Design Skiplist - Solution & Explanation

In the first approach, we'll implement a basic Skiplist with explicit support for multiple levels. Each node holds multiple forward pointers, one for each level, and we'll decide to promote nodes to higher levels based on a random mechanism.

The Skiplist maintains multiple levels in a hierarchical manner. Each element has a 50% chance to be promoted to the next level to efficiently handle search, insertion, and deletion operations.

This solution implements a Skiplist class using a head Node initialized with the maximum level of the Skiplist. The methods randomLevel, search, add, and erase utilize the structure's properties to efficiently handle elements.

1. search: Starts from the topmost level and moves laterally until the target is found or surpassed.

2. add: Uses an update array to keep track of pointers that need redirection upon adding a new node with a randomly assigned level.

3. erase: Similar to search, it finds the target node and adjusts the pointers to remove it if present.

This approach further refines the previous implementation by introducing a configurable probability factor for determining node promotion to higher levels. This allows flexibility in tuning the space used versus time efficiency trade-offs.

This Java solution inculcates configurable probability support, influencing how frequently nodes are elevated up levels. The Skiplist class holds a constant probability P, tuned to the desired performance vs space trade-offs.

The rest of the structure and logic adhere similarly to the previous foundational approach but offers finer control over node promotions through the probabilistic function.

This approach involves creating a skiplist using multiple levels of linked lists. Each element has a chance to appear in the higher-level lists. This approach gives an average time complexity of O(log n) for search, insert, and delete operations.

We use random coin flips (or a function) to determine the level of each new node. This randomness ensures the balance of the skiplist dynamically.

Steps:

1. Design Node and Skiplist classes.
2. Implement search(), add(), and erase() functions.
3. Use randomness to decide the node's level while inserting.

This solution uses a Node class to represent each node in the skiplist and a Skiplist class to manage the operations. The random_level method randomly determines the level of each node, effectively balancing the skiplist by distributing nodes over multiple levels.

The search function looks for a given target by traversing from the topmost level down to level 0, taking advantage of the skiplist's structure to skip over large segments.

The add function not only inserts the new node at the found position but also handles node insertion at all appropriate levels according to the determined node level. The erase function removes the node if it exists, updating all pointers affected by the removal.

In this method, a deterministic approach is followed that emulates random choice but gives a semblance of control. Here, instead of true randomness, fixed levels for new nodes can be decided after every operation by ensuring a balance factor (based on a pre-defined coin probability) is maintained.

This approach ensures that skiplists retain efficient time complexity while assisting in balancing node distributions across levels.

This solution implements a Javascript skiplist utilizing a Node class with array-based forward pointers. The skiplist operations manipulate the pointers strategically, following a fixed structure for level determination maintained by rehashed 'random' probabilities.

It effectively handles search, insert, and removal operations while maintaining overall balance across levels under deterministic configurations.

The core idea of a skip list is to use multiple "levels" to store data, with each level acting as an index. Data starts from the bottom level linked list and gradually rises to higher levels, eventually forming a multi-level linked list structure. Each level's nodes only contain part of the data, allowing for jumps to reduce search time.

In this problem, we use a Node class to represent the nodes of the skip list. Each node contains a val field and a next array. The length of the array is level, indicating the next node at each level. We use a Skiplist class to implement the skip list operations.

The skip list contains a head node head and the current maximum level level. The head node's value is set to -1 to mark the starting position of the list. We use a dynamic array next to store pointers to successor nodes.

For the search operation, we start from the highest level of the skip list and traverse downwards until we find the target node or determine that the target node does not exist. At each level, we use the find_closest method to jump to the node closest to the target.

For the add operation, we first randomly decide the level of the new node. Then, starting from the highest level, we find the node closest to the new value at each level and insert the new node at the appropriate position. If the level of the inserted node is greater than the current maximum level of the skip list, we need to update the level of the skip list.

For the erase operation, similar to the search operation, we traverse each level of the skip list to find and delete the target node. When deleting a node, we need to update the next pointers at each level. If the highest level of the skip list has no nodes, we need to decrease the level of the skip list.

Additionally, we define a random_level method to randomly decide the level of the new node. This method generates a random number between [1, max_level] until the generated random number is greater than or equal to p. We also have a find_closest method to find the node closest to the target value at each level.

The time complexity of the above operations is O(log n), where n is the number of nodes in the skip list. The space complexity is O(n).