leetcode 55 jump game

Okay, let's break down LeetCode problem 55, "Jump Game," with a comprehensive tutorial. We'll cover the problem statement, strategies for solving it, code implementation in Python, and analysis of its complexity.

**Problem Statement:**

You are given an integer array `nums`. You are initially positioned at the **first index** of the array.

Each element `nums[i]` represents your maximum jump length from that position.

Determine if you are able to reach the last index.

**Example 1:**

**Example 2:**

**Constraints:**

* `0 = nums[i] = 10^5`

**Understanding the Problem:**

The core idea is to simulate jumps through the array. At each position `i`, you can jump a distance of up to `nums[i]` positions forward. The question is whether you can reach the last index of the array starting from the first.

**Strategies for Solving:**

Several approaches can be used to solve the Jump Game problem:

1. **Backtracking (Brute Force):** Recursively try all possible jump combinations from each position. This will lead to exponential time complexity and is not efficient for larger arrays. While good for understanding the problem, it's not practical for LeetCode due to time limit exceed.

2. **Dynamic Programming (Top-Down with Memoization):** Improve upon backtracking by storing the results of subproblems to avoid recomputation. This still involves recursion but is more efficient.

3. **Dynamic Programming (Bottom-Up):** Iterate through the array from the end to the beginning. At each index, determine whether it is "good" (can reach the end) or "bad".

4. **Greedy Approach (Most Efficient):** Iterate through the array from left to right. Keep track of the furthest reachable index. If the current index is beyond the furthest reachable index, it means you cannot reach that index. This is generally the most efficient approach.

**Greedy Approach in Detail:**

The greedy approach is the most efficient and elegant solution. Here ...

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