Skip List - Efficient Search, Insert and Delete in Linked List
Last Updated :
23 Jul, 2025
A skip list is a data structure that allows for efficient search, insertion and deletion of elements in a sorted list. It is a probabilistic data structure, meaning that its average time complexity is determined through a probabilistic analysis.
In a skip list, elements are organized in layers, with each layer having a smaller number of elements than the one below it. The bottom layer is a regular linked list, while the layers above it contain "skipping" links that allow for fast navigation to elements that are far apart in the bottom layer. The idea behind this is to allow for quick traversal to the desired element, reducing the average number of steps needed to reach it.
Skip lists are implemented using a technique called "coin flipping." In this technique, a random number is generated for each insertion to determine the number of layers the new element will occupy. This means that, on average, each element will be in log(n) layers, where n is the number of elements in the bottom layer.
Skip lists have an average time complexity of O(log n) for search, insertion and deletion, which is similar to that of balanced trees, such as AVL trees and red-black trees, but with the advantage of simpler implementation and lower overhead.
In summary, skip lists provide a simple and efficient alternative to balanced trees for certain use cases, particularly when the average number of elements in the list is large.
Can we search in a sorted linked list better than O(n) time? The worst-case search time for a sorted linked list is O(n) as we can only linearly traverse the list and cannot skip nodes while searching. For a Balanced Binary Search Tree, we skip almost half of the nodes after one comparison with the root. For a sorted array, we have random access and we can apply Binary Search on arrays. Can we augment sorted linked lists to search faster? The answer is Skip List. The idea is simple, we create multiple layers so that we can skip some nodes. See the following example list with 16 nodes and two layers. The upper layer works as an "express lane" that connects only the main outer stations, and the lower layer works as a "normal lane" that connects every station. Suppose we want to search for 50, we start from the first node of the "express lane" and keep moving on the "express lane" till we find a node whose next is greater than 50. Once we find such a node (30 is the node in the following example) on "express lane", we move to "normal lane" using a pointer from this node, and linearly search for 50 on "normal lane". In the following example, we start from 30 on the "normal lane" and with linear search, we find 50. 
What is the time complexity with two layers? worst-case time complexity is several nodes on the "express lane" plus several nodes in a segment (A segment is several "normal lane" nodes between two "express lane" nodes) of the "normal lane". So if we have n nodes on "normal lane", ?n (square root of n) nodes on "express lane" and we equally divide the "normal lane", then there will be ?n nodes in every segment of "normal lane". ?n is an optimal division with two layers. With this arrangement, the number of nodes traversed for a search will be O(?n). Therefore, with O(?n) extra space, we can reduce the time complexity to O(?n).
Advantages of Skip List:
- The skip list is solid and trustworthy.
- To add a new node to it, it will be inserted extremely quickly.
- Easy to implement compared to the hash table and binary search tree
- The number of nodes in the skip list increases, and the possibility of the worst-case decreases
- Requires only ?(logn) time in the average case for all operations.
- Finding a node in the list is relatively straightforward.
Disadvantages of Skip List:
- It needs a greater amount of memory than the balanced tree.
- Reverse search is not permitted.
- Searching is slower than a linked list
- Skip lists are not cache-friendly because they don't optimize the locality of reference
Can we do better? The time complexity of skip lists can be reduced further by adding more layers. The time complexity of the search, the insert, and delete can become O(Logn) in an average cases with O(n) extra space. We will soon be publishing more posts on Skip Lists. References MIT Video Lecture on Skip Lists https://en.wikipedia.org/wiki/Skip_list
Sure, here are some related subtopics with details:
- Coin flipping technique: The coin flipping technique is used to randomly determine the number of layers an element will occupy in a skip list. When a new element is inserted, a random number is generated and compared to a predetermined threshold. If the random number is less than the threshold, the element is inserted into the next layer. This process is repeated until the random number is greater than the threshold, at which point the element is inserted into the bottom layer. The coin flipping technique is what gives skip lists their probabilistic nature and allows for their efficient average time complexity.
- Search operation: The search operation in a skip list involves traversing the layers from the top to bottom, skipping over elements that are not of interest, until the desired element is found or it is determined that it does not exist in the list. The skipping links in each layer allow for quick navigation to the desired element, reducing the average number of steps needed to reach it. The average time complexity of search in a skip list is O(log n), where n is the number of elements in the bottom layer.
- Insertion operation: The insertion operation in a skip list involves generating a random number to determine the number of layers the new element will occupy, as described in the coin flipping technique, and then inserting the element into the appropriate layers. The insertion operation preserves the sorted order of the elements in the list. The average time complexity of insertion in a skip list is O(log n), where n is the number of elements in the bottom layer.
- Deletion operation: The deletion operation in a skip list involves finding the element to be deleted and then removing it from all layers in which it appears. The deletion operation preserves the sorted order of the elements in the list. The average time complexity of deletion in a skip list is O(log n), where n is the number of elements in the bottom layer.
- Complexity analysis: The average time complexity of search, insertion, and deletion in a skip list is O(log n), where n is the number of elements in the bottom layer. This is due to the use of the coin flipping technique, which allows for quick navigation to the desired element, and the probabilistic nature of skip lists, which ensures that the average number of steps needed to reach an element is logarithmic in the number of elements in the bottom layer. However, the worst-case time complexity of these operations can be O(n), which occurs in the rare case where all elements are inserted into only the bottom layer.
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem