A recursive object (e.g. function or data structure) is defined using itself.

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How many ways to find a sum totalling n using only certain Integers?

Using an infinite supply of integers of a set S, how many ways are there to reach a sum of n? Clarification: The Integers are arbitrary, positive, and may not include 1. At first I thought it was ...
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34 views

Finding number of numbers <= N, containing atleast one of the digits 2,4,6,8

Given an integer $N$, I want to find the number of numbers $\le N$, that contain at least one of the digits from the set $\{2, 4, 6, 8\}$. How do I go about solving this problem? I was thinking of ...
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34 views

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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1answer
88 views

Space complexity analysis of binary recursive sum algorithm

I was reading page 147 of Goodrich and Tamassia, Data Structures and Algorithms in Java, 3rd Ed. (Google books). It gives example of linear sum algorithm which uses linear recursion to calculate sum ...
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42 views

Recursive equations vs. inference rules

It seems to me that recursive equations can always be presented as inference rules. For the forward direction, an example is addition over Peono numerals (built from $O$ and $S(\_)$) $$ ...
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37 views

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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37 views

Smarter recursion to compute #tilings of $m \times n$ board with $2 \times 1$ dominoes?

So I was thinking about how to computationally (e.g., with recursion) obtain the number of tilings of an $m \times n$ board with $2 \times 1$ dominoes. If $m \leq n$, then we can use recursion on $n$ ...
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44 views

Understanding the reason behind the μ (mu) operator [closed]

What is the purpose of the $\mu$ operator? Is there a real world example? Is it correct that it can create partial functions out of total functions and it makes a function $g$ with k parameters out ...
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39 views

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 $n\in\mathbb{N}$ ...
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158 views

How Dynamic programming can be used for Coin Change problem?

As far as I can unserstand Dynamic programming stands simply for memoization (which is a fancy name for lazy evaluation or plain "caching"). Now, I read that there is we can reduce complexity of ...
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103 views

Analysis of a recursive algorithm, where running time strongly depends on input

I want to find the worst-case running time of an algorithm, which follows the following recurrence equation: The worst-case running time is $\Theta(n^2) + T(n, 2, n)$, where $T(x, i, y) = ...
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72 views

Recurrence Equation in Algorithm [duplicate]

Can anyone help me in solving this complex recurrence? \begin{eqnarray} T(n) &=& n +\sum_{k-1}^n T(n-k)+T(k) & Where& T(1) = 1. \end{eqnarray} although this topic will already ...
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57 views

Unrolling multi-variable mu (μ) expressions in type theory

Unrolling an iso-recursive μ-type expression such as, say, one isomorphic to natural numbers: μα.1+α using ...
5
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1answer
110 views

What does Tarski's Fixed-Point theorem give us that that Y-Combinator does't

I'm taking a graduate course on the theory of functional programming, based on Paul Taylor's "Practical Foundations of Mathematics." I understand the statement of Tarski's theorem about how for any ...
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92 views

What are efficient ways to compute the derivatives of iterated functions?

The derivatives of iterated functions at a fixed point $z_0$ are useful in constructing a Taylors series of iterated analytic functions - in other words, the Taylors series of a dynamical system ...
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64 views

How many times can you divide a list of n elements in 1/2 [closed]

I am trying to wrap my head around recursion and divide and conquer algorithms. Can someone provide a proof and explanation of how many times a list of n elements can be divided in 1/2 on both sides.. ...
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39 views

Barnes-Hut algorithm and recursion limit [closed]

I'm running the Barnes-Hut simulation algorithm for an $n$-body simulation. If while distributing each particles to their corresponding nodes, two particles comes closer to a level smaller than the ...
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152 views

The minimization operator is an effective operator

Assume $\{f_i^{(n)}\}_{i=0}^\infty$ is a Gödel enumeration of the $\mu$-recursive functions of $n$ arguments, such that the $S^m_n$ theorem and the universal function theorem hold. Denote the set of ...
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45 views

Can indirect recursion also be tail recursive? [closed]

Consider the following function definitions: ...
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1answer
1k views

CLRS 4.4-3 Height of recursion tree for T(N) = 4T(n/2 +2) + n

I'm having a hard time with the following question: Use a recursion tree to determine a good asymptotic upper bound on the recurrence $T(n) = 4T(n/2 + 2) + n$. Use the substitution method to verify ...
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2answers
125 views

Solution to recurrence $T(n) = T(n/2) + n^2$

I am getting confused with the solution to this recurrence - $T(n) = T(n/2) + n^2$ Recursion tree - ...
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81 views

Would adding recursive named functions to Simply typed lambda calculus make it Turing complete?

Say I have Simply typed lambda calculus, and add an assignment rule: <identifier> : <type> = <abstraction> Where ...
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120 views

Viterbi algorithm recursive justification

I have a question regarding recursion in Viterbi algorithm. Define $\pi(k; u; v)$ which is the maximum probability for any sequence of length $k$, ending in the tag bigram $(u; v)$. The base case ...
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401 views

single algorithm to work on both directed and undirected graph to detect cycles?

I have been trying to implement an algorithm to detect cycles (probably how many of them) in a directed and undirected graph. That is the code should apply for both ...
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260 views

Finding growth of “inter-recursive” functions

consider following code int f(int x) { if(x<1) return 1; else return f(x-1)+g(x); } int g(int x) { if(x<2) return 1; else return f(x-1)+g(x/2); } ...
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Can a tree be traversed without recursion, stack, or queue, and just a handful of pointers?

Half a decade ago I was sitting in a data structures class where the professor offered extra credit if anyone could traverse a tree without using recursion, a stack, queue, etc. (or any other similar ...
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173 views

Algorithm to determine if recursion was breadth first or depth first

Given a tree $T$ and a sequence of nodes $S$, with the only constraint on $S$ being that it's done through some type of recursion - that is, a node can only appear in $S$ if all of its ancestors have ...
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66 views

Give a recursive function $r$ on $A$ that reverses a string

I really need help with this task here. Im stuck at it and I really would appreciate your help Here is the task: Give a recursive function $r$ on $A$ that reverses a string. For instance, ...
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409 views

Recursive equation for complexity: T(n) = log(n) * T(log(n)) + n

For analyzing the running time of an algorithm , I'm stuck with this recursive equation : $$ T(n) = \log(n) \cdot T(\log n) + n $$ Obviously this can't be handled with the use of the Master Theorem, ...
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3answers
2k views

Printing The Longest Path from Root to Leaf in Binary Tree [duplicate]

I am stumped as to how to print the longest path from the root of a binary tree to a leaf, essentially traversing the height of the tree. I've got the following for finding the height of a binary ...
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1answer
293 views

Complexity of a recursive bignum multiplication algorithm

We have started learning about analysis of recursive algorithms and I got the gist of it. However there are some questions, like the one I'm going to post, that confuse me a little. The exercise ...
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75 views

Tight asymptotic bound for recursive algorithm

I have this algorithm where: $$ T(n) = \begin{cases} 1 & \text{if}\; n \le 1 \\ T(n/2) + 1 & \text{otherwise} \\ \end{cases} $$ So, evaluating for $T(0), T(1), T(2), T(3), \ldots, T(n)$, ...
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87 views

Ordering a list of lists subject to constraints

I have encountered a surprisingly challenging problem arranging a matrix-like (List of Lists) of values subject to the following constraints (or deciding it is not possible): A matrix of m randomly ...
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1answer
339 views

Recursive definition of sum of two numbers in terms of the successor function

This is a question from the book Data structures using C and C++ by Tenenbaum. Not a homework problem but self-study. Recursive definition of a+b, where a and b ...
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191 views

Inductive vs. recursive definition

When should I call a definition recursive and when should I call it inductive? I have read Carl Mummert's nice answer on MSE. So if I understand correctly we refer to definitions of objects like ...
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Iterative and/or tail-recursive implementations of merge sort?

I recently learned how to implement merge-sort, using a standard recursive algorithm. Can the algorithm be implemented in a way that allows for a tail-recursive implementation? Can it be implemented ...
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196 views

Complexity of recursive solution to coin change

How do you go about analysing coin change recursive solution. i.e, T(N,K) = T(N,K-1) + T(N-1,K) for K denominations that add up to amount N. You can find the ...
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332 views

Can we create recursive functions only by using if-else statements?

I have to show whether a program containing only if-else statements but no loops is able to calculate the following type of functions: $f^n(x)$. The function $f$ is applied $n$ times to $x$, so I ...
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67 views

Time complexity of mutually recursive functions

Suppose I have two mutually recursive functions like this: ...
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128 views

Partial recursive function and Turing machine

The wikipedia article about primitive recursion states that An equivalent definition states that a partial recursive function is one that can be computed by a Turing machine. My question is how ...
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253 views

Register Machine code for Fibonacci Numbers

I am not sure whether this is the right place to ask this question. I would like to write a register machine code which when given an input of n in register 1, returns (also in register 1) the nth ...
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142 views

Particularly Tricky Recurrence Relation (Master's Theorem)

Master's theorem is shown below, The recursive function to be solved is shown below, I understand that a refers to the number of recursive calls in this ...
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517 views

Towers of Hanoi but with arbitrary initial and final configuration

Recently, I came across this problem, a variation of towers of hanoi. Problem statement: Consider the folowing variation of the well know problem Towers of Hanoi: We are given $n$ towers ...
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81 views

Resolving this recurrence equation [duplicate]

I have this recurrence equation: $T(n) = T(n/4) + T(3n/4) + \mathcal{O}(n)$ $T(1) = 1$ I know that the result is $\mathcal{O}(n \log n)$ but i don't know how to proceed.
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233 views

Proving correctness of the algorithm for convex polygon minimum cost triangulation

I have read many solutions for the minimum cost of triangulation problem and intuitively get the idea , however I am struggling to figure out how to prove it formally. I kind of feel that it has to be ...
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111 views

Can a recurrence relation be translated to a composite function of itself?

Perhaps this is a question for stackoverflow because its practical nature, but I am not aware of any general method to relate recurrence relations and recursive functions. Having as an example this ...
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25 views

The use of master theorem appriopriately [duplicate]

I have a recurrence relation and trying to use master theorem to solve it. The recurrence relation is: $T(n) = 3T(n/5) + n^{0.5}$ Can I use the master theorem in that relation? If so, can I say that ...
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300 views

Need help about solving a recurrence relation

I have a recurrence relation which is like the following: $T(n) = 2T(\frac{n}{2}) + \log_{2}n$ I am using recursion tree method to solve this. And at the end, i came up with the following equation: ...