Minimize Manhattan Distances

// Accepted solution for LeetCode #3102: Minimize Manhattan Distances
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;
    }
}

// Accepted solution for LeetCode #3102: Minimize Manhattan Distances
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
}

# Accepted solution for LeetCode #3102: Minimize Manhattan Distances
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

// Accepted solution for LeetCode #3102: Minimize Manhattan Distances
// Rust example auto-generated from java reference.
// Replace the signature and local types with the exact LeetCode harness for this problem.
impl Solution {
    pub fn rust_example() {
        // Port the logic from the reference block below.
    }
}

// Reference (java):
// // Accepted solution for LeetCode #3102: Minimize Manhattan Distances
// 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;
//     }
// }

// Accepted solution for LeetCode #3102: Minimize Manhattan Distances
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);
    }
}