Mathematical questions about Algorithms, including the Analysis of Algorithms, Combinatorial Algorithms, proofs of correctness, invariants, and semantic analyses

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5 views

Count arrangment such that each person wear different tshirt

Few friends are going to a party. Each person has his own collection of T-Shirts. There are 100 different kind of T-Shirts. Each T-Shirt has a unique id between 1 and 100. No person has two T-Shirts ...
2
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1answer
51 views

Finding an $n$ such that $n^2 \equiv -1 \mod p$

What is an efficient algorithm to find the first number $n$ such that $n^2 \equiv -1 \mod p$ for a prime $p$, if such an $n$ exists? Is there anything better than the brute-force approach up to $p-1 ...
2
votes
1answer
25 views

How to measure trajectory regularity?

I have two animal running trajectories. A regular one with repeated back and forth running between point A and B, like the one on top in the figure. The other one is very irregular, animal paused and ...
1
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1answer
40 views

How can i find if a given number occurs in a custom Fibonacci sequence?

Its a recent interview question from Amazon. For e.g. let starting numbers be $a$ and $b$, then third number will be $a+b$ and so on: forming recursion like: $F(n)=F(n-1)+F(n-2) , n\ge 2$ $F(1)=a$ ...
0
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1answer
28 views
0
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0answers
22 views

Number of unique ways to edge-label a complete graph with $k$ distinct labels.

Given $k$ distinct labels, how many unique ways to label the edges of a complete graph with $n$ nodes (nodes are not labeled). For example, to label a complete graph with 3 nodes using 4 distinct ...
1
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2answers
30 views

Algorithm of order $O(n)$

Design an $O(n)$ algorithm, that given a real number $x$ and a sorted array $S$ of $n$ numbers, determine whether there exists any two elements in $S$ whose sum is exactly $x$. The solution I have ...
3
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2answers
74 views

Fibonaaci Recurrence

This is an interesting question where we are trying to solve another recursion which has same tree structure as the given recursion and also has term similarities Given Data in question ...
0
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0answers
30 views

sorting young's tableau better than n^3

Young's tableau takes $O(n^3)$ to sort. http://en.wikipedia.org/wiki/Young_tableau Simply it is a matrix sorted by rows and columns. However, if we build heap and then sort it we can do better than ...
3
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2answers
38 views

Largest Equilateral Triangle in a Polygon

Is there an algorithm to determine the largest equilateral triangle in a convex polygon?
1
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0answers
13 views

Sparse matrix algorithms involving data-driven or random access / walk

I am looking for some well-known algorithms in which sparse matrix elements are accessed in a non-structured way, i.e. row/column depends on a value of another (sparse) matrix/vector element or some ...
0
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0answers
20 views

Solving a simple Recurrence in summation form(very special case)

I have a bit confusing recursion form $\sum_{n=2}^{\infty}\{f(n)\frac{n}{n-1}\}=C, \tag 1$ $f(0)=b,f(1)= a,f(2)=c$ and $C$ are constants. Could you help me to solve this recursion or help me to ...
0
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0answers
16 views

Recurrence relation-Summation of a series [on hold]

Sir, I have a Converging recurrence relation given as below, $-(\psi(n-1)) ...
1
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0answers
47 views

Solving the recursion $F(n)=K_0F(n-1)/(n-1)+K_1F(n-2)/(n-2)$

Please help me in solving the recursion $F(n)=K_0\frac{F(n-1)}{n-1}+K_1\frac{F(n-2)}{n-2}$, preferably using power series for the values of $F(n)$ in terms of $n$. Here $K_1$ and $K_2$ are ...
-1
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0answers
32 views

Construction of a Huffman tree

My task is the following: Provide an example for the following: A complete Huffmann tree with $n=5, q=2$, lengths $l_1,......,l_5$ and $l_1>l_2>l_3>l_4$. (Draw the tree and give weights ...
1
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3answers
94 views

Solution of a recurrence with varying coefficient

Please help me to find the solution of the recurrence in terms of n(implies $(f(n))$ and also the summation of the recurrence up to infinity ($sum = \sum_{n=0}^\infty f(n)$) . ...
1
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0answers
18 views

Fast way to find the sum of LCM of the given range of numbers?

I want to find the sum of LCM of a given range of integers.For example: Input: 5 Output: LCM(1,5)+LCM(2,5)+LCM(3,5)+......LCM(5,5)// ie. 55 The method I use ...
1
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0answers
29 views

Maximal flow in flow-networks

I want to do the task (b),(c) and (d)in the picture above. I have done (b) correctly. For (c) I only found one (s-t) augmenting path, namely (s,1),(1,3),(3,2),(2,4),(4,t) and I only can push one ...
0
votes
2answers
20 views

$O(n \log k)$ for merging of $k$ lists with total of $n$ elements

Is it possible to merge $K$ sorted list into one sorted list where n is the total number of elements in all the input lists in $O(n \log k)$ time? What I tried to do is, taking the list, and adding ...
0
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0answers
37 views

Miscellaneous questions about trees

I want to know which of the following claims are true: 1) Let T be a minimal spanning tree in G for a weight function w. Then T is also a minimal spanning tree for the weight function obtained from w ...
0
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1answer
20 views

Proof about spanning tress in graphs

Let $G=(V,E)$ be a graph and $T_i=(V,F_i),i=1,2$ two disjoint spanning trees in $G$. Let $f_1 \in F_1$. Prove that there is $f_2\in F_2 $ such that $T:=T_1-f_1+f_2$ is a spanning tree.
2
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1answer
17 views

How to efficiently compute the pareto front in a >2 dimensional multi-objective case?

I'm currently working on an optimization problem with 4 different objective functions and need an algorithm to compute the pareto frontier from several "solutions" to that problem. I already found ...
6
votes
3answers
202 views

Solving recurrence relation: Product form

Please help in finding the solution of this recursion. $$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$ where $ f(1)=1$ and $f(2)=2$.
0
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0answers
34 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
0
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3answers
24 views

A property regarding intervals

While I was solving a problem on TopCoder I used the following assumption. I have n intervals: $ [a_1,b_1], [a_2,b_2],...,[a_n,b_n]$ and a number $T$ such that: $$ a_1 + a_2 + ... + a_n \leq T \leq ...
1
vote
1answer
22 views

How Jaccard similarity can be approximated with minhash similarity?

In Page 81 of this book, Mining of Massive Data Sets. It says the following: Now, consider the probability that h(S1) = h(S2). If we imagine the rows permuted randomly, and we proceed from the top, ...
0
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0answers
3 views

SPSA loss function

I need a simple implementation (in Matlab) of the most basic SPSA algorithm for a friend of mine. I have found this code from J.C. Spall but I don't know where to find an implementation of an ...
1
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1answer
63 views

Find the pair of values $a[i]$, $a[j]$ such that $a[i]\,\&\,a[j]$ is maximum

Given a list of positive integers, find the largest possible value of $a[i]$ $\&$ $a[j]$, where $i$, $j$ are indices of the list. $ i\ne j $, $a[i]\,\&\,a[j]$ is bitwise AND of $a[i]$ and ...
-1
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0answers
53 views

Maximise and operation [duplicate]

Given an array of $n$ non-negative integers $A_1, A_2, \dots, A_N$, find a pair of integers $A_u$, $A_v$, where $1 \leq u < v \leq N$, such that the bitwise-and ($A_u$ and $A_v$) is as large as ...
2
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1answer
50 views

Why is the sentence true?

I am looking at the algorithm of the insertion sort: ...
0
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1answer
42 views

Best approach or algorithm to solve equation with multiple variables?

I have an equation : $A^6x_1 + A^5x_2 + A^4x_3 + A^3x_4 + A^2x_5 + A^1x_6 + x_7 = B$ What can be the best algorithm/approach I can use to crack this? $A$ and $B$ are constants. $x_1,x_2...x_7$ are ...
0
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0answers
20 views

How does inserting N objects one at a time into an ordered AVL tree yield an efficient sorting algorithim

If we assume reblalancing an AVL tree of height n after an insertion or deletion takes O(n) operations. How does inserting N objects one at a time into an ordered AVL tree yield an efficient sorting ...
1
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0answers
29 views

Lowest sum of 2 sets of number pairs

I am given a set of unique integers $n$. I need to compute the smallest sum $s$ such that there are two different pairs of integers $(x1, x2)$ and $(y1, y2)$ where $x1 < x2$ and $y1 < y2$ and ...
1
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0answers
38 views

Finding whether a sum of numbers in a set generate another number

I have a set of numbers {a1....an} and another number k. I need to find whether sum of any combination of numbers in the set ...
2
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2answers
31 views

Flip cards to get maximum sum

Given N cards where if ith card has number x on its front side then it will have -x on back side and a single operation that can be done only once that is to flip any number of cards in consecutive ...
0
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1answer
56 views

Algorithm for a counter [closed]

I am looking for an accurate algorithm for calculation required in any kind of counter, Like for example I have two values a: max 48 bits b: max 30 bits x: (a << 30) | (b); Now, if value of b ...
1
vote
1answer
41 views

Find extra work done by Bob

Alice has challenegd Bob game of N puzzle.N puzzle is played on N*N grid with each cell containing distinct numbered tile from 1 to N*N-1 Except one which is empty cell and represented as 0. Move ...
0
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0answers
14 views

Strassens Matrix Multiplication Algorithm to compute product of 2 4X4 Matrices

Im trying to learn starssens matrix multiplication Algorithm.So far i know that it uses 7 multiplications and replaces a multiplication by several additions and subtractions,to achieve better ...
0
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2answers
22 views

Algorithm to find all possible base26 numbers of a 32-bit cropped value

I have following formula: base26value = 26(26(26(26(26(26a + b) + c) + d) + e) + f) + g The variables a..g have a range ...
0
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1answer
31 views

Kolmogorov complexity of an algorithm?

I've read that Kolmogorov comlexity is about calculating the least number of bits needed to describe a string or other mathematical objects. Does 'other mathematical objects' include algorithms too? ...
1
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0answers
27 views

directed simple graph, all paths from node $ v_0 $ to an other node $ v $, MATLAB

consider a directed simple graph $ G=(V,E) $ with $ V=\lbrace v_0,v_1,\ldots,v_k \rbrace $ and adjacency matrix $ A=(a_{ij}) $, where $ a_{ij}=1 $ means, that there is an arc from node $ v_i $ to node ...
-3
votes
1answer
41 views

Best/Worst Test Cases

For the two functions, would their worst/best Big O cases change compared to each others? If so, how would they change. I believe that when dealing with Big O, order is the only thing that matters, so ...
-1
votes
1answer
19 views

Determine fraction length of fixed-point binary

How to determine fraction length of fixed-point binary so that distinct entries of a group of decimal numbers (for example: 1, 0.456, 0.444) remain distinct after converting them from decimal to ...
0
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0answers
32 views

Can someone help me with the following math question/dilemma?

I have a pool of objects that are randomly selected from a global object database. The objects certain numeric attributes: The objects from the pool are fed to users in real time Users will either ...
0
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2answers
107 views

Finding all possible pairs of integers $(a,b)$ such that $a^b=n$.

Given a large integer $n$ (could be as large as $10^{18}$), how can I find all possible pairs of integers $(a,b)$ such that $$a^b=n.$$ A fast algorithm is preferable. The question How to quickly ...
2
votes
1answer
23 views

The probability of getting a certain image by random pixelation

Well, seeing that I'm terribly bad at math I don't know how to solve this, I'll try to explain, excuse me if I sound dumb. Just suppose that I've got a photo/image with 320x240 resolution and 24 bit ...
1
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0answers
24 views

Solving tridiagonal matrices where the top left element is zero

If I have a matrix like this: $$ \left[\begin{array}{rrrrrrrrr|r} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & ...
0
votes
1answer
30 views

Find a kernel in a directed graph.

It's a question from a sample exam I'm trying to solve but with no success yet. Let $G(V, E)$ be a directed graph. set $A \subseteq V$ is a kernel if: i. $\forall u,v\in A \implies (u, v), ...
2
votes
1answer
31 views

Show that the height of the heap is $\lfloor \lg n \rfloor$

I want to show that a heap with $n$ elements has the height $ \lfloor \lg n\rfloor$. That's what I have tried: The "best" case is a complete binary tree,and then it is of the form: So,the height ...
1
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2answers
18 views

Asymptotic Notations in limits

Can the asymptotic notations, like Big O, be defined using limits? example:- Lim x->infi (f(n)/g(n))=c for defining f(n)=O(g(n)) If not, why??