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1131. Maximum of Absolute Value Expression
Description
Given two arrays of integers with equal lengths, return the maximum value of:
|arr1[i] - arr1[j]| + |arr2[i] - arr2[j]| + |i - j|
where the maximum is taken over all 0 <= i, j < arr1.length.
Example 1:
Input: arr1 = [1,2,3,4], arr2 = [-1,4,5,6] Output: 13
Example 2:
Input: arr1 = [1,-2,-5,0,10], arr2 = [0,-2,-1,-7,-4] Output: 20
Constraints:
2 <= arr1.length == arr2.length <= 40000-10^6 <= arr1[i], arr2[i] <= 10^6
Solutions
Solution 1: Mathematics + Enumeration
Let’s denote $x_i = arr1[i]$, $y_i = arr2[i]$. Since the size relationship between $i$ and $j$ does not affect the value of the expression, we can assume $i \ge j$. Then the expression can be transformed into:
\[| x_i - x_j | + | y_i - y_j | + i - j = \max \begin{cases} (x_i + y_i) - (x_j + y_j) \\ (x_i - y_i) - (x_j - y_j) \\ (-x_i + y_i) - (-x_j + y_j) \\ (-x_i - y_i) - (-x_j - y_j) \end{cases} + i - j\\ = \max \begin{cases} (x_i + y_i + i) - (x_j + y_j + j) \\ (x_i - y_i + i) - (x_j - y_j + j) \\ (-x_i + y_i + i) - (-x_j + y_j + j) \\ (-x_i - y_i + i) - (-x_j - y_j + j) \end{cases}\]Therefore, we just need to find the maximum value $mx$ and the minimum value $mi$ of $a \times x_i + b \times y_i + i$, where $a, b \in {-1, 1}$. The answer is the maximum value among all $mx - mi$.
The time complexity is $O(n)$, where $n$ is the length of the array. The space complexity is $O(1)$.
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class Solution { public int maxAbsValExpr(int[] arr1, int[] arr2) { int[] dirs = {1, -1, -1, 1, 1}; final int inf = 1 << 30; int ans = -inf; int n = arr1.length; for (int k = 0; k < 4; ++k) { int a = dirs[k], b = dirs[k + 1]; int mx = -inf, mi = inf; for (int i = 0; i < n; ++i) { mx = Math.max(mx, a * arr1[i] + b * arr2[i] + i); mi = Math.min(mi, a * arr1[i] + b * arr2[i] + i); ans = Math.max(ans, mx - mi); } } return ans; } } -
class Solution { public: int maxAbsValExpr(vector<int>& arr1, vector<int>& arr2) { int dirs[5] = {1, -1, -1, 1, 1}; const int inf = 1 << 30; int ans = -inf; int n = arr1.size(); for (int k = 0; k < 4; ++k) { int a = dirs[k], b = dirs[k + 1]; int mx = -inf, mi = inf; for (int i = 0; i < n; ++i) { mx = max(mx, a * arr1[i] + b * arr2[i] + i); mi = min(mi, a * arr1[i] + b * arr2[i] + i); ans = max(ans, mx - mi); } } return ans; } }; -
class Solution: def maxAbsValExpr(self, arr1: List[int], arr2: List[int]) -> int: dirs = (1, -1, -1, 1, 1) ans = -inf for a, b in pairwise(dirs): mx, mi = -inf, inf for i, (x, y) in enumerate(zip(arr1, arr2)): mx = max(mx, a * x + b * y + i) mi = min(mi, a * x + b * y + i) ans = max(ans, mx - mi) return ans -
func maxAbsValExpr(arr1 []int, arr2 []int) int { dirs := [5]int{1, -1, -1, 1, 1} const inf = 1 << 30 ans := -inf for k := 0; k < 4; k++ { a, b := dirs[k], dirs[k+1] mx, mi := -inf, inf for i, x := range arr1 { y := arr2[i] mx = max(mx, a*x+b*y+i) mi = min(mi, a*x+b*y+i) ans = max(ans, mx-mi) } } return ans } -
function maxAbsValExpr(arr1: number[], arr2: number[]): number { const dirs = [1, -1, -1, 1, 1]; const inf = 1 << 30; let ans = -inf; for (let k = 0; k < 4; ++k) { const [a, b] = [dirs[k], dirs[k + 1]]; let mx = -inf; let mi = inf; for (let i = 0; i < arr1.length; ++i) { const [x, y] = [arr1[i], arr2[i]]; mx = Math.max(mx, a * x + b * y + i); mi = Math.min(mi, a * x + b * y + i); ans = Math.max(ans, mx - mi); } } return ans; }