Why Jump Game Problem in DSA Rust?

The Big Idea

What if you could know instantly if you can finish a tricky jump path without trying every step?

The Scenario

Imagine you are playing a board game where each square tells you how many steps you can jump forward. You want to reach the last square, but you have to decide carefully which jumps to make.

The Problem

Trying every possible jump by hand is slow and confusing. You might jump too far or too short, and it's easy to get stuck or miss the best path. Checking all options manually wastes time and causes mistakes.

The Solution

The Jump Game Problem uses a smart way to check if you can reach the end by tracking the farthest place you can jump so far. This way, you don't try every jump but only keep the best reach, making the problem simple and fast to solve.

Before vs After

✗ Before

fn can_reach_end(jumps: &[usize]) -> bool {
    // Try all jumps recursively (slow)
    fn try_jump(position: usize, jumps: &[usize]) -> bool {
        if position >= jumps.len() - 1 { return true; }
        for step in 1..=jumps[position] {
            if try_jump(position + step, jumps) { return true; }
        }
        false
    }
    try_jump(0, jumps)
}

✓ After

fn can_reach_end(jumps: &[usize]) -> bool {
    let mut max_reach = 0;
    for i in 0..jumps.len() {
        if i > max_reach { return false; }
        max_reach = max_reach.max(i + jumps[i]);
    }
    true
}

What It Enables

This concept lets you quickly decide if you can reach the end without checking every jump, saving time and effort.

Real Life Example

Think of planning a road trip where each gas station tells you how far you can drive next. You want to know if you can reach your destination without running out of fuel.

Key Takeaways

Manual checking of all jumps is slow and confusing.

Tracking the farthest reachable position simplifies the problem.

This method quickly tells if the end is reachable or not.

Practice

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

easy

A. The minimum number of jumps to reach the last index

B. The maximum jump length from the first index

C. If you can reach the last index starting from the first 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 it is possible to move from the first index to the last index by jumping according to the values in the array.

2. Step 2: Identify the goal

   The goal is to check if the last index is reachable, not the number of jumps or sums.

3. Final Answer:

   If you can reach the last index starting from the first index -> Option C

4. Quick Check:

   Jump Game = Reach last index [OK]

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

Common Mistakes:

• Confusing reachability with minimum jumps
• Thinking about total jump sums
• Assuming jumps can be backward

2. Which of the following Rust code snippets correctly declares a vector representing jump lengths for the Jump Game problem?

easy

A. let jumps = vec![2, 3, 1, 1, 4];

B. let jumps = [2, 3, 1, 1, 4];

C. let jumps = (2, 3, 1, 1, 4);

D. let jumps = {2, 3, 1, 1, 4};

Solution

1. Step 1: Identify Rust vector syntax

   In Rust, vectors are declared using vec![elements], which creates a growable array.

2. Step 2: Check each option

   let jumps = vec![2, 3, 1, 1, 4]; uses vec![], which is correct. let jumps = [2, 3, 1, 1, 4]; is an array, not a vector. let jumps = (2, 3, 1, 1, 4); is a tuple. let jumps = {2, 3, 1, 1, 4}; uses braces which is invalid syntax.

3. Final Answer:

   let jumps = vec![2, 3, 1, 1, 4]; -> Option A

4. Quick Check:

   vec! macro creates vectors [OK]

Hint: Use vec![] for vectors, not [] or () [OK]

Common Mistakes:

• Using array syntax instead of vec! macro
• Confusing tuples with vectors
• Using braces {} which is invalid

3. What is the output of the following Rust code snippet for the Jump Game problem?

fn can_jump(nums: Vec<usize>) -> bool {
    let mut max_reach = 0;
    for i in 0..nums.len() {
        if i > max_reach {
            return false;
        }
        max_reach = max_reach.max(i + nums[i]);
    }
    true
}

fn main() {
    let jumps = vec![2, 3, 1, 1, 4];
    println!("{}", can_jump(jumps));
}

medium

A. Runtime panic

B. false

C. Compilation error

D. true

Solution

1. Step 1: Trace the loop and max_reach updates

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

2. Step 2: Check if any index is unreachable

   At each i, i <= max_reach, so no false return. Loop completes and returns true.

3. Final Answer:

   true -> Option D

4. Quick Check:

   Max reach covers all indices [OK]

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

Common Mistakes:

• Not updating max_reach correctly
• Returning false too early
• Confusing max_reach with jump count

4. Identify the bug in the following Rust function for the Jump Game problem and select the fix.

fn can_jump(nums: Vec<usize>) -> bool {
    let mut max_reach = 0;
    for i in 0..nums.len() {
        if i >= max_reach {
            return false;
        }
        max_reach = max_reach.max(i + nums[i]);
    }
    true
}

medium

A. Change 'max_reach.max(i + nums[i])' to 'max_reach.min(i + nums[i])'

B. Change 'if i >= max_reach' to 'if i > max_reach'

C. Change 'for i in 0..nums.len()' to 'for i in 1..nums.len()'

D. Add 'max_reach += nums[i]' inside the loop

Solution

1. Step 1: Analyze the condition for unreachable index

   The condition 'if i >= max_reach' returns false even when i equals max_reach, which is incorrect because reaching exactly max_reach is allowed.

2. Step 2: Correct the condition

   Changing it to 'if i > max_reach' allows reaching max_reach index and only returns false if i is beyond max_reach.

3. Final Answer:

   Change 'if i >= max_reach' to 'if i > max_reach' -> Option B

4. Quick Check:

   Allow index equal to max_reach [OK]

Hint: Use strict greater than to check unreachable index [OK]

Common Mistakes:

• Using >= instead of > causes early false
• Using min instead of max for max_reach
• Skipping index 0 in loop

5. Given the array vec![3, 2, 1, 0, 4], what is the correct output of the Jump Game problem using the standard greedy approach in Rust?

hard

A. false

B. true

C. Compilation error

D. Runtime panic

Solution

1. Step 1: Trace max_reach for each index

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

2. Step 2: Understand why last index is unreachable

   Index 4 cannot be reached because max_reach never extends beyond 3 due to zero jump at index 3.

3. Final Answer:

   false -> Option A

4. Quick Check:

   Zero jump blocks progress [OK]

Hint: Zero jump can block reaching the end [OK]

Common Mistakes:

• Assuming always true if first jump > 0
• Ignoring zero jump blocking path
• Confusing max_reach with jump count