Questions tagged [discrete-mathematics]

Questions about discrete mathematics, the study of mathematical structures that are fundamentally discrete rather than continuous.

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29
votes
2answers
12k views

Counting binary trees

(I'm a student with some mathematical background and I'd like to know how to count the number of a specific kind of binary trees.) Looking at Wikipedia page for Binary Trees, I've noticed this ...
6
votes
2answers
138 views

Invertible function that randomizes order

I am looking for an invertible discrete function $f:\{0,1,2,\dots,n-1\} \to \{0,1,2,\dots,n-1\}$ for some given integer $n$. I want $f(0),f(1),\dots,f(n-1)$ to return all the integers in range $[0..n)...
10
votes
3answers
51k views

Why is the minimum height of a binary tree $\log_2(n+1) - 1$?

In my Java class, we are learning about complexity of different types of collections. Soon we will be discussing binary trees, which I have been reading up on. The book states that the minimum height ...
20
votes
4answers
9k views

What is an intuitive way to explain and understand De Morgan's Law?

De Morgan's Law is often introduced in an introductory mathematics for computer science course, and I often see it as a way to turn statements from AND to OR by negating terms. Is there a more ...
18
votes
6answers
3k views

Real life examples of negative weight edges in graphs

I am unable to relate to any real life examples of negative weight edges in graphs. Distances between cities cannot be negative. Time taken to travel from one point to another cannot be negative. ...
6
votes
1answer
626 views

How to find a subset of potentially maximal vectors (of numbers) in a set of vectors

I have a set S (so no duplicates) of d-dimensional vectors of non-negative real numbers (or if you would prefer, floats). I say a vector u "covers" a vector v if, in every dimension 1..d, u[i] >= v[i]...
2
votes
1answer
797 views

Counting elements that are greater than the median of medians

Short version: I want to know where the $-2$ comes from in the formula on p. 221 of CLRS 3rd edition. Long version: CLRS (3rd ed.) give an algorithm for $O(n)$ worst case arbitrary order statistic of ...
38
votes
6answers
7k views

What use are groups, monoids, and rings in database computations?

Why would a company like Twitter be interested in algebraic concepts like groups, monoids and rings? See their repository at github:twitter/algebird. All I could find is: Implementations of ...
20
votes
3answers
11k views

How hard is finding the discrete logarithm?

The discrete logarithm is the same as finding $b$ in $a^b=c \bmod N$, given $a$, $c$, and $N$. I wonder what complexity groups (e.g. for classical and quantum computers) this is in, and what ...
10
votes
2answers
924 views

What mathematics can be interesting for these CS areas?

For my CS degree I have had most of the "standard" mathematical background: Calculus: differential, integral, complex numbers Algebra: pretty much the concepts up until fields. Number Theory: XGCD ...
10
votes
5answers
29k views

How much math does one need to know to understand discrete math/structures for computer science?

Normally universities teach discrete math / discrete structure. My question is, how much math does one need to know to understand this area? Is calculus required or will precalculus do just fine? Does ...
16
votes
1answer
229 views

Polytime and polyspace algorithm for determining the leading intersection of n discrete monotonic functions

Some frontmatter: I'm a recreational computer scientist and employed software engineer. So, pardon if this prompt seems somewhat out of left field -- I routinely play with mathematical simulcra and ...
8
votes
3answers
448 views

Algorithm to shrink a DFA by introducing nondeterminism?

This is somewhat related to another question I asked, but I feel it's different enough to warrant its own question. I'm doing research where I'm trying to find the structure of complements of a ...
7
votes
1answer
2k views

Proof of Ramsey's theorem: the number of cliques or anti cliques in a graph

Ramsey's theorem states that every graph with $n$ nodes contains either a clique or an independent set with at least $\frac{1}{2}\log_2 n$ nodes. I tried to look it up at a few places (including ...
6
votes
1answer
3k views

What is the maximum number of shortest paths between any pair of vertices in a chordal graph?

A graph $G$ is chordal if it doesn't have induced cycles of length 4 or more. Chordal graphs are precisely the class of graphs that admit a clique tree representation. A clique tree $T$ of $G$ is a ...
4
votes
2answers
158 views

Min path cover for a three-layer graph with all paths traversing all layers

Best to start with an example. I want to design fictional fruits. The fruits have three attributes: color, taste and smell. There are $c$ possible colors, $t$ possible tastes and $s$ possible smells. ...
2
votes
2answers
212 views

Is discrete math enough for computer science ? Or there other Math topics that I should also learn With it?

I want to learn computer science, SO is discrete math enough for computer science ? Or there other Math topics that I should also learn With it ? I don’t have specific topic that I care more about ...
1
vote
1answer
153 views

What makes an MILP problem solvable?

Knapsack problems, Assignment problems can all be expressed as (MILP) mixed integer linear programs. MILP is NP-complete. But Knapsack problem is solvable in pseudo-polynomial time using dynamic ...
0
votes
1answer
95 views

Do all the numbers belong to same slot in the Hashtable?

I was reading the CLRS. In the Hashing Chapter on page 262 a statement says: "For example, if we know that the keys are random real numbers $k$ independently and uniformly distributed in the range $0 \...
3
votes
2answers
567 views

How to check if a specific ILP problem can be solved in polynomial time or not?

How can we know that a specific ILP problem is solvable in polynomial time or not given the constraints?
2
votes
1answer
4k views

Average height of a BST with n Nodes

I have to find the maximum, minimum, and average height of a BST with n nodes. After doing some researching I found that the maximum height is $n-1$ and the minimum height is $\log_2(n+1)-1$. My ...
2
votes
1answer
60 views

Prove that there is a sequence of k minimum spaning trees between two distinct minimum spanning trees that each one is different in only 1 edge [duplicate]

I'm pracitcing exams towards finals, Given an undirected graph $G(V,E)$ , we denote 2 MST $T,T'$ neighbours if by deleting one edge from $T$ and add another one we get $T'$. Prove : for every 2 ...
2
votes
2answers
716 views

How to prove with induction [duplicate]

So far I have learned how to write proofs by induction and it went fine until I got this recursive problem, which I'm not quite sure how to begin and how to prove that with induction. ...
2
votes
1answer
2k views

Calculating Binet's formula for Fibonacci numbers with arbitrary precision

Binet's formula for the nth Fibonacci numbers is remarkable because the equation "converts" via a few arithmetic operations an irrational number $\phi$ into an integer sequence. However, using finite ...
1
vote
1answer
4k views

Is the reverse postorder of a digraph's reverse the same as the postorder of the digraph?

I've been reading Sedgewick's intro to algorithms book, and he says that the reverse postorder of a digraph's reverse is not the same as the postorder of the digraph, however in both cases it seems ...
1
vote
2answers
573 views

Finding the maximum of a bitonic sequence

Given to us is an array $B[1],\ldots,B[n]$ as input, which satisfies the following property: there exists a special index $i^* ∈ \{1, \dots , n\}$ such that $$B[1] < B[2] < \dots < B[i^*− 1] &...
0
votes
1answer
129 views

How to exclude all points adjacent to a given point from the feasible region of IP

Consider a basic integer program such as: $$\begin{align} \min_x & \quad c^Tx \\ \text{s.t.} & \quad Ax \leq b \\ &\quad x_i \in \{-100,\ldots,100\} \end{align} $$ where $x \in \mathbb{...
0
votes
1answer
85 views

What is the difference in 'logical array blocked' and array list B, and what do they represent?

In Johnson's 1975 Paper 'Finding All the Elementary Circuits of a Directed Graph', his psuedocode refers to two separate data structures, logical array blocked and list array B. What is the difference ...