算法导论 6-3 Young氏矩阵
一、题目
二、思考
最小Young氏矩阵和最小堆的思想差不多，可以通过比较两者的同异来理解Young氏矩阵
不同点：
| min-Heap | min-Young | |
| 堆顶（最小值） | H[1] | Y[i][j] |
| 最后一个元素的位置 | H[N] | Y[N][N] |
| 最后一个元素 | 不一定是最大值 | 一定是最大值 |
| parent | H[i]的parent是H[i/2] | Y[i][j]的parent是Y[i-1][j]和Y[i][j-1] |
| child | H[i]的child是H[i*2]和H[i*2+1] | Y[i][j]的child是Y[i+1][j]和Y[i][j+1] |
| find(x) | 从H[1]开始遍历 | 从西南角开始，或当前值小于key，则向东走，否则向北走 |
相同点：
1.堆顶是最小值所在的位置
2.parent的值<=child的值
3.空条件：堆顶元素值为INIT
4.满条件：最后一个元素的值不为INIT
5.delete：插入到最后一个元素的位置，然后向parent调整
6.extract：提取堆顶元素，将堆顶元素置为INIT，然后child调整
三、练习
a）不唯一 2 3 4 5 8 9 12 14 16 c） 提取Y[1][1]，并用ox7FFFFFFF填充。然后向右下调整 YOUNG-EXTRACR-MIN(Y) 1 if Y[1][1] == 0X7FFFFFFF 2 then error "heap underflow" 3 min <- Y[1][1] 4 A[1][1] <- 0X7FFFFFFF 5 MAX-HEAPIFY(Y, 1, 1) 6 return min //递归 MIN-YOUNGIFY(Y, i, j) 1 min <- Y[i][j] 2 mini <- i 3 minj <- j 4 if i < m and Y[i+1][j] < min 5 mini <- i+1 6 minj <- j 7 min <- Y[i+1][j] 8 if j < n and Y[i+1][j+1] < min 9 mini <- i 10 minj <- j+1 11 min <- Y[i][j+1] 12 if i != mini or j != minj 13 exchange Y[i][j] <-> Y[mini][minj] 14 MIN-YOUNGIFY(Y, mini, minj) d） //若表未满，插入到右下角，然后向左上调整 MIN-YOUNG-INSERT(Y, key) 1 if Y[m][n] < 0x7fffffff 2 then error "young overflow" 3 Y[m][n] <- key 4 flag <- true 5 i <- m 6 j <- n 7 max <- key 8 maxi <- i 9 maxj <- j 10 while true 11 if i > 1 and max < Y[i-1][j] 12 maxi <- i - 1 13 maxj <- j 14 max <- Y[i-1][j] 15 if j > 1 and max < Y[i][j-1] 16 maxi <- i 17 maxj <- j-1 18 max <- Y[i][j-1] 19 if max == Y[i][j] 20 break 21 exchange Y[maxi][maxj] <-> Y[i][j] 22 i <- maxi 23 j <- maxj 24 max <- Y[i][j] e）排序过程为： 取下矩阵Y[1][1]元素，O(1) 将Y[1][1]置为0x7FFFFFFF，O(1) 调整矩阵，O(n) 对n^2个结点各执行一次，因此时间为O(n^3) f） 从左下角开找，若比它小，则向上，则比它大，则向右找 FIND-YOUNG-KEY(Y, key) 1 i <- m 2 j <- 0 3 while i >= 1 and j <= n 4 if Y[i][j] = key 5 then return true 6 else if Y[i][j] < key 7 then j++ 8 else if Y[i][j] > key 9 then i-- 10 return false
四、代码（旧版）
（1）Young.h
//头文件 #include <iostream> #include <stdio.h> using namespace std; //宏定义 #define N 100 class Young { public: //成员变量 int Y[N+1][N+1]; int m, n; //构造与析构 Young(){} Young(int a, int b); //Heap(int size):length(size),heap_size(size){} ~Young(){} //功能函数 void Min_Youngify(int i, int j); void Build_Min_Young(); void YoungSort(); bool Find_Young_key(int key); //优先队列函数 void Young_Decrease_Key(int i, int j, int key); void Min_Young_Insert(int key); int Young_Minimum(); int Young_Extract_Min(); void Young_Delete(int i, int j); //辅助函数 void Print(); }; Young::Young(int a, int b):m(a),n(b) { int i, j; for(i = 1; i <= m; i++) { for(j = 1; j <= n; j++) Y[i][j] = 0x7fffffff; } } void Young::Build_Min_Young() { int i, j, s; for(s = m + n; s >= 2; s--) { for(i = 1; i <= m; i++) { j = s - i; if(j > n)continue; if(j < 1)break; Min_Youngify(i, j); } } } int Young::Young_Extract_Min() { int Min = Y[1][1]; Y[1][1] = 0x7fffffff; Min_Youngify(1, 1); return Min; } void Young::Min_Youngify(int i, int j) { int Min = Y[i][j], mini = i, minj = j; if(i+1 < m && Y[i+1][j] < Min) { mini = i + 1; minj = j; Min = Y[i+1][j]; } if(j+1 < n && Y[i][j+1] < Min) { mini = i; minj = j + 1; Min = Y[i][j+1]; } if(i != mini || j != minj) { swap(Y[i][j], Y[mini][minj]); Min_Youngify(mini, minj); } } void Young::Min_Young_Insert(int key) { if(Y[m][n] < 0x7fffffff) { cout<<"error:young overflow"<<endl; return ; } Y[m][n] = key; int i = m, j = n, Max = key, maxi = i, maxj = j; while(true) { if(i > 1 && Max < Y[i-1][j]) { maxi = i - 1; maxj = j; Max = Y[i-1][j]; } if(j > 1 && Max < Y[i][j-1]) { maxi = i; maxj = j - 1; Max = Y[i][j-1]; } if(Max == Y[i][j]) break; swap(Y[maxi][maxj], Y[i][j]); i = maxi; j = maxj; Max = Y[i][j]; } } bool Young::Find_Young_key(int key) { int i = m, j = 0; while(i >= 1 && j <= n) { if(Y[i][j] == key) return true; else if(Y[i][j] < key) j++; else if(Y[i][j] > key) i--; } return false; } void Young::Print() { int i, j; for(i = 1; i <= m; i++) { for(j = 1; j <= n; j++) cout<<Y[i][j]<<' '; cout<<endl; } } void Young::Young_Delete(int i, int j) { Y[i][j] = 0x7fffffff; Min_Youngify(i, j); } void Young::Young_Decrease_Key(int i, int j, int key) { if(key > Y[i][j]) { cout<<"error:new key is larger than current key"<<endl; return ; } Y[i][j] = key; int Max = key, maxi, maxj; while(true) { if(i > 1 && Max < Y[i-1][j]) { maxi = i - 1; maxj = j; Max = Y[i-1][j]; } if(j > 1 && Max < Y[i][j-1]) { maxi = i; maxj = j - 1; Max = Y[i][j-1]; } if(Max == key) break; swap(Y[maxi][maxj], Y[i][j]); i = maxi; j = maxj; } } int Young::Young_Minimum() { return Y[1][1]; } void Young::YoungSort() { while(Y[1][1] != 0x7fffffff) { cout<<Young_Extract_Min()<<' '; } cout<<endl; }
#include <iostream> #include "Young.h" using namespace std; int main() { int m, n, x; char c; cin>>m>>n; //输入随机测试数据 Young *Y = new Young(m, n); int i, j; for(i = 1; i <= m; i++) { for(j = 1; j <= n; j++) Y->Y[i][j] = rand() % 100; } Y->Print(); //建堆 Y->Build_Min_Young(); Y->Print(); //测试成员函数 while(cin>>c) { switch(c) { case 'E': cout<<Y->Young_Extract_Min()<<endl; break; case 'P': Y->Print(); break; case 'I': x = rand() % 100; cout<<x<<endl; Y->Min_Young_Insert(x); break; case 'F': cin>>x; cout<<Y->Find_Young_key(x)<<endl; break; } } return 0; }
五、代码（新版）
按照clean code的方式重构了代码，并使用gtest做测试
（1）Young.h
#include <vector> using namespace std; #define SIZE_OF_YOUNG 100 #define INIT_DATA 0x7fffffff enum errorType { SUCCESS = 0, SIZE_OVERFLOW, OUT_OF_SIZE, PARAM_ERROR, }; struct structCoordinate { int h_index; int w_index; structCoordinate(int h, int w):h_index(h),w_index(w){} }; class Young { int Y[SIZE_OF_YOUNG+1][SIZE_OF_YOUNG+1]; int height, width; bool isInRange(structCoordinate position); void build(); void YoungifyWithSon(structCoordinate position); void YoungifyWithParent(structCoordinate position); void setDataInPosition(structCoordinate position, int data); int getDataFromPosition(structCoordinate position); void swapDatainPosition(structCoordinate c1, structCoordinate c2); void Print(); public: Young(int h = SIZE_OF_YOUNG, int w = SIZE_OF_YOUNG); void fillData(int *data, int size); int extractMin(); bool findKey(int key); errorType insert(int key); errorType decreaseKey(structCoordinate position, int key); int getMinimum(); vector<int> sort(); errorType deleteData(structCoordinate position); };
#include "Young.h" #include <iostream> using namespace std; enum directionType { EAST = 0, SOUTH, WEST, NORTH }; structCoordinate directionStep[4] = {structCoordinate(0, 1), structCoordinate(1, 0), structCoordinate(0, -1), structCoordinate(-1, 0)}; structCoordinate getNeibourCoordinate(directionType direction, structCoordinate currentPosition) { currentPosition.h_index += directionStep[direction].h_index; currentPosition.w_index += directionStep[direction].w_index; return currentPosition; } Young::Young(int h, int w):height(h),width(w) { if(height < 1 || height > SIZE_OF_YOUNG) height = SIZE_OF_YOUNG; if(width < 1 || width > SIZE_OF_YOUNG) width = SIZE_OF_YOUNG; for(int i = 1; i <= height; i++) for(int j = 1; j <= width; j++) Y[i][j] = INIT_DATA; } void Young::fillData(int *data, int size) { int i, j, cnt = 0; for(i = 1; i <= height; i++) { if(cnt >= size) break; for(j = 1; j <= width; j++) { if(cnt >= size) break; Y[i][j] = data[cnt]; cnt++; } } build(); } int Young::extractMin() { structCoordinate minDataPosition(1, 1); int minData = getDataFromPosition(minDataPosition); setDataInPosition(minDataPosition, INIT_DATA); YoungifyWithSon(minDataPosition); return minData; } bool Young::findKey(int key) { structCoordinate currentPosition(height, 1); while(isInRange(currentPosition) == true) { if(getDataFromPosition(currentPosition) == key) return true; else if(getDataFromPosition(currentPosition) < key) currentPosition = getNeibourCoordinate(EAST, currentPosition); else if(getDataFromPosition(currentPosition) > key) currentPosition = getNeibourCoordinate(NORTH, currentPosition); } return false; } errorType Young::insert(int key) { structCoordinate currentPosition(height, width); if(getDataFromPosition(currentPosition) < INIT_DATA) return SIZE_OVERFLOW; setDataInPosition(currentPosition, key); YoungifyWithParent(currentPosition); return SUCCESS; } errorType Young::decreaseKey(structCoordinate position, int key) { if(false == isInRange(position)) return OUT_OF_SIZE; if(getDataFromPosition(position) < key) return PARAM_ERROR; setDataInPosition(position, key); YoungifyWithParent(position); return SUCCESS; } int Young::getMinimum() { return Y[1][1]; } vector<int> Young::sort() { vector<int> ret; while(getMinimum() != INIT_DATA) { ret.push_back(extractMin()); } return ret; } errorType Young::deleteData(structCoordinate position) { if(false == isInRange(position)) return OUT_OF_SIZE; if(getDataFromPosition(position) == INIT_DATA) return PARAM_ERROR; setDataInPosition(position, INIT_DATA); YoungifyWithSon(position); } bool Young::isInRange(structCoordinate position) { if(position.h_index < 1 || position.w_index < 1) return false; if(position.h_index > height || position.w_index > width) return false; return true; } void Young::build() { int h, w, step; int maxHeight = height, minHeight = 1; int maxWidth = width, minWidth = 1; int maxStep = maxHeight + maxWidth; int minStep = minHeight + minWidth; for(step = maxStep; step >= minStep; step--) { for(h = minHeight; h <= maxHeight; h++) { w = step - h; if(w > maxWidth)continue; if(w < minWidth)break; YoungifyWithSon(structCoordinate(h, w)); } } } void Young::YoungifyWithSon(structCoordinate position) { int Min = getDataFromPosition(position); int i = position.h_index, j = position.w_index; structCoordinate minDataPosition = position; structCoordinate southNeibour = getNeibourCoordinate(SOUTH, position); if(isInRange(southNeibour) && getDataFromPosition(southNeibour) < getDataFromPosition(minDataPosition)) { minDataPosition = southNeibour; } structCoordinate eastNeibour = getNeibourCoordinate(EAST, position); if(isInRange(eastNeibour) && getDataFromPosition(eastNeibour) < getDataFromPosition(minDataPosition)) { minDataPosition = eastNeibour; } if(getDataFromPosition(position) != getDataFromPosition(minDataPosition)) { swapDatainPosition(position, minDataPosition); YoungifyWithSon(minDataPosition); } } void Young::YoungifyWithParent(structCoordinate position) { structCoordinate maxDataPosition = position; while(true) { structCoordinate northNeibour = getNeibourCoordinate(NORTH, position); if(isInRange(northNeibour) && getDataFromPosition(maxDataPosition) < getDataFromPosition(northNeibour)) { maxDataPosition = northNeibour; } structCoordinate westNeibour = getNeibourCoordinate(WEST, position); if(isInRange(westNeibour) && getDataFromPosition(maxDataPosition) < getDataFromPosition(westNeibour)) { maxDataPosition = westNeibour; } if(getDataFromPosition(maxDataPosition) == getDataFromPosition(position)) break; swapDatainPosition(maxDataPosition, position); position = maxDataPosition; } } void Young::setDataInPosition(structCoordinate position, int data) { Y[position.h_index][position.w_index] = data; } int Young::getDataFromPosition(structCoordinate position) { return Y[position.h_index][position.w_index]; } void Young::swapDatainPosition(structCoordinate c1, structCoordinate c2) { int data1 = getDataFromPosition(c1); int data2 = getDataFromPosition(c2); setDataInPosition(c1, data2); setDataInPosition(c2, data1); } void Young::Print() { for(int i = 1; i <= height; i++) { for(int j = 1; j <= width; j++) cout<<Y[i][j]<<' '; cout<<endl; } }
