**π‘ Problem Formulation:** The topic explores how to determine if a number `k`

is sufficient to reach the last index of a given array. This problem assumes an array where each element indicates the maximum you can jump forward from that position, and `k`

is the starting jump power. The goal is to confirm if using `k`

is adequate to jump from the beginning to the end of the array. For instance, given the array [2,3,1,1,4] and `k=2`

, the expected output is “True” since the jumps can cover the array length.

## Method 1: Iterative Step-through

This method involves sequentially moving through the array, decrementing `k`

at each step and increasing it by the value of the current element if it’s larger. It checks if we can reach or exceed the array length before `k`

becomes zero. This is robust for all input sizes and patterns.

Here’s an example:

def can_reach_end(arr, k): for i in range(len(arr)): if k <= 0: return False k = max(k - 1, arr[i]) return True # Example usage: print(can_reach_end([2, 3, 1, 1, 4], 2))

Output: True

This code defines a function `can_reach_end`

which iterates through the array, checking if each step can be made with the current `k`

. It decrements `k`

unless the array provides a bigger boost. The function returns True if we can get through the array before `k`

runs outβillustrating a straightforward way of solving the problem iteratively.

## Method 2: Backtracking

The backtracking method tries to reach the end of the array from the beginning by exploring all potential paths. It tests if, at each index, the jump can be made with the remaining `k`

. While it is comprehensive, it’s more computational heavy, especially on large inputs.

Here’s an example:

def can_jump(index, arr, k): if index >= len(arr) - 1: return True return any(can_jump(index + step, arr, k - step) for step in range(1, min(k, arr[index]) + 1)) # Example usage: print(can_jump(0, [2, 3, 1, 1, 4], 2))

Output: True

The `can_jump`

function recursively explores all jump possibilities at each array index from 1 to the minimum of `k`

or the current array value. It returns True if any of these paths succeed to reach the end. However, due to its recursive nature, it can be quite slow on large arrays and suffer from stack overflow for deep recursion.

## Method 3: Dynamic Programming

Dynamic Programming can be utilized by creating a memoization mechanism that records the states we’ve previously calculated. This technique avoids redundant calculations seen in plain recursion, thereby optimizing performance.

Here’s an example:

def can_reach(arr, k, index=0, memo=None): if memo is None: memo = {} if index in memo: return memo[index] if index >= len(arr) - 1: return True for step in range(1, min(k, arr[index]) + 1): if can_reach(arr, k - step, index + step, memo): memo[index] = True return True memo[index] = False return False # Example usage: print(can_reach([2, 3, 1, 1, 4], 2))

Output: True

The `can_reach`

function utilizes memoization to cache already computed results hence skipping repetitive states. The dictionary `memo`

holds these states. This method improves efficiency when compared to plain recursion and is more advisable for larger inputs.

## Method 4: Greedy Jumping

Greedy jumping algorithm attempts to make the farthest jump at every step, aiming to reach the end in as few jumps as possible. This method can be highly efficient and provide an optimal solution without exhaustively checking every possibility.

Here’s an example:

def can_reach_greedily(arr, k): i, reach = 0, 0 while i <= reach and reach = len(arr) - 1 # Example usage: print(can_reach_greedily([2, 3, 1, 1, 4], 2))

Output: True

The function `can_reach_greedily`

uses a greedy algorithm, jumping to the farthest reachable index at each step, based on the current index `i`

and the available jump power `k`

. It stops when the reach becomes equal or greater than the last index or when `i`

exceeds the current reach. This method is optimal for most cases and less complex.

## Bonus One-Liner Method 5: Using Reduce

Python’s `functools.reduce`

function can be leveraged to apply a cumulative operation that combines elements of the array, leading to a one-liner solution that’s both compact and elegant.

Here’s an example:

from functools import reduce # Example usage: print(reduce(lambda acc, n: max(acc - 1, n), [2, 3, 1, 1, 4], 2) >= 0)

Output: True

This code snippet uses `reduce`

with a lambda function that traverses the array, updating the accumulator `acc`

(initially set to `k`

) with the greater value between `acc-1`

and the current number. It efficiently checks if the accumulator never dips below zero, indicating that the end can be reached.

## Summary/Discussion

**Method 1: Iterative Step-through.**Strong in simplicity and reliability. May not be the most efficient for sparse arrays with lots of zeroes.**Method 2: Backtracking.**Exhaustive and guarantees a solution if one exists. Computationally intensive and not suitable for large inputs.**Method 3: Dynamic Programming.**Offers optimization over backtracking with memoization. Can handle larger inputs but incurs space complexity.**Method 4: Greedy Jumping.**Fast and efficient. Works best when the goal is to minimize the number of jumps or for arrays with consistent jump values.**Bonus Method 5: Using Reduce.**Provides a concise and elegant one-liner solution. May be less intuitive to understand and debug for those unfamiliar with`reduce`

.