solves perfect squares

Author: anishLearnsToCode

README.md

@@ -228,7 +228,7 @@
 | 276  | 🔒 [Paint Fence](https://leetcode.com/problems/paint-fence)                                                                                               |                                                                                                                                                                                                                                          |                                                                                                                                   |
 | 277  | 🔒 [Find The Celebrity](https://leetcode.com/problems/find-the-celebrity)                                                                                 |                                                                                                                                                                                                                                          |                                                                                                                                   |
 | 278  | [First Bad Version](https://leetcode.com/problems/first-bad-version)                                                                                      |                                                             [![Java](assets/java.png)](src/FirstBadVersion.java) [![Python](assets/python.png)](python/first_bad_version.py)                                                             |                                                                                                                                   |
-| 279  | [Perfect Squares](https://leetcode.com/problems/perfect-squares)                                                                                          |                                                                                                                                                                                                                                          |                                                                                                                                   |
+| 279  | [Perfect Squares](https://leetcode.com/problems/perfect-squares)                                                                                          |                                                                                                                 [![Java](assets/java.png)](src/PerfectSquares.java)                                                                                                                         |                                                                                                                                   |
 | 280  | 🔒 [Wiggle Sort](https://leetcode.com/problems/wiggle-sort)                                                                                               |                                                                                                                                                                                                                                          |                                                                                                                                   |
 | 281  | 🔒 [Zigzag Iterator](https://leetcode.com/problems/zigzag-iterator)                                                                                       |                                                                                                                                                                                                                                          |                                                                                                                                   |
 | 283  | [Move Zeroes](https://leetcode.com/problems/move-zeroes)                                                                                                  |                                                                   [![Java](assets/java.png)](src/MoveZeros.java) [![Python](assets/python.png)](python/move_zeroes.py)                                                                   |                                                                                                                                   |

src/PerfectSquares.java

@@ -1,32 +1,44 @@
 // https://leetcode.com/problems/perfect-squares
-// T: O(N * k) k = size of perfect squares array which is constant
-// S: O(N + k)
+// T: O(k + sqrt(n)) k = size of perfect squares set which is constant
+// S: O(k)
+
+import java.util.Set;
 
 public class PerfectSquares {
-    private static final int[] PERFECT_SQUARES = {
+    private static final Set<Integer> PERFECT_SQUARES = Set.of(
             1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576,
             625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849,
             1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3600, 3721,
             3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241,
             6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216, 9409,
-            9604, 9801, 10000, 10_201
-    };
+            9604, 9801, 10_000, 10_201
+    );
 
-    public int numSquares(int n) {
-        final int[] dp = new int[n + 1];
-        dp[0] = dp[1] = 1;
-        for (int i = 2 ; i < dp.length ; i++) {
-            dp[i] = numSquares(i, dp);
-        }
-        return dp[dp.length - 1];
+    private boolean isSquare(int n) {
+        return PERFECT_SQUARES.contains(n);
     }
 
-    private int numSquares(int n, int[] dp) {
-        int result = Integer.MAX_VALUE, square;
-        for (int i = 0 ; PERFECT_SQUARES[i] <= n ; i++) {
-            square = PERFECT_SQUARES[i];
-            result = Math.min(result, dp[square] + dp[n - square]);
+    // Based on Lagrange's Four Square theorem, there
+    // are only 4 possible results: 1, 2, 3, 4.
+    int numSquares(int n) {
+        // If n is a perfect square, return 1.
+        if (isSquare(n)) return 1;
+
+        // The result is 4 if and only if n can be written in the
+        // form of 4^k*(8*m + 7). Please refer to
+        // Legendre's three-square theorem.
+        while (n % 4 == 0) {
+            n >>= 2;
         }
-        return result;
+        if (n % 8 == 7) return 4;
+
+        // Check whether 2 is the result. if number can be composed as a^2 + b^2
+        for(int i = 1; i * i <= n; i++) {
+            if (isSquare(n - i * i)) {
+                return 2;
+            }
+        }
+
+        return 3;
     }
 }