Friday, November 27, 2015

[itint5]直角路线遍历棋盘 - 阿牧遥 - 博客园


[itint5]直角路线遍历棋盘 - 阿牧遥 - 博客园
https://github.com/zdzapple/itint5/blob/master/%E7%9B%B4%E8%A7%92%E8%B7%AF%E7%BA%BF%E9%81%8D%E5%8E%86%E6%A3%8B%E7%9B%98.cpp
无向图的欧拉路径:只有0或者2个度数为奇数的点
欧拉回路:每个顶点的度数都是偶数
有向图的欧拉路径:起始顶点的出度(从这个顶点发出的有向边的数量)比入度(指向这个顶点的有向边的数量)多1,结束顶点的出度比入度少1,而其它顶点的出度和入度都相等。
欧拉回路:以下两个之一:
每个顶点的出度和入度都相等;
存在一系列的(有向)环,使得图里的每一条边都恰好属于某一个环。

具体的转换方式为:n,m的棋盘,建一个包含n+m个顶点的图G(为了方便说明,类似二分图将其分为两列,左边n个顶点,右边m个顶点,分别代表n行和n列)。对于目标格子(i,j),左边第i个顶点和右边第j个顶点连一条边。最后的问题其实就是问转换之后的图G是否存在欧拉欧拉回路或者欧拉路径。
证明:相邻两步为直角,其实就是从某一行变到某一列。访问图G中的一条边,意味着访问棋盘中的一个目标点。由于图G中的边只连接左边的点(代表某一行)和右边的点(代表某一列),因此访问一条边就意味着从某一行变到了某一列,也就是转直角了。
所以问题变为能否从一点出发访问G中的所有边有且仅有一次。这个就是欧拉回路问题了。

//如果存在满足条件的遍历,返回true,否则返回false
bool existPath(vector<int> &x, vector<int> &y)
{
int n = x.size();
if (x.empty())
return true;
int rowNum = x[0];
int colNum = y[0];
int i, j;
for (i = 0; i < n; ++  i)
{
if (rowNum < x[i])
rowNum = x[i];
if (colNum < y[i])
colNum = y[i];
}
// the graoh should be connected graph
//判断连通性
    bool xa[50] = {false};
    bool ya[50] = {false};
    xa[x[0]] = true;
    ya[y[0]] = true;
    i = 1, j = x.size() - 1;
    while (i <= j)
{
        if(xa[x[i]] || ya[y[i]]) {
            xa[x[i]] = true;
            ya[y[i]] = true;
            i ++;
            j = x.size() - 1;
        }
        else {
            int xt = x[j], yt = y[j];
            x[j] = x[i];
            y[j] = y[i];
            x[i] = xt;
            y[i]= yt;
            j--;
        }
    }
    if (i!=x.size())
return false;


// the number of odd degresses should be 0 or 2
vector<int> leftDegress(rowNum + 1, 0);
vector<int> rightDegress(colNum + 1, 0);
for (i = 0; i < n; ++ i)
{
leftDegress[x[i]] ++;
rightDegress[y[i]] ++;
}
int oddDegressCount = 0;
for (i = 0; i <= rowNum; ++ i)
{
if (leftDegress[i] % 2)
oddDegressCount ++;
}
for (i = 0; i <= colNum; ++ i)
{
if (rightDegress[i] % 2)
oddDegressCount ++;
}
return oddDegressCount == 0 || oddDegressCount == 2;
}

这题一开始直接用暴力的DFS来做,果然到25的规模就挂了.
vector<bool> visited(50, false);
vector<vector<int> > vec_row(50);
vector<vector<int> > vec_col(50);
 
bool findPath(vector<int> &x, vector<int> &y, int idx, int depth, int direction) {
    if (depth == x.size()) return true;
    visited[idx] = true;
    if (direction == 0) {
        int row = x[idx];
        for (int i = 0; i < vec_row[row].size(); i++) {
            if (!visited[vec_row[row][i]]) {
                if (findPath(x, y, vec_row[row][i], depth+1, 1))
                    return true;
            }
        }
    else {
        int col = y[idx];
        for (int i = 0; i < vec_col[col].size(); i++) {
            if (!visited[vec_col[col][i]]) {
                if (findPath(x, y, vec_col[col][i], depth+1, 0))
                    return true;
            }
        }
    }
     
    visited[idx] = false;
    return false;
}
 
 
//如果存在满足条件的遍历,返回true,否则返回false
bool existPath(vector<int> &x, vector<int> &y) {
    int k = x.size();
    if (k == 0) return true;
    for (int i = 0; i < k; i++) {
        vec_row[x[i]].push_back(i);
        vec_col[y[i]].push_back(i);
    }
    for (int i = 0; i < k; i++) {
        if (findPath(x, y, i, 1, 0) ||
            findPath(x, y, i, 1, 1)) {
                return true;
            }
    }
    return false;
}

这题可以直接转换为欧拉回路(路径)问题,这样,如果有解的时候要输出遍历路径的时候,也比较好办了。
具体的转换方式为:n,m的棋盘,建一个包含n+m个顶点的图G(为了方便说明,类似二分图将其分为两列,左边n个顶点,右边m个顶点,分别代表n行和n列)。对于目标格子(i,j),左边第i个顶点和右边第j个顶点连一条边。最后的问题其实就是问转换之后的图G是否存在欧拉欧拉回路或者欧拉路径。
证明:相邻两步为直角,其实就是从某一行变到某一列。访问图G中的一条边,意味着访问棋盘中的一个目标点。由于图G中的边只连接左边的点(代表某一行)和右边的点(代表某一列),因此访问一条边就意味着从某一行变到了某一列,也就是转直角了。
所以问题变为能否从一点出发访问G中的所有边有且仅有一次。这个就是欧拉回路问题了。
 所以欧拉路径是:1.连通;2.奇点为2,为0时是欧拉回路。
 这里的连通我用并查集来做。注意写并查集的merge时,要先找到根,再merge。
vector<int> djset;
 
int find(int i) {
    int x = i;
    while (djset[x] != x) {
        x = djset[x];
    }
    djset[i] = x;
    return x;
}
 
void merge(int i, int j) {
    djset[find(i)] = djset[find(j)];
}
 
//如果存在满足条件的遍历,返回true,否则返回false
bool existPath(vector<int> &x, vector<int> &y) {
    vector<int> axis(100, 0);
    // 计算奇点数目
    for (int i = 0; i < x.size(); i++) {
        axis[x[i]] = !axis[x[i]]; // 奇偶变换
    }
    for (int i = 0; i < y.size(); i++) {
        axis[y[i]+50] = !axis[y[i]+50];
    }
    int count = 0;
    for (int i = 0; i < axis.size(); i++) {
        if (axis[i]) count++;
    }
    if (count != 0 && count != 2) return false;
     
    djset.resize(x.size());
    for (int i = 0; i < djset.size(); i++) {
        djset[i] = i;
    }
     
    // 判断连通性
    for (int i = 0; i < x.size(); i++) {
        for (int j = i+1; j < x.size(); j++) {
            if (x[i] == x[j]) {
                merge(i, j);
            }
        }
    }
    for (int i = 0; i < y.size(); i++) {
        for (int j = i+1; j < y.size(); j++) {
            if (y[i] == y[j]) {
                merge(i, j);
            }
        }
    }
     
    for (int i = 0; i < x.size(); i++) {
        if (find(i) != find(0)) {
            return false;
        }
    }
    return true;
}
Read full article from [itint5]直角路线遍历棋盘 - 阿牧遥 - 博客园

No comments:

Post a Comment

Labels

GeeksforGeeks (1107) LeetCode (934) Algorithm (805) Review (697) to-do (622) LeetCode - Review (448) Classic Algorithm (330) Classic Interview (296) Dynamic Programming (290) Google Interview (241) Tree (145) POJ (139) Difficult Algorithm (134) EPI (127) LeetCode - Phone (121) Different Solutions (119) Bit Algorithms (116) Lintcode (114) Cracking Coding Interview (110) Smart Algorithm (109) Math (104) HackerRank (89) Binary Search (79) Graph Algorithm (74) Greedy Algorithm (69) Binary Tree (66) DFS (62) Interview Corner (61) List (58) LeetCode - Extended (57) Advanced Data Structure (55) Codility (54) Algorithm Interview (53) BFS (53) ComProGuide (52) Geometry Algorithm (48) USACO (46) Trie (44) Stack (43) Binary Search Tree (42) Mathematical Algorithm (42) ACM-ICPC (41) Data Structure (40) Interval (40) Jobdu (39) Knapsack (39) Recursive Algorithm (38) Space Optimization (38) String Algorithm (38) Matrix (37) Codeforces (36) Introduction to Algorithms (36) Must Known (36) Backtracking (35) 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) to-do-must (28) Random (27) Graph (26) GeeksQuiz (25) Logic Thinking (25) Palindrome (25) hihocoder (25) Company-Facebook (23) High Frequency (23) Pre-Sort (23) Queue (23) TopCoder (23) Algorithm Game (22) Company - LinkedIn (22) Hash (22) Lintcode - Review (22) Priority Queue (22) Binary Indexed Trees (21) DFS + Review (21) Brain Teaser (20) CareerCup (20) Company - Twitter (20) Merge Sort (20) Bisection Method (19) Follow Up (19) Post-Order Traverse (19) UVA (19) LeetCode Hard (18) Probabilities (18) Company-Uber (17) Topological Sort (17) Codercareer (16) Game Theory (16) Heap (16) Ordered Stack (16) Shortest Path (16) String Search (16) Tree Traversal (16) itint5 (16) Difficult (15) Iterator (15) O(N) (15) Binary Search - Bisection (14) Number (14) Number Theory (14) Time Complexity (14) Two Pointers (14) Amazon Interview (13) BST (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) Modify Tree (12) Reconstruct Tree (12) Reservoir Sampling (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) Reverse Thinking (11) Tree DP (11) X Sum (11) 挑战程序设计竞赛 (11) Coin Change (10) Company - Microsoft (10) Facebook Hacker Cup (10) HackerRank Easy (10) Rolling Hash (10) SPOJ (10) Theory (10) Tutorialhorizon (10) Bucket Sort (9) DFS+Cache (9) DP-Space Optimization (9) Divide and Conquer (9) Kadane - Extended (9) Mathblog (9) Max-Min Flow (9) O(1) Space (9) Prefix Sum (9) Quick Sort (9) Stack Overflow (9) Stock (9) System Design (9) Use XOR (9) Book Notes (8) Bottom-Up (8) Company-Amazon (8) DFS+BFS (8) DP-Multiple Relation (8) LeetCode - DP (8) Longest Common Subsequence(LCS) (8) Prime (8) Suffix Tree (8) Tech-Queries (8) 穷竭搜索 (8) Algorithm Problem List (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) Linked List (7) Math-Divisible (7) Probability DP (7) Quick Select (7) Radix Sort (7) Simulation (7) TreeMap (7) Xpost (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) Dutch Flag (6) How To (6) Interviewstreet (6) Knapsack - MultiplePack (6) Manacher (6) Minimum Spanning Tree (6) Multiple Data Structures (6) One Pass (6) Programming Pearls (6) Rabin-Karp (6) Randomized Algorithms (6) Sampling (6) Schedule (6) Suffix Array (6) Threaded (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) DP-Include vs Exclude (5) Fast Slow Pointers (5) Graph Cycle (5) Hash Strategy (5) Immutability (5) Java (5) Left and Right Array (5) Matrix Chain Multiplication (5) Maze (5) Microsoft Interview (5) Morris Traversal (5) Pre-Sum (5) Pruning (5) Quadtrees (5) Quick Partition (5) Quora (5) SPFA(Shortest Path Faster Algorithm) (5) Subarray Sum (5) Sweep Line (5) Traversal Once (5) TreeSet (5) Word Search (5) jiuzhang (5) 单调栈 (5) 树形DP (5) 1point3acres (4) Abbreviation (4) Anagram (4) Approximate Algorithm (4) Backtracking-Include vs Exclude (4) Brute Force - Enumeration (4) Chess Game (4) Consistent Hash (4) Cycle (4) Dijkstra (4) Distributed (4) Eulerian Cycle (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) Stream (4) Subset Sum (4) Subsets (4) Sudoku (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) Anagrams (3) B Tree (3) Big Data Algorithm (3) Binary Search - Smart (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) Edit Distance (3) Expression (3) Finite Automata (3) Forward && Backward Scan (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) Online Algorithm (3) Pascal's Triangle (3) Pattern Match (3) Project Euler (3) Rectangle (3) Scala (3) SegmentFault (3) Sieve of Eratosthenes (3) Stack - Smart (3) State Machine (3) Streaming Algorithm (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) 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) 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) 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) Knuth Shuffle (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) Palindromic (2) Parent-Only Tree (2) Parentheses (2) Parser (2) Peak (2) Programming (2) Proof (2) Range Minimum Query (2) Reuse Forward Backward (2) Robot (2) Rosettacode (2) Scan from right (2) Search (2) Shuffle (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) 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 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-Snapchat (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 - Easy (1) Interview - Facebook (1) Isomorphic (1) JDK8 (1) K Dimensional Tree (1) Knapsack - Fractional (1) Knapsack - ZeroOnePack (1) Knight (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) 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 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) Regular Expression (1) Resources (1) Reverse Inorder Traversal (1) Robin (1) Selection (1) Self Balancing BST (1) Similarity (1) Sort && Binary Search (1) Square (1) Stac (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) Thread (1) TimSort (1) Top-Down (1) Tournament (1) Tournament Tree (1) Transform Tree in Place (1) Tree Diameter (1) Tree Rotate (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