFS, Flood Fill (2) DP - DFS (2) DP-3D Table (2) DP-Classical (2) DP-Output Solution (2) DP-Slide Window Gap (2) DP-i-k-j (2) DP-树形 (2) Distributed Algorithms (2) Divide and Conqure (2) Doubly Linked List (2) Edit Distance (2) Factor (2) Forward && Backward Scan (2) GoHired (2) Graham Scan (2) Graph BFS+DFS (2) Graph Coloring (2) Graph-Cut Vertices (2) Hamiltonian Cycle (2) Huffman Tree (2) In-order Traverse (2) Include or Exclude Last Element (2) Information Retrieval (2) Interview - Linkedin (2) Invariant (2) Islands (2) Linked Interview (2) Linked List Sort (2) Longest SubArray (2) Lucene-Solr (2) Math-Remainder Queue (2) Matrix Power (2) Median (2) Minimum Vertex Cover (2) Negative All Values (2) Number Each Digit (2) Numerical Method (2) Object Design (2) Order Statistic Tree (2) Parent-Only Tree (2) Parentheses (2) Parser (2) Peak (2) Programming (2) Range Minimum Query (2) Regular Expression (2) Return Multiple Values (2) Reuse Forward Backward (2) Robot (2) Rosettacode (2) Scan from right (2) Search (2) SimHash (2) Simple Algorithm (2) Skyline (2) Spatial Index (2) Strongly Connected Components (2) Summary (2) TV (2) Tile (2) Traversal From End (2) Tree Sum (2) Tree Traversal Return Multiple Values (2) Word Break (2) Word Graph (2) Word Trie (2) Yahoo Interview (2) Young Tableau (2) 剑指Offer (2) 数位DP (2) 1-X (1) 51Nod (1) Akka (1) Algorithm - New (1) Algorithm Series (1) Algorithms Part I (1) Analysis of Algorithm (1) Array-Element Index Negative (1) Array-Rearrange (1) Augmented BST (1) Auxiliary Array (1) Auxiliary Array: Inc&Dec (1) BACK (1) BK-Tree (1) BZOJ (1) Basic (1) Bayes (1) Beauty of Math (1) Big Integer (1) Big Number (1) Binary (1) Binary Sarch Tree (1) Binary String (1) Binary Tree Variant (1) Bipartite (1) Bit-Missing Number (1) BitMap (1) BitMap index (1) BitSet (1) Bug Free Code (1) BuildIt (1) C/C++ (1) CC Interview (1) Cache (1) Calculate Height at Same Recusrion (1) Cartesian tree (1) Check Tree Property (1) Chinese (1) Circular Buffer (1) Cloest (1) Clone (1) Code Quality (1) Codesolutiony (1) Company - Alibaba (1) Company - Palantir (1) Company - WalmartLabs (1) Company-Apple (1) Company-Epic (1) Company-Salesforce (1) Company-Yelp (1) Compression Algorithm (1) Concurrency (1) Cont Improvement (1) Convert BST to DLL (1) Convert DLL to BST (1) Custom Sort (1) Cyclic Replacement (1) DFS-Matrix (1) DP - Probability (1) DP Fill Diagonal First (1) DP-Difficult (1) DP-End with 0 or 1 (1) DP-Fill Diagonal First (1) DP-Graph (1) DP-Left and Right Array (1) DP-MaxMin (1) DP-Memoization (1) DP-Node All Possibilities (1) DP-Optimization (1) DP-Preserve Previous Value (1) DP-Print All Solution (1) Database (1) Detect Negative Cycle (1) Diagonal (1) Directed Graph (1) Do Two Things at Same Recusrion (1) Domino (1) Dr Dobb's (1) Duplicate (1) Equal probability (1) External Sort (1) FST (1) Failure Function (1) Fraction (1) Front End Pointers (1) Funny (1) Fuzzy String Search (1) Game (1) Generating Function (1) Generation (1) Genetic algorithm (1) GeoHash (1) Geometry - Orientation (1) Google APAC (1) Graph But No Graph (1) Graph Transpose (1) Graph Traversal (1) Graph-Coloring (1) Graph-Longest Path (1) Gray Code (1) HOJ (1) Hanoi (1) Hard Algorithm (1) How Hash (1) How to Test (1) Improve It (1) In Place (1) Inorder-Reverse Inorder Traverse Simultaneously (1) Interpolation search (1) Interview (1) Interview - Facebook (1) Isomorphic (1) JDK8 (1) K Dimensional Tree (1) Knapsack - Fractional (1) Knapsack - ZeroOnePack (1) Knight (1) Knuth Shuffle (1) Kosaraju’s algorithm (1) Kruskal (1) Kth Element (1) Least Common Ancestor (1) LeetCode - Binary Tree (1) LeetCode - Coding (1) LeetCode - Detail (1) LeetCode - Related (1) Linked List Reverse (1) Linkedin (1) Linkedin Interview (1) Local MinMax (1) Logic Pattern (1) Longest Common Subsequence (1) Longest Common Substring (1) Longest Prefix Suffix(LPS) (1) Machine Learning (1) Maintain State (1) Manhattan Distance (1) Map && Reverse Map (1) Math - Induction (1) Math-Multiply (1) Math-Sum Of Digits (1) Matrix - O(N+M) (1) Matrix BFS (1) Matrix Graph (1) Matrix Search (1) Matrix+DP (1) Matrix-Rotate (1) Max Min So Far (1) Memory-Efficient (1) MinHash (1) MinMax Heap (1) Monotone Queue (1) Monto Carlo (1) Multi-End BFS (1) Multi-Reverse (1) Multiple DFS (1) Multiple Tasks (1) Next Element (1) Next Successor (1) Offline Algorithm (1) PAT (1) Parenthesis (1) Partition (1) Path Finding (1) Patience Sort (1) Persistent (1) Pigeon Hole Principle (1) Power Set (1) Pratical Algorithm (1) PreProcess (1) Probabilistic Data Structure (1) Python (1) Queue & Stack (1) RSA (1) Ranking (1) Rddles (1) ReHash (1) Realtime (1) Recurrence Relation (1) Recursive DFS (1) Recursive to Iterative (1) Red-Black Tree (1) Region (1) Resources (1) Reverse Inorder Traversal (1) Robin (1) Selection (1) Self Balancing BST (1) Similarity (1) Sort && Binary Search (1) Square (1) Streaming Algorithm (1) String Algorithm. Symbol Table (1) String DP (1) String Distance (1) SubMatrix (1) Subsequence (1) System of Difference Constraints(差分约束系统) (1) TSP (1) Ternary Search Tree (1) Test (1) Test Cases (1) Thread (1) TimSort (1) Top-Down (1) Tournament (1) Tournament Tree (1) Transform Tree in Place (1) Tree Diameter (1) Tree Rotate (1) Tree Without Tree Predefined (1) Trie and Heap (1) Trie vs Hash (1) Trie vs HashMap (1) Triplet (1) Two Data Structures (1) Two Stacks (1) USACO - Classical (1) USACO - Problems (1) UyHiP (1) Valid Tree (1) Vector (1) Virtual Matrix (1) Wiggle Sort (1) Wikipedia (1) ZOJ (1) ZigZag (1) baozitraining (1) codevs (1) cos126 (1) javabeat (1) jum (1) namic Programming (1) sqrt(N) (1) 两次dijkstra (1) 九度 (1) 二进制枚举 (1) 夹逼法 (1) 归一化 (1) 折半枚举 (1) 枚举 (1) 状态压缩DP (1) 男人八题 (1) 英雄会 (1) 逆向思维 (1)

Popular Posts