Jump Game Problem in DSA Typescript - Time & Space Complexity

Time Complexity: Jump Game Problem

O(n)

Understanding Time Complexity

We want to understand how the time needed to solve the Jump Game problem changes as the input grows.

Specifically, how the number of steps to check jumps grows with the size of the input array.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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;
}
    

This code checks if you can jump from the start to the end of the array using the maximum reachable index at each step.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: A single loop that goes through each element of the array once.
• How many times: The loop runs exactly once for each element, so n times if the array length is n.

How Execution Grows With Input

As the array gets longer, the number of steps grows directly with its length.

Input Size (n)	Approx. Operations
10	About 10 checks
100	About 100 checks
1000	About 1000 checks

Pattern observation: The work grows in a straight line with input size; doubling the input doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to decide if you can jump to the end grows directly with the size of the input array.

Common Mistake

[X] Wrong: "Because we check jumps from each position, the time is much bigger than n, maybe n squared."

[OK] Correct: The code only moves forward once and updates the farthest reachable index without nested loops, so it never repeats work for the same positions.

Interview Connect

Understanding this linear time solution shows you can find efficient ways to solve problems by tracking progress smartly, a skill valued in many coding challenges.

Self-Check

"What if we changed the code to try all possible jumps from each position instead of tracking the max reach? How would the time complexity change?"

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