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

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What Exactly is “Improvement” in Seed AI?

From LessWrongWiki, a seed AI [I]mproves itself by recursively rewriting its own source code without human intervention. Some would even say this could bring about an Intelligence Explosion. ...
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13 views

regression tree- greedy recursive partitioning (square error)

I have the following dataset $\{((1,1),9),((1,2),-4),((1,3),2),((2,2),4),((2,3),2)\}$. I would like to find the regression tree corresponding to the greedy recursive partitioning procedure with ...
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0answers
42 views

Recursive Algorithm design for finding the permutation of a string

Recently I am reading about induction and found that recursion and induction are the sides of the same coin. For example induction states that: $$ R_{n} = R_{n-1} + n$$ i.e. the nth solution can be ...
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0answers
17 views

What is this recurrence doing [on hold]

I'm having a hard time understand what is going on here. It seems like there are two indices i and j and some kind of minimum is found recursively? I'm also not sure what role k plays. ...
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0answers
17 views

Recursion Type in Grammar Productions [closed]

The grammar G0 is defined by the productions P= xP|y which type of recursion is it left , central , right or indirect recursion
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3answers
206 views
+50

Why are loops faster than recursion?

In practice I understand that any recursion can be written as a loop (and vice versa(?)) and if we measure with actual computers we find that loops are faster than recursion for the same problem. But ...
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1answer
24 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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0answers
22 views

3-Sum Algorithm Recursion

HOW can we implement the 3-sum problem where the sum of the 3 numbers of the array is zero using only recursion.I figured that there are 3 recursive calls but I have not figured out the base case and ...
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2answers
489 views

How is the complexity of recursive algorithms calculated and do they admit better complexity than non-recursive algorithms?

How are asymptotical time complexities calculated for recursive algorithms? Recursive algorithms call themselves and therefore take up more space compared to non-recursive algorithms. But are they ...
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1answer
103 views

Trying to understand this Dynamic Programming solution

The problem is as follows. Minimize the sum of absolute differences between a matching of $n$ values from two sets, $A=\{a_1,a_2,\cdots, a_n\}$ and the set $B=\{b_1, b_2,\cdots, b_m \}$, with $n\leq ...
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2answers
71 views

Why does the recurrence equation for QuickSort considers all the elements in the array?

I have been taught that QuickSort has the following recurrence equation in the best case: $T(n) = \begin{cases} c & \text{if } n=1 \\ 2\ T(\frac{n}{2}) + c ...
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2answers
61 views

How to calculate the mergesort time complexity?

Recently while reading a book I came across the following statement: Mergesort works by dividing nodes in half at each level until the number of nodes becomes 1 hence total number of times we ...
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0answers
28 views

Problems understanding proof of smn theorem using Church-Turing thesis

I am reading Barry Cooper's Computability Theory and he states the following as the s-m-n theorem: Let $f:\mathbb{N}^2\mapsto\mathbb{N}$ be a (partial) recursive function. Then there exists a ...
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37 views

Longest double increasing subsequence (LIS variant)

I'll start with the definitions:Let $S = s_1s_2...s_n$ be a sequence of $n$ integers. A double increasing subsequence of $S$ is a sequence $P=p_1p_2...p_k$ (not necessarily continuous) where for each ...
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1answer
36 views

Revisiting Fixed Point: What does it mean in the world of computer science?

A while ago I asked Fixed point, what does it mean in the world of computer science? While the answers did help me to understand what Fixed point meant, the answers left me in a murky world when ever ...
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1answer
29 views

How to treat $\epsilon$ and '$' in top-down predictive parsing (predict table)?

How to treat $\epsilon$ and '\$' in top-down parser using predict table? The construction of the predict table Given a product $X \rightarrow w$, row $X$ and column $t$ -Mark $X \rightarrow w$ ...
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1answer
103 views

Complexity of dynamic card game algorithm

Consider the following dynamic card game with a regular deck of 26 red cards and 26 black cards. A dealer draws the unturned cards one by one, and we can ask him to stop at any time. For every red ...
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1answer
60 views

Complexity and Recurrence relation for Lowest Common Ancestor Binary Tree

I have written this solution for finding the Lowest Common Ancestor in a Binary Tree. Now I wanted to find the time complexity of this problem by solving via recurrence relation. Can someone suggest ...
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1answer
29 views

Print Binary Tree Diameter Path [duplicate]

Diameter of the tree is defined as a long path or route, between any two nodes in a tree. The path may or may not goes through the ROOT. Print the Longest leaf to leaf path in a binary tree and its ...
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1answer
47 views

primitive recursion in the lambda calculus

I am having trouble finding out what a primitive subset of the lambda calculus would look like. I reference primitive recursion as shown here: ...
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0answers
27 views

Poly-variadic Y combinator

I have written a lambda calculus interpreter, and it seems to work. I cant find the combinator for something I want though. I want to be able to define an arbitrary number of mutually recursive ...
7
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1answer
131 views

What is the running time of this recursive algorithm?

I made the following (ungolfed) Haskell program for the code golf challenge of computing the first $n$ values of A229037. This is my proposed solution to compute the $n$th value: ...
2
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2answers
78 views

Recurrence relations when function call is made inside loop

int fun (int n) { int x=1, k; if (n==1) return x; for (k=1; k<n; ++k) x = x + fun(k) * fun(n – k); return x; } What is the value of fun(5)? I ...
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0answers
17 views

What would be the time complexity if a recursive function is inside a loop? [duplicate]

I am confused in calculating the time complexity for a recursive function inside a loop. ...
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2answers
99 views

How many recursive calls does it take to compute binomial probability weights?

I was given an algorithm and I was asked to estimate how often it would be called if I was trying to calculate ${100 \choose 50}0,25^{50}0.75^{50}$, the Binomial distribution of $50$ elements chosen ...
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1answer
86 views

How can the class of tail recursive functions be compared to the classes of PR and R?

How can the class of tail recursive functions (TR) be compared to the classes of primitive recursive functions (PR) and recursive functions (R)? The computation of a PR function always halts. This ...
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15 views

Time complexity with flooring of nested function calls [duplicate]

Prepping for an exam and wondering whether I correctly calculated the time complexity. Function is given as: $function XYZ(n:integer)\\ begin for\ i:=1 \ do \ 2*n^2 \ do;\\ ...
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1answer
39 views

Upper bound of algorithm with flooring

Would much appreciate someone explaining how they managed to get to the upper bound of this algorithm. $$T(n) = 2T(\lfloor \sqrt n\rfloor) + ln (n)$$ $$T(1) = 0$$ Solution is given as: $$T(n) = ...
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1answer
65 views

How to show all possible implied parenthesis?

Can I use recursion to find out the possible parenthesis we can add to this expression: 2*3-4*5 ? (2*(3-(4*5))) = -34 ((2*3)-(4*5)) = -14 ((2*(3-4))*5) = -10 (2*((3-4)*5)) = -10 (((2*3)-4)*5) = ...
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0answers
19 views

How to apply recursion in this problem [duplicate]

Problem Statement : You are situated in an N dimensional grid at position (x1,x2,...,xN). The dimensions of the grid are (D1,D2,...DN). In one step, you can walk one step ahead or behind in any one ...
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3answers
77 views

Value returned by recursive function

(pseudocode) ...
2
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1answer
53 views

Induction proof of alpha-beta search

Is there a functional specification of alpha-beta search that makes it easy to prove by induction that the algorithm works? My first thought is that the algorithm introduces an $[\alpha,\beta]$ ...
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0answers
9 views

recursion trees and big theta bounds [duplicate]

Draw recursion trees and use them to find big theta bounds on the solutions to the following recurrences. For each, assume that T(1) = 1 and that n is a power of the appropriate integer. ex) T(n) = ...
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3answers
59 views

Recursive function problem

This might seem very simple but it's been giving me lots of headaches. For example if I run f(12345) the result is 5310135. I understand why it would return 5310, the rest of 135 is what is giving ...
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1answer
51 views

Recursive definition of the set of strings over an alphabet [closed]

The problem is the following: Write the recursive definition for the set of all Strings Q over the alphabet {a, b} (i.e. the set of strings consisting of a's and b', for example a, baba, ...
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0answers
18 views

Why do we not store the min in any of the recursive clusters in a Van Emde Boas tree?

I was reading the chapter of van Emde Boas in CLRS (page 547 section 20.3 3rd edition) and it says: Furthermore, the element stored in min does not appear in any of the recursive $vEB( ...
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5answers
300 views

Do recursive algorithms generally perform better than their for-loop counterpart?

I'm sure this is not a challenge for you but it remains an open question for me: Is it wise to prefer a recursive algorithm over its for-loop counterpart? E.g. take the evaluation of the natural ...
4
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1answer
162 views

Recursive definition of a language given the regular expression

Consider the language: $$ L_1 = \{ x \in \Sigma^* : x \text{ does not contain the substring } 110\} $$ I know that there is a DFA that accepts this language, and furthermore, that the regular ...
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1answer
102 views

Help needed with lesson on recursion

I'm studying CS online, and I'm reading this lecture on recursion, see "3.2. A Mathematical Example". I understood the beginning and I even made a program that calculates $X$ to the power of $N$ ...
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86 views

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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3answers
270 views

If recursive Fibonacci is $O(2^N)$ then why do I get 15 calls for N=5?

I learned that recursive Fibonacci is $O(2^N)$. However, when I implement it and print out the recursive calls that were made, I only get 15 calls for N=5. What I am missing? Should it not be 32 or ...
4
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1answer
52 views

Proof by induction over rules for mutually recursive relations

Consider the (big-step) semantics of a language ($a, e$ terms, $v$ values), defined by two mutually recursive relations, $\downarrow$ and $\Downarrow$, given by a set of rule-schemata (simplified): ...
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1answer
45 views

Proving correctness of a recursive algorithm using induction

For the program mean(A,n) if n = 1 then return A[n] else return A[n]/n+mean(A,n-1)*(n-1)/n end Show that if the recursive call to ...
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1answer
79 views

Using dynamic programming to find the number ofl increasing subsequences [closed]

I got this question today and I'm nowhere near the solution, Given a sequence of real numbers (X1, X2, ..,Xn). write an algorithm as efficient there is, that finds the number of strictly increasing ...
3
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1answer
191 views

Since we need space for recursive calls, is the space complexity of the recursive factorial is n?

As Wikipedia says, quickSort needs O(log n) extra space when the following conditions are met: In-place partitioning is used. This unstable partition requires O(1) space. After partitioning, the ...
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30 views

Recursive macros and termination of assembler

A debate has arisen in the course of using an assembler (which does a bit of preprocessing first, but is mainly a static assembler): This particular assembler allows recursive macros to be defined, ...
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0answers
21 views

Solving $T(n)=2T(n/2) + n \lg n$ , For ex: Counting inversions implemented with full mergesort [duplicate]

How to solve the recurrence equation $T(n)=2T(n/2) + n \lg n$ For ex: I implemented "Counting inversions" with a full mergesort instead of just using merge part, So the outer complexity will be $n ...
2
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1answer
55 views

Undecidable definition of pure function?

I am trying to come up with a formal definition for functional purity in a simple programming language (think JavaScript). What I've got so far is this: DEFINITION: A statement is impure if ...
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88 views

What is the complexity of this recursive merge of two ordered Python lists?

This is not an assignment, but it is related to my Data Structures class. I just wrote this Python code to merge two ordered python lists. I do know that I could do something like this: list1 + ...