In an infinite chess board with coordinates from -infinity to +infinity, you have a knight at square [0, 0].
A knight has 8 possible moves it can make, as illustrated below. Each move is two squares in a cardinal direction, then one square in an orthogonal direction.
Return the minimum number of steps needed to move the knight to the square [x, y]. It is guaranteed the answer exists.
Example 1:
Input: x = 2, y = 1 Output: 1 Explanation: [0, 0] → [2, 1]
Example 2:
Input: x = 5, y = 5 Output: 4 Explanation: [0, 0] → [2, 1] → [4, 2] → [3, 4] → [5, 5]
Constraints:
-300 <= x, y <= 3000 <= |x| + |y| <= 300
This problem can be solved using the BFS shortest path model. The search space for this problem is not large, so we can directly use the naive BFS. The solution below also provides the code for bidirectional BFS for reference.
Bidirectional BFS is a common optimization method for BFS. The main implementation ideas are as follows:
- Create two queues, q1 and q2, for "start -> end" and "end -> start" search directions, respectively.
- Create two hash maps, m1 and m2, to record the visited nodes and their corresponding expansion times (steps).
- During each search, prioritize the queue with fewer elements for search expansion. If a node visited from the other direction is found during the expansion, it means the shortest path has been found.
- If one of the queues is empty, it means that the search in the current direction cannot continue, indicating that the start and end points are not connected, and there is no need to continue the search.
class Solution:
def minKnightMoves(self, x: int, y: int) -> int:
q = deque([(0, 0)])
ans = 0
vis = {(0, 0)}
dirs = ((-2, 1), (-1, 2), (1, 2), (2, 1), (2, -1), (1, -2), (-1, -2), (-2, -1))
while q:
for _ in range(len(q)):
i, j = q.popleft()
if (i, j) == (x, y):
return ans
for a, b in dirs:
c, d = i + a, j + b
if (c, d) not in vis:
vis.add((c, d))
q.append((c, d))
ans += 1
return -1class Solution {
public int minKnightMoves(int x, int y) {
x += 310;
y += 310;
int ans = 0;
Queue<int[]> q = new ArrayDeque<>();
q.offer(new int[] {310, 310});
boolean[][] vis = new boolean[700][700];
vis[310][310] = true;
int[][] dirs = {{-2, 1}, {-1, 2}, {1, 2}, {2, 1}, {2, -1}, {1, -2}, {-1, -2}, {-2, -1}};
while (!q.isEmpty()) {
for (int k = q.size(); k > 0; --k) {
int[] p = q.poll();
if (p[0] == x && p[1] == y) {
return ans;
}
for (int[] dir : dirs) {
int c = p[0] + dir[0];
int d = p[1] + dir[1];
if (!vis[c][d]) {
vis[c][d] = true;
q.offer(new int[] {c, d});
}
}
}
++ans;
}
return -1;
}
}class Solution {
public:
int minKnightMoves(int x, int y) {
x += 310;
y += 310;
int ans = 0;
queue<pair<int, int>> q;
q.push({310, 310});
vector<vector<bool>> vis(700, vector<bool>(700));
vis[310][310] = true;
vector<vector<int>> dirs = {{-2, 1}, {-1, 2}, {1, 2}, {2, 1}, {2, -1}, {1, -2}, {-1, -2}, {-2, -1}};
while (!q.empty()) {
for (int k = q.size(); k > 0; --k) {
auto p = q.front();
q.pop();
if (p.first == x && p.second == y) return ans;
for (auto& dir : dirs) {
int c = p.first + dir[0], d = p.second + dir[1];
if (!vis[c][d]) {
vis[c][d] = true;
q.push({c, d});
}
}
}
++ans;
}
return -1;
}
};func minKnightMoves(x int, y int) int {
x, y = x+310, y+310
ans := 0
q := [][]int{{310, 310}}
vis := make([][]bool, 700)
for i := range vis {
vis[i] = make([]bool, 700)
}
dirs := [][]int{{-2, 1}, {-1, 2}, {1, 2}, {2, 1}, {2, -1}, {1, -2}, {-1, -2}, {-2, -1}}
for len(q) > 0 {
for k := len(q); k > 0; k-- {
p := q[0]
q = q[1:]
if p[0] == x && p[1] == y {
return ans
}
for _, dir := range dirs {
c, d := p[0]+dir[0], p[1]+dir[1]
if !vis[c][d] {
vis[c][d] = true
q = append(q, []int{c, d})
}
}
}
ans++
}
return -1
}use std::collections::VecDeque;
const DIR: [(i32, i32); 8] = [
(-2, 1),
(2, 1),
(-1, 2),
(1, 2),
(2, -1),
(-2, -1),
(1, -2),
(-1, -2),
];
impl Solution {
#[allow(dead_code)]
pub fn min_knight_moves(x: i32, y: i32) -> i32 {
// The original x, y are from [-300, 300]
// Let's shift them to [0, 600]
let x: i32 = x + 300;
let y: i32 = y + 300;
let mut ret = -1;
let mut vis: Vec<Vec<bool>> = vec![vec![false; 618]; 618];
// <X, Y, Current Steps>
let mut q: VecDeque<(i32, i32, i32)> = VecDeque::new();
q.push_back((300, 300, 0));
while !q.is_empty() {
let (i, j, s) = q.front().unwrap().clone();
q.pop_front();
if i == x && j == y {
ret = s;
break;
}
Self::enqueue(&mut vis, &mut q, i, j, s);
}
ret
}
#[allow(dead_code)]
fn enqueue(
vis: &mut Vec<Vec<bool>>,
q: &mut VecDeque<(i32, i32, i32)>,
i: i32,
j: i32,
cur_step: i32
) {
let next_step = cur_step + 1;
for (dx, dy) in DIR {
let x = i + dx;
let y = j + dy;
if Self::check_bounds(x, y) || vis[x as usize][y as usize] {
// This <X, Y> pair is either out of bound, or has been visited before
// Just ignore this pair
continue;
}
// Otherwise, add the pair to the queue
// Also remember to update the vis vector
vis[x as usize][y as usize] = true;
q.push_back((x, y, next_step));
}
}
#[allow(dead_code)]
fn check_bounds(i: i32, j: i32) -> bool {
i < 0 || i > 600 || j < 0 || j > 600
}
}class Solution:
def minKnightMoves(self, x: int, y: int) -> int:
def extend(m1, m2, q):
for _ in range(len(q)):
i, j = q.popleft()
step = m1[(i, j)]
for a, b in (
(-2, 1),
(-1, 2),
(1, 2),
(2, 1),
(2, -1),
(1, -2),
(-1, -2),
(-2, -1),
):
x, y = i + a, j + b
if (x, y) in m1:
continue
if (x, y) in m2:
return step + 1 + m2[(x, y)]
q.append((x, y))
m1[(x, y)] = step + 1
return -1
if (x, y) == (0, 0):
return 0
q1, q2 = deque([(0, 0)]), deque([(x, y)])
m1, m2 = {(0, 0): 0}, {(x, y): 0}
while q1 and q2:
t = extend(m1, m2, q1) if len(q1) <= len(q2) else extend(m2, m1, q2)
if t != -1:
return t
return -1class Solution {
private int n = 700;
public int minKnightMoves(int x, int y) {
if (x == 0 && y == 0) {
return 0;
}
x += 310;
y += 310;
Map<Integer, Integer> m1 = new HashMap<>();
Map<Integer, Integer> m2 = new HashMap<>();
m1.put(310 * n + 310, 0);
m2.put(x * n + y, 0);
Queue<int[]> q1 = new ArrayDeque<>();
Queue<int[]> q2 = new ArrayDeque<>();
q1.offer(new int[] {310, 310});
q2.offer(new int[] {x, y});
while (!q1.isEmpty() && !q2.isEmpty()) {
int t = q1.size() <= q2.size() ? extend(m1, m2, q1) : extend(m2, m1, q2);
if (t != -1) {
return t;
}
}
return -1;
}
private int extend(Map<Integer, Integer> m1, Map<Integer, Integer> m2, Queue<int[]> q) {
int[][] dirs = {{-2, 1}, {-1, 2}, {1, 2}, {2, 1}, {2, -1}, {1, -2}, {-1, -2}, {-2, -1}};
for (int k = q.size(); k > 0; --k) {
int[] p = q.poll();
int step = m1.get(p[0] * n + p[1]);
for (int[] dir : dirs) {
int x = p[0] + dir[0];
int y = p[1] + dir[1];
if (m1.containsKey(x * n + y)) {
continue;
}
if (m2.containsKey(x * n + y)) {
return step + 1 + m2.get(x * n + y);
}
m1.put(x * n + y, step + 1);
q.offer(new int[] {x, y});
}
}
return -1;
}
}typedef pair<int, int> PII;
class Solution {
public:
int n = 700;
int minKnightMoves(int x, int y) {
if (x == 0 && y == 0) return 0;
x += 310;
y += 310;
unordered_map<int, int> m1;
unordered_map<int, int> m2;
m1[310 * n + 310] = 0;
m2[x * n + y] = 0;
queue<PII> q1;
queue<PII> q2;
q1.push({310, 310});
q2.push({x, y});
while (!q1.empty() && !q2.empty()) {
int t = q1.size() <= q2.size() ? extend(m1, m2, q1) : extend(m2, m1, q2);
if (t != -1) return t;
}
return -1;
}
int extend(unordered_map<int, int>& m1, unordered_map<int, int>& m2, queue<PII>& q) {
vector<vector<int>> dirs = {{-2, 1}, {-1, 2}, {1, 2}, {2, 1}, {2, -1}, {1, -2}, {-1, -2}, {-2, -1}};
for (int k = q.size(); k > 0; --k) {
auto p = q.front();
q.pop();
int i = p.first, j = p.second;
int step = m1[i * n + j];
for (auto& dir : dirs) {
int x = i + dir[0], y = j + dir[1];
if (m1.count(x * n + y)) continue;
if (m2.count(x * n + y)) return step + 1 + m2[x * n + y];
m1[x * n + y] = step + 1;
q.push({x, y});
}
}
return -1;
}
};func minKnightMoves(x int, y int) int {
if x == 0 && y == 0 {
return 0
}
n := 700
x, y = x+310, y+310
q1, q2 := []int{310*700 + 310}, []int{x*n + y}
m1, m2 := map[int]int{310*700 + 310: 0}, map[int]int{x*n + y: 0}
dirs := [][]int{{-2, 1}, {-1, 2}, {1, 2}, {2, 1}, {2, -1}, {1, -2}, {-1, -2}, {-2, -1}}
extend := func() int {
for k := len(q1); k > 0; k-- {
p := q1[0]
q1 = q1[1:]
i, j := p/n, p%n
step := m1[i*n+j]
for _, dir := range dirs {
x, y := i+dir[0], j+dir[1]
t := x*n + y
if _, ok := m1[t]; ok {
continue
}
if v, ok := m2[t]; ok {
return step + 1 + v
}
m1[t] = step + 1
q1 = append(q1, t)
}
}
return -1
}
for len(q1) > 0 && len(q2) > 0 {
if len(q1) <= len(q2) {
q1, q2 = q2, q1
m1, m2 = m2, m1
}
t := extend()
if t != -1 {
return t
}
}
return -1
}use std::collections::VecDeque;
use std::collections::HashMap;
const DIR: [(i32, i32); 8] = [
(-2, 1),
(2, 1),
(-1, 2),
(1, 2),
(2, -1),
(-2, -1),
(1, -2),
(-1, -2),
];
impl Solution {
#[allow(dead_code)]
pub fn min_knight_moves(x: i32, y: i32) -> i32 {
if x == 0 && y == 0 {
return 0;
}
// Otherwise, let's shift <X, Y> from [-300, 300] -> [0, 600]
let x = x + 300;
let y = y + 300;
let mut ret = -1;
// Initialize the two hash map, used to track if a node has been visited
let mut map_to: HashMap<i32, i32> = HashMap::new();
let mut map_from: HashMap<i32, i32> = HashMap::new();
// Input the original status
map_to.insert(601 * 300 + 300, 0);
map_from.insert(601 * x + y, 0);
let mut q_to: VecDeque<(i32, i32)> = VecDeque::new();
let mut q_from: VecDeque<(i32, i32)> = VecDeque::new();
// Initialize the two queue
q_to.push_back((300, 300));
q_from.push_back((x, y));
while !q_to.is_empty() && !q_from.is_empty() {
let step = if q_to.len() < q_from.len() {
Self::extend(&mut map_to, &mut map_from, &mut q_to)
} else {
Self::extend(&mut map_from, &mut map_to, &mut q_from)
};
if step != -1 {
ret = step;
break;
}
}
ret
}
#[allow(dead_code)]
fn extend(
map_to: &mut HashMap<i32, i32>,
map_from: &mut HashMap<i32, i32>,
cur_q: &mut VecDeque<(i32, i32)>
) -> i32 {
let n = cur_q.len();
for _ in 0..n {
let (i, j) = cur_q.front().unwrap().clone();
cur_q.pop_front();
// The cur_step here must exist
let cur_step = map_to
.get(&(601 * i + j))
.unwrap()
.clone();
for (dx, dy) in DIR {
let x = i + dx;
let y = j + dy;
// Check if this node has been visited
if map_to.contains_key(&(601 * x + y)) {
// Just ignore this node
continue;
}
// Check if this node has been visited by the other side
if map_from.contains_key(&(601 * x + y)) {
// We found the node
return (
cur_step +
1 +
map_from
.get(&(601 * x + y))
.unwrap()
.clone()
);
}
// Otherwise, update map_to and push the new node to queue
map_to.insert(601 * x + y, cur_step + 1);
cur_q.push_back((x, y));
}
}
-1
}
}