一二维前缀和差分公式辨析
2026/6/10 18:42:39 网站建设 项目流程

一维前缀和:

s[i] = s[i-1] + a[i]

一维前缀和区间查询公式:

sum(l, r) = s[r] - s[l-1]

二维前缀和构建:

s[i][j] = a[i][j] + s[i-1][j] + s[i][j-1] - s[i-1][j-1]

求两点内面积:

sum = s[x2][y2] - s[x1-1][y2] - s[x2][y1-1] + s[x1-1][y1-1]

一维差分:

d[i] = a[i] - a[i-1]

区间增值:

d[l] += c d[r+1] -= c

一维还原:

a[i] = a[i-1] + d[i]

二维差分构建:

d[i][j] = a[i][j] - a[i-1][j] - a[i][j-1] + a[i-1][j-1]

矩阵增值(顺序无所谓):

d[x1][y1] += c d[x2+1][y1] -= c d[x1][y2+1] -= c d[x2+1][y2+1] += c

二维还原:

a[i][j] = d[i][j] + a[i-1][j] + a[i][j-1] - a[i-1][j-1]

对比记忆最佳

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