Jump Game IV

This problem can be approached by modeling the array as a graph and then performing a breadth-first search (BFS) to find the shortest path to the last index. Here’s a step-by-step explanation of how to solve this problem:

Create a Graph: Treat each index i as a node. There will be an edge from i to i+1, i-1, and to all indices j where arr[i] == arr[j] and i != j.
BFS: Perform a BFS starting from index 0 and ending at the last index of the array. Keep track of the number of steps taken to reach each node.
Visited Set: Use a set to store the visited indices to avoid redundant computations.
Result: The minimum number of steps to reach the last index will be the answer.

Here’s the code implementing the above approach:

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from collections import defaultdict, deque

class Solution:
    def minJumps(self, arr: List[int]) -> int:
        # Create a graph using a dictionary to store indices with the same value
        graph = defaultdict(list)
        for i, val in enumerate(arr):
            graph[val].append(i)

        # Initialize BFS queue with the start index and visited set
        queue = deque([(0, 0)]) # (index, steps)
        visited = set([0])

        while queue:
            idx, steps = queue.popleft()
            # If the last index is reached, return the steps
            if idx == len(arr) - 1:
                return steps

            # Jump to i + 1
            if idx + 1 < len(arr) and idx + 1 not in visited:
                visited.add(idx + 1)
                queue.append((idx + 1, steps + 1))

            # Jump to i - 1
            if idx - 1 >= 0 and idx - 1 not in visited:
                visited.add(idx - 1)
                queue.append((idx - 1, steps + 1))

            # Jump to all j where arr[i] == arr[j] and i != j
            while graph[arr[idx]]:
                j = graph[arr[idx]].pop()
                if j != idx and j not in visited:
                    visited.add(j)
                    queue.append((j, steps + 1))

        return -1 # This line should never be reached

This code has a time complexity of O(n) and a space complexity of O(n), where n is the length of the array arr. It handles the constraints well and correctly computes the minimum number of steps to reach the last index of the array.

Identifying Problem Isomorphism

“Jump Game IV” has an approximate isomorphism to “Word Ladder”.

In “Jump Game IV”, you are given an array of integers, and you can jump to any index i where arr[i] equals arr[curr] where curr is your current position. You can also jump to arr[curr + 1] or arr[curr - 1] if it is within bounds. The goal is to reach the end of the array using the minimum number of jumps.

Similarly, in “Word Ladder”, you are given a start word, an end word, and a list of words. From the current word, you can jump to any word in the list that can be formed by changing one letter of the current word. The goal is to reach the end word using the least number of jumps.

The reason behind this mapping is that both problems involve finding the shortest path to a target in a graph-like structure where the possible steps from a current position/state are defined by some conditions.

“Word Ladder” is simpler than “Jump Game IV” as it involves only one type of jump - changing one letter to reach another valid word in the list. “Jump Game IV” involves different types of jumps which adds complexity to the problem.

10 Prerequisite LeetCode Problems

“Jump Game IV” (LeetCode Problem #1345) involves understanding of Breadth-First Search (BFS), graph representation, and handling of complex edge cases. Here are 10 problems to build these foundational skills:

“Number of Islands” (LeetCode Problem #200): This problem introduces the basics of BFS on a grid, which is a key concept for “Jump Game IV”.
“Word Ladder” (LeetCode Problem #127): This problem requires understanding of BFS in the context of transforming one word into another, which can help you understand the concept of minimum steps to reach a goal.
“01 Matrix” (LeetCode Problem #542): This problem involves BFS traversal on a grid to find the minimum distance, similar to finding the minimum steps in “Jump Game IV”.
“Perfect Squares” (LeetCode Problem #279): This problem helps to understand BFS in the context of finding minimum steps to reach a target.
“Open the Lock” (LeetCode Problem #752): This problem involves BFS traversal to find the minimum number of moves, similar to “Jump Game IV”.
“Shortest Path in Binary Matrix” (LeetCode Problem #1091): This problem involves BFS on a grid, similar to “Jump Game IV”.
“Jump Game II” (LeetCode Problem #45): This problem helps to understand the concept of taking minimum steps to achieve a goal, which is similar to “Jump Game IV”.
“Sliding Puzzle” (LeetCode Problem #773): This problem requires BFS to find the minimum number of moves to solve a puzzle, similar to “Jump Game IV”.
“Rotting Oranges” (LeetCode Problem #994): This problem involves BFS traversal on a grid to find the minimum time, which is related to the concept of minimum steps in “Jump Game IV”.
“Minimum Knight Moves” (LeetCode Problem #1197): This problem requires BFS to find the minimum number of moves, similar to “Jump Game IV”.

Problem Classification

Problem Statement: Given an array of integers arr, you are initially positioned at the first index of the array.

In one step you can jump from index i to index:

i + 1 where: i + 1 < arr.length. i - 1 where: i - 1 >= 0. j where: arr[i] == arr[j] and i != j. Return the minimum number of steps to reach the last index of the array.

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

Example 1:

Input: arr = [100,-23,-23,404,100,23,23,23,3,404] Output: 3 Explanation: You need three jumps from index 0 –> 4 –> 3 –> 9. Note that index 9 is the last index of the array.

Example 2:

Input: arr = [7] Output: 0 Explanation: Start index is the last index. You do not need to jump.

Example 3:

Input: arr = [7,6,9,6,9,6,9,7] Output: 1 Explanation: You can jump directly from index 0 to index 7 which is last index of the array.

Constraints:

1 <= arr.length <= 5 * 104 -108 <= arr[i] <= 108

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Alternatively, if you’re working on a specific problem, you might ask something like:

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.