Maximum Number of Jumps to Reach the Last Index

#1573 Most AskedMedium

Maximum Number of Jumps to Reach the Last Index #1573 Most Asked

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ArraysDynamic Programming

You are given a 0-indexed array nums of n integers and an integer target.

You are initially positioned at index 0. In one step, you can jump from index i to any index j such that:

0 <= i < j < n
-target <= nums[j] - nums[i] <= target

Return the maximum number of jumps you can make to reach index n - 1.

If there is no way to reach index n - 1, return -1.

Example 1:

Input: nums = [1,3,6,4,1,2], target = 2
Output: 3
Explanation: To go from index 0 to index n - 1 with the maximum number of jumps, you can perform the following jumping sequence:
- Jump from index 0 to index 1. 
- Jump from index 1 to index 3.
- Jump from index 3 to index 5.
It can be proven that there is no other jumping sequence that goes from 0 to n - 1 with more than 3 jumps. Hence, the answer is 3.

Example 2:

Input: nums = [1,3,6,4,1,2], target = 3
Output: 5
Explanation: To go from index 0 to index n - 1 with the maximum number of jumps, you can perform the following jumping sequence:
- Jump from index 0 to index 1.
- Jump from index 1 to index 2.
- Jump from index 2 to index 3.
- Jump from index 3 to index 4.
- Jump from index 4 to index 5.
It can be proven that there is no other jumping sequence that goes from 0 to n - 1 with more than 5 jumps. Hence, the answer is 5.

Example 3:

Input: nums = [1,3,6,4,1,2], target = 0
Output: -1
Explanation: It can be proven that there is no jumping sequence that goes from 0 to n - 1. Hence, the answer is -1.

Constraints:

2 <= nums.length == n <= 1000
-10⁹ <= nums[i] <= 10⁹
0 <= target <= 2 * 10⁹

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

Can the array contain negative jump values or zero? What is the expected behavior in these cases?
What is the expected return value if it's impossible to reach the last index?
What is the maximum size of the input array? Is memory usage a significant concern?
If there are multiple paths with the same minimum number of jumps, is any one of them acceptable?
Is the input array guaranteed to be non-empty?

Brute Force Solution

Approach

Imagine you're playing a game where you need to hop to the end, and each spot tells you how far you can jump. The brute force method is like trying every single possible jump combination to see if you can reach the end. If you can, you note how many jumps it took and keep going until you've tested them all.

Here's how the algorithm would work step-by-step:

Start at the beginning.
Try jumping just one spot forward, then two, then three, all the way up to the maximum distance the starting spot allows.
For each of those jumps, repeat the process from the new spot: try jumping one, two, or more spots from there.
Keep doing this for every single jump until you either reach the end or can't jump any further.
If you reach the end, remember how many jumps it took to get there.
Go back and try a different initial jump from the starting spot.
Continue trying every single possible path from the beginning until you've explored every single option.
Once you've tried every path, find the path that reached the end with the fewest jumps. If no path reached the end, then there's no solution.

Code Implementation

def max_jumps_brute_force(jump_lengths):
    minimum_jumps = float('inf')

    def solve(current_position, number_of_jumps):
        nonlocal minimum_jumps

        # If we reached the end, update min jumps
        if current_position >= len(jump_lengths) - 1:
            minimum_jumps = min(minimum_jumps, number_of_jumps)
            return

        maximum_jump = jump_lengths[current_position]

        # Iterate through all possible jump lengths
        for jump_length in range(1, maximum_jump + 1):

            next_position = current_position + jump_length

            # Recursively call solve for each possible jump
            solve(next_position, number_of_jumps + 1)

    solve(0, 0)

    # Return -1 if no solution found
    if minimum_jumps == float('inf'):
        return -1
    else:
        return minimum_jumps

Big(O) Analysis

Time Complexity

O(n^n) – The described brute force approach explores all possible jump combinations. In the worst case, from each of the n positions, we might have up to n possible jumps (although limited by the array value at that position). This leads to a recursive branching process where each node can have up to n children, resulting in a potential exploration of n^n possible paths. Therefore, the time complexity is O(n^n) reflecting the exponential growth of the search space.

Space Complexity

O(N) – The brute force approach, as described, explores all possible jump combinations using recursion. In the worst-case scenario, the recursion depth can reach N, where N is the length of the input array, representing each position visited. Each recursive call adds a new frame to the call stack. Thus, the auxiliary space used by the recursion stack can grow linearly with the input size, leading to O(N) space complexity.

Optimal Solution

Approach

The goal is to find the fewest jumps to the end. The efficient approach focuses on maximizing the reach of each jump, rather than exploring all jump combinations. It's like always trying to jump to the farthest reachable spot, hoping it gets us closer to the end in fewer steps.

Here's how the algorithm would work step-by-step:

Start at the beginning position.
Look at the current position and determine the farthest point reachable from here.
Among all the places reachable from the current position, choose the one that allows you to go the farthest in the next jump. It's like picking the trampoline that sends you the highest.
Make the jump to that best spot.
Repeat this process of finding the farthest reachable spot and jumping there until you reach the end.
If, at any point, you cannot move forward (you can't jump to any new locations), then reaching the end is impossible.

Code Implementation

def maximum_jumps(numbers):
    array_length = len(numbers)
    current_position = 0
    number_of_jumps = 0

    while current_position < array_length - 1:
        maximum_reachable_index = current_position + numbers[current_position]

        # If we can reach the end from here, jump and return
        if maximum_reachable_index >= array_length - 1:
            number_of_jumps += 1
            return number_of_jumps

        next_jump_index = -1
        maximum_next_reach = -1

        # Need to find the best next jump within our reach
        for i in range(current_position + 1, maximum_reachable_index + 1):

            # Prioritize jumps that extend furthest
            if i + numbers[i] > maximum_next_reach:
                maximum_next_reach = i + numbers[i]
                next_jump_index = i

        # If we can't find a valid next jump, it's impossible
        if next_jump_index == -1:
            return -1

        current_position = next_jump_index
        number_of_jumps += 1

    return number_of_jumps

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the array once, using a single loop of size n, where n is the number of elements in the input array. Inside the loop, we are finding the farthest reachable position. Though this looks like it might be another loop, we are simply keeping track of the best position so far. Thus, the dominant operation is the single pass through the array, making the time complexity linear.

Space Complexity

O(1) – The provided plain English explanation describes an iterative process that primarily involves updating the current position and determining the farthest reachable position. There's no explicit mention of creating any auxiliary data structures like lists, hash maps, or sets to store intermediate results, visited locations, or gathered items. Only a few variables are used to track the current position and the farthest reachable index. Therefore, the algorithm's space usage remains constant irrespective of the input array's size (N), leading to O(1) space complexity.

Edge Cases

Case	How to Handle
Null or empty input array	Return 0 if the input array is null or empty as there are no jumps to make.
Array with only one element	Return 0 if the array has only one element because we are already at the end.
First element is zero when array length > 1	Return 0 immediately since you cannot make any jumps if starting position is blocked.
Cannot reach the end of the array	Return 0 if, at any point, the reachable range becomes smaller or equal to the current index indicating an impossible solution.
Large array with small jump values (lots of small steps needed)	The algorithm should efficiently traverse the array, making sure the jump count is incremented correctly without timing out.
Array with a single large jump that reaches the end	The algorithm should correctly identify this as a single jump to reach the last index.
Integer overflow for jump count in extremely large arrays	Use a data type (like long) that can accommodate a potentially large number of jumps.
Maximum sized input array that causes a memory error	If memory is a concern, consider iterative approaches that minimize memory usage instead of recursive solutions.

Null or empty input array

How to Handle:

Return 0 if the input array is null or empty as there are no jumps to make.

Array with only one element

How to Handle:

Return 0 if the array has only one element because we are already at the end.

First element is zero when array length > 1

How to Handle:

Return 0 immediately since you cannot make any jumps if starting position is blocked.

Cannot reach the end of the array

How to Handle:

Return 0 if, at any point, the reachable range becomes smaller or equal to the current index indicating an impossible solution.

Large array with small jump values (lots of small steps needed)

How to Handle:

The algorithm should efficiently traverse the array, making sure the jump count is incremented correctly without timing out.

Array with a single large jump that reaches the end

How to Handle:

The algorithm should correctly identify this as a single jump to reach the last index.

Integer overflow for jump count in extremely large arrays

How to Handle:

Use a data type (like long) that can accommodate a potentially large number of jumps.

Maximum sized input array that causes a memory error

How to Handle:

If memory is a concern, consider iterative approaches that minimize memory usage instead of recursive solutions.