Description
Given an m x n matrix matrix and an integer k, return the max sum of a rectangle in the matrix such that its sum is no larger than k.
It is guaranteed that there will be a rectangle with a sum no larger than k.
Example 1:
Input: matrix = [[1,0,1],[0,-2,3]], k = 2 Output: 2 Explanation: Because the sum of the blue rectangle [[0, 1], [-2, 3]] is 2, and 2 is the max number no larger than k (k = 2).
Example 2:
Input: matrix = [[2,2,-1]], k = 3 Output: 3
Constraints:
m == matrix.lengthn == matrix[i].length1 <= m, n <= 100-100 <= matrix[i][j] <= 100-105 <= k <= 105
Follow up: What if the number of rows is much larger than the number of columns?
Solutions
Solution 1: Enumerate Boundaries + Ordered Set
We can enumerate the upper and lower boundaries i and j of the rectangle, then calculate the sum of the elements in each column within this boundary, and record it in the array nums. The problem is transformed into how to find the maximum subarray sum not exceeding k in the array nums.
We can use an ordered set to quickly find the maximum value less than or equal to x, thereby obtaining a subarray with the maximum subarray sum not exceeding k.
The time complexity is O(m^2 × n × log n), and the space complexity is O(n).
PythonJavaC++GoTypeScript
class Solution: def maxSumSubmatrix(self, matrix: List[List[int]], k: int) -> int: m, n = len(matrix), len(matrix[0]) ans = -inf for i in range(m): nums = [0] * n for j in range(i, m): for h in range(n): nums[h] += matrix[j][h] s = 0 ts = SortedSet([0]) for x in nums: s += x p = ts.bisect_left(s - k) if p != len(ts): ans = max(ans, s - ts[p]) ts.add(s) return ans(code-box)
class Solution { public int maxSumSubmatrix(int[][] matrix, int k) { int m = matrix.length; int n = matrix[0].length; final int inf = 1 << 30; int ans = -inf; for (int i = 0; i < m; ++i) { int[] nums = new int[n]; for (int j = i; j < m; ++j) { for (int h = 0; h < n; ++h) { nums[h] += matrix[j][h]; } int s = 0; TreeSet<Integer> ts = new TreeSet<>(); ts.add(0); for (int x : nums) { s += x; Integer y = ts.ceiling(s - k); if (y != null) { ans = Math.max(ans, s - y); } ts.add(s); } } } return ans; } }(code-box)
class Solution { public: int maxSumSubmatrix(vector<vector<int>>& matrix, int k) { int m = matrix.size(), n = matrix[0].size(); const int inf = 1 << 30; int ans = -inf; for (int i = 0; i < m; ++i) { vector<int> nums(n); for (int j = i; j < m; ++j) { for (int h = 0; h < n; ++h) { nums[h] += matrix[j][h]; } set<int> ts; int s = 0; ts.insert(0); for (int x : nums) { s += x; auto it = ts.lower_bound(s - k); if (it != ts.end()) { ans = max(ans, s - *it); } ts.insert(s); } } } return ans; } };(code-box)
func maxSumSubmatrix(matrix [][]int, k int) int { m, n := len(matrix), len(matrix[0]) const inf = 1 << 30 ans := -inf for i := 0; i < m; i++ { nums := make([]int, n) for j := i; j < m; j++ { for h := 0; h < n; h++ { nums[h] += matrix[j][h] } s := 0 rbt := redblacktree.NewWithIntComparator() rbt.Put(0, nil) for _, x := range nums { s += x if y, ok := rbt.Ceiling(s - k); ok { ans = max(ans, s-y.Key.(int)) } rbt.Put(s, nil) } } } return ans }(code-box)
function maxSumSubmatrix(matrix: number[][], k: number): number { const m = matrix.length; const n = matrix[0].length; let ans = -Infinity; for (let i = 0; i < m; ++i) { const nums: number[] = new Array(n).fill(0); for (let j = i; j < m; ++j) { for (let h = 0; h < n; ++h) { nums[h] += matrix[j][h]; } let s = 0; const ts: TreeSet<number> = new TreeSet<number>(); ts.add(0); for (const x of nums) { s += x; const p = ts.ceil(s - k); if (p !== undefined) { ans = Math.max(ans, s - p); } ts.add(s); } } } return ans; } type Compare<T> = (lhs: T, rhs: T) => number; class RBTreeNode<T = number> { data: T; count: number; left: RBTreeNode<T> | null; right: RBTreeNode<T> | null; parent: RBTreeNode<T> | null; color: number; constructor(data: T) { this.data = data; this.left = this.right = this.parent = null; this.color = 0; this.count = 1; } sibling(): RBTreeNode<T> | null { if (!this.parent) return null; // sibling null if no parent return this.isOnLeft() ? this.parent.right : this.parent.left; } isOnLeft(): boolean { return this === this.parent!.left; } hasRedChild(): boolean { return ( Boolean(this.left && this.left.color === 0) || Boolean(this.right && this.right.color === 0) ); } } class RBTree<T> { root: RBTreeNode<T> | null; lt: (l: T, r: T) => boolean; constructor(compare: Compare<T> = (l: T, r: T) => (l < r ? -1 : l > r ? 1 : 0)) { this.root = null; this.lt = (l: T, r: T) => compare(l, r) < 0; } rotateLeft(pt: RBTreeNode<T>): void { const right = pt.right!; pt.right = right.left; if (pt.right) pt.right.parent = pt; right.parent = pt.parent; if (!pt.parent) this.root = right; else if (pt === pt.parent.left) pt.parent.left = right; else pt.parent.right = right; right.left = pt; pt.parent = right; } rotateRight(pt: RBTreeNode<T>): void { const left = pt.left!; pt.left = left.right; if (pt.left) pt.left.parent = pt; left.parent = pt.parent; if (!pt.parent) this.root = left; else if (pt === pt.parent.left) pt.parent.left = left; else pt.parent.right = left; left.right = pt; pt.parent = left; } swapColor(p1: RBTreeNode<T>, p2: RBTreeNode<T>): void { const tmp = p1.color; p1.color = p2.color; p2.color = tmp; } swapData(p1: RBTreeNode<T>, p2: RBTreeNode<T>): void { const tmp = p1.data; p1.data = p2.data; p2.data = tmp; } fixAfterInsert(pt: RBTreeNode<T>): void { let parent = null; let grandParent = null; while (pt !== this.root && pt.color !== 1 && pt.parent?.color === 0) { parent = pt.parent; grandParent = pt.parent.parent; /* Case : A Parent of pt is left child of Grand-parent of pt */ if (parent === grandParent?.left) { const uncle = grandParent.right; /* Case : 1 The uncle of pt is also red Only Recoloring required */ if (uncle && uncle.color === 0) { grandParent.color = 0; parent.color = 1; uncle.color = 1; pt = grandParent; } else { /* Case : 2 pt is right child of its parent Left-rotation required */ if (pt === parent.right) { this.rotateLeft(parent); pt = parent; parent = pt.parent; } /* Case : 3 pt is left child of its parent Right-rotation required */ this.rotateRight(grandParent); this.swapColor(parent!, grandParent); pt = parent!; } } else { /* Case : B Parent of pt is right child of Grand-parent of pt */ const uncle = grandParent!.left; /* Case : 1 The uncle of pt is also red Only Recoloring required */ if (uncle != null && uncle.color === 0) { grandParent!.color = 0; parent.color = 1; uncle.color = 1; pt = grandParent!; } else { /* Case : 2 pt is left child of its parent Right-rotation required */ if (pt === parent.left) { this.rotateRight(parent); pt = parent; parent = pt.parent; } /* Case : 3 pt is right child of its parent Left-rotation required */ this.rotateLeft(grandParent!); this.swapColor(parent!, grandParent!); pt = parent!; } } } this.root!.color = 1; } delete(val: T): boolean { const node = this.find(val); if (!node) return false; node.count--; if (!node.count) this.deleteNode(node); return true; } deleteAll(val: T): boolean { const node = this.find(val); if (!node) return false; this.deleteNode(node); return true; } deleteNode(v: RBTreeNode<T>): void { const u = BSTreplace(v); // True when u and v are both black const uvBlack = (u === null || u.color === 1) && v.color === 1; const parent = v.parent!; if (!u) { // u is null therefore v is leaf if (v === this.root) this.root = null; // v is root, making root null else { if (uvBlack) { // u and v both black // v is leaf, fix double black at v this.fixDoubleBlack(v); } else { // u or v is red if (v.sibling()) { // sibling is not null, make it red" v.sibling()!.color = 0; } } // delete v from the tree if (v.isOnLeft()) parent.left = null; else parent.right = null; } return; } if (!v.left || !v.right) { // v has 1 child if (v === this.root) { // v is root, assign the value of u to v, and delete u v.data = u.data; v.left = v.right = null; } else { // Detach v from tree and move u up if (v.isOnLeft()) parent.left = u; else parent.right = u; u.parent = parent; if (uvBlack) this.fixDoubleBlack(u); // u and v both black, fix double black at u else u.color = 1; // u or v red, color u black } return; } // v has 2 children, swap data with successor and recurse this.swapData(u, v); this.deleteNode(u); // find node that replaces a deleted node in BST function BSTreplace(x: RBTreeNode<T>): RBTreeNode<T> | null { // when node have 2 children if (x.left && x.right) return successor(x.right); // when leaf if (!x.left && !x.right) return null; // when single child return x.left ?? x.right; } // find node that do not have a left child // in the subtree of the given node function successor(x: RBTreeNode<T>): RBTreeNode<T> { let temp = x; while (temp.left) temp = temp.left; return temp; } } fixDoubleBlack(x: RBTreeNode<T>): void { if (x === this.root) return; // Reached root const sibling = x.sibling(); const parent = x.parent!; if (!sibling) { // No sibiling, double black pushed up this.fixDoubleBlack(parent); } else { if (sibling.color === 0) { // Sibling red parent.color = 0; sibling.color = 1; if (sibling.isOnLeft()) this.rotateRight(parent); // left case else this.rotateLeft(parent); // right case this.fixDoubleBlack(x); } else { // Sibling black if (sibling.hasRedChild()) { // at least 1 red children if (sibling.left && sibling.left.color === 0) { if (sibling.isOnLeft()) { // left left sibling.left.color = sibling.color; sibling.color = parent.color; this.rotateRight(parent); } else { // right left sibling.left.color = parent.color; this.rotateRight(sibling); this.rotateLeft(parent); } } else { if (sibling.isOnLeft()) { // left right sibling.right!.color = parent.color; this.rotateLeft(sibling); this.rotateRight(parent); } else { // right right sibling.right!.color = sibling.color; sibling.color = parent.color; this.rotateLeft(parent); } } parent.color = 1; } else { // 2 black children sibling.color = 0; if (parent.color === 1) this.fixDoubleBlack(parent); else parent.color = 1; } } } } insert(data: T): boolean { // search for a position to insert let parent = this.root; while (parent) { if (this.lt(data, parent.data)) { if (!parent.left) break; else parent = parent.left; } else if (this.lt(parent.data, data)) { if (!parent.right) break; else parent = parent.right; } else break; } // insert node into parent const node = new RBTreeNode(data); if (!parent) this.root = node; else if (this.lt(node.data, parent.data)) parent.left = node; else if (this.lt(parent.data, node.data)) parent.right = node; else { parent.count++; return false; } node.parent = parent; this.fixAfterInsert(node); return true; } find(data: T): RBTreeNode<T> | null { let p = this.root; while (p) { if (this.lt(data, p.data)) { p = p.left; } else if (this.lt(p.data, data)) { p = p.right; } else break; } return p ?? null; } *inOrder(root: RBTreeNode<T> = this.root!): Generator<T, undefined, void> { if (!root) return; for (const v of this.inOrder(root.left!)) yield v; yield root.data; for (const v of this.inOrder(root.right!)) yield v; } *reverseInOrder(root: RBTreeNode<T> = this.root!): Generator<T, undefined, void> { if (!root) return; for (const v of this.reverseInOrder(root.right!)) yield v; yield root.data; for (const v of this.reverseInOrder(root.left!)) yield v; } } class TreeSet<T = number> { _size: number; tree: RBTree<T>; compare: Compare<T>; constructor( collection: T[] | Compare<T> = [], compare: Compare<T> = (l: T, r: T) => (l < r ? -1 : l > r ? 1 : 0), ) { if (typeof collection === 'function') { compare = collection; collection = []; } this._size = 0; this.compare = compare; this.tree = new RBTree(compare); for (const val of collection) this.add(val); } size(): number { return this._size; } has(val: T): boolean { return !!this.tree.find(val); } add(val: T): boolean { const successful = this.tree.insert(val); this._size += successful ? 1 : 0; return successful; } delete(val: T): boolean { const deleted = this.tree.deleteAll(val); this._size -= deleted ? 1 : 0; return deleted; } ceil(val: T): T | undefined { let p = this.tree.root; let higher = null; while (p) { if (this.compare(p.data, val) >= 0) { higher = p; p = p.left; } else { p = p.right; } } return higher?.data; } floor(val: T): T | undefined { let p = this.tree.root; let lower = null; while (p) { if (this.compare(val, p.data) >= 0) { lower = p; p = p.right; } else { p = p.left; } } return lower?.data; } higher(val: T): T | undefined { let p = this.tree.root; let higher = null; while (p) { if (this.compare(val, p.data) < 0) { higher = p; p = p.left; } else { p = p.right; } } return higher?.data; } lower(val: T): T | undefined { let p = this.tree.root; let lower = null; while (p) { if (this.compare(p.data, val) < 0) { lower = p; p = p.right; } else { p = p.left; } } return lower?.data; } first(): T | undefined { return this.tree.inOrder().next().value; } last(): T | undefined { return this.tree.reverseInOrder().next().value; } shift(): T | undefined { const first = this.first(); if (first === undefined) return undefined; this.delete(first); return first; } pop(): T | undefined { const last = this.last(); if (last === undefined) return undefined; this.delete(last); return last; } *[Symbol.iterator](): Generator<T, void, void> { for (const val of this.values()) yield val; } *keys(): Generator<T, void, void> { for (const val of this.values()) yield val; } *values(): Generator<T, undefined, void> { for (const val of this.tree.inOrder()) yield val; return undefined; } /** * Return a generator for reverse order traversing the set */ *rvalues(): Generator<T, undefined, void> { for (const val of this.tree.reverseInOrder()) yield val; return undefined; } } class TreeMultiSet<T = number> { _size: number; tree: RBTree<T>; compare: Compare<T>; constructor( collection: T[] | Compare<T> = [], compare: Compare<T> = (l: T, r: T) => (l < r ? -1 : l > r ? 1 : 0), ) { if (typeof collection === 'function') { compare = collection; collection = []; } this._size = 0; this.compare = compare; this.tree = new RBTree(compare); for (const val of collection) this.add(val); } size(): number { return this._size; } has(val: T): boolean { return !!this.tree.find(val); } add(val: T): boolean { const successful = this.tree.insert(val); this._size++; return successful; } delete(val: T): boolean { const successful = this.tree.delete(val); if (!successful) return false; this._size--; return true; } count(val: T): number { const node = this.tree.find(val); return node ? node.count : 0; } ceil(val: T): T | undefined { let p = this.tree.root; let higher = null; while (p) { if (this.compare(p.data, val) >= 0) { higher = p; p = p.left; } else { p = p.right; } } return higher?.data; } floor(val: T): T | undefined { let p = this.tree.root; let lower = null; while (p) { if (this.compare(val, p.data) >= 0) { lower = p; p = p.right; } else { p = p.left; } } return lower?.data; } higher(val: T): T | undefined { let p = this.tree.root; let higher = null; while (p) { if (this.compare(val, p.data) < 0) { higher = p; p = p.left; } else { p = p.right; } } return higher?.data; } lower(val: T): T | undefined { let p = this.tree.root; let lower = null; while (p) { if (this.compare(p.data, val) < 0) { lower = p; p = p.right; } else { p = p.left; } } return lower?.data; } first(): T | undefined { return this.tree.inOrder().next().value; } last(): T | undefined { return this.tree.reverseInOrder().next().value; } shift(): T | undefined { const first = this.first(); if (first === undefined) return undefined; this.delete(first); return first; } pop(): T | undefined { const last = this.last(); if (last === undefined) return undefined; this.delete(last); return last; } *[Symbol.iterator](): Generator<T, void, void> { yield* this.values(); } *keys(): Generator<T, void, void> { for (const val of this.values()) yield val; } *values(): Generator<T, undefined, void> { for (const val of this.tree.inOrder()) { let count = this.count(val); while (count--) yield val; } return undefined; } /** * Return a generator for reverse order traversing the multi-set */ *rvalues(): Generator<T, undefined, void> { for (const val of this.tree.reverseInOrder()) { let count = this.count(val); while (count--) yield val; } return undefined; } }(code-box)
