Jump Game

 

1. Clarify the problem: Before diving into solving the problem, let's clarify the requirements:

  • We are given an array of non-negative integers representing the maximum jump length at each position.
  • We need to determine if it is possible to reach the last index of the array starting from the first index.

2. Analyze the problem: To solve this problem, we can use a greedy approach. We start from the first index and iterate through the array, keeping track of the maximum index we can reach. If at any point the maximum index is less than the current index, it means we cannot proceed further and we return false. If we successfully iterate through the array and the maximum index is greater than or equal to the last index, we return true.

3. Design an algorithm: Here is the algorithm to solve the problem:

  1. Initialize the maximum index variable to 0.
  2. Iterate through the array from the first index to the second-to-last index:
    • If the maximum index is less than the current index, return false.
    • Update the maximum index if the sum of the current index and its corresponding jump length is greater than the current maximum index.
  3. Return true if the maximum index is greater than or equal to the last index, otherwise return false.

4. Explain your approach: The approach involves iterating through the array and keeping track of the maximum index we can reach. If at any point the maximum index is less than the current index, it means we cannot proceed further and we return false. If we successfully iterate through the array and the maximum index is greater than or equal to the last index, we return true.

5. Write clean and readable code:

python
def canJump(nums):
    max_index = 0

    for i in range(len(nums)-1):
        if max_index < i:
            return False

        max_index = max(max_index, i + nums[i])

    return max_index >= len(nums)-1

6. Test your code: Let's test the code with some test cases:

  • Test case 1:

    • nums = [2,3,1,1,4]
    • The expected output is True because we can jump from index 0 to index 4.

  • Test case 2:

    • nums = [3,2,1,0,4]
    • The expected output is False because we cannot jump from index 0 to the last index.
python
# Test case 1
nums1 = [2, 3, 1, 1, 4]
print(canJump(nums1))  # Expected output: True

# Test case 2
nums2 = [3, 2, 1, 0, 4]
print(canJump(nums2))  # Expected output: False

7. Optimize if necessary: The current solution is based on a greedy approach and doesn't have any obvious optimization opportunities.

8. Handle error cases: The code assumes that the input array is non-empty. If the input array is empty, the behavior of the code may not be as expected. We can add a check at the beginning to handle this case and return False.

9. Discuss complexity analysis: The time complexity of the solution is O(n), where n is the length of the input array. We need to iterate through the array once. The space complexity is O(1) as we are using a constant amount of extra space.

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