How to analyse Complexity of Recurrence Relation
Last Updated :
23 Jul, 2025
The analysis of the complexity of a recurrence relation involves finding the asymptotic upper bound on the running time of a recursive algorithm. This is usually done by finding a closed-form expression for the number of operations performed by the algorithm as a function of the input size, and then determining the order of growth of the expression as the input size becomes large.
Here are the general steps to analyze the complexity of a recurrence relation:
Substitute the input size into the recurrence relation to obtain a sequence of terms.
Identify a pattern in the sequence of terms, if any, and simplify the recurrence relation to obtain a closed-form expression for the number of operations performed by the algorithm.
Determine the order of growth of the closed-form expression by using techniques such as the Master Theorem, or by finding the dominant term and ignoring lower-order terms.
Use the order of growth to determine the asymptotic upper bound on the running time of the algorithm, which can be expressed in terms of big O notation.
It's important to note that the above steps are just a general outline and that the specific details of how to analyze the complexity of a recurrence relation can vary greatly depending on the specific recurrence relation being analyzed.
In the previous post, we discussed the analysis of loops. Many algorithms are recursive. When we analyze them, we get a recurrence relation for time complexity. We get running time on an input of size n as a function of n and the running time on inputs of smaller sizes. For example in Merge Sort, to sort a given array, we divide it into two halves and recursively repeat the process for the two halves. Finally, we merge the results. Time complexity of Merge Sort can be written as T(n) = 2T(n/2) + cn. There are many other algorithms like Binary Search, Tower of Hanoi, etc.
need of solving recurrences:
The solution of recurrences is important because it provides information about the running time of a recursive algorithm. By solving a recurrence, we can determine the asymptotic upper bound on the number of operations performed by the algorithm, which is crucial for evaluating the efficiency and scalability of the algorithm.
Here are some of the reasons for solving recurrences:
Algorithm Analysis: Solving a recurrence is an important step in analyzing the time complexity of a recursive algorithm. This information can then be used to determine the best algorithm for a particular problem, or to optimize an existing algorithm.
Performance Comparison: By solving recurrences, it is possible to compare the running times of different algorithms, which can be useful in selecting the best algorithm for a particular use case.
Optimization: By understanding the running time of an algorithm, it is possible to identify bottlenecks and make optimizations to improve the performance of the algorithm.
Research: Solving recurrences is an essential part of research in computer science, as it helps to understand the behavior of algorithms and to develop new algorithms.
Overall, solving recurrences plays a crucial role in the analysis, design, and optimization of algorithms, and is an important topic in computer science.
There are mainly three ways of solving recurrences:
Substitution Method:
We make a guess for the solution and then we use mathematical induction to prove the guess is correct or incorrect.
For example consider the recurrence T(n) = 2T(n/2) + n
We guess the solution as T(n) = O(nLogn). Now we use induction to prove our guess.
We need to prove that T(n) <= cnLogn. We can assume that it is true for values smaller than n.
T(n) = 2T(n/2) + n
<= 2cn/2Log(n/2) + n
= cnLogn - cnLog2 + n
= cnLogn - cn + n
<= cnLogn
Recurrence Tree Method:
In this method, we draw a recurrence tree and calculate the time taken by every level of the tree. Finally, we sum the work done at all levels. To draw the recurrence tree, we start from the given recurrence and keep drawing till we find a pattern among levels. The pattern is typically arithmetic or geometric series.
For example, consider the recurrence relation
T(n) = T(n/4) + T(n/2) + cn2
cn2
/ \
T(n/4) T(n/2)
If we further break down the expression T(n/4) and T(n/2),
we get the following recursion tree.
cn2
/ \
c(n2)/16 c(n2)/4
/ \ / \
T(n/16) T(n/8) T(n/8) T(n/4)
Breaking down further gives us following
cn2
/ \
c(n2)/16 c(n2)/4
/ \ / \
c(n2)/256 c(n2)/64 c(n2)/64 c(n2)/16
/ \ / \ / \ / \
To know the value of T(n), we need to calculate the sum of tree
nodes level by level. If we sum the above tree level by level,
we get the following series T(n) = c(n^2 + 5(n^2)/16 + 25(n^2)/256) + ....
The above series is a geometrical progression with a ratio of 5/16.
To get an upper bound, we can sum the infinite series. We get the sum as (n2)/(1 - 5/16) which is O(n2)
Master Method:
Master Method is a direct way to get the solution. The master method works only for the following type of recurrences or for recurrences that can be transformed into the following type.
T(n) = aT(n/b) + f(n) where a >= 1 and b > 1
There are the following three cases:
- If f(n) = O(nc) where c < Logba then T(n) = ?(nLogba)
- If f(n) = ?(nc) where c = Logba then T(n) = ?(ncLog n)
- If f(n) = ?(nc) where c > Logba then T(n) = ?(f(n))
How does this work?
The master method is mainly derived from the recurrence tree method. If we draw the recurrence tree of T(n) = aT(n/b) + f(n), we can see that the work done at the root is f(n), and work done at all leaves is ?(nc) where c is Logba. And the height of the recurrence tree is Logbn
In the recurrence tree method, we calculate the total work done. If the work done at leaves is polynomially more, then leaves are the dominant part, and our result becomes the work done at leaves (Case 1). If work done at leaves and root is asymptotically the same, then our result becomes height multiplied by work done at any level (Case 2). If work done at the root is asymptotically more, then our result becomes work done at the root (Case 3).
Examples of some standard algorithms whose time complexity can be evaluated using the Master Method
- Merge Sort: T(n) = 2T(n/2) + ?(n). It falls in case 2 as c is 1 and Logba] is also 1. So the solution is ?(n Logn)
- Binary Search: T(n) = T(n/2) + ?(1). It also falls in case 2 as c is 0 and Logba is also 0. So the solution is ?(Logn)
Notes:
- It is not necessary that a recurrence of the form T(n) = aT(n/b) + f(n) can be solved using Master Theorem. The given three cases have some gaps between them. For example, the recurrence T(n) = 2T(n/2) + n/Logn cannot be solved using master method.
- Case 2 can be extended for f(n) = ?(ncLogkn)
If f(n) = ?(ncLogkn) for some constant k >= 0 and c = Logba, then T(n) = ?(ncLogk+1n)
For more details, please refer: Design and Analysis of Algorithms.
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