(暴力枚举) $O(N^3)$

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;

public class Main {
    private static BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
    private static PrintWriter writer = new PrintWriter(System.out);
    public static void main(String[] args) throws IOException{
        int n = Integer.parseInt(reader.readLine());
        reader.close();
        for (int a = 0; a * a * 4 <= n; a ++)
            for (int b = a; b * b * 3 <= n - a * a; b ++)
                for (int c = b; c * c * 2 <= n - a * a - b * b; c ++) {
                    int d = (int) Math.sqrt(n - a * a - b * b - c * c);
                    if (n == a * a + b * b + c * c + d * d) {
                        if (c > d) {
                            int temp = c;
                            c = d;
                            d = temp;
                        }
                        writer.println(a + " " + b + " " + c + " " + d);
                        writer.flush();
                        writer.close();
                        return;
                    }
                }
    }

}

(二分) $O(N^2logN)$

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.List;

public class Main {
    private static BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
    private static PrintWriter writer = new PrintWriter(System.out);

    static class Sum {
        int s, c, d;
        public Sum(int s, int c, int d) {
            this.s = s;
            this.c = c;
            this.d = d;
        }
    }

    private static List<Sum> list = new ArrayList<Sum>();

    public static void main(String[] args) throws IOException{
        int n = Integer.parseInt(reader.readLine());
        reader.close();

        for (int c = 0; c * c <= n; c ++)
            for (int d = 0; c * c + d * d <= n; d ++)
                list.add(new Sum(c * c + d * d, c, d));

        // 字典序排序
        list.sort((o1, o2) -> {
            if (o1.s != o2.s) return o1.s - o2.s;
            if (o1.c != o2.c) return o1.c - o2.c;
            return o1.d - o2.d;
        });

        // 枚举符合条件的前两项
        for (int a = 0; a * a <= n; a ++) {
            for (int b = a; a * a + b * b <= n; b ++) {
                int x = n - a * a - b * b;
                // 二分查到x是否存在后两项中
                int l = 0, r = list.size() - 1;
                while (l < r) {
                    int mid = l + r >> 1;
                    if (list.get(mid).s >= x) r = mid;
                    else l = mid + 1;
                }

                // 如果找到的话
                if (list.get(l).s == x) {
                    int c = list.get(l).c;
                    int d = list.get(l).d;
                    writer.println(a + " " + b + " " + c + " " + d);
                    writer.flush();
                    writer.close();
                    return;
                }
            }
        }
    }

}

(哈希表) $O(N^2)$

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.HashMap;

public class Main {
    private static BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
    private static PrintWriter writer = new PrintWriter(System.out);

    static class Node {
        int c, d;
        public Node(int c, int d) {
            this.c = c;
            this.d = d;
        }
    }

    private static HashMap<Integer, Node> S = new HashMap<>();

    public static void main(String[] args) throws IOException{
        int n = Integer.parseInt(reader.readLine());
        reader.close();

        for (int c = 0; c * c <= n; c ++) {
            for (int d = 0; c * c + d * d <= n; d ++) {
                int t = c * c + d * d;
                if (!S.containsKey(t)) {
                    S.put(t, new Node(c, d));
                }
            }
        }

        for (int a = 0; a * a <= n; a ++) {
            for (int b = 0; a * a + b * b <= n; b ++) {
                int t = n - a * a - b * b;
                if (S.containsKey(t)) {
                    writer.println(a + " " + b + " " + S.get(t).c + " " + S.get(t).d);
                    writer.flush();
                    writer.close();
                    return;
                }
            }
        }

    }

}