Jump Game Problem in DSA Typescript

Mental Model

You want to jump through an array from start to end using steps allowed by each position's number. The key is to check if you can reach the last position by jumping forward.

Analogy: Imagine stepping stones across a river. Each stone tells you how far you can jump next. You want to see if you can reach the last stone without falling in the water.

Start -> [3] -> 2 -> 1 -> 0 -> 4 -> End
Each number shows max jump length from that stone.

Dry Run Walkthrough

Input: array: [3, 2, 1, 0, 4]

Goal: Determine if you can jump from the first index to the last index following jump rules.

Step 1: Start at index 0 with jump length 3, max reachable index is 0 + 3 = 3

maxReach = 3, currentIndex = 0

Why: We can jump up to index 3 from start, so far so good.

Step 2: Move to index 1 with jump length 2, update maxReach to max(3, 1+2) = 3

maxReach = 3, currentIndex = 1

Why: From index 1, max jump is 3, no improvement.

Step 3: Move to index 2 with jump length 1, update maxReach to max(3, 2+1) = 3

maxReach = 3, currentIndex = 2

Why: From index 2, max jump is still 3, no improvement.

Step 4: Move to index 3 with jump length 0, maxReach stays 3

maxReach = 3, currentIndex = 3

Why: At index 3, jump length is 0, can't move forward from here.

Step 5: Try to move to index 4 but maxReach is 3, so index 4 is unreachable

maxReach = 3, currentIndex = 4 (unreachable)

Why: We cannot reach the last index because maxReach is less than 4.

Result:

Cannot reach last index: maxReach = 3 < lastIndex = 4

Annotated Code

DSA Typescript

class JumpGame {
  canJump(nums: number[]): boolean {
    let maxReach = 0;
    for (let i = 0; i < nums.length; i++) {
      if (i > maxReach) {
        return false; // can't reach this position
      }
      maxReach = Math.max(maxReach, i + nums[i]);
    }
    return true;
  }
}

const game = new JumpGame();
const input = [3, 2, 1, 0, 4];
const result = game.canJump(input);
console.log(result ? "Can reach last index" : "Cannot reach last index");

if (i > maxReach) {

Check if current index is beyond max reachable index, if yes, return false

maxReach = Math.max(maxReach, i + nums[i]);

Update max reachable index based on current position and jump length

Output

Cannot reach last index

Complexity Analysis

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

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

vs Alternative: Compared to recursive or backtracking approaches that try all paths (exponential), this greedy method is much faster and simpler.

Edge Cases

Empty array

Returns true because we are already at the last index

DSA Typescript

for (let i = 0; i < nums.length; i++) {

Single element array [0]

Returns true because start is the last index

DSA Typescript

for (let i = 0; i < nums.length; i++) {

Array with zero at start [0, 1]

Returns false because can't move anywhere

DSA Typescript

if (i > maxReach) {

When to Use This Pattern

When you see a problem asking if you can reach the end of an array by jumping steps defined at each position, use the greedy maxReach pattern to track how far you can go.

Common Mistakes

Mistake: Not checking if current index is beyond max reachable index before updating maxReach
Fix: Add a condition to return false immediately if current index is unreachable

Mistake: Trying to simulate all jump paths instead of tracking max reachable index
Fix: Use a single pass greedy approach updating maxReach to avoid exponential complexity

Summary

Determines if you can jump from start to end of an array using allowed steps at each position.

Use when you need to check reachability in jump or step-based array problems.

Keep track of the farthest index you can reach and fail early if stuck.

Practice

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

easy

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

B. The maximum jump length you can make from any position in the array.

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 key 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 by jumping according to the values. -> Option A

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 needed
• Thinking the problem asks for maximum jump length
• Assuming sum of jumps is the answer

2. Which of the following TypeScript function signatures correctly represents a function solving the Jump Game problem?

easy

A. function canJump(nums: number[]): boolean

B. function canJump(nums: number): number[]

C. function canJump(nums: string[]): boolean

D. function canJump(nums: number[][]): boolean

Solution

1. Step 1: Analyze the input and output types

   The input is an array of numbers representing jump lengths, so type is number[]. The output is a boolean indicating if last index is reachable.

2. Step 2: Match the function signature

   function canJump(nums: number[]): boolean matches input type number[] and output boolean, which fits the problem.

3. Final Answer:

   function canJump(nums: number[]): boolean -> Option A

4. Quick Check:

   Input array, output boolean = function canJump(nums: number[]): boolean [OK]

Hint: Input array of numbers, output boolean for reachability [OK]

Common Mistakes:

• Using wrong input types like string or 2D arrays
• Returning array instead of boolean
• Incorrect function parameter types

3. What is the output of the following TypeScript code?

function canJump(nums: number[]): boolean {
  let maxReach = 0;
  for (let i = 0; i < nums.length; i++) {
    if (i > maxReach) return false;
    maxReach = Math.max(maxReach, i + nums[i]);
  }
  return true;
}

console.log(canJump([2,3,1,1,4]));

medium

A. Runtime error

B. false

C. undefined

D. true

Solution

1. Step 1: Trace the loop and maxReach updates

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

2. Step 2: Check if loop completes without returning false

   Since i never exceeds maxReach, loop completes and returns true.

3. Final Answer:

   true -> Option D

4. Quick Check:

   All indices reachable, so true [OK]

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

Common Mistakes:

• Not updating maxReach correctly
• Returning false too early
• Confusing output with minimum jumps

4. Identify the bug in this TypeScript code for the Jump Game problem:

function canJump(nums: number[]): boolean {
  let maxReach = 0;
  for (let i = 0; i <= nums.length; i++) {
    if (i > maxReach) return false;
    maxReach = Math.max(maxReach, i + nums[i]);
  }
  return true;
}

medium

A. The return type should be number, not boolean.

B. The loop condition should be i < nums.length, not i <= nums.length.

C. maxReach should be initialized to 1 instead of 0.

D. The function should return false at the end instead of true.

Solution

1. Step 1: Check loop boundary conditions

   The loop runs from i=0 to i <= nums.length, which means it accesses nums[nums.length] causing undefined access.

2. Step 2: Correct the loop condition

   Changing to i < nums.length prevents out-of-bounds access and fixes the bug.

3. Final Answer:

   The loop condition should be i < nums.length, not i <= nums.length. -> Option B

4. Quick Check:

   Loop index must stay within array bounds [OK]

Hint: Loop index must be less than array length [OK]

Common Mistakes:

• Using <= instead of < in loop condition
• Ignoring out-of-bounds array access
• Incorrect initialization of maxReach

5. Given the array [3,2,1,0,4], why does the standard greedy Jump Game algorithm return false?

hard

A. Because the array length is less than 5, so it always returns false.

B. Because the maximum jump length at index 0 is too large.

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

D. Because the algorithm only checks the first two elements.

Solution

1. Step 1: Analyze jump reachability step-by-step

   Start maxReach=0. i=0: maxReach=max(0,0+3)=3; i=1: 1<=3, maxReach=max(3,1+2)=3; i=2: 2<=3, maxReach=max(3,2+1)=3; i=3: 3<=3, maxReach=max(3,3+0)=3; i=4: 4>3, return false.

2. Step 2: Understand why last index is unreachable

   The zero at index 3 prevents extending maxReach 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 C

4. Quick Check:

   Zero jump blocks progress = false [OK]

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

Common Mistakes:

• Ignoring zeros that block jumps
• Assuming array length affects result
• Misunderstanding maxReach updates