Paint House
There are a row of n houses, each house can be painted with one of the three colors: red, blue or green. The cost of painting each house with a certain color is different. You have to paint all the houses such that no two adjacent houses have the same color.
The cost of painting each house with a certain color is represented by a n x 3 cost matrix. For example, costs[0][0] is the cost of painting house 0 with color red; costs[1][2] is the cost of painting house 1 with color green, and so on... Find the minimum cost to paint all houses.
Note:
All costs are positive integers.
分析：
典型动态规划，通过遍历所有情况可以弥补前面的选择对后面的影响。时间复杂度为O(n*3*3) = O(n)；利用滚动数组，空间复杂度为O(3*2) = O(1)。
代码：
int minCost2(vector<vector<int> > cost) {
vector<int> opt(3, 0), temp(3, 0);
for(int i = 0; i < cost.size(); i++) {
for(int j = 0; j < 3; j++) {
temp[j] = INT_MAX;
for(int k = 0; k < 3; k++) {
if(j != k)
temp[j] = min(temp[j], opt[k] + cost[i][j]);
}
}
opt.swap(temp);
}
int minc = INT_MAX;
for(int i : opt)
minc = min(minc ,i);
return minc;
}
Paint House II
There are a row of n houses, each house can be painted with one of the k colors. The cost of painting each house with a certain color is different. You have to paint all the houses such that no two adjacent houses have the same color.
The cost of painting each house with a certain color is represented by a n x k cost matrix. For example, costs[0][0] is the cost of painting house 0 with color 0; costs[1][2]is the cost of painting house 1 with color 2, and so on... Find the minimum cost to paint all houses.
Note:
All costs are positive integers.
Follow up:
Could you solve it in O(nk) runtime?
分析：
如果采用I中的方法，时间复杂度为O(n*k*k)，空间复杂度为O(k*2)。为了降低时间复杂度，可以通过减少两重k循环里的大量重复计算来使得O(k*k)的复杂度变为O(k)。此题中，对于j = j1, j2两种情况，它们的内部k循环有k-2次是重复比较了cost[i][j1] + x和cost[i][j2] + x的大小的，可以通过一次cost[i][j1]和cost[i][j2]的比较替代；扩展到j = 1...k种情况，只需找到小的cost[i][j], j = 1...k.
解法：
动态规划，第i轮的最小代价是j = j1时，假设第i + 1轮中，j = j1是默认剔除的，很简单，前一轮的结果后一轮的最小结果都应该使用的，那么第i + 1轮的最小代价是min(cost[i+1][j])的j的取值时；然而j = j1并不是默认剔除的，故在第i + 1轮是cost[i + 1][j1]是无法使用上一轮的最小结果的，但它应该使用第二小的结果，故只需要比较第i + 1轮中，j == j1和j != j1两种情况的值即可。时间复杂度为O(n*(k + 常数)) = O(nk)，空间复杂度，利用滚动值存储中间结果，为O(1)
代码：
int minCost(vector<vector<int> > cost) {
int min1 = 0, min2 = 0, record = -1, c = INT_MAX;
for(int i = 0; i < cost.size(); i++) {
int minval1 = INT_MAX, minval2 = INT_MAX, last = -1;
for(int j = 0; j < cost[0].size(); j++) {
if(record != j) {
if(minval1 > cost[i][j]) {
minval2 = minval1;
minval1 = cost[i][j];
last = j;
}
else
minval2 = min(minval2, cost[i][j]);
}
}
int a = minval1 + min1, b = minval2 + min1;
if(record != -1)
c = cost[i][record] + min2;
if(a < c) {
record = last;
min1 = a;
min2 = min(b, c);
}
else {
min1 = c;
min2 = a;
}
}
cout<<endl;
return min1;
}