Questions tagged [recursion]

Questions about objects such as functions, algorithms or data structures that are expressed using "smaller" instances of themselves.

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Distinct Binary Heaps

I have $n$ elements out of $n-1$ are distinct. The repeated element is either minimum or maximum element. I need to figure out how many distinct heaps can be made from it. My analysis : I started ...
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3answers
53 views

How can any non-primitive-recursive function like the Ackermann function be implemented on hardware?

If for-loops and function calls both boil down to jump instructions when implemented on a real machine, then how is "The Ackermann function isn't implementable with for-loops" a meaningful phrase?
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14 views

Complexity of iterative exponentiation

I've watched multiple videos and read articles about recursion but I'm still confused. I've got this problem here but I'm unsure how to answer it: The following function calculates $x^n$ ...
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1answer
271 views

Finding the longest overlapping interval pair

Say I have a list of $n$ integral intervals $[a,b]$ where each represents a set $S = \{a, a+1, \ldots, b\}$. An overlap is defined as $|S_1 \cap S_2|$. Example: $[3,6]$ and $[5,9]$ overlap on $[5,6]$ ...
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Is it still called “recursion” when you're using call-stack as a stack?

The most obvious way to solve a problem of balancing parentheses, like https://leetcode.com/problems/valid-parentheses/, is through a stack (a last-in-first-out (LIFO) datastructure). However, if you ...
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1answer
61 views

How to find the Big-O for finding combinations of balanced parentheses?

Given n pairs of parentheses, a function which returns the total number of all combinations well-formed parentheses could be: ...
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1answer
41 views

Is this lambda abstraction created as a generator of a recursive function?

In lambda calculus, a recursive function $f$ is obtained by $$ f = Y g $$ where $Y$ is the Y combinator and $g$ is the generator of $f$ i.e. $f$ is a fixed point of $g$ i.e. $f == g f$. In The ...
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Towers of Hanoi through ITERATION [closed]

instead of recursion i tried my own version of solution...can anyone tell me the difference between my solution and that of standard iteration solution ...
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1answer
534 views

Prove Recursive formula (Dynamic programming) N(C,i)

I've been asked to prove the correctness of the following recursive formula. The formula is trying to define, how many ways you can spend your money C on the i amount of beers. I did the following ...
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1answer
215 views

Understanding Martin Farach's suffix tree algorithm

I feel stuck at this point. I have spent several days trying to get my head around the algorithm, but both resources I have [1] [2] seems to skip over whatever details that would make me comfortable ...
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2answers
101 views

Pancake Sorting Graph Recursive Definition

I'm having trouble understanding exactly how the graph for Pn (where n = number of pancakes) is defined recursively for n>= 4. I can see obviously that, in the case of n=4, there will be 4 rough ...
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What is the running time of the following procedure? [duplicate]

Please help me calculate the time complexity of the following procedure. Power(n) 1:If n=0 then 2: return 1 3:else if 4: return Power(n-1) + Power(n-1) Explain ...
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What recursive T(N) function typically can conclude the algorithm is O(n ^ 2), O(n log n), O(n), and O(log n)?

Is it true that some common forms of recursive T(n) can give the following conclusions? When T(n) = T(n/c) + b where c is a constant > 1, b is any constant ...
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1answer
39 views

Recursion Time Complexity (Half n' Half)

This is my solution for Leetcode 395, and I'm wondering how I can come up with its time complexity: Input: string $s = s_1,\ldots,s_n$, integer $k$ Go over all symbols $s_1,\ldots,s_n$, one by one ...
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1answer
476 views

What is the formal justification for the correctness of the second formulation of rod cutting DP solution

CLRS on section 15.1 3rd edition has a good discussion of the rod cutting problem. I will add a description at the end of the question for reference. Define $r_j$ to be the optimal way to cut a rod ...
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4answers
3k views

Teaching Recursion

I'm a teacher assistant in my university and my next topic is recursion. what way is the best to teach recursion so that the student can grasp the concept easily and can think recursively? I was ...
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2answers
336 views

Making a recursive formula for finding amount of ways to spend money on beer

So far, i've only made recursive formulas for finding simple patterns such as fibonacci, however i can't seem to get my head around this. The information available is that there are $n$ different ...
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1answer
168 views

Are all foldable data structures also recursive?

I was checking what Wikipedia has to say on reduce. It says: In functional programming, fold (also termed reduce, accumulate, aggregate, compress, or inject) refers to a family of higher-order ...
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2answers
864 views

Does the Y combinator contradict the Curry-Howard correspondence?

The Y combinator has the type $(a \rightarrow a) \rightarrow a$. By the Curry-Howard Correspondence, because the type $(a \rightarrow a) \rightarrow a$ is inhabited, it must correspond to a true ...
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31 views

Transforming an immutable binary tree without recursion [closed]

I'm struggling on this one. I have a Binary Decision Diagram, which is pretty much tree-like. Each node has a hi and lo node. I need to recurse into the tree, and if some conditions are the case ...
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1answer
30 views

Understanding proof of upper bound on complexity of recursive computation of graph chromatic polynomial

This question is about section 2.3 of Wilf's ``Algorithms and Complexity'' https://www.math.upenn.edu/~wilf/AlgoComp.pdf in which he analyses the complexity of a recursive computation of the ...
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2answers
74 views

Is there any recursive function f whose code is unique?

I am doing some reviewing for the term final on computability and found out this simple exercise. I am very fresh on theoretical computer science so if you do have an answer please make it simple. ...
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How to approach backtracking when using immutable types (Python)? [closed]

In Python when we are building a recursive algorithm that uses backtracking a mutable type such as a list is great to use. It can be modified at each call in our recursion tree, then returned back to ...
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2answers
47 views

Iteration vs Recursion question in Lisp method

I am curious if the following method would be called iterative or recursive: ...
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1answer
24 views

Turing Machine equivalence in MinTM proof

The proof with contradiction that $MIN_{\mathrm{TM}}$ is not Turing-recognizable from Michael Sipser's textbook "Introduction to the Theory of Computation" (Theorem 6.7) is as follows: $C=$ "On ...
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2answers
87 views

Prove that $T(n) \leq 8n^2$ or find value of $n$ when statement is not true (recurrence relation)

We have a function $T: \mathbb{N}\to\mathbb{N}$ defined recurrently: $$T(n)=\begin{cases} 0 &\text{ if } n=0,\\ 3T(\lfloor{n/2}\rfloor) + 2n^2 &\text{otherwise.} \end{cases}$$ Prove that for ...
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1answer
49 views

Count the number of ways numbers 1,2,…,n can be divided into two sets of equal sum

count the number of ways numbers 1,2,…,n can be divided into two sets of equal sum. This is my recursive algorithm, what is wrong here?: ...
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2answers
39 views

How to use Master Theorem with strange format of $b$ parameter?

I have a funcion $T: \mathbb{N}\to\mathbb{N}$ defined as: $$T(n)=\begin{cases} 6 &\text{ if } n=0,\\ T(n-1) + 6n + 6 &\text{otherwise.} \end{cases}$$ How can I apply the Master Theorem to ...
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1answer
77 views

Given price and number of pages of each book, What is the maximum number of pages you can buy?

You are in a book shop which sells n different books. You know the price and number of pages of each book. You have decided that the total price of your purchases will be at most x. What is the ...
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1answer
150 views

Merge sort and quciksort recursion tree depth

1) I need to determine recursion tree depth for strings composed of 10, 100 and 1000 elements when using merge sort. For the 10 elements one I can do it on a paper, just drawing tree, but what about ...
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1answer
61 views

Assigning $m$ balls to $n$ buckets - recursive algorithm

I came across the following problem and the answer to that problem: Given $m$ balls and $n$ bins, find out how many ways to assign the balls to the bins. Notice the bins have no order: for example, ...
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37 views

How to solve 2 variable recursion?

T(m,n) = T(m-1,n) + T(floor(m/2), n-1) Base conditions T(m,n) = 1 when n = 0 T(m,n) = 0 when m < n Edited: Below is the code for which I want to know the time complexity in terms of m and n. <...
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2answers
104 views

What is the height of a tree with recursion formula: $T(n) = T(n - \sqrt{n})$

I know if the time complexity of an algorithm is given with the above formula, then the algorithm works in constant time but my question is that what will be the height of the recursion tree for this ...
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1answer
199 views

Handling epsilon productions in recursive descent parsing

I am working on a recursive descent parser for lambda calculus. In my grammar, after removing left-recursion, I am left with the following two productions: ...
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1answer
53 views

Number of Function Calls In Recursive Code

I am new to recursion. I am doing some practice questions and I was wondering what the technique is for going from some recursive code to identifying the number of function calls it makes. ...
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1answer
55 views

What is the closed-form expression for $T_n = \left(\sum_{i=1}^{n-1}7 T_i\right) + 1$ where $T_1 = 1 ?$ [closed]

Problem: Find the closed-form expression for$$ T_n = \left(\sum_{i=1}^{n-1}7 T_i\right) + 1 \tag{1} $$where $T_1 = 1 .$ Calculating this sum I came up with the following result: $$ T_n = 8^{\left(...
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1answer
40 views

Is there purely recursive functions? [duplicate]

Is there any problem that can be only solved with recursion, and not with iteration? (haven't been able to find anything online). If there isn't any, is there a reason why? Thanks in advance!
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1answer
29 views

Computability: Proving a predicate is not recursively enumerable

Let P(p) <=> for each x, comp(p,x) is defined. Can anyone explain to me how to prove that P is not RE (recursively enumerable) ?
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1answer
47 views

Multiple choices for a single case in the recursive formula of a Dynamic Programming algorithm

I am developing a Dynamic Programming algorithm for a problem in scheduling. In the recursive formula, I have three cases: (1) $t_{i-1} = int$ (2) $t_{i-1} = app \quad \& \quad r(j) \leq p $ and (...
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1answer
35 views

Recursive definition for the length of a string?

I found a couple of answers online but I don't quite understand why the answer is right: The length of a string is: If a string has no characters, then its length is 0. Otherwise, the length of the ...
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66 views

Write the Brute Force Recursive Code to generate the longest substring containing k distinct vowels

Given a string s we have to find the length of the longest substring of s which contain exactly K distinct vowels. This is the problem statment given on geeksforgeeks Input : s = “artyebui”, k = 2 ...
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1answer
208 views

Worst-case input for median-of-medians with groups of size 3

Typically, median of medians algorithm is written with groups of size $5$ or $7$ to ensure worst-case linear performance. The argument against groups of size $k=3$ is typically that we get a ...
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1answer
75 views

Avoiding “side effects” in recursive functions

I am writing a function to find the intersection between two sets. The non-functional requirements of the assignment include avoiding "side effects". ...
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2answers
180 views

prove by induction that the complete recursion tree for computing the nth Fibonacci number has n leaves

I have referenced this similar question: Prove correctness of recursive Fibonacci algorithm, using proof by induction *Edit: my professor had a significant typo in this assignment, I have attempted ...
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T(n) = T(n-1) +3n^2 - 2n +1 [duplicate]

I was wondering how do I solve this, I've been trying any possible way to but I've failed: $$\begin{align*} T(n) &= T(n-1) +3n^2 - 2n +1 ,& n \ge 1 \\ T(0) &=2 &\\ \end{align*}$$
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1answer
166 views

Can memoization be applied to any recursive algorithm?

I am new to the concepts of recursion, backtracking and dynamic programming. I am having a hard time understanding if at all I can apply memoization to a particular recursive algorithm and if there ...
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2answers
498 views

What property of cons allows elimination of tail recursion modulo cons?

I'm familiar with the idea of basic tail recursion elimination, where functions that return the direct result of a call to themselves can be rewritten as iterative loops. ...
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50 views

Is McCarthy Formalism first ever formalism for defining functions recursively in computer science?

McCarthy formalism is a formalism for defining functions recursively, first introduced in classic paper Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I (1960). ...