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08.03. Magic Index

Description

A magic index in an array A[0...n-1] is defined to be an index such that A[i] = i. Given a sorted array of distinct integers, write a method to find a magic index, if one exists, in array A. If not, return -1. If there are more than one magic index, return the smallest one.

Example1:


 Input: nums = [0, 2, 3, 4, 5]

 Output: 0

Example2:


 Input: nums = [1, 1, 1]

 Output: 1

Note:

  1. 1 <= nums.length <= 1000000

Solutions

We design a function $dfs(i, j)$ to find the magic index in the array $nums[i, j]$. If found, return the value of the magic index, otherwise return $-1$. So the answer is $dfs(0, n-1)$.

The implementation of the function $dfs(i, j)$ is as follows:

  1. If $i > j$, return $-1$.
  2. Otherwise, we take the middle position $mid = (i + j) / 2$, then recursively call $dfs(i, mid-1)$. If the return value is not $-1$, it means that the magic index is found in the left half, return it directly. Otherwise, if $nums[mid] = mid$, it means that the magic index is found, return it directly. Otherwise, recursively call $dfs(mid+1, j)$ and return.

In the worst case, the time complexity is $O(n)$, and the space complexity is $O(n)$. Where $n$ is the length of the array $nums$.

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class Solution:
    def findMagicIndex(self, nums: List[int]) -> int:
        def dfs(i: int, j: int) -> int:
            if i > j:
                return -1
            mid = (i + j) >> 1
            l = dfs(i, mid - 1)
            if l != -1:
                return l
            if nums[mid] == mid:
                return mid
            return dfs(mid + 1, j)

        return dfs(0, len(nums) - 1)

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class Solution {
    public int findMagicIndex(int[] nums) {
        return dfs(nums, 0, nums.length - 1);
    }

    private int dfs(int[] nums, int i, int j) {
        if (i > j) {
            return -1;
        }
        int mid = (i + j) >> 1;
        int l = dfs(nums, i, mid - 1);
        if (l != -1) {
            return l;
        }
        if (nums[mid] == mid) {
            return mid;
        }
        return dfs(nums, mid + 1, j);
    }
}

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class Solution {
public:
    int findMagicIndex(vector<int>& nums) {
        function<int(int, int)> dfs = [&](int i, int j) {
            if (i > j) {
                return -1;
            }
            int mid = (i + j) >> 1;
            int l = dfs(i, mid - 1);
            if (l != -1) {
                return l;
            }
            if (nums[mid] == mid) {
                return mid;
            }
            return dfs(mid + 1, j);
        };
        return dfs(0, nums.size() - 1);
    }
};

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func findMagicIndex(nums []int) int {
    var dfs func(i, j int) int
    dfs = func(i, j int) int {
        if i > j {
            return -1
        }
        mid := (i + j) >> 1
        if l := dfs(i, mid-1); l != -1 {
            return l
        }
        if nums[mid] == mid {
            return mid
        }
        return dfs(mid+1, j)
    }
    return dfs(0, len(nums)-1)
}

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function findMagicIndex(nums: number[]): number {
    const dfs = (i: number, j: number): number => {
        if (i > j) {
            return -1;
        }
        const mid = (i + j) >> 1;
        const l = dfs(i, mid - 1);
        if (l !== -1) {
            return l;
        }
        if (nums[mid] === mid) {
            return mid;
        }
        return dfs(mid + 1, j);
    };
    return dfs(0, nums.length - 1);
}

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impl Solution {
    fn dfs(nums: &Vec<i32>, i: usize, j: usize) -> i32 {
        if i >= j || nums[j - 1] < 0 {
            return -1;
        }
        let mid = (i + j) >> 1;
        if nums[mid] >= (i as i32) {
            let l = Self::dfs(nums, i, mid);
            if l != -1 {
                return l;
            }
        }
        if nums[mid] == (mid as i32) {
            return mid as i32;
        }
        Self::dfs(nums, mid + 1, j)
    }

    pub fn find_magic_index(nums: Vec<i32>) -> i32 {
        Self::dfs(&nums, 0, nums.len())
    }
}

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/**
 * @param {number[]} nums
 * @return {number}
 */
var findMagicIndex = function (nums) {
    const dfs = (i, j) => {
        if (i > j) {
            return -1;
        }
        const mid = (i + j) >> 1;
        const l = dfs(i, mid - 1);
        if (l !== -1) {
            return l;
        }
        if (nums[mid] === mid) {
            return mid;
        }
        return dfs(mid + 1, j);
    };
    return dfs(0, nums.length - 1);
};

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class Solution {
    func findMagicIndex(_ nums: [Int]) -> Int {
        return find(nums, 0, nums.count - 1)
    }

    private func find(_ nums: [Int], _ i: Int, _ j: Int) -> Int {
        if i > j {
            return -1
        }
        let mid = (i + j) >> 1
        let l = find(nums, i, mid - 1)
        if l != -1 {
            return l
        }
        if nums[mid] == mid {
            return mid
        }
        return find(nums, mid + 1, j)
    }
}

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