Jump Game Problem in DSA Python

Mental Model

You can jump forward from each position up to a certain number of steps. The goal is to see if you can reach the end by making these jumps.

Analogy: Imagine standing on stepping stones across a river. Each stone lets you jump forward a few stones ahead. You want to know if you can reach the last stone without falling in the water.

Index: 0   1   2   3   4
Array: [2] [3] [1] [1] [4]
You start at index 0 and can jump up to 2 steps forward ->

Dry Run Walkthrough

Input: array: [2, 3, 1, 1, 4]

Goal: Determine if you can jump from the start (index 0) to the last index (index 4)

Step 1: Start at index 0, max reachable index is 0

Position: 0, MaxReach: 0

Why: We begin at the first stone and can jump up to 2 steps from here

Step 2: From index 0, jump up to 2 steps, update max reachable to max(0, 0+2)=2

Position: 0, MaxReach: 2

Why: We can now reach up to index 2 from the start

Step 3: Move to index 1, since 1 <= max reachable (2), update max reachable to max(2, 1+3)=4

Position: 1, MaxReach: 4

Why: From index 1, we can jump up to 3 steps, reaching the end

Step 4: Move to index 2, since 2 <= max reachable (4), update max reachable to max(4, 2+1)=4

Position: 2, MaxReach: 4

Why: From index 2, jump 1 step, max reachable stays 4

Step 5: Move to index 3, since 3 <= max reachable (4), update max reachable to max(4, 3+1)=4

Position: 3, MaxReach: 4

Why: From index 3, jump 1 step, max reachable stays 4

Step 6: Move to index 4, since 4 <= max reachable (4), we reached the last index

Position: 4, MaxReach: 4

Why: We can reach the last stone, so the answer is true

Result:

1 -> 2 -> 3 -> 4 -> 5 (indices) reachable with max jump 4, can reach last index: true

Annotated Code

DSA Python

from typing import List

class JumpGame:
    def can_jump(self, nums: List[int]) -> bool:
        max_reach = 0
        for i in range(len(nums)):
            if i > max_reach:
                return False  # can't reach this position
            max_reach = max(max_reach, i + nums[i])
            if max_reach >= len(nums) - 1:
                return True  # can reach or pass last index
        return False

if __name__ == '__main__':
    game = JumpGame()
    arr = [2, 3, 1, 1, 4]
    result = game.can_jump(arr)
    print(str(result).lower())

if i > max_reach:

Check if current position is reachable from previous jumps

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

Update max reachable index from current position

if max_reach >= len(nums) - 1:

If we can reach or pass the last index, return true early

Output

true

Complexity Analysis

Time: O(n) because we scan the array once updating the max reachable index

Space: O(1) because we use only a few variables regardless of input size

vs Alternative: Compared to a backtracking approach that tries all jumps (exponential), this greedy method is much faster and uses less memory

Edge Cases

empty array

returns true because we are already at the last index (no jumps needed)

DSA Python

for i in range(len(nums)):

single element array [0]

returns true because start is also the last index

DSA Python

for i in range(len(nums)):

array with zero blocking progress, e.g. [3,2,1,0,4]

returns false because cannot jump past zero at index 3

DSA Python

if i > max_reach:

When to Use This Pattern

When you see a problem asking if you can reach the end by jumping variable steps, use the greedy max reachable index pattern to quickly check reachability.

Common Mistakes

Mistake: Trying to simulate every jump path instead of tracking max reachable index
Fix: Use a single variable to track the farthest reachable index and check if current position is reachable

Mistake: Not checking if current index is beyond max reachable, leading to incorrect true result
Fix: Add a condition to return false immediately if current index is not reachable

Summary

Determines if you can jump from start to end of array using max jump lengths at each position.

Use when you need to check reachability in a sequence with variable jump lengths.

Keep track of the farthest reachable index and verify each position is reachable to solve efficiently.

Practice

1. What does the Jump Game problem ask you to determine given an array nums of non-negative integers?

easy

A. The maximum jump length possible from the first index.

B. If you can reach the last index starting from the first index by jumping according to the values in nums.

C. The minimum number of jumps needed to reach the last index.

D. The sum of all jump lengths in the array.

Solution

1. Step 1: Understand the problem statement

   The Jump Game problem asks if you can start at the first index and reach the last index by jumping forward using the numbers in the array as maximum jump lengths.

2. Step 2: Identify the goal

   The goal is to check if the last index is reachable, not how many jumps it takes or the sum of jumps.

3. Final Answer:

   If you can reach the last index starting from the first index by jumping according to the values in nums. -> Option B

4. Quick Check:

   Jump Game = Reach last index [OK]

Hint: Focus on reachability, not jump count or sums [OK]

Common Mistakes:

• Confusing reachability with minimum jumps needed
• Thinking the problem asks for maximum jump length
• Assuming sum of jumps is required

2. Which of the following Python code snippets correctly checks if you can jump to the last index in the Jump Game problem?

easy

A. max_reach = 0
for i in range(len(nums)):
if i > max_reach:
return False
max_reach = max(max_reach, i + nums[i])
return True

B. for i in range(len(nums)): if i < max_reach: return False max_reach = max(max_reach, i + nums[i]) return True

C. for i in range(len(nums)): max_reach = min(max_reach, i + nums[i]) return max_reach >= len(nums)-1

D. for i in range(len(nums)): if i > max_reach: return True max_reach = max(max_reach, i + nums[i]) return False

Solution

1. Step 1: Understand the logic of max_reach

   We track the farthest index reachable so far. If current index is beyond max_reach, we cannot proceed.

2. Step 2: Analyze the code snippets

   for i in range(len(nums)): if i > max_reach: return False max_reach = max(max_reach, i + nums[i]) return True correctly returns False if current index is beyond max_reach, updates max_reach, and returns True if loop finishes.

3. Final Answer:

   Option A code snippet correctly implements the jump check logic. -> Option A

4. Quick Check:

   Check i > max_reach to fail early [OK]

Hint: Check if current index exceeds max reachable index [OK]

Common Mistakes:

• Using < instead of > to compare index and max_reach
• Returning True or False too early inside loop
• Using min instead of max for max_reach

3. What is the output of the following code for nums = [2,3,1,1,4]?

def can_jump(nums):
    max_reach = 0
    for i in range(len(nums)):
        if i > max_reach:
            return False
        max_reach = max(max_reach, i + nums[i])
    return True

print(can_jump(nums))

medium

A. IndexError

B. False

C. True

D. None

Solution

1. Step 1: Trace the loop and max_reach updates

   Start max_reach=0. i=0: 0 <= 0, max_reach = max(0,0+2)=2. i=1: 1 <= 2, max_reach = max(2,1+3)=4. i=2: 2 <= 4, max_reach = max(4,2+1)=4. i=3: 3 <= 4, max_reach = max(4,3+1)=4. i=4: 4 <= 4, max_reach = max(4,4+4)=8.

2. Step 2: Confirm loop completes and returns True

   Since loop never returns False, function returns True at end.

3. Final Answer:

   True -> Option C

4. Quick Check:

   max_reach covers last index, so True [OK]

Hint: Update max_reach and check if current index is reachable [OK]

Common Mistakes:

• Stopping early without checking all indices
• Confusing max and min when updating max_reach
• Assuming False if max_reach equals current index

4. The following code is intended to solve the Jump Game problem but has a bug. What is the bug?

def can_jump(nums):
    max_reach = 0
    for i in range(len(nums)):
        if i >= max_reach:
            return False
        max_reach = max(max_reach, i + nums[i])
    return True

medium

A. The loop should start from i = 1, not 0.

B. The max_reach update should use min instead of max.

C. The function should return False at the end, not True.

D. The condition should be 'if i > max_reach' instead of 'if i >= max_reach'.

Solution

1. Step 1: Analyze the condition 'if i >= max_reach'

   This condition incorrectly returns False even when i equals max_reach, which is a reachable index.

2. Step 2: Correct the condition to 'if i > max_reach'

   Changing to '>' allows the current index to be equal to max_reach, meaning it is reachable and should not return False prematurely.

3. Final Answer:

   The condition should be 'if i > max_reach' instead of 'if i >= max_reach'. -> Option D

4. Quick Check:

   Use strict greater than to detect unreachable index [OK]

Hint: Use '>' not '>=' to check if index is beyond reach [OK]

Common Mistakes:

• Using '>=' causes false negatives
• Updating max_reach incorrectly
• Returning False too early

5. Given the array nums = [3,2,1,0,4], why does the standard greedy Jump Game solution return False?

hard

A. Because the zero at index 3 blocks any further jumps, making the last index unreachable.

B. Because the first element is too large, causing an overflow error.

C. Because the array length is less than 5, so the problem is invalid.

D. Because the last element is greater than the sum of previous elements.

Solution

1. Step 1: Trace jumps and max_reach

   Start max_reach=0. i=0: max_reach = max(0,0+3)=3. i=1: 1 <= 3, max_reach = max(3,1+2)=3. i=2: 2 <= 3, max_reach = max(3,2+1)=3. i=3: 3 <= 3, max_reach = max(3,3+0)=3. i=4: 4 > 3, so return False.

2. Step 2: Understand why last index is unreachable

   The zero at index 3 prevents extending max_reach beyond 3, so index 4 cannot be reached.

3. Final Answer:

   Because the zero at index 3 blocks any further jumps, making the last index unreachable. -> Option A

4. Quick Check:

   Zero blocks jump, last index unreachable [OK]

Hint: Zeros can block progress if max_reach can't pass them [OK]

Common Mistakes:

• Ignoring zeros that block jumps
• Assuming large first element guarantees success
• Misunderstanding array length relevance