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(, ), temp(, );

    for(int i = ; i < cost.size(); i++) {

        for(int j = ; j < ; j++) {

            temp[j] = INT_MAX;

            for(int k = ; k < ; 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 = , min2 = , record = -, c = INT_MAX;

    for(int i = ; i < cost.size(); i++) {

        int minval1 = INT_MAX, minval2 = INT_MAX, last = -;

        for(int j = ; j < cost[].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 != -)

            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;

}