All Questions
Tagged with co.combinatorics computer-science
128 questions
5
votes
1
answer
470
views
Arrangement of integers 1..k^2 in k*k grid to minimize energy function
Question arises from considering cache oblivious algorithms.
What is the optimal way arrange the numbers $1$ to $k^2$ in a grid, to minimize to average difference between any two neighbouring squares?...
- 51
5
votes
1
answer
2k
views
Decomposition of a complete graph into maximal matching subgraphs
Is there a general way to decompose a complete graph $K_n$ into an union of maximal matching subgraphs such that no two subgraphs share an edge?
For example, consider $K_4$ with vertices $V=${1,2,3,4}...
- 377
2
votes
1
answer
227
views
Recoving an unknown tree graph with knowledge of root node to leaf node distances
Imagine I have an unknown (undirected) tree graph, $G$, with some unknown number of nodes $||V||$. However, I know the edge-length between nodes is of fixed size, $L_{edge} = 1$, and I have access to ...
- 329
1
vote
3
answers
501
views
Operator probability in a RPN string
Consider the set $S_n$ of all strings of length $n$ ($n$ integer, $n \geq 3$)
representing an expression in RPN
( http://en.wikipedia.org/wiki/Reverse_Polish_notation. )
Assumptions (to simplify):
...
- 31
22
votes
9
answers
17k
views
Fast evaluation of polynomials
Hello everybody !
I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer ...
- 1,654
3
votes
2
answers
830
views
Partition a square into sub-rectangles with restrictions
Is there an algorithm to generate all partitions of given square by using $n$ vertical and $n$ horizontal lines into sub-rectangles under the following restrictions:
1- No vertical line crosses any ...
- 2,993
0
votes
1
answer
182
views
the maximal length of a special dicksonian sequence
First, we define a sequence $t_{1},t_{2},\cdots,t_{k}$ of n-tuples dicksonian, if $\forall 1\leq i < j\leq k,$ there does not exist a non-negative n-tuple t such that
$t_{i}+t=t_{j}.$ For example, ...
- 1,528
5
votes
1
answer
700
views
What is the pathwidth of the 3D-grid (mesh or lattice) with sidelength k?
This question is now also on https://cstheory.stackexchange.com/questions/4081/what-is-the-pathwidth-of-the-3d-grid-mesh-or-lattice-with-sidelength-k, where a discussion started, and one reference ...
- 151
4
votes
2
answers
2k
views
Coloring edges on a graph s.t. the set of edges for any two vertices have no more than 'k' colors in common
Please imagine the case where one has a planar graph, $G$, with a set of $|V|$ vertices, $(v_1, ..., v_{|V|}) \in V$, and $|E|$ edges, $(e_1, ..., e_{|E|}) \in E$. Now, provided a total of $N$ colors,...
3
votes
3
answers
390
views
Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?
I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
- 33
5
votes
1
answer
362
views
Drawing graphs on circles
Please consider the following problem:
Given: a simple graph (without self-loops and without multiple edges) $G$ on $n$ vertices.
Task: place equidistantly the vertices of $G$ on a circle of unit ...
user3409
30
votes
1
answer
3k
views
An edge partitioning problem on cubic graphs
Hello everyone,
I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...
- 1,164
0
votes
3
answers
402
views
boolean functions and averaging / counting
Hey guys,
I have a slightly imprecise question. I would like say something about a whole set of binary strings evaluated by a binary function by just looking at some type of average. The easiest ...
- 95
3
votes
4
answers
2k
views
Enumerative algorithm through inclusion-exclusion
Hello everybody !
I wondered, without really knowing where to search, whether there was a "smart" way to enumerate/iterate over all the elements of a set which can be counted by inclusion-exclusion. ...
- 1,654
8
votes
1
answer
2k
views
Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice
Please imagine a discrete random walk on an N-dimensional rectangular lattice with dimensional lengths $(l_1, ..., l_N) \in L$ and total lattice points $P = \prod{l_i}$, for $i = 1, ..., N$. At each ...
- 599
2
votes
1
answer
307
views
Approximating a recursively-defined function
Let $$f(k) := \frac{2k-1}{k}\bigl(1-\sum\limits_{i\lt k}\frac{i\ f(i)}{k+i-1}\bigr)$$ for $k\in\mathbb{N}^{+}$.
So $f(1) = 1$, $f(2) = 3/4$, $f(3) = 35/72$, etc.
(This function arises when ...
- 2,896
1
vote
0
answers
576
views
Minimizing quadratic form over permutations
Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem:
$\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$,
where $S_n$ ...
- 1,839
6
votes
2
answers
1k
views
Bijective proof of weak form of Stirling's approximation
There are short and sweet proofs of various forms of Stirling's approximation. But even the sweetest among them don't instill the same conviction in the reader as a direct bijective proof.
Computer ...
- 2,071
2
votes
2
answers
249
views
Indexing schemes of binary sequences
I am looking for "low-complexity" indexing methods to enumerate binary sequences of a given length and a given weight.
Formally, let $T_k^n = \{x_1^n \in \{0,1\}^n: \sum_{i=1}^n x_i = k\}$. How to ...
- 1,839
7
votes
1
answer
357
views
How long are the certificates produced by the Zeilberger and WZ methods for solving combinatorial sums (A=B)?
In the book "A = B" by Petkovesk, Wilf, and Zeilberger, (downloadable here), the authors provide several algorithmic methods for finding closed forms or recurrences for sums involving e.g. binomial ...
- 23k
11
votes
1
answer
696
views
I am searching for the name of a partition (if it already exists)
I derived this definition by searching for a representation of a family of sets. I am quite sure that someone should have thought to this before, because it seems to be quite straightforward given a ...
- 143
9
votes
0
answers
759
views
Finding a set with the maximum number of finite alphabet strings within a fixed Levenshtein distance of one-another
Please consider the set of all possible strings of some finite size $M$ alphabet $\Sigma$, $\alpha$ $= a_1, a_2, ..., a_k, ..., a_n$, of length $|\alpha| = L$. The Levenshtein distance (or 'edit ...
7
votes
1
answer
805
views
Counting Eulerian Orientation in a 4-regular undirected graph
We would like to know how hard it is to count Eulerian orientation in an undirected 4-regular graph. For a given edge orientation to be Eulerian, we mean that every vertex has 2 in-edges and 2 out-...
11
votes
1
answer
860
views
Counting colored rook configurations in the cube - when is it even?
Informal Statement
In the $n\times n \times n$ grid, we can places rooks (those from chess) such that no two rooks can attack each other. One way to achieve this is to place a rook in position $(i,j,...
- 1,088
13
votes
1
answer
799
views
Bipartite Nim-Geography
Two players are playing a game on a bipartite graph where all of the edges are nim-heaps of various sizes. A token starts on one of the vertices, and on your turn you must move the token over an edge ...
- 8,688
4
votes
6
answers
751
views
Reconstructing an ordering of a multiset from its consecutive submultisets
We have a multiset $S$ of size $t$ with $r$ distinct elements, where $t$ is much larger than $r$. We want to reconstruct an ordering $s_1, s_2, ... s_t$ of the elements of $S$ given the values of $t$ ...
- 599
12
votes
1
answer
1k
views
Characterization of Boolean-valued functions on the discrete cube based on its Fourier coefficients.
Consider functions on the discrete cube $\{-1,1\}^n$.
We consider the Discrete Fourier Transform of such functions. Suppose we denote the parity function on a subset $S \subseteq [n]$ of co-...
20
votes
5
answers
1k
views
Is there a natural family of languages whose generating functions are holonomic (i.e. D-finite)?
Let $L$ be a language on a finite alphabet and let $L_n$ be the number of words of length $n$. Let $f_L(x) = \sum_{n \ge 0} L_n x^n$. The following are well-known:
If $L$ is regular, then $f_L$ is ...
- 118k