887. Super Egg Drop

Description

You are given k identical eggs and you have access to a building with n floors labeled from 1 to n.

You know that there exists a floor f where 0 <= f <= n such that any egg dropped at a floor higher than f will break, and any egg dropped at or below floor f will not break.

Each move, you may take an unbroken egg and drop it from any floor x (where 1 <= x <= n). If the egg breaks, you can no longer use it. However, if the egg does not break, you may reuse it in future moves.

Return the minimum number of moves that you need to determine with certainty what the value of f is.

 

Example 1:

Input: k = 1, n = 2
Output: 2
Explanation: 
Drop the egg from floor 1. If it breaks, we know that f = 0.
Otherwise, drop the egg from floor 2. If it breaks, we know that f = 1.
If it does not break, then we know f = 2.
Hence, we need at minimum 2 moves to determine with certainty what the value of f is.

Example 2:

Input: k = 2, n = 6
Output: 3

Example 3:

Input: k = 3, n = 14
Output: 4

 

Constraints:

  • 1 <= k <= 100
  • 1 <= n <= 104

Solutions

Solution 1

Python Code
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class Solution:
    def superEggDrop(self, k: int, n: int) -> int:
        @cache
        def dfs(i: int, j: int) -> int:
            if i < 1:
                return 0
            if j == 1:
                return i
            l, r = 1, i
            while l < r:
                mid = (l + r + 1) >> 1
                a = dfs(mid - 1, j - 1)
                b = dfs(i - mid, j)
                if a <= b:
                    l = mid
                else:
                    r = mid - 1
            return max(dfs(l - 1, j - 1), dfs(i - l, j)) + 1

        return dfs(n, k)

Java Code
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class Solution {
    private int[][] f;

    public int superEggDrop(int k, int n) {
        f = new int[n + 1][k + 1];
        return dfs(n, k);
    }

    private int dfs(int i, int j) {
        if (i < 1) {
            return 0;
        }
        if (j == 1) {
            return i;
        }
        if (f[i][j] != 0) {
            return f[i][j];
        }
        int l = 1, r = i;
        while (l < r) {
            int mid = (l + r + 1) >> 1;
            int a = dfs(mid - 1, j - 1);
            int b = dfs(i - mid, j);
            if (a <= b) {
                l = mid;
            } else {
                r = mid - 1;
            }
        }
        return f[i][j] = Math.max(dfs(l - 1, j - 1), dfs(i - l, j)) + 1;
    }
}

C++ Code
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class Solution {
public:
    int superEggDrop(int k, int n) {
        int f[n + 1][k + 1];
        memset(f, 0, sizeof(f));
        function<int(int, int)> dfs = [&](int i, int j) -> int {
            if (i < 1) {
                return 0;
            }
            if (j == 1) {
                return i;
            }
            if (f[i][j]) {
                return f[i][j];
            }
            int l = 1, r = i;
            while (l < r) {
                int mid = (l + r + 1) >> 1;
                int a = dfs(mid - 1, j - 1);
                int b = dfs(i - mid, j);
                if (a <= b) {
                    l = mid;
                } else {
                    r = mid - 1;
                }
            }
            return f[i][j] = max(dfs(l - 1, j - 1), dfs(i - l, j)) + 1;
        };
        return dfs(n, k);
    }
};

Go Code
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func superEggDrop(k int, n int) int {
	f := make([][]int, n+1)
	for i := range f {
		f[i] = make([]int, k+1)
	}
	var dfs func(i, j int) int
	dfs = func(i, j int) int {
		if i < 1 {
			return 0
		}
		if j == 1 {
			return i
		}
		if f[i][j] != 0 {
			return f[i][j]
		}
		l, r := 1, i
		for l < r {
			mid := (l + r + 1) >> 1
			a, b := dfs(mid-1, j-1), dfs(i-mid, j)
			if a <= b {
				l = mid
			} else {
				r = mid - 1
			}
		}
		f[i][j] = max(dfs(l-1, j-1), dfs(i-l, j)) + 1
		return f[i][j]
	}
	return dfs(n, k)
}

TypeScript Code
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function superEggDrop(k: number, n: number): number {
    const f: number[][] = new Array(n + 1).fill(0).map(() => new Array(k + 1).fill(0));
    const dfs = (i: number, j: number): number => {
        if (i < 1) {
            return 0;
        }
        if (j === 1) {
            return i;
        }
        if (f[i][j]) {
            return f[i][j];
        }
        let l = 1;
        let r = i;
        while (l < r) {
            const mid = (l + r + 1) >> 1;
            const a = dfs(mid - 1, j - 1);
            const b = dfs(i - mid, j);
            if (a <= b) {
                l = mid;
            } else {
                r = mid - 1;
            }
        }
        return (f[i][j] = Math.max(dfs(l - 1, j - 1), dfs(i - l, j)) + 1);
    };
    return dfs(n, k);
}

Solution 2

Python Code
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class Solution:
    def superEggDrop(self, k: int, n: int) -> int:
        f = [[0] * (k + 1) for _ in range(n + 1)]
        for i in range(1, n + 1):
            f[i][1] = i
        for i in range(1, n + 1):
            for j in range(2, k + 1):
                l, r = 1, i
                while l < r:
                    mid = (l + r + 1) >> 1
                    a, b = f[mid - 1][j - 1], f[i - mid][j]
                    if a <= b:
                        l = mid
                    else:
                        r = mid - 1
                f[i][j] = max(f[l - 1][j - 1], f[i - l][j]) + 1
        return f[n][k]

Java Code
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class Solution {
    public int superEggDrop(int k, int n) {
        int[][] f = new int[n + 1][k + 1];
        for (int i = 1; i <= n; ++i) {
            f[i][1] = i;
        }
        for (int i = 1; i <= n; ++i) {
            for (int j = 2; j <= k; ++j) {
                int l = 1, r = i;
                while (l < r) {
                    int mid = (l + r + 1) >> 1;
                    int a = f[mid - 1][j - 1];
                    int b = f[i - mid][j];
                    if (a <= b) {
                        l = mid;
                    } else {
                        r = mid - 1;
                    }
                }
                f[i][j] = Math.max(f[l - 1][j - 1], f[i - l][j]) + 1;
            }
        }
        return f[n][k];
    }
}

C++ Code
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class Solution {
public:
    int superEggDrop(int k, int n) {
        int f[n + 1][k + 1];
        memset(f, 0, sizeof(f));
        for (int i = 1; i <= n; ++i) {
            f[i][1] = i;
        }
        for (int i = 1; i <= n; ++i) {
            for (int j = 2; j <= k; ++j) {
                int l = 1, r = i;
                while (l < r) {
                    int mid = (l + r + 1) >> 1;
                    int a = f[mid - 1][j - 1];
                    int b = f[i - mid][j];
                    if (a <= b) {
                        l = mid;
                    } else {
                        r = mid - 1;
                    }
                }
                f[i][j] = max(f[l - 1][j - 1], f[i - l][j]) + 1;
            }
        }
        return f[n][k];
    }
};

Go Code
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func superEggDrop(k int, n int) int {
	f := make([][]int, n+1)
	for i := range f {
		f[i] = make([]int, k+1)
	}
	for i := 1; i <= n; i++ {
		f[i][1] = i
	}
	for i := 1; i <= n; i++ {
		for j := 2; j <= k; j++ {
			l, r := 1, i
			for l < r {
				mid := (l + r + 1) >> 1
				a, b := f[mid-1][j-1], f[i-mid][j]
				if a <= b {
					l = mid
				} else {
					r = mid - 1
				}
			}
			f[i][j] = max(f[l-1][j-1], f[i-l][j]) + 1
		}
	}
	return f[n][k]
}

TypeScript Code
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function superEggDrop(k: number, n: number): number {
    const f: number[][] = new Array(n + 1).fill(0).map(() => new Array(k + 1).fill(0));
    for (let i = 1; i <= n; ++i) {
        f[i][1] = i;
    }
    for (let i = 1; i <= n; ++i) {
        for (let j = 2; j <= k; ++j) {
            let l = 1;
            let r = i;
            while (l < r) {
                const mid = (l + r + 1) >> 1;
                const a = f[mid - 1][j - 1];
                const b = f[i - mid][j];
                if (a <= b) {
                    l = mid;
                } else {
                    r = mid - 1;
                }
            }
            f[i][j] = Math.max(f[l - 1][j - 1], f[i - l][j]) + 1;
        }
    }
    return f[n][k];
}