演算法導論 6-3 Young氏矩陣

一、題目二、思考

a）不唯一234589121416c）提取Y[1][1]，並用ox7FFFFFFF填充。然後向右下調整YOUNG-EXTRACR-MIN(Y)1    if  Y[1][1] == 0X7FFFFFFF2        then error "heap underflow"3    min <- Y[1][1]4    A[1][1] <- 0X7FFFFFFF5    MAX-HEAPIFY(Y, 1, 1)6    return min//遞迴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 <- i10        minj <- j+111        min <- Y[i][j+1]12    if i != mini or j != minj13        exchange Y[i][j] <-> Y[mini][minj]14        MIN-YOUNGIFY(Y, mini, minj)d）//若表未滿，插入到右下角，然後向左上調整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 <- j10    while true11        if i > 1 and max < Y[i-1][j]12            maxi <- i - 113            maxj <- j14            max <- Y[i-1][j]15        if j > 1 and max < Y[i][j-1]16            maxi <- i17            maxj <- j-118            max <- Y[i][j-1]19        if max == Y[i][j]20            break21        exchange Y[maxi][maxj] <-> Y[i][j]22        i <- maxi23        j <- maxj24        max <- Y[i][j]e）排序過程為：取下矩陣Y[1][1]元素，O(1)將Y[1][1]置為0x7FFFFFFF，O(1)調整矩陣，O(n)對n^2個結點各執行一次，因此時間為O(n^3)f）從左下角開找，若比它小，則向上，則比它大，則向右找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

三、代碼（1）Young.h

//標頭檔#include <iostream>#include <stdio.h>using namespace std;//宏定義#define N 100class Young{public://成員變數int Y[N+1][N+1];int m, n;//構造與析構Young(){}Young(int a, int b);//Heap(int size):length(size),heap_size(size){}~Young(){}//功能函數void Min_Youngify(int i, int j);void Build_Min_Young();void YoungSort();bool Find_Young_key(int key);//優先隊列函數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);//輔助函數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 < m && Y[i+1][j] < Min){mini = i + 1;minj = j;Min = Y[i+1][j];}if(j < 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;}

（2）main.cpp

#include <iostream>#include "Young.h"using namespace std;int main(){int m, n, x;char c;cin>>m>>n;//輸入隨機測試資料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();//測試成員函數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;}