Jump Game Problem in DSA Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the Jump Game problem?

To determine if you can reach the last index of an array starting from the first index, given that each element represents the maximum jump length from that position.

beginner

In the Jump Game problem, what does each element in the input array represent?

Each element represents the maximum number of steps you can jump forward from that position.

intermediate

What is a simple greedy approach to solve the Jump Game problem?

Keep track of the farthest reachable index while iterating through the array. If at any point the current index is beyond the farthest reachable index, return False. If you reach or pass the last index, return True.

intermediate

Why does the greedy approach work for the Jump Game problem?

Because at each step, you only need to know the farthest position you can reach so far. If you can always move forward without getting stuck, you can reach the end.

beginner

What is the time complexity of the greedy solution for the Jump Game problem?

O(n), where n is the length of the input array, because it requires a single pass through the array.

In the Jump Game problem, what does a zero in the array mean?

A You cannot jump forward from that position

B You can jump any number of steps from that position

C You must jump exactly one step from that position

D You can jump backward from that position

A zero means you cannot move forward from that position, which might block progress if you land there.

What should you do if your current index is greater than the farthest reachable index in the greedy approach?

A Return True

B Continue iterating

C Return False

D Restart from the beginning

If current index is beyond the farthest reachable index, it means you cannot move forward to this position, so return False.

What is the initial value of the farthest reachable index in the greedy solution?

A 1

B 0

C Length of array

D -1

We start at index 0, so the farthest reachable index initially is 0.

If the input array is [2,3,1,1,4], can you reach the last index?

A Yes

B No

C Only if you jump backward

D Only if you skip some elements

You can jump from index 0 to 1 (max 2 steps), then from 1 to 4, reaching the last index.

What is the space complexity of the greedy solution for the Jump Game problem?

A O(n^2)

B O(n)

C O(log n)

D O(1)

The greedy solution uses constant extra space, only variables to track indices.

Explain how the greedy algorithm solves the Jump Game problem step-by-step.

Think about tracking the farthest point you can jump to as you move forward.

Describe a scenario where you cannot reach the last index in the Jump Game problem.

Imagine stepping into a hole you cannot jump out of.

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