Count Number of Sequences
algorithm,math,combinations,permutation,discrete-mathematics
Inversions The key to this type of problem (of swapping adjacent elements) is to consider the number of inversions in the permutation after each swap. Two elements a[i] and a[j] form an inversion if a[i] > a[j] and i < j. The number of inversions in a permutation is the...
Fastest k nearest neighbor with arbitrary metric?
algorithm,math,discrete-mathematics,nearest-neighbor
I would advice you to go for Locality-sensitive hashing (LSH), which is in trend right now. It reduces the dimensionality of high-dimensional data, but I am not sure if your dimension will go well with that algorithm. See the Wikipedia page for more. You can use your own metric, but...
Generating all lexicographical permutations without comparisons of elements
algorithm,permutation,discrete-mathematics,lexicographic
You can do it this way: // n is the number of elements in the given sequence p = all permutations of [0, 1, ..., n - 1] in lexicographical order for permutation in p: for i = 0 ... n - 1 print(s[permutation[i]]) // s is the given sequence...
How to know the lowest value that is not in a group of numbers
java,algorithm,discrete-mathematics
You should decompose your problem. Global algorithm IMO, you should start at n = 0, and go up. For each value of n, check if the sequence of digits needed to represent it is allowed. As soon as you can't, n is your answer. Smaller part The new problem is...
Planar graph and Finding Triangles
algorithm,math,graph,discrete-mathematics
Let V be the number of vertices, E be the number of edges, F3 be the number of triangles, and F4 be the number of quads. We have equations V = 16 # given 4 V = 2 E # number of vertex-edge incidences 3 F3 + 4 F4 =...
Prove that (x+1)! is not O(x!) [closed]
algorithm,big-o,discrete-mathematics,proof
Your proof is 100% correct. An alternative one is using limits to infinity. lim_(x->infinity) (x+1)!/x! when x->infinity = lim_(x->infinity) x = infinity The above contradicts (x+1)! for being in O(x!)...
Apparent error in Matlab's perms function
matlab,math,permutation,discrete-mathematics,elementary-functions
Yes, this seems to be a bug. Good catch! But probably a bug in the documentation, rather than in the function. If you type open perms to see the source code, you'll see the following description in the first lines: %PERMS All possible permutations. % PERMS(1:N), or PERMS(V) where V...
Combine data into smaller discrete intervals
f#,mapping,intervals,discrete-mathematics
let interval = 10 let index = [6;12;18;24] let value =[101;102;103;104] let intervals = List.map (fun e -> e/interval) index let keys = List.map2(fun e1 e2 -> (e1,e2)) intervals value let skeys = Seq.ofList keys let result = skeys |>Seq.groupBy (fun p -> fst p) |>Seq.map (fun p -> snd...
PHP: how to input binary operators + - * / % in a form?
php,user-input,discrete-mathematics
Try this <?php $n = $_POST['c1']; $p = trim(str_replace(' ', '', $_POST['c2'])); // Remove spaces eval("\$result = {$p};"); echo $result; ?> ...
How to sum sequence?
c++,algorithm,math,discrete-mathematics
This is Dirichlet's divisor summatory function D(x). Using the following formula (source) where gives the following O(sqrt(n)) psuedo-code (that happens to be valid Python): def seq_sum(n): sqrtn = int(math.sqrt(n)) return sum(n // k for k in range(1, sqrtn + 1)) * 2 - sqrtn ** 2 Notes: The // operator...
Proving optimality for a new algorithm that finds spanning tree with contradiction
algorithm,graph,graph-theory,discrete-mathematics,minimum-spanning-tree
Yes, the algorithm always finds a spanning tree (if such exist, i.e. the graph has a source). Not sure if it finds a minimal spanning tree though. You have already proven there are no cycles in the resulting graph, so we only need to show connectivity. Proof that the resulting...
Graph coloring upper bound
graph,discrete-mathematics,graph-coloring
in here it is so clear,by the way the sentence is In an optimal coloring there must be at least one of the graph’s m edges between every pair of color classes I'll break it down and make every part more clear. In an optimal coloring : this means that...
F(n) = F(n-1) - F(n-2)
math,sequence,discrete-mathematics,cyclic
Another way to approach this problem. Note that F(n) = F(n - 1) - F(n - 2) is the same as F(n) - F(n - 1) + F(n - 2) = 0 which makes it a linear difference equation. Such equations have fundamental solutions a^n where a is a root...
Proving breadth-first traversal on graphs
algorithm,graph-algorithm,discrete-mathematics,proof
Disclaimer: I don't know how much formal you want your proof to be and I'm not familiar with formal proofs. induction on the while loop: Is it true at the beginning? Does it remain true after a step (quite simple path property)? same idea, induction on k (why k+1???): Is...
What is a good algorithm to check whether or not a number exist in multiple sets without searching them all?
database,algorithm,math,sharding,discrete-mathematics
One data structure suitable for this application is a Bloom filter. A Bloom filter is a probabilistic data structure which allows you test whether an item is already in a set. If the test returning false then the item is definitely not in the set (0% false negatives), if true...
Roster/Timetable generation
discrete-mathematics,solver,simulated-annealing,rostering,tabu-search
If you go with OptaPlanner and don't want to follow the Employee Rostering design of assigning 8 hours Shifts (planning entities) to Employees (planning value), because of your 2th constraint, then you could try to follow the Cheap Time Example design, something like this: @PlanningEntity public class WorkAssignment { Employee...
Placing squares on a Cartesian plane and working out their X,Y coords based on sequence number
math,sequence,discrete-mathematics,cartesian
(On edit: I added a second function which enables the Cartesian coordinate version to be obtained directly.) I got this far before my head exploded. It is closed form in the sense that it gives the coordinates of say the one millionth square without needing to place them one by...
How to find all strings that do not contain substring palindromes
string,algorithm,probability,combinatorics,discrete-mathematics
Suppose you build up palindrome-free strings one letter at a time. For the first letter, you have M choices, and for the second, you have M-1, since you can't use the first letter. This much is obvious. For every letter after the first two, you can't use the previous letter,...
How to find the coordinates of a line in HTML5 Canvas
javascript,html5,canvas,geometry,discrete-mathematics
Here's one way: Save all your line segments (walls) in an array. var walls=[]; var wall={x0:50,y0:50,x1:150,y1:150}; walls.push(wall); When you are dragging your window into place, compare the mouse position to the nearest point on every line segment (wall). Place the window on whichever wall is closest to the mouse. This...
I can't figure out this sequence - 11110000111000110010
python,algorithm,math,sequence,discrete-mathematics
There are many possible solutions to this problem. Here's a reusable solution that simply decrements from 4 to 1 and adds the expected number of 1's and 0's. Loops used : 1 def sequence(n): string = "" for i in range(n): string+='1'*(n-i) string+='0'*(n-i) return string print sequence(4) There's another single-line...
High value among some values disturbs the average…How to avoid the effect?
math,statistics,discrete-mathematics,moving-average,usage-statistics
The classical solution to this kind of problem is using the median instead of the average. You can also combine these two ideas: discard a certain amount of values (e.g. the upper and the lower 25%) and compute the mean of the remainder. This is called a truncated mean.
Algorithmic big o order of growth code
algorithm,discrete-mathematics
The loops in fact only go up to the cube root of N. (i^3 < n, etc.) The 3 nested loops of this length, give O(cube root of N, cubed). This O(N) Of note, if you were correct and they each went to one third of N, then cubing this...
Algorithm on List and Maximum Product
c++,algorithm,math,data-structures,discrete-mathematics
I don't know about the first statement, but the second statement can be said false by the following argument: Since there are sqrt(n) sequences, with n elements each, the total number of elements in n*sqrt(n). In the worst case, you would need to check every element at least once to...