Jump Game

class Solution {
public:
    bool canReachEnd(vector<int>& jumps) {
        int goal = jumps.size() - 1;
        for (int idx = jumps.size() - 1; idx >= 0; idx--) {
            if (idx + jumps[idx] >= goal) {
                goal = idx;
            }
        }
        return goal == 0;
    }
};

The given C++ program defines a solution to determine if one can reach the end of an array where each element specifies the maximum jump length from that position. Analyze the provided code to comprehend the strategy employed:

• Initialize goal with the last index of the jumps array, which represents the target to reach.
• Iterate backwards from the end of the array (jumps.size() - 1) to the beginning.
• During each iteration, check if the sum of the current index (idx) and the jump length at that index (jumps[idx]) reaches or exceeds the current goal.
• If true, update the goal to the current index as this becomes the new target to achieve.
• Continue the process until you reach the start of the array.
• Return true if you can reduce the goal to 0 (start of the array), meaning it's possible to jump to the end. Otherwise, return false.

This approach utilizes a greedy algorithm that efficiently navigates the array in a backward manner, minimizing the need to explore all possible jump paths. Adjustments or optimizations might be necessary for specific variations of the problem or different inputs.

public class Solution {
    public boolean jumpCheck(int[] steps) {
        int reach = steps.length - 1;
        for (int position = steps.length - 1; position >= 0; position--) {
            if (position + steps[position] >= reach) {
                reach = position;
            }
        }
        return reach == 0;
    }
}

This Java-based implementation checks if it's possible to jump to the end of an array by starting at the first element and using each element as the maximum jump length from that position. Here's a straightforward guide on how it operates:

• Declare a reach variable initialized to the last index of the steps array.
• Execute a loop that iterates backward from the end of the array to the start.
• In each iteration, check if the sum of the current position and its jump length can reach or surpass the value held by reach. Update reach to the current position if the condition is met.
• Return true if reach becomes zero, indicating that the start of the array can successfully reach the end. If not, false is returned at the end of the function.

The approach utilizes a greedy algorithm that efficiently determines the feasibility of reaching the end of the array from the start, taking into account the varying jump lengths at each index.

bool isReachable(int* arr, int len) {
    int finalPos = len - 1;
    for (int idx = len - 1; idx >= 0; idx--) {
        if (idx + arr[idx] >= finalPos) {
            finalPos = idx;
        }
    }
    return finalPos == 0;
}

The "Jump Game" solution provided is implemented in C and focuses on determining if it's possible to reach the last index of an array starting from the first index. The elements in the array represent the maximum jump length at that position.

• The function isReachable takes an integer array arr and its length len as parameters.
• It initializes finalPos to the last index of the array.
• A reverse loop starts from the last element to the first. This loop checks if the current position combined with the allowed jump at that position can at least reach or exceed the position currently considered as the final reachable position (finalPos).
• If so, the finalPos gets updated to the current index, indicating a new possible earlier start for the last jump.
• The loop continues until all elements are checked or the start of the array is confirmed reachable (finalPos == 0).
• The function returns true if the start of the array (index 0) is reachable, otherwise it returns false.

This approach is optimal as it reduces the problem to a few linear operations, utilizing reverse iteration and greedy updating of reachable positions, making it computationally efficient.

var canReachEnd = function (positions) {
    let finalPosition = positions.length - 1;
    for (let index = positions.length - 1; index >= 0; index--) {
        if (index + positions[index] >= finalPosition) {
            finalPosition = index;
        }
    }
    return finalPosition === 0;
};

The Jump Game problem involves determining if it's possible to reach the end of an array from the beginning, given an array of non-negative integers where each element represents the maximum number of steps that can be taken from that position. The provided code snippet in JavaScript implements a solution to this problem using an efficient approach.

Here is a concise breakdown of the function canReachEnd:

• Initialize finalPosition to the last index of the positions array.
• Iterate backwards through the array starting from the last element.
• For each element, check if the sum of the current index and the value at that index (maximum jump from the position) is greater than or equal to finalPosition. If true, update finalPosition to be the current index because this shows that from this point, the end can be potentially reached.
• At the end of the loop, if finalPosition is zero, return true indicating the start of the array connects to the end via valid jumps. If not, return false.

This reverse iteration allows efficient checking by consolidating potential endpoints progressively backward until the start of the array is reached or detected as unreachable. This greedy algorithm has a time complexity of ( O(n) ), where ( n ) is the number of elements in the array, because it goes through the elements a single time.

class Solution:
    def jump_possible(self, jumps: List[int]) -> bool:
        target = len(jumps) - 1
        for pos in range(len(jumps) - 1, -1, -1):
            if pos + jumps[pos] >= target:
                target = pos
        return target == 0

This Python solution defines a method jump_possible to determine if it's feasible to jump to the last index of an array starting from the first index, based on the integers provided in the array jumps. Each integer represents the maximum number of indices you can jump forward from that position.

• The method initializes target to the last index of the array.
• Iterates backward through the array:
  • For each position, it checks if, by utilizing the max jump from that position, the target can either be reached or surpassed.
  • If possible, the target is updated to the current position, meaning, from this position, you can eventually reach or pass the original last index.
• At the end, if the iteration has managed to set the target to 0, return True, indicating successful traversal; otherwise, False.

This method effectively determines jump feasibility using a greedy approach, updating a backtrack target to see if the start of the list can connect to the end through the allowed jumps.