Welcome to Subscribe On Youtube
3102. Minimize Manhattan Distances
Description
You are given a array points representing integer coordinates of some points on a 2D plane, where points[i] = [xi, yi].
The distance between two points is defined as their Manhattan distance.
Return the minimum possible value for maximum distance between any two points by removing exactly one point.
Example 1:
Input: points = [[3,10],[5,15],[10,2],[4,4]]
Output: 12
Explanation:
The maximum distance after removing each point is the following:
- After removing the 0th point the maximum distance is between points (5, 15) and (10, 2), which is
\|5 - 10\| + \|15 - 2\| = 18. - After removing the 1st point the maximum distance is between points (3, 10) and (10, 2), which is
\|3 - 10\| + \|10 - 2\| = 15. - After removing the 2nd point the maximum distance is between points (5, 15) and (4, 4), which is
\|5 - 4\| + \|15 - 4\| = 12. - After removing the 3rd point the maximum distance is between points (5, 15) and (10, 2), which is
\|5 - 10\| + \|15 - 2\| = 18.
12 is the minimum possible maximum distance between any two points after removing exactly one point.
Example 2:
Input: points = [[1,1],[1,1],[1,1]]
Output: 0
Explanation:
Removing any of the points results in the maximum distance between any two points of 0.
Constraints:
3 <= points.length <= 105points[i].length == 21 <= points[i][0], points[i][1] <= 108
Solutions
Solution 1
-
class Solution { public int minimumDistance(int[][] points) { TreeMap<Integer, Integer> tm1 = new TreeMap<>(); TreeMap<Integer, Integer> tm2 = new TreeMap<>(); for (int[] p : points) { int x = p[0], y = p[1]; tm1.merge(x + y, 1, Integer::sum); tm2.merge(x - y, 1, Integer::sum); } int ans = Integer.MAX_VALUE; for (int[] p : points) { int x = p[0], y = p[1]; if (tm1.merge(x + y, -1, Integer::sum) == 0) { tm1.remove(x + y); } if (tm2.merge(x - y, -1, Integer::sum) == 0) { tm2.remove(x - y); } ans = Math.min( ans, Math.max(tm1.lastKey() - tm1.firstKey(), tm2.lastKey() - tm2.firstKey())); tm1.merge(x + y, 1, Integer::sum); tm2.merge(x - y, 1, Integer::sum); } return ans; } } -
class Solution { public: int minimumDistance(vector<vector<int>>& points) { multiset<int> st1; multiset<int> st2; for (auto& p : points) { int x = p[0], y = p[1]; st1.insert(x + y); st2.insert(x - y); } int ans = INT_MAX; for (auto& p : points) { int x = p[0], y = p[1]; st1.erase(st1.find(x + y)); st2.erase(st2.find(x - y)); ans = min(ans, max(*st1.rbegin() - *st1.begin(), *st2.rbegin() - *st2.begin())); st1.insert(x + y); st2.insert(x - y); } return ans; } }; -
from sortedcontainers import SortedList class Solution: def minimumDistance(self, points: List[List[int]]) -> int: sl1 = SortedList() sl2 = SortedList() for x, y in points: sl1.add(x + y) sl2.add(x - y) ans = inf for x, y in points: sl1.remove(x + y) sl2.remove(x - y) ans = min(ans, max(sl1[-1] - sl1[0], sl2[-1] - sl2[0])) sl1.add(x + y) sl2.add(x - y) return ans -
func minimumDistance(points [][]int) int { st1 := redblacktree.New[int, int]() st2 := redblacktree.New[int, int]() merge := func(st *redblacktree.Tree[int, int], x, v int) { c, _ := st.Get(x) if c+v == 0 { st.Remove(x) } else { st.Put(x, c+v) } } for _, p := range points { x, y := p[0], p[1] merge(st1, x+y, 1) merge(st2, x-y, 1) } ans := math.MaxInt for _, p := range points { x, y := p[0], p[1] merge(st1, x+y, -1) merge(st2, x-y, -1) ans = min(ans, max(st1.Right().Key-st1.Left().Key, st2.Right().Key-st2.Left().Key)) merge(st1, x+y, 1) merge(st2, x-y, 1) } return ans } -
function minimumDistance(points: number[][]): number { const st1 = new TreapMultiSet<number>(); const st2 = new TreapMultiSet<number>(); for (const [x, y] of points) { st1.add(x + y); st2.add(x - y); } let ans = Infinity; for (const [x, y] of points) { st1.delete(x + y); st2.delete(x - y); ans = Math.min(ans, Math.max(st1.last() - st1.first(), st2.last() - st2.first())); st1.add(x + y); st2.add(x - y); } return ans; } type CompareFunction<T, R extends 'number' | 'boolean'> = ( a: T, b: T, ) => R extends 'number' ? number : boolean; interface ITreapMultiSet<T> extends Iterable<T> { add: (...value: T[]) => this; has: (value: T) => boolean; delete: (value: T) => void; bisectLeft: (value: T) => number; bisectRight: (value: T) => number; indexOf: (value: T) => number; lastIndexOf: (value: T) => number; at: (index: number) => T | undefined; first: () => T | undefined; last: () => T | undefined; lower: (value: T) => T | undefined; higher: (value: T) => T | undefined; floor: (value: T) => T | undefined; ceil: (value: T) => T | undefined; shift: () => T | undefined; pop: (index?: number) => T | undefined; count: (value: T) => number; keys: () => IterableIterator<T>; values: () => IterableIterator<T>; rvalues: () => IterableIterator<T>; entries: () => IterableIterator<[number, T]>; readonly size: number; } class TreapNode<T = number> { value: T; count: number; size: number; priority: number; left: TreapNode<T> | null; right: TreapNode<T> | null; constructor(value: T) { this.value = value; this.count = 1; this.size = 1; this.priority = Math.random(); this.left = null; this.right = null; } static getSize(node: TreapNode<any> | null): number { return node?.size ?? 0; } static getFac(node: TreapNode<any> | null): number { return node?.priority ?? 0; } pushUp(): void { let tmp = this.count; tmp += TreapNode.getSize(this.left); tmp += TreapNode.getSize(this.right); this.size = tmp; } rotateRight(): TreapNode<T> { // eslint-disable-next-line @typescript-eslint/no-this-alias let node: TreapNode<T> = this; const left = node.left; node.left = left?.right ?? null; left && (left.right = node); left && (node = left); node.right?.pushUp(); node.pushUp(); return node; } rotateLeft(): TreapNode<T> { // eslint-disable-next-line @typescript-eslint/no-this-alias let node: TreapNode<T> = this; const right = node.right; node.right = right?.left ?? null; right && (right.left = node); right && (node = right); node.left?.pushUp(); node.pushUp(); return node; } } class TreapMultiSet<T = number> implements ITreapMultiSet<T> { private readonly root: TreapNode<T>; private readonly compareFn: CompareFunction<T, 'number'>; private readonly leftBound: T; private readonly rightBound: T; constructor(compareFn?: CompareFunction<T, 'number'>); constructor(compareFn: CompareFunction<T, 'number'>, leftBound: T, rightBound: T); constructor( compareFn: CompareFunction<T, any> = (a: any, b: any) => a - b, leftBound: any = -Infinity, rightBound: any = Infinity, ) { this.root = new TreapNode<T>(rightBound); this.root.priority = Infinity; this.root.left = new TreapNode<T>(leftBound); this.root.left.priority = -Infinity; this.root.pushUp(); this.leftBound = leftBound; this.rightBound = rightBound; this.compareFn = compareFn; } get size(): number { return this.root.size - 2; } get height(): number { const getHeight = (node: TreapNode<T> | null): number => { if (node == null) return 0; return 1 + Math.max(getHeight(node.left), getHeight(node.right)); }; return getHeight(this.root); } /** * * @complexity `O(logn)` * @description Returns true if value is a member. */ has(value: T): boolean { const compare = this.compareFn; const dfs = (node: TreapNode<T> | null, value: T): boolean => { if (node == null) return false; if (compare(node.value, value) === 0) return true; if (compare(node.value, value) < 0) return dfs(node.right, value); return dfs(node.left, value); }; return dfs(this.root, value); } /** * * @complexity `O(logn)` * @description Add value to sorted set. */ add(...values: T[]): this { const compare = this.compareFn; const dfs = ( node: TreapNode<T> | null, value: T, parent: TreapNode<T>, direction: 'left' | 'right', ): void => { if (node == null) return; if (compare(node.value, value) === 0) { node.count++; node.pushUp(); } else if (compare(node.value, value) > 0) { if (node.left) { dfs(node.left, value, node, 'left'); } else { node.left = new TreapNode(value); node.pushUp(); } if (TreapNode.getFac(node.left) > node.priority) { parent[direction] = node.rotateRight(); } } else if (compare(node.value, value) < 0) { if (node.right) { dfs(node.right, value, node, 'right'); } else { node.right = new TreapNode(value); node.pushUp(); } if (TreapNode.getFac(node.right) > node.priority) { parent[direction] = node.rotateLeft(); } } parent.pushUp(); }; values.forEach(value => dfs(this.root.left, value, this.root, 'left')); return this; } /** * * @complexity `O(logn)` * @description Remove value from sorted set if it is a member. * If value is not a member, do nothing. */ delete(value: T): void { const compare = this.compareFn; const dfs = ( node: TreapNode<T> | null, value: T, parent: TreapNode<T>, direction: 'left' | 'right', ): void => { if (node == null) return; if (compare(node.value, value) === 0) { if (node.count > 1) { node.count--; node?.pushUp(); } else if (node.left == null && node.right == null) { parent[direction] = null; } else { // 旋到根节点 if ( node.right == null || TreapNode.getFac(node.left) > TreapNode.getFac(node.right) ) { parent[direction] = node.rotateRight(); dfs(parent[direction]?.right ?? null, value, parent[direction]!, 'right'); } else { parent[direction] = node.rotateLeft(); dfs(parent[direction]?.left ?? null, value, parent[direction]!, 'left'); } } } else if (compare(node.value, value) > 0) { dfs(node.left, value, node, 'left'); } else if (compare(node.value, value) < 0) { dfs(node.right, value, node, 'right'); } parent?.pushUp(); }; dfs(this.root.left, value, this.root, 'left'); } /** * * @complexity `O(logn)` * @description Returns an index to insert value in the sorted set. * If the value is already present, the insertion point will be before (to the left of) any existing values. */ bisectLeft(value: T): number { const compare = this.compareFn; const dfs = (node: TreapNode<T> | null, value: T): number => { if (node == null) return 0; if (compare(node.value, value) === 0) { return TreapNode.getSize(node.left); } else if (compare(node.value, value) > 0) { return dfs(node.left, value); } else if (compare(node.value, value) < 0) { return dfs(node.right, value) + TreapNode.getSize(node.left) + node.count; } return 0; }; return dfs(this.root, value) - 1; } /** * * @complexity `O(logn)` * @description Returns an index to insert value in the sorted set. * If the value is already present, the insertion point will be before (to the right of) any existing values. */ bisectRight(value: T): number { const compare = this.compareFn; const dfs = (node: TreapNode<T> | null, value: T): number => { if (node == null) return 0; if (compare(node.value, value) === 0) { return TreapNode.getSize(node.left) + node.count; } else if (compare(node.value, value) > 0) { return dfs(node.left, value); } else if (compare(node.value, value) < 0) { return dfs(node.right, value) + TreapNode.getSize(node.left) + node.count; } return 0; }; return dfs(this.root, value) - 1; } /** * * @complexity `O(logn)` * @description Returns the index of the first occurrence of a value in the set, or -1 if it is not present. */ indexOf(value: T): number { const compare = this.compareFn; let isExist = false; const dfs = (node: TreapNode<T> | null, value: T): number => { if (node == null) return 0; if (compare(node.value, value) === 0) { isExist = true; return TreapNode.getSize(node.left); } else if (compare(node.value, value) > 0) { return dfs(node.left, value); } else if (compare(node.value, value) < 0) { return dfs(node.right, value) + TreapNode.getSize(node.left) + node.count; } return 0; }; const res = dfs(this.root, value) - 1; return isExist ? res : -1; } /** * * @complexity `O(logn)` * @description Returns the index of the last occurrence of a value in the set, or -1 if it is not present. */ lastIndexOf(value: T): number { const compare = this.compareFn; let isExist = false; const dfs = (node: TreapNode<T> | null, value: T): number => { if (node == null) return 0; if (compare(node.value, value) === 0) { isExist = true; return TreapNode.getSize(node.left) + node.count - 1; } else if (compare(node.value, value) > 0) { return dfs(node.left, value); } else if (compare(node.value, value) < 0) { return dfs(node.right, value) + TreapNode.getSize(node.left) + node.count; } return 0; }; const res = dfs(this.root, value) - 1; return isExist ? res : -1; } /** * * @complexity `O(logn)` * @description Returns the item located at the specified index. * @param index The zero-based index of the desired code unit. A negative index will count back from the last item. */ at(index: number): T | undefined { if (index < 0) index += this.size; if (index < 0 || index >= this.size) return undefined; const dfs = (node: TreapNode<T> | null, rank: number): T | undefined => { if (node == null) return undefined; if (TreapNode.getSize(node.left) >= rank) { return dfs(node.left, rank); } else if (TreapNode.getSize(node.left) + node.count >= rank) { return node.value; } else { return dfs(node.right, rank - TreapNode.getSize(node.left) - node.count); } }; const res = dfs(this.root, index + 2); return ([this.leftBound, this.rightBound] as any[]).includes(res) ? undefined : res; } /** * * @complexity `O(logn)` * @description Find and return the element less than `val`, return `undefined` if no such element found. */ lower(value: T): T | undefined { const compare = this.compareFn; const dfs = (node: TreapNode<T> | null, value: T): T | undefined => { if (node == null) return undefined; if (compare(node.value, value) >= 0) return dfs(node.left, value); const tmp = dfs(node.right, value); if (tmp == null || compare(node.value, tmp) > 0) { return node.value; } else { return tmp; } }; const res = dfs(this.root, value) as any; return res === this.leftBound ? undefined : res; } /** * * @complexity `O(logn)` * @description Find and return the element greater than `val`, return `undefined` if no such element found. */ higher(value: T): T | undefined { const compare = this.compareFn; const dfs = (node: TreapNode<T> | null, value: T): T | undefined => { if (node == null) return undefined; if (compare(node.value, value) <= 0) return dfs(node.right, value); const tmp = dfs(node.left, value); if (tmp == null || compare(node.value, tmp) < 0) { return node.value; } else { return tmp; } }; const res = dfs(this.root, value) as any; return res === this.rightBound ? undefined : res; } /** * * @complexity `O(logn)` * @description Find and return the element less than or equal to `val`, return `undefined` if no such element found. */ floor(value: T): T | undefined { const compare = this.compareFn; const dfs = (node: TreapNode<T> | null, value: T): T | undefined => { if (node == null) return undefined; if (compare(node.value, value) === 0) return node.value; if (compare(node.value, value) >= 0) return dfs(node.left, value); const tmp = dfs(node.right, value); if (tmp == null || compare(node.value, tmp) > 0) { return node.value; } else { return tmp; } }; const res = dfs(this.root, value) as any; return res === this.leftBound ? undefined : res; } /** * * @complexity `O(logn)` * @description Find and return the element greater than or equal to `val`, return `undefined` if no such element found. */ ceil(value: T): T | undefined { const compare = this.compareFn; const dfs = (node: TreapNode<T> | null, value: T): T | undefined => { if (node == null) return undefined; if (compare(node.value, value) === 0) return node.value; if (compare(node.value, value) <= 0) return dfs(node.right, value); const tmp = dfs(node.left, value); if (tmp == null || compare(node.value, tmp) < 0) { return node.value; } else { return tmp; } }; const res = dfs(this.root, value) as any; return res === this.rightBound ? undefined : res; } /** * @complexity `O(logn)` * @description * Returns the last element from set. * If the set is empty, undefined is returned. */ first(): T | undefined { const iter = this.inOrder(); iter.next(); const res = iter.next().value; return res === this.rightBound ? undefined : res; } /** * @complexity `O(logn)` * @description * Returns the last element from set. * If the set is empty, undefined is returned . */ last(): T | undefined { const iter = this.reverseInOrder(); iter.next(); const res = iter.next().value; return res === this.leftBound ? undefined : res; } /** * @complexity `O(logn)` * @description * Removes the first element from an set and returns it. * If the set is empty, undefined is returned and the set is not modified. */ shift(): T | undefined { const first = this.first(); if (first === undefined) return undefined; this.delete(first); return first; } /** * @complexity `O(logn)` * @description * Removes the last element from an set and returns it. * If the set is empty, undefined is returned and the set is not modified. */ pop(index?: number): T | undefined { if (index == null) { const last = this.last(); if (last === undefined) return undefined; this.delete(last); return last; } const toDelete = this.at(index); if (toDelete == null) return; this.delete(toDelete); return toDelete; } /** * * @complexity `O(logn)` * @description * Returns number of occurrences of value in the sorted set. */ count(value: T): number { const compare = this.compareFn; const dfs = (node: TreapNode<T> | null, value: T): number => { if (node == null) return 0; if (compare(node.value, value) === 0) return node.count; if (compare(node.value, value) < 0) return dfs(node.right, value); return dfs(node.left, value); }; return dfs(this.root, value); } *[Symbol.iterator](): Generator<T, any, any> { yield* this.values(); } /** * @description * Returns an iterable of keys in the set. */ *keys(): Generator<T, any, any> { yield* this.values(); } /** * @description * Returns an iterable of values in the set. */ *values(): Generator<T, any, any> { const iter = this.inOrder(); iter.next(); const steps = this.size; for (let _ = 0; _ < steps; _++) { yield iter.next().value; } } /** * @description * Returns a generator for reversed order traversing the set. */ *rvalues(): Generator<T, any, any> { const iter = this.reverseInOrder(); iter.next(); const steps = this.size; for (let _ = 0; _ < steps; _++) { yield iter.next().value; } } /** * @description * Returns an iterable of key, value pairs for every entry in the set. */ *entries(): IterableIterator<[number, T]> { const iter = this.inOrder(); iter.next(); const steps = this.size; for (let i = 0; i < steps; i++) { yield [i, iter.next().value]; } } private *inOrder(root: TreapNode<T> | null = this.root): Generator<T, any, any> { if (root == null) return; yield* this.inOrder(root.left); const count = root.count; for (let _ = 0; _ < count; _++) { yield root.value; } yield* this.inOrder(root.right); } private *reverseInOrder(root: TreapNode<T> | null = this.root): Generator<T, any, any> { if (root == null) return; yield* this.reverseInOrder(root.right); const count = root.count; for (let _ = 0; _ < count; _++) { yield root.value; } yield* this.reverseInOrder(root.left); } }