[LeetCode] Largest Rectangle in Histogram 解题报告
2015-02-24 01:33
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Given n non-negative integers representing the histogram's bar height where the width of each bar is 1, find the area of largest rectangle in the histogram.Above is a histogram where width of each bar is 1, given height =
» Solve this problem[解题思路]对于每一个height,遍历前面所有的height,求取面积最大值。时间复杂度是O(n*n)
这就意味着,可以维护一个递增的栈,每次比较栈顶与当前元素。如果当前元素小于栈顶元素,则入站,否则合并现有栈,直至栈顶元素小于当前元素。结尾入站元素0,重复合并一次。思路很巧妙。代码实现如下, 大数据 76ms过。
[2,1,5,6,2,3].The largest rectangle is shown in the shaded area, which has area =
10unit.For example,Given height =
[2,1,5,6,2,3],return
10.
» Solve this problem[解题思路]对于每一个height,遍历前面所有的height,求取面积最大值。时间复杂度是O(n*n)
[code]1: int largestRectangleArea(vector<int> &height) { 2: // Start typing your C/C++ solution below 3: // DO NOT write int main() function 4: //int result[height.size()]; 5: int maxV = 0; 6: for(int i =0; i< height.size(); i++) 7: { 8: int minV = height[i]; 9: for(int j =i; j>=0; j--) 10: { 11: minV = std::min(minV, height[j]); 12: int area = minV*(i-j+1); 13: if(area > maxV) 14: maxV = area; 15: } 16: } 17: return maxV; 18: }可以过小数据,但是大数据超时。可以理解,这个解法包含了很多重复计算。一个简单的改进,是只对合适的右边界(峰顶),往左遍历面积。
1: int largestRectangleArea(vector<int> &height) { 2: int maxV = 0; 3: for(int i =0; i< height.size(); i++) 4: { 5: if(i+1 < height.size() 6: && height[i] <= height[i+1]) // if not peak node, skip it 7: continue; 8: int minV = height[i]; 9: for(int j =i; j>=0; j--) 10: { 11: minV = std::min(minV, height[j]); 12: int area = minV*(i-j+1); 13: if(area > maxV) 14: maxV = area; 15: } 16: } 17: return maxV; 18: }这样的话,就可以通过大数据。但是这个优化只是比较有效的剪枝,算法仍然是O(n*n).想了半天,也想不出来O(n)的解法,于是上网google了一下。如下图所示,从左到右处理直方,i=4时,小于当前栈顶(及直方3),于是在统计完区间[2,3]的最大值以后,消除掉阴影部分,然后把红线部分作为一个大直方插入。因为,无论后面还是前面的直方,都不可能得到比目前栈顶元素更高的高度了。
这就意味着,可以维护一个递增的栈,每次比较栈顶与当前元素。如果当前元素小于栈顶元素,则入站,否则合并现有栈,直至栈顶元素小于当前元素。结尾入站元素0,重复合并一次。思路很巧妙。代码实现如下, 大数据 76ms过。
1: int largestRectangleArea(vector<int> &height) { 2: // Start typing your C/C++ solution below 3: // DO NOT write int main() function 4: int stack[height.size()+1], width[height.size()+1]; 5: if(height.size() == 0) return 0; 6: int top = 0, area = INT_MIN; 7: stack[0] = 0; 8: width[0] = 0; 9: int newHeight; 10: for(int i =0; i<= height.size(); i++) 11: { 12: if(i < height.size()) newHeight = height[i]; 13: else newHeight = -1; 14: if(newHeight>= stack[top]) 15: { 16: stack[++top] = newHeight; 17: width[top] = 1; 18: } 19: else 20: { 21: int minV = INT_MAX; 22: int wid= 0; 23: while(stack[top] > newHeight) 24: { 25: minV = min(minV, stack[top]); 26: wid += width[top]; 27: area = max(area, minV*(wid)); 28: top--; 29: } 30: stack[++top] = newHeight; 31: width[top] = wid+1; 32: } 33: } 34: return area; 35: }[总结]这道题蛮有意思。引入stack的做法相当巧妙。Update: Refactor code 5/7/2013评论中Zhongwen Ying的code写的比我post的code简洁多了。把他的code format一下集成进来。
1: int largestRectangleArea(vector<int> &h) { 2: stack<int> S; 3: h.push_back(0); 4: int sum = 0; 5: for (int i = 0; i < h.size(); i++) { 6: if (S.empty() || h[i] > h[S.top()]) S.push(i); 7: else { 8: int tmp = S.top(); 9: S.pop(); 10: sum = max(sum, h[tmp]*(S.empty()? i : i-S.top()-1)); 11: i--; 12: } 13: } 14: return sum; 15: }
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