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What books are there to learn to implement these graph algorithms? |
|graphs| |
I'm a math background student and not familiar with algorithms and coding. I read [this post][1] on reddit where the author used label propagation, Fruchterman-Reingold algorithm, edge betweenness clustering to classify math subjects based on the data of published articles in 2018 on arxiv.org. I think being able to im... |
I'm a math background student and not familiar with algorithms and coding. I read [this post][1] on reddit where the author used label propagation, Fruchterman-Reingold algorithm, edge betweenness clustering to classify math subjects based on the data of published articles in 2018 on arxiv.org. I think being able to im... |
The standard/usual/common definition of a push-down automaton allows instructions of the form $(p,a,A,q,\alpha)$ meaning in state $p$ reading $a$ from the input and $A$ on top of the stack, we move to state $q$ and replace $A$ by $\alpha$ on the stack.
Thus, we always combine both a pop $A$ and a push $\alpha$.
[... |
null |
I saw a post on Reddit (https://www.reddit.com/r/math/comments/ci50d3/visualizing_mathematical_subjects/) that utilizes label propagation, Fruchterman-Reingold algorithm, and edge betweenness clustering to classify mathematical subjects based on papers in 2018 on arxiv.org.
In a math book, definitions and theorems a... |
I saw a post on Reddit (https://www.reddit.com/r/math/comments/ci50d3/visualizing_mathematical_subjects/) that utilizes label propagation, Fruchterman-Reingold algorithm, and edge betweenness clustering to classify mathematical subjects based on papers in 2018 on arxiv.org.
In a math book, definitions and theorems a... |
> Auxiliary Space: The extra space that is taken by an algorithm
> temporarily to finish its work
>
> Space Complexity: Space complexity is the total space taken by the
> algorithm with respect to the input size plus the auxiliary space that
> the algorithm uses.
Referred from [here][1]
i.e., space complexi... |
null |
For question 1, I suggest that you start with basic programming courses. If you understand how to program, and you understand the specific algorithms you are referring to, you should be able to implement them. You should not need a specific course or textbook that is specifically on implementing those particular algo... |
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In the simply typed lambda calculus we have type algebra - types can be added, multiplied and exponentiated, where addition corresponds to the sum type, multiplication to the product type, and functions to the exponential type (where A -> B corresponds to B^A). Due to the commutativity and associativity of the sum type... |
How is the direct product of the functions (A -> B) * (C -> D) equivalent to the function (A * C) -> (B * D)? Is there an error here? |
|lambda-calculus|type-theory|sets| |
Let's say, I have a set of cats $C = \{ a, b, c, ... \}$. Each cat has a tail. I can say, that the lengths of the tails of cats are denoted as $t_a$, $t_b$, etc. Otherwise, I can say that they are denoted as $t(a)$, $t(b)$, etc. Which option is preferable and when? |
When to use subscript and when to make it a function? |
|notation| |
|graphs|data-mining|knowledge-representation| |
null |
Is it so that for each virtual adress there exists an adress in the hard disk? But the hard disk can be larger than the virtual memory? |
Is there an adress in the hard disk for each virtual adress? |
|memory-management|virtual-memory| |
Is it so that for each virtual adress there exists an adress in the hard disk? But the hard disk can be larger than the virtual memory?
EDIT: Or can some virtual adresses always map to physical, and some map to hard drive which then is loaded to physical adresses? |
A minimum weight perfect matching finds a min-weight disjoint set of edges that pair up all nodes in a graph. It can be solved efficiently by using the [blossom algorithm](https://en.wikipedia.org/wiki/Blossom_algorithm).
A *marked* minimum weight perfect matching problem finds a set of paths that pair up the marked... |
Known algorithm for marked matching with degree 3 edges? |
|reference-request|matching|error-correcting-codes| |
There is a well-known Curry-Howard-Lambek correspondence between certain type systems, proof calculi and categories.
Some variants of Barendregt's pure type systems have the property of strong normalization, i.e. any term can be reduced to a normal form after a finite number of reduction steps, the other of them don... |
I am currently in the process of trying to understand the original proof of NP-completeness of SAT given in the seminal paper by Cook [[COOK71]](http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=805047) and have struggled with a few of the details of the proof of Theorem 1 presented in the paper. Specifically, ... |
Problems with proof of NP-completness of SAT following Cooks original paper |
|complexity-theory|np-complete|np-hard|satisfiability| |
A minimum weight perfect matching finds a min-weight disjoint set of edges that pair up all nodes in a graph. It can be solved efficiently by using the [blossom algorithm](https://en.wikipedia.org/wiki/Blossom_algorithm).
A *marked* minimum weight perfect matching problem finds a set of paths that pair up the marked... |
A minimum weight perfect matching finds a min-weight disjoint set of edges that pair up all nodes in a graph. It can be solved efficiently by using the [blossom algorithm](https://en.wikipedia.org/wiki/Blossom_algorithm).
A *marked* minimum weight perfect matching problem finds a set of paths that pair up the marked... |
As a newcomer to the field of parallel computation, is it accurate to say that CPUs are essentially "thread" calculators? My current understanding is that a "process" is made up of multiple "threads" and a single core of a CPU can execute instructions from one "thread" at a time, sometimes a bit more with hyper-threadi... |
Is it correct to say that CPUs only compute threads? |
**Problem :** Consider a random vector $v$ which is uniformly distributed over the sample space $S = \{v \in \mathbb{Z}^{n} : 1^Tv = a , v \ge 0\}$ . How to efficiently generate such random vector ?
note : I'm actually interested in $n=6,a=30$ .
**Attempt :** The size of the sample space [1] is the coefficient $... |
Generate uniform random vectors |
|algorithms|trees|combinatorics|probability-theory|random-number-generator| |
In a typical modern operating system, a page frame of RAM can essentially be one of four things:
- Free. Unallocated. Available for use at any time.
- Used by the operating system for its own internal purposes. An OS has data structures, and they need to live somewhere. Page tables might also fall under this gene... |
A minimum weight perfect matching finds a min-weight disjoint set of edges that pair up all nodes in a graph. It can be solved efficiently by using the [blossom algorithm](https://en.wikipedia.org/wiki/Blossom_algorithm).
A *marked* minimum weight perfect matching problem finds a set of paths that pair up the marked... |
A minimum weight perfect matching finds a min-weight disjoint set of edges that pair up all nodes in a graph. It can be solved efficiently by using the [blossom algorithm](https://en.wikipedia.org/wiki/Blossom_algorithm).
A *marked* minimum weight perfect matching problem finds a set of paths that pair up the marked... |
null |
You are correct. They are not equivalent. For example, the function $f$ defined as $f(a,c)=(1,1)$ if $a=c$, otherwise $f(a,c)=(0,0)$, is in (A * C) -> (B * D) (assuming $0,1 \in $ B,D), but has no corresponding element in (A -> B) * (C -> D). As you say, the number of elements in those two sets is different, so the... |
Both are valid, and the choice is based on personal preference and taste. One benefit of the notation $t(a),\dots$ is that it allows you to refer to the function $t$ that maps from $C$ to lengths, which may be useful in some contexts. One benefit of the notation $t_a,\dots$ is that if $C$ is finite and small, in some... |
Note that the number of vectors $v \in \mathbb{Z}^n$ such that $v \ge 0$ and $1^\top v = a$ is (by a stars-and-bars argument) exactly
$${a+n-1 \choose n}.$$
The number of such vectors with $v_1=b$ is exactly
$${a-b+n-2 \choose n-1}.$$
Therefore, the probability that $v_1=c$ is exactly
$${{a-b+n-2 \choose... |
does finite automata have memory? |
null |
null |
null |
We define the set S as {(s1, f1), (s2, f2), ..., (si, fi)}, where each si is the frequency that it is repeated in the multiset T. How many ways can we partition the multiset T into different partitions, where each partition includes distinct part/segments?
For example, given S = {(s1, 1), (s2, 2), (s3, 1)} results i... |
How many ways we can partition a multiset, where each part/segment in the partition has distinct elements? |
|algorithms|combinatorics|recursion|permutations| |
We define the set S as {(s1, f1), (s2, f2), ..., (si, fi)}, where each si is the frequency that it is repeated in the multiset T. How many ways can we partition the multiset T into different partitions, where each partition includes distinct part/segments?
For example, given S = {(s1, 1), (s2, 2), (s3, 1)} results i... |
null |
We define the set S as {(s1, f1), (s2, f2), ..., (si, fi)}, where each si is the frequency that it is repeated in the multiset T. How many ways can we partition the multiset T into different partitions, where each partition includes distinct part/segments?
For example, given S = {(s1, 1), (s2, 2), (s3, 1)} results i... |
We define the set S as {(s1, f1), (s2, f2), ..., (si, fi)}, where each si is the frequency that it is repeated in the multiset T. How many ways can we partition the multiset T into different partitions, where each partition includes distinct part/segments?
For example, given S = {(s1, 1), (s2, 2), (s3, 1)} results i... |
We define the set S as {(s1, f1), (s2, f2), ..., (si, fi)}, where each si is the frequency that it is repeated in the multiset T. How many ways can we partition the multiset T into different partitions, where each partition includes distinct part/segments?
For example, given S = {(s1, 1), (s2, 2), (s3, 1)} results i... |
We define the set S as $\{(s_1, f_1), (s_2, f_2), ..., (s_i, f_i)\}$, where each si is the frequency that it is repeated in the multiset T. How many ways can we partition the multiset T into different partitions, where each partition includes distinct part/segments?
For example, given $S = \{(s_1, 1), (s_2, 2), (s_3... |
null |
Bubble Sort's time complexity of $\Theta(n^2)$ is not good — it is a common worst-case complexity class for simple sorting algorithms that do not use divide & conquer methods to partition the dataset. These include eg. Selection Sort and Insertion Sort, each of which generally out-performs Bubble Sort due to making far... |
I don't know which algorithm is more efficient for sorting, but I want to use Bubble sort and how to implement this algorithm.
|
Is the bubble sort algorithm more efficient for sorting and how to implement it? |
|algorithms|sorting| |
null |
I think a good starting point is to look at a concrete sorting problem and then check the most voted solutions on LeetCode.
If you want to know how good bubble sort works on the problem you can just [filter the submitted solutions for "bubble sort"](https://leetcode.com/problems/sort-colors/solutions/?q=bubble+sort&... |
These are two different things that have two different goals.
The two-phase locking protocol is designed to guarantee serializability for transactions that access concurrently a single, *centralized* database, using a locking mechanism. So its goal is to maintain the consistency of a centralized database, avoiding “... |
No, it is not efficient at all. Its main advantage is that it is short to write, but that's all.
See [here](https://en.wikipedia.org/wiki/Sorting_algorithm#Comparison_sorts) for a comparison on comparison sorts.
If you still want to implement Bubble Sort, check [here](https://en.wikipedia.org/wiki/Bubble_sort#Imp... |
I'm looking for an algorithm to solve a modified version of the assignment problem. It differs from the [standard assignment problem][1] in that the modified version has groups of tasks instead of just tasks. Similarly to the standard problem, each of the *n* workers is assigned one task, and there are *n* groups of ta... |
Algorithm for a modified worker-task assignment problem with groups of tasks and substitutability between tasks within groups |
|algorithms|optimization|constraint-satisfaction|assignment-problem| |
null |
We have an array of integers of length N(even). A and B play a game where A selects a number followed by B without replacement until the array becomes empty. The goal is for A to have a sum which is (i) even, (ii) odd. In both cases, B plays against A and both players play optimally. We have to find who wins in (i) and... |
Another Game Theory Problem |
|algorithms|time-complexity|data-structures|game-theory| |
I am trying to analyze the algorithm Product to multiply two n-bits binary numbers x and y.
```
function Product(x, y);
1. prod = 0;
2. while y not eq 0
3. y = y -1; 4. prod = prod + x;
return prod;
```
Assume: n = |x| = |y|
Give the worst case time T(n) function for algorithm Product.
Express T(n) in ... |
How can we analyze the Product to multiply two n-bits binary numbers x and y? |
|algorithms| |
Note that the number of vectors $v \in \mathbb{Z}^n$ such that $v \ge 0$ and $1^\top v = a$ is (by a stars-and-bars argument) exactly
$${a+n-1 \choose n -1}.$$
The number of such vectors with $v_1=b$ is exactly
$${a-b+n-2 \choose n-2}.$$
Therefore, the probability that $v_1=b$ is exactly
$${{a-b+n-2 \cho... |
null |
Let's say there are four sets containing integers. No set has any duplicates within itself, but duplicates exist across sets.
```
A = [1, 2, 3, 4]
B = [2, 3, 4, 5]
C = [3, 4, 5, 6]
D = [4, 5, 6, 7]
```
The goal is to store each unique element in a resulting set `R`. The naïve single-threaded solution would be ... |
null |
We have an array of integers of length N(even). A and B play a game where A selects a number followed by B without replacement until the array becomes empty(Both A and B select N/2 elements each). The goal is for A to have a sum (of the N/2 elements A chooses) which is (i) even, (ii) odd. In both cases, B plays against... |
I am trying to analyze the algorithm Product to multiply two n-bits binary numbers x and y.
```
function Product(x, y);
1. prod = 0;
2. while y not eq 0
3. y = y -1; 4. prod = prod + x;
return prod;
```
Assume: n = |x| = |y|
Give the worst case time T(n) function for algorithm Product.
Express T(n) in ... |
|algorithms|time-complexity|data-structures|dynamic-programming|game-theory| |
You can solve this problem as follows. Build a bipartite graph with one vertex per worker and one vertex per group of tasks (one vertex for the entire group; not one vertex per task). The cost of assigning worker $w$ to group $g$ is the smallest cost of assigning $w$ to some task in $g$ (i.e., for each task $t$ in $g... |
null |
A simple question, but I can't find an answer in quite the form I'm looking for:
Assume $\mathsf{P} \neq \mathsf{NP}$, and thus $\mathsf{P} \subsetneq \mathsf{NP}$. If we have $L \in \mathsf{NP}$ and $L \notin \mathsf{P}$, does that imply that there is no algorithm for $L$ which runs in $O(n^k)$ for all inputs of si... |
Time complexity of $\mathsf{NP}$ problem under assumption of $\mathsf{P} \neq \mathsf{NP}$ |
|complexity-theory|time-complexity|np| |
A simple question, but I can't find an answer in quite the form I'm looking for:
Assume $\mathsf{P} \neq \mathsf{NP}$, and thus $\mathsf{P} \subsetneq \mathsf{NP}$. If we have $L \in \mathsf{NP}$ and $L \notin \mathsf{P}$, does that imply that there is no algorithm for $L$ which runs in $O(n^k)$ steps for all inputs... |
In a typical modern operating system, a page frame of RAM can essentially be one of four things:
- Free. Unallocated. Available for use at any time.
- Used by the operating system for its own internal purposes. An OS has data structures, and they need to live somewhere. Page tables might also fall under this gene... |
I've been struggling with this analysis of algorithms notion for a few hours now in an introductory course. Let us define a function $N(n)$ for positive integer $n$ as the number of times performing $n=log_7(n)$ yields $\leq 1$. So $N(1)=0, N(7)=1, N(7^7)=2$, et cetera.
I'm asked to find a good algorithm for computi... |
Bit model vs RAM model confusion |
|algorithms|algorithm-analysis| |
null |
It's often said that State Machine Replication and Byzantine broadcast are equivalent (e.g., the abstract of [1]), but it seems only obvious how to reduce SMR to BB. I.e., give BB one can construct an SMR by rotating the leader.
What does it even mean to reduce BB to SMR then? In SMR, there are clients and servers, ... |
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