LeetCode #1306 — MEDIUM

Jump Game III

Move from brute-force thinking to an efficient approach using array strategy.

The Problem

Problem Statement

Given an array of non-negative integers arr, you are initially positioned at start index of the array. When you are at index i, you can jump to i + arr[i] or i - arr[i], check if you can reach any index with value 0.

Notice that you can not jump outside of the array at any time.

Example 1:

Input: arr = [4,2,3,0,3,1,2], start = 5
Output: true
Explanation: 
All possible ways to reach at index 3 with value 0 are: 
index 5 -> index 4 -> index 1 -> index 3 
index 5 -> index 6 -> index 4 -> index 1 -> index 3

Example 2:

Input: arr = [4,2,3,0,3,1,2], start = 0
Output: true 
Explanation: 
One possible way to reach at index 3 with value 0 is: 
index 0 -> index 4 -> index 1 -> index 3

Example 3:

Input: arr = [3,0,2,1,2], start = 2
Output: false
Explanation: There is no way to reach at index 1 with value 0.

Constraints:

1 <= arr.length <= 5 * 10⁴
0 <= arr[i] < arr.length
0 <= start < arr.length

Step 01

Brute Force Baseline

Problem summary: Given an array of non-negative integers arr, you are initially positioned at start index of the array. When you are at index i, you can jump to i + arr[i] or i - arr[i], check if you can reach any index with value 0. Notice that you can not jump outside of the array at any time.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array

Example 1

[4,2,3,0,3,1,2]
5

Example 2

[4,2,3,0,3,1,2]
0

Example 3

[3,0,2,1,2]
2

Core Insight

What unlocks the optimal approach

Think of BFS to solve the problem.
When you reach a position with a value = 0 then return true.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #1306: Jump Game III
class Solution {
    public boolean canReach(int[] arr, int start) {
        Deque<Integer> q = new ArrayDeque<>();
        q.offer(start);
        while (!q.isEmpty()) {
            int i = q.poll();
            if (arr[i] == 0) {
                return true;
            }
            int x = arr[i];
            arr[i] = -1;
            for (int j : List.of(i + x, i - x)) {
                if (j >= 0 && j < arr.length && arr[j] >= 0) {
                    q.offer(j);
                }
            }
        }
        return false;
    }
}

// Accepted solution for LeetCode #1306: Jump Game III
func canReach(arr []int, start int) bool {
	q := []int{start}
	for len(q) > 0 {
		i := q[0]
		q = q[1:]
		if arr[i] == 0 {
			return true
		}
		x := arr[i]
		arr[i] = -1
		for _, j := range []int{i + x, i - x} {
			if j >= 0 && j < len(arr) && arr[j] >= 0 {
				q = append(q, j)
			}
		}
	}
	return false
}

# Accepted solution for LeetCode #1306: Jump Game III
class Solution:
    def canReach(self, arr: List[int], start: int) -> bool:
        q = deque([start])
        while q:
            i = q.popleft()
            if arr[i] == 0:
                return True
            x = arr[i]
            arr[i] = -1
            for j in (i + x, i - x):
                if 0 <= j < len(arr) and arr[j] >= 0:
                    q.append(j)
        return False

// Accepted solution for LeetCode #1306: Jump Game III
struct Solution;
use std::collections::VecDeque;

impl Solution {
    fn can_reach(arr: Vec<i32>, start: i32) -> bool {
        let n = arr.len();
        let mut visited = vec![false; n];
        let mut queue = VecDeque::new();
        queue.push_back(start);
        visited[start as usize] = true;
        while let Some(i) = queue.pop_front() {
            if arr[i as usize] == 0 {
                return true;
            } else {
                let l = i - arr[i as usize];
                let r = i + arr[i as usize];
                if l >= 0 && !visited[l as usize] {
                    visited[l as usize] = true;
                    queue.push_back(l);
                }
                if r < n as i32 && !visited[r as usize] {
                    visited[r as usize] = true;
                    queue.push_back(r);
                }
            }
        }
        false
    }
}

#[test]
fn test() {
    let arr = vec![4, 2, 3, 0, 3, 1, 2];
    let start = 5;
    let res = true;
    assert_eq!(Solution::can_reach(arr, start), res);
    let arr = vec![4, 2, 3, 0, 3, 1, 2];
    let start = 0;
    let res = true;
    assert_eq!(Solution::can_reach(arr, start), res);
    let arr = vec![3, 0, 2, 1, 2];
    let start = 2;
    let res = false;
    assert_eq!(Solution::can_reach(arr, start), res);
}

// Accepted solution for LeetCode #1306: Jump Game III
function canReach(arr: number[], start: number): boolean {
    const q = [start];
    for (const i of q) {
        if (arr[i] === 0) {
            return true;
        }
        if (arr[i] === -1 || arr[i] === undefined) {
            continue;
        }
        q.push(i + arr[i], i - arr[i]);
        arr[i] = -1;
    }
    return false;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n)

Space

O(n)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED

O(n) time

O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.