# 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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### The evolution of the term "recursive" from Goedel to Church to present day

I'm currently studying some of the history of computation / computability, in the early days known as recursion theory. I see Goedel's definition of recursive functions seems significant in his paper,...
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### Why can't a programming language be both fully recursive and polymorphic

In my theory of computation class last Spring my professor said in passing that a programming language cannot be both fully recursive and polymorphic. I didn't think much of it till now? What does it ...
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### Facility location on a tree

Question: Given a tree representing a neighbourhood where each node is a house. Assign an antenna to each node such that the whole tree is covered. An antenna of strength 0 can only ...
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### Termination of deterministic term rewriting

Consider a simple language: $$t ::= plus ~ t ~ t ~ | ~ gen ~ t ~ | ~ except ~ N ~ t ~ | ~ N$$ with N constructors plus, gen and except, N being the natural numbers, and $G = \{t_n\}$ a finite, ...
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### Axiomatisation in the presence of recursion

I read Klaus Havelund's thesis on the Fork Calculus: http://havelund.com/Publications/thesis.ps He develops the Fork calculus for reasoning about concurrent functional programs, the motivation being ...
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### Implementation of the divide-and-conquer principle for a specific summation formula

I have found two formulas in the work on pages 5 and 6, of which I am trying to develop a recursive implementation. The similarity to the DFT or FFT might be useful here. I divide this question into ...
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### How do you write a python\pseudo code that generates all pair permutations?

What would be a good pseudo code or Python 3 code for the following permutations problem? Let us define a n-permutation as a bijective function $\pi: \{0,...,n-1\}\rightarrow \{0,...,n-1\}$ and ...
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### How to prove νX. A × X ≅ (μX. 1 + X) -> A?

How can we prove Stream A = νX. A × X is isomorphic to Nat -> A = (μX. 1 + X) -> A ? In programming sense, ...
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### Reference request: Leaf-heavy master theorem algorithms

I know many algorithms that can be analyzed using master theorem, but the only algorithm I know where the time is dominated by the leaves is fast matrix multiplication. Are there other recursive ...
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### number lesser than a given number composed of only binary numbers

Give a decimal number, such as 123. Need to figure out all smaller numbers lesser than 123 made up by ...
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### Help with deterministic selection algorithm

All we know what is Deterministic Selection Algorithm: Line up elements in groups of five (this number $5$ is not important, it could be e.g. $7$ without changing the algorithm much). Call each group ...
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### Recursive definitions, How it is done?

I read that recursive definitions, refer to the definition of a function in that function body, cannot be done in $\lambda$-calculus, but recursion can be achieved by using $Y$ combinator. As I know, ...
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### Maximum Move in a maze

Given a N * N maze, and string of N,E,W,S denoting positions to move to. I need to determine how many moves are possible in sequence out of a string (containing these 4 letters only) from each cell as ...
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### Structural induction in non-local program transformation

Assume a functional language and a specialization operation (pulling out sub-expressions): let f x y = (h 23 x) + (g 42 y) becomes ...
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### Smarter recursion to compute #tilings of $m \times n$ board with small shapes that fit in $2 \times 2$ square?

This is a generalization of another question I posted because I wasn't clear that I cared about more than $2 \times 1$ dominoes (it's just a special case), and there is an explicit tractable formula ...
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### Runtime complexity of permutation function

I am trying to find the asymptotic run time complexity of the following function which will return a list of all permutations of nums. ...
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### Bottom-up well-balanced mergesort

If you implement mergesort top-down you can always split the input of length $n$ into one of length $\lfloor n / 2\rfloor$ and one of length $\lceil n / 2 \rceil$. This ensures that all merges are ...
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### Diffucuty in understanding code after a recursive call

This is an example algorithm of a recursive insertion sort I'm trying to understand. I've have tried understanding this with the help of print statements (which I've commented). ...
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### 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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### 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). ...
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### Longest increasing subsequence (Dynamic Programming)

I have written the following recursive structure for finding length of longest increasing subsequence. ...
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### Knight's Tour Parberry algorithm: 4 knight's tour merge procedure

I'm implementing Parberry's algorithm for closed Knight's tour problem. Brief idea of the algorithm: split the board in $4$ parts, find the tour on them recursively then delete $1$ edge in each part ...
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### Finding the $k$th smallest element in union of two sorted arrays

I know that this problem is solvable in linear time with a merge but I want to get a sub-linear algorithm. What I came up is that, if a[k] < b[k] then the $k$th ...
1 vote
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### Can the Sieve of Eratosthenes be adapted to find twin primes

The Sieve of Eratosthenes is an algorithm generate the prime numbers, $2,3,5,7,11,13,...$ by drawing a list of numbers crossing out multiples of the smallest number in the list. Is there anyway to ...
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### Using the μ (mu) operator

Problem I've got this function: $f(x,y)=(6-3\cdot x)\cdot(y+2)$, with $(x,y)\in\mathbb{N}^2$ Now I have to find $g=\mu f$. Proposed solution My solution was to find the smallest $n\in\mathbb{N}$ to ...
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### How to generate tree variants of a tree using recursion?

I have a tree T, I need to generate all possible variants of T by permuting all its child nodes(please refer the following figure). how can I generate all variants, T, using recursion? any help is ...
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### Can the algorithm be optimized?

I am new to backtracking and recursion. I have seen numerous explanations on how on to find the minimum number of coins needed to make a particular amount. This involves a top down dynamic approach ...
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### Calculating Runtime Complexity: Recursion + Memoization vs Dynamic Programming (with example)

For cases where recursion is used as well as memoization (so that a number of subtrees of what would otherwise be the overall recursive call tree are each replaced to be ...
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