3102. Minimize Manhattan Distances

Description

You are given an array points representing integer coordinates of some points on a 2D plane, where points[i] = [x_i, y_i].

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 0^th point the maximum distance is between points (5, 15) and (10, 2), which is |5 - 10| + |15 - 2| = 18.
After removing the 1^st point the maximum distance is between points (3, 10) and (10, 2), which is |3 - 10| + |10 - 2| = 15.
After removing the 2^nd point the maximum distance is between points (5, 15) and (4, 4), which is |5 - 4| + |15 - 4| = 12.
After removing the 3^rd 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 <= 10⁵
points[i].length == 2
1 <= points[i][0], points[i][1] <= 10⁸

Solutions

Solution 1: Ordered Set

For two points $(x_1, y_1)$ and $(x_2, y_2)$, their Manhattan distance is $|x_1 - x_2| + |y_1 - y_2|$. We can transform it into $\max(x_1 - x_2, x_2 - x_1) + \max(y_1 - y_2, y_2 - y_1)$, which is:

$$ |x_1 - x_2| + |y_1 - y_2| = \max \begin{cases} x_1 - x_2 + y_1 - y_2 \ x_2 - x_1 + y_2 - y_1 \ x_1 - x_2 + y_2 - y_1 \ x_2 - x_1 + y_1 - y_2 \end{cases} $$

This can be simplified to:

$$ |x_1 - x_2| + |y_1 - y_2| = \max \begin{cases} (x_1 + y_1) - (x_2 + y_2) \ (x_2 + y_2) - (x_1 + y_1) \ (x_1 - y_1) - (x_2 - y_2) \ (x_2 - y_2) - (x_1 - y_1) \end{cases} $$

Here, the first two cases can be represented as $\max(\max(x_1 + y_1, x_2 + y_2) - \min(x_1 + y_1, x_2 + y_2))$, and the last two cases can be represented as $\max(\max(x_1 - y_1, x_2 - y_2) - \min(x_1 - y_1, x_2 - y_2))$.

Therefore, we can store all points according to the values of $x + y$ and $x - y$ in two ordered sets, respectively. Then, for each point, we remove it, update the values in the ordered sets, calculate the difference between the maximum and minimum values, and take the minimum value.

The time complexity is $O(n \log n)$, and the space complexity is $O(n)$. Here, $n$ is the number of points.

Python3JavaC++GoTypeScript

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

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;
    }
};

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);
    }
}

3102. Minimize Manhattan Distances

Description

Solutions

Solution 1: Ordered Set

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