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I would like to understand how to solve the Codility ArrayRecovery challenge, but I don't even know what branch of knowledge to consult. Is it combinatorics, optimization, computer science, set theory, or something else?
Edit:
The branch of knowledge to consult is constraint programming, particularly constraint propagation. You also need some combinatorics to know that if you take k numbers at a time from the range [1..n], with the restriction that no number can be bigger than the one before it, that works out to be
(n+k-1)!/k!(n-1)! possible combinations
which is the same as the number of combinations with replacements of n things taken k at a time, which has the mathematical notation . You can read about why it works out like that here.
Peter Norvig provides an excellent example of how to solve this kind of problem with his Sudoku solver.
You can read the full description of the ArrayRecovery problem via the link above. The short story is that there is an encoder that takes a sequence of integers in the range 1 up to some given limit (say 100 for our purposes) and for each element of the input sequence outputs the most recently seen integer that is smaller than the current input, or 0 if none exists.
input 1, 2, 3, 4 => output 0, 1, 2, 3
input 2, 4, 3 => output 0, 2, 2
The full task is, given the output (and the range of allowable input), figure out how many possible inputs could have generated it. But before I even get to that calculation, I'm not confident about how to even approach formulating the equation. That is what I am asking for help with. (Of course a full solution would be welcome, too, if it is explained.)
I just look at some possible outputs and wonder. Here are some sample encoder outputs and the inputs I can come up with, with * meaning any valid input and something like > 4 meaning any valid input greater than 4. If needed, inputs are referred to as A1, A2, A3, ... (1-based indexing)
Edit #2
Part of the problem I was having with this challenge is that I did not manually generate the exactly correct sets of possible inputs for an output. I believe the set below is correct now. Look at this answer's edit history if you want to see my earlier mistakes.
output #1: 0, 0, 0, 4
possible inputs: [>= 4, A1 >= * >= 4, 4, > 4]
output #2: 0, 0, 0, 2, 3, 4 # A5 ↴ See more in discussion below
possible inputs: [>= 2, A1 >= * >=2, 2, 3, 4, > 4]
output #3: 0, 0, 0, 4, 3, 1
possible inputs: none # [4, 3, 1, 1 >= * > 4, 4, > 1] but there is no number 1 >= * > 4
The second input sequence is very tightly constrained compared to the first just by adding 2 more outputs. The third sequence is so constrained as to be impossible.
But the set of constraints on A5 in example #2 is a bit harder to articulate. Of course A5 > O5, that is the basic constraint on all the inputs. But any output > A4 and after O5 has to appear in the input after A4, so A5 has to be an element of the set of numbers that comes after A5 that is also > A4. Since there is only 1 such number (A6 == 4), A5 has to be it, but it gets more complicated if there is a longer string of numbers that follow. (Editor's note: actually it doesn't.)
As the output set gets longer, I worry these constraints just get more complicated and harder to get right. I cannot think of any data structures for efficiently representing these in a way that leads to efficiently calculating the number of possible combinations. I also don't quite see how to algorithmically add constraint sets together.
Here are the constraints I see so far for any given An
An > On
An <= min(Set of other possible numbers from O1 to n-1 > On). How to define the set of possible numbers greater than On?
Numbers greater than On that came after the most recent occurrence of On in the input
An >= max(Set of other possible numbers from O1 to n-1 < On). How to define the set of possible numbers less than On?
Actually this set is empty because On is, by definition, the largest possible number from the previous input sequence. (Which it not to say it is strictly the largest number from the previous input sequence.)
Any number smaller than On that came before the last occurrence of it in the input would be ineligible because of the "nearest" rule. No numbers smaller that On could have occurred after the most recent occurrence because of the "nearest" rule and because of the transitive property: if Ai < On and Aj < Ai then Aj < On
Then there is the set theory:
An must be an element of the set of unaccounted-for elements of the set of On+1 to Om, where m is the smallest m > n such that Om < On. Any output after such Om and larger than Om (which An is) would have to appear as or after Am.
An element is unaccounted-for if it is seen in the output but does not appear in the input in a position that is consistent with the rest of the output. Obviously I need a better definition than this in order to code and algorithm to calculate it.
It seems like perhaps some kind of set theory and/or combinatorics or maybe linear algebra would help with figuring out the number of possible sequences that would account for all of the unaccounted-for outputs and fit the other constraints. (Editor's note: actually, things never get that complicated.)
The code below passes all of Codility's tests. The OP added a main function to use it on the command line.
The constraints are not as complex as the OP thinks. In particular, there is never a situation where you need to add a restriction that an input be an element of some set of specific integers seen elsewhere in the output. Every input position has a well-defined minimum and maximum.
The only complication to that rule is that sometimes the maximum is "the value of the previous input" and that input itself has a range. But even then, all the values like that are consecutive and have the same range, so the number of possibilities can be calculated with basic combinatorics, and those inputs as a group are independent of the other inputs (which only serve to set the range), so the possibilities of that group can be combined with the possibilities of other input positions by simple multiplication.
Algorithm overview
The algorithm makes a single pass through the output array updating the possible numbers of input arrays after every span, which is what I am calling repetitions of numbers in the output. (You might say maximal subsequences of the output where every element is identical.) For example, for output 0,1,1,2 we have three spans: 0, 1,1 and 2. When a new span begins, the number of possibilities for the previous span is calculated.
This decision was based on a few observations:
For spans longer than 1 in length, the minimum value of the input
allowed in the first position is whatever the value is of the input
in the second position. Calculating the number of possibilities of a
span is straightforward combinatorics, but the standard formula
requires knowing the range of the numbers and the length of the span.
Every time the value of the
output changes (and a new span beings), that strongly constrains the value of the previous span:
When the output goes up, the only possible reason is that the previous input was the value of the new, higher output and the input corresponding to the position of the new, higher output, was even higher.
When an output goes down, new constraints are established, but those are a bit harder to articulate. The algorithm stores stairs (see below) in order to quantify the constraints imposed when the output goes down
The aim here was to confine the range of possible values for every span. Once we do that accurately, calculating the number of combinations is straightforward.
Because the encoder backtracks looking to output a number that relates to the input in 2 ways, both smaller and closer, we know we can throw out numbers that are larger and farther away. After a small number appears in the output, no larger number from before that position can have any influence on what follows.
So to confine these ranges of input when the output sequence decreased, we need to store stairs - a list of increasingly larger possible values for the position in the original array. E.g for 0,2,5,7,2,4 stairs build up like this: 0, 0,2, 0,2,5, 0,2,5,7, 0,2, 0,2,4.
Using these bounds we can tell for sure that the number in the position of the second 2 (next to last position in the example) must be in (2,5], because 5 is the next stair. If the input were greater than 5, a 5 would have been output in that space instead of a 2. Observe, that if the last number in the encoded array was not 4, but 6, we would exit early returning 0, because we know that the previous number couldn't be bigger than 5.
The complexity is O(n*lg(min(n,m))).
Functions
CombinationsWithReplacement - counts number of combinations with replacements of size k from n numbers. E.g. for (3, 2) it counts 3,3, 3,2, 3,1, 2,2, 2,1, 1,1, so returns 6 It is the same as choose(n - 1 + k, n - 1).
nextBigger - finds next bigger element in a range. E.g. for 4 in sub-array 1,2,3,4,5 it returns 5, and in sub-array 1,3 it returns its parameter Max.
countSpan (lambda) - counts how many different combinations a span we have just passed can have. Consider span 2,2 for 0,2,5,7,2,2,7.
When curr gets to the final position, curr is 7 and prev is the final 2 of the 2,2 span.
It computes maximum and minimum possible values of the prev span. At this point stairs consist of 2,5,7 then maximum possible value is 5 (nextBigger after 2 in the stair 2,5,7). A value of greater than 5 in this span would have output a 5, not a 2.
It computes a minimum value for the span (which is the minimum value for every element in the span), which is prev at this point, (remember curr at this moment equals to 7 and prev to 2). We know for sure that in place of the final 2 output, the original input has to have 7, so the minimum is 7. (This is a consequence of the "output goes up" rule. If we had 7,7,2 and curr would be 2 then the minimum for the previous span (the 7,7) would be 8 which is prev + 1.
It adjusts the number of combinations. For a span of length L with a range of n possibilities (1+max-min), there are possibilities, with k being either L or L-1 depending on what follows the span.
For a span followed by a larger number, like 2,2,7, k = L - 1 because the last position of the 2,2 span has to be 7 (the value of the first number after the span).
For a span followed by a smaller number, like 7,7,2, k = L because
the last element of 7,7 has no special constraints.
Finally, it calls CombinationsWithReplacement to find out the number of branches (or possibilities), computes new res partial results value (remainder values in the modulo arithmetic we are doing), and returns new res value and max for further handling.
solution - iterates over the given Encoder Output array. In the main loop, while in a span it counts the span length, and at span boundaries it updates res by calling countSpan and possibly updates the stairs.
If the current span consists of a bigger number than the previous one, then:
Check validity of the next number. E.g 0,2,5,2,7 is invalid input, becuase there is can't be 7 in the next-to-last position, only 3, or 4, or 5.
It updates the stairs. When we have seen only 0,2, the stairs are 0,2, but after the next 5, the stairs become 0,2,5.
If the current span consists of a smaller number then the previous one, then:
It updates stairs. When we have seen only 0,2,5, our stairs are 0,2,5, but after we have seen 0,2,5,2 the stairs become 0,2.
After the main loop it accounts for the last span by calling countSpan with -1 which triggers the "output goes down" branch of calculations.
normalizeMod, extendedEuclidInternal, extendedEuclid, invMod - these auxiliary functions help to deal with modulo arithmetic.
For stairs I use storage for the encoded array, as the number of stairs never exceeds current position.
#include <algorithm>
#include <cassert>
#include <vector>
#include <tuple>
const int Modulus = 1'000'000'007;
int CombinationsWithReplacement(int n, int k);
template <class It>
auto nextBigger(It begin, It end, int value, int Max) {
auto maxIt = std::upper_bound(begin, end, value);
auto max = Max;
if (maxIt != end) {
max = *maxIt;
}
return max;
}
auto solution(std::vector<int> &B, const int Max) {
auto res = 1;
const auto size = (int)B.size();
auto spanLength = 1;
auto prev = 0;
// Stairs is the list of numbers which could be smaller than number in the next position
const auto stairsBegin = B.begin();
// This includes first entry (zero) into stairs
// We need to include 0 because we can meet another zero later in encoded array
// and we need to be able to find in stairs
auto stairsEnd = stairsBegin + 1;
auto countSpan = [&](int curr) {
const auto max = nextBigger(stairsBegin, stairsEnd, prev, Max);
// At the moment when we switch from the current span to the next span
// prev is the number from previous span and curr from current.
// E.g. 1,1,7, when we move to the third position cur = 7 and prev = 1.
// Observe that, in this case minimum value possible in place of any of 1's can be at least 2=1+1=prev+1.
// But if we consider 7, then we have even more stringent condition for numbers in place of 1, it is 7
const auto min = std::max(prev + 1, curr);
const bool countLast = prev > curr;
const auto branchesCount = CombinationsWithReplacement(max - min + 1, spanLength - (countLast ? 0 : 1));
return std::make_pair(res * (long long)branchesCount % Modulus, max);
};
for (int i = 1; i < size; ++i) {
const auto curr = B[i];
if (curr == prev) {
++spanLength;
}
else {
int max;
std::tie(res, max) = countSpan(curr);
if (prev < curr) {
if (curr > max) {
// 0,1,5,1,7 - invalid because number in the fourth position lies in [2,5]
// and so in the fifth encoded position we can't something bigger than 5
return 0;
}
// It is time to possibly shrink stairs.
// E.g if we had stairs 0,2,4,9,17 and current value is 5,
// then we no more interested in 9 and 17, and we change stairs to 0,2,4,5.
// That's because any number bigger than 9 or 17 also bigger than 5.
const auto s = std::lower_bound(stairsBegin, stairsEnd, curr);
stairsEnd = s;
*stairsEnd++ = curr;
}
else {
assert(curr < prev);
auto it = std::lower_bound(stairsBegin, stairsEnd, curr);
if (it == stairsEnd || *it != curr) {
// 0,5,1 is invalid sequence because original sequence lloks like this 5,>5,>1
// and there is no 1 in any of the two first positions, so
// it can't appear in the third position of the encoded array
return 0;
}
}
spanLength = 1;
}
prev = curr;
}
res = countSpan(-1).first;
return res;
}
template <class T> T normalizeMod(T a, T m) {
if (a < 0) return a + m;
return a;
}
template <class T> std::pair<T, std::pair<T, T>> extendedEuclidInternal(T a, T b) {
T old_x = 1;
T old_y = 0;
T x = 0;
T y = 1;
while (true) {
T q = a / b;
T t = a - b * q;
if (t == 0) {
break;
}
a = b;
b = t;
t = x; x = old_x - x * q; old_x = t;
t = y; y = old_y - y * q; old_y = t;
}
return std::make_pair(b, std::make_pair(x, y));
}
// Returns gcd and Bezout's coefficients
template <class T> std::pair<T, std::pair<T, T>> extendedEuclid(T a, T b) {
if (a > b) {
if (b == 0) return std::make_pair(a, std::make_pair(1, 0));
return extendedEuclidInternal(a, b);
}
else {
if (a == 0) return std::make_pair(b, std::make_pair(0, 1));
auto p = extendedEuclidInternal(b, a);
std::swap(p.second.first, p.second.second);
return p;
}
}
template <class T> T invMod(T a, T m) {
auto p = extendedEuclid(a, m);
assert(p.first == 1);
return normalizeMod(p.second.first, m);
}
int CombinationsWithReplacement(int n, int k) {
int res = 1;
for (long long i = n; i < n + k; ++i) {
res = res * i % Modulus;
}
int denom = 1;
for (long long i = k; i > 0; --i) {
denom = denom * i % Modulus;
}
res = res * (long long)invMod(denom, Modulus) % Modulus;
return res;
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Only the above is needed for the Codility challenge. Below is to run on the command line.
//
// Compile with: gcc -std=gnu++14 -lc++ -lstdc++ array_recovery.cpp
//
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
#include <string.h>
// Usage: 0 1 2,3, 4 M
// Last arg is M, the max value for an input.
// Remaining args are B (the output of the encoder) separated by commas and/or spaces
// Parentheses and brackets are ignored, so you can use the same input form as Codility's tests: ([1,2,3], M)
int main(int argc, char* argv[]) {
int Max;
std::vector<int> B;
const char* delim = " ,[]()";
if (argc < 2 ) {
printf("Usage: %s M 0 1 2,3, 4... \n", argv[0]);
return 1;
}
for (int i = 1; i < argc; i++) {
char* parse;
parse = strtok(argv[i], delim);
while (parse != NULL)
{
B.push_back(atoi(parse));
parse = strtok (NULL, delim);
}
}
Max = B.back();
B.pop_back();
printf("%d\n", solution(B, Max));
return 0;
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//
// Only the above is needed for the Codility challenge. Below is to run on the command line.
//
// Compile with: gcc -std=gnu++14 -lc++ -lstdc++ array_recovery.cpp
//
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
#include <string.h>
// Usage: M 0 1 2,3, 4
// first arg is M, the max value for an input.
// remaining args are B (the output of the encoder) separated by commas and/or spaces
int main(int argc, char* argv[]) {
int Max;
std::vector<int> B;
const char* delim = " ,";
if (argc < 3 ) {
printf("Usage: %s M 0 1 2,3, 4... \n", argv[0]);
return 1;
}
Max = atoi(argv[1]);
for (int i = 2; i < argc; i++) {
char* parse;
parse = strtok(argv[i], delim);
while (parse != NULL)
{
B.push_back(atoi(parse));
parse = strtok (NULL, delim);
}
}
printf("%d\n", solution(B, Max));
return 0;
}
Let's see an example:
Max = 5
Array is
0 1 3 0 1 1 3
1
1 2..5
1 3 4..5
1 3 4..5 1
1 3 4..5 1 2..5
1 3 4..5 1 2..5 >=..2 (sorry, for a cumbersome way of writing)
1 3 4..5 1 3..5 >=..3 4..5
Now count:
1 1 2 1 3 2 which amounts to 12 total.
Here's an idea. One known method to construct the output is to use a stack. We pop it while the element is greater or equal, then output the smaller element if it exists, then push the greater element onto the stack. Now what if we attempted to do this backwards from the output?
First we'll demonstrate the stack method using the c∅dility example.
[2, 5, 3, 7, 9, 6]
2: output 0, stack [2]
5: output 2, stack [2,5]
3: pop 5, output, 2, stack [2,3]
7: output 3, stack [2,3,7]
... etc.
Final output: [0, 2, 2, 3, 7, 3]
Now let's try reconstruction! We'll use stack both as the imaginary stack and as the reconstituted input:
(Input: [2, 5, 3, 7, 9, 6])
Output: [0, 2, 2, 3, 7, 3]
* Something >3 that reached 3 in the stack
stack = [3, 3 < *]
* Something >7 that reached 7 in the stack
but both of those would've popped before 3
stack = [3, 7, 7 < x, 3 < * <= x]
* Something >3, 7 qualifies
stack = [3, 7, 7 < x, 3 < * <= x]
* Something >2, 3 qualifies
stack = [2, 3, 7, 7 < x, 3 < * <= x]
* Something >2 and >=3 since 3 reached 2
stack = [2, 2 < *, 3, 7, 7 < x, 3 < * <= x]
Let's attempt your examples:
Example 1:
[0, 0, 0, 2, 3, 4]
* Something >4
stack = [4, 4 < *]
* Something >3, 4 qualifies
stack = [3, 4, 4 < *]
* Something >2, 3 qualifies
stack = [2, 3, 4, 4 < *]
* The rest is non-increasing with lowerbound 2
stack = [y >= x, x >= 2, 2, 3, 4, >4]
Example 2:
[0, 0, 0, 4]
* Something >4
stack [4, 4 < *]
* Non-increasing
stack = [z >= y, y >= 4, 4, 4 < *]
Calculating the number of combinations is achieved by multiplying together the possibilities for all the sections. A section is either a bounded single cell; or a bound, non-increasing subarray of one or more cells. To calculate the latter we use the multi-choose binomial, (n + k - 1) choose (k - 1). Consider that we can express the differences between the cells of a bound, non-increasing sequence of 3 cells as:
(ub - cell_3) + (cell_3 - cell_2) + (cell_2 - cell_1) + (cell_1 - lb) = ub - lb
Then the number of ways to distribute ub - lb into (x + 1) cells is
(n + k - 1) choose (k - 1)
or
(ub - lb + x) choose x
For example, the number of non-increasing sequences between
(3,4) in two cells is (4 - 3 + 2) choose 2 = 3: [3,3] [4,3] [4,4]
And the number of non-increasing sequences between
(3,4) in three cells is (4 - 3 + 3) choose 3 = 4: [3,3,3] [4,3,3] [4,4,3] [4,4,4]
(Explanation attributed to Brian M. Scott.)
Rough JavaScript sketch (the code is unreliable; it's only meant to illustrate the encoding. The encoder lists [lower_bound, upper_bound], or a non-increasing sequence as [non_inc, length, lower_bound, upper_bound]):
function f(A, M){
console.log(JSON.stringify(A), M);
let i = A.length - 1;
let last = A[i];
let s = [[last,last]];
if (A[i-1] == last){
let d = 1;
s.splice(1,0,['non_inc',d++,last,M]);
while (i > 0 && A[i-1] == last){
s.splice(1,0,['non_inc',d++,last,M]);
i--
}
} else {
s.push([last+1,M]);
i--;
}
if (i == 0)
s.splice(0,1);
for (; i>0; i--){
let x = A[i];
if (x < s[0][0])
s = [[x,x]].concat(s);
if (x > s[0][0]){
let [l, _l] = s[0];
let [lb, ub] = s[1];
s[0] = [x+1, M];
s[1] = [lb, x];
s = [[l,_l], [x,x]].concat(s);
}
if (x == s[0][0]){
let [l,_l] = s[0];
let [lb, ub] = s[1];
let d = 1;
s.splice(0,1);
while (i > 0 && A[i-1] == x){
s =
[['non_inc', d++, lb, M]].concat(s);
i--;
}
if (i > 0)
s = [[l,_l]].concat(s);
}
}
// dirty fix
if (s[0][0] == 0)
s.splice(0,1);
return s;
}
var a = [2, 5, 3, 7, 9, 6]
var b = [0, 2, 2, 3, 7, 3]
console.log(JSON.stringify(a));
console.log(JSON.stringify(f(b,10)));
b = [0,0,0,4]
console.log(JSON.stringify(f(b,10)));
b = [0,2,0,0,0,4]
console.log(JSON.stringify(f(b,10)));
b = [0,0,0,2,3,4]
console.log(JSON.stringify(f(b,10)));
b = [0,2,2]
console.log(JSON.stringify(f(b,4)));
b = [0,3,5,6]
console.log(JSON.stringify(f(b,10)));
b = [0,0,3,0]
console.log(JSON.stringify(f(b,10)));
A vampire number is defined here https://en.wikipedia.org/wiki/Vampire_number. A number V is a vampire number if:
It can be expressed as X*Y such that X and Y have N/2 digits each where N is the number of digits in V
Both X & Y should not have trailing zeros
X & Y together should have the same digits as V
I came up with a solution,
strV = sort(toString(V))
for factor <- pow(10, N/2) to sqrt(V)
if factor divides V
X <- factor
Y <- V/factor
if X and Y have trailing zeros
continue
checkStr = sort(toString(X) + toString(Y))
if checkStr equals strV return true
Another possible solution is to permute the string represented by V and split it into half and check if its a vampire number. Which one is the best way to do so?
The algorithm I propose here will not go through all permutations of digits. It will eliminate possibilities as fast as possible so that only a fraction of permutations will actually be tested.
Algorithm explained by example
Here is how it works based on example number 125460. If you are fine with reading the code directly, then you can skip this (long) part:
At first the two fangs (i.e. vampire factors) are obviously not known, and the problem can be represented as follows:
?**
X ?**
-------
=125460
For the left most digit of the first factor (marked with ?) we could choose any of the digits 0,1,2,5,4, or 6. But on closer analysis 0 would not be a viable possibility, as the product would never reach more than a 5-digit number. So it would be a waste of time to go through all permutations of digits that start with a zero.
For the left most digit of the second factor (also marked with ?), the same is true. However, when looking at the combinations, we can again filter out some pairs that cannot contribute to reaching the target product. For instance, this combination should be discarded:
1**
X 2**
-------
=125460
The greatest number that can be achieved with these digits is 199x299 = 59501 (ignoring the fact that we don't even have a 9), which is not even half of the desired number. So we should reject the combination (1, 2). For the same reason, the pair (1, 5) can be discarded for taking these positions. Similarly, the pairs (4, 5), (4, 6), and (5, 6) can be rejected as well, because they yield a too large product (>= 200000). I will call this kind of a test -- where it is determined whether the target number is within reach for a certain chosen digit pair, the "range test".
At this stage there is no difference between the first and the second fang, so we should also not have to investigate pairs where the second digit is smaller than the first, because they mirror a pair that would already have been investigated (or rejected).
So of all the possible pairs that could take up this first position (there are 30 possibilities to take 2 digits from a set of 6 digits), only the following 4 need to be investigated:
(1, 6), (2, 4), (2, 5), (2, 6)
In a more elaborate notation this means we are limiting the search to these number patterns:
1** 2** 2** 2**
X 6** X 4** X 5** X 6**
------- ------- ------- -------
=125460 =125460 =125460 =125460
A B C D
It is clear that this reduction of possibilities before even looking at the other positions greatly reduces the search tree.
The algorithm will take each of these 4 possibilities in order, and for each will check the possibilities for the next digit position. So first configuration A is analysed:
1?*
X 6?*
-------
=125460
The pairs that are available for the ?-marked positions are these 12:
(0, 2), (0, 4), (0, 5)
(2, 0), (2, 4), (2, 5)
(4, 0), (4, 2), (4, 5)
(5, 0), (5, 2), (5, 4)
Again, we can eliminate pairs by applying the range test. Let's take for instance the pair (5, 4). This would mean we had factors 15* and 64* (where * is an unknown digit at this point). The product of these two will be maximised with 159 * 649, i.e. 103191 (again ignoring the fact we do not even have a 9 available): this is too low for reaching the target, so this pair can be ignored. By further applying the range test, all these 12 pairs can be discarded, and so the search within configuration A stops here: there is no solution there.
Then the algorithm moves to configuration B:
2?*
X 4?*
-------
=125460
Again, the range test is applied to the possible pairs for the second position, and again it turns out none of these pairs passes the test: for instance (5, 6) can never represent a greater product than 259 * 469 = 121471, which is (only just) too small.
Then the algorithm moves to option C:
2?*
X 5?*
-------
=125460
Of all 12 possible pairs, only the following survive the range test: (4, 0), (4, 1), (6, 0), (6, 1). So now we have the following second-level configurations:
24* 24* 26* 26*
X 50* X 51* X 50* X 51*
------- ------- ------- -------
=125460 =125460 =125460 =125460
Ca Cb Cc Cd
In configuration Ca, there is no pair that passes the range test.
In configuration Cb, the pair (6, 0) passes, and leads to a solution:
246
X 510
-------
=125460
At this point the algorithm stops searching. The outcome is clear. In total the number of configurations looked at is very small compared to a brute force permutation checking algorithm. Here is a visualisation of the search tree:
*-+-- (1, 6)
|
+-- (2, 4)
|
+-- (2, 5) -+-- (4, 0)
| |
| +-- (4, 1) ---- (6, 0) = success: 246 * 510
/ /
| +-- (6, 0)
| |
| +-- (6, 1)
|
+-- (2, 6) ---- (0, 1) ---- (4, 5) = success: 204 * 615
The variants below / are only for showing what else the algorithm would have done, if there had not been a solution found. But in this actual case, that part of the search tree was actually never followed.
I have no clear idea of the time complexity, but it seems to run quite well for larger numbers, showing that the elimination of digits at an early stage makes the width of the search tree quite narrow.
Here is a live JavaScript implementation, which also runs some test cases when it it is activated (and it has a few other optimisations -- see code comments).
/*
Function: vampireFangs
Arguments:
vampire: number to factorise into two fangs, if possible.
Return value:
Array with two fangs if indeed the argument is a vampire number.
Otherwise false (not a vampire number) or null (argument too large to
compute)
*/
function vampireFangs(vampire) {
/* Function recurse: for the recursive part of the algorithm.
prevA, prevB: partial, potential fangs based on left-most digits of the given
number
counts: array of ten numbers representing the occurrence of still
available digits
divider: power of 100, is divided by 100 each next level in the search tree.
Determines the number of right-most digits of the given number that
are ignored at first in the algorithm. They will be considered in
deeper levels of recursion.
*/
function recurse(vampire, prevA, prevB, counts, divider) {
if (divider < 1) { // end of recursion
// Product of fangs must equal original number and fangs must not both
// end with a 0.
return prevA * prevB === vampire && (prevA % 10 + prevB % 10 > 0)
? [prevA, prevB] // Solution found
: false; // It's not a solution
}
// Get left-most digits (multiple of 2) of potential vampire number
var v = Math.floor(vampire/divider);
// Shift decimal digits of partial fangs to the left to make room for
// the next digits
prevA *= 10;
prevB *= 10;
// Calculate the min/max A digit that can potentially contribute to a
// solution
var minDigA = Math.floor(v / (prevB + 10)) - prevA;
var maxDigA = prevB ? Math.floor((v + 1) / prevB) - prevA : 9;
if (maxDigA > 9) maxDigA = 9;
for (var digA = minDigA; digA <= maxDigA; digA++) {
if (!counts[digA]) continue; // this digit is not available
var fangA = prevA + digA;
counts[digA]--;
// Calculate the min/max B digit that can potentially contribute to
// a solution
var minDigB = Math.floor(v / (fangA + 1)) - prevB;
var maxDigB = fangA ? (v + 1) / fangA - prevB : 9;
// Don't search mirrored A-B digits when both fangs are equal until now.
if (prevA === prevB && digA > minDigB) minDigB = digA;
if (maxDigB > 9) maxDigB = 9;
for (var digB = minDigB; digB <= Math.min(maxDigB, 9); digB++) {
if (!counts[digB]) continue; // this digit is not available
var fangB = prevB + digB;
counts[digB]--;
// Recurse by considering the next two digits of the potential
// vampire number, for finding the next digits to append to
// both partial fangs.
var result = recurse(vampire, fangA, fangB, counts, divider / 100);
// When one solution is found: stop searching & exit search tree.
if (result) return result; // solution found
// Restore counts
counts[digB]++;
}
counts[digA]++;
}
}
// Validate argument
if (typeof vampire !== 'number') return false;
if (vampire < 0 || vampire % 1 !== 0) return false; // not positive and integer
if (vampire > 9007199254740991) return null; // beyond JavaScript precision
var digits = vampire.toString(10).split('').map(Number);
// A vampire number has an even number of digits
if (!digits.length || digits.length % 2 > 0) return false;
// Register per digit (0..9) the frequency of that digit in the argument
var counts = [0,0,0,0,0,0,0,0,0,0];
for (var i = 0; i < digits.length; i++) {
counts[digits[i]]++;
}
return recurse(vampire, 0, 0, counts, Math.pow(10, digits.length - 2));
}
function Timer() {
function now() { // try performance object, else use Date
return performance ? performance.now() : new Date().getTime();
}
var start = now();
this.spent = function () { return Math.round(now() - start); }
}
// I/O
var button = document.querySelector('button');
var input = document.querySelector('input');
var output = document.querySelector('pre');
button.onclick = function () {
var str = input.value;
// Convert to number
var vampire = parseInt(str);
// Measure performance
var timer = new Timer();
// Input must be valid number
var result = vampire.toString(10) !== str ? null
: vampireFangs(vampire);
output.textContent = (result
? 'Vampire number. Fangs are: ' + result.join(', ')
: result === null
? 'Input is not an integer or too large for JavaScript'
: 'Not a vampire number')
+ '\nTime spent: ' + timer.spent() + 'ms';
}
// Tests (numbers taken from wiki page)
var tests = [
// Negative test cases:
[1, 999, 126000, 1023],
// Positive test cases:
[1260, 1395, 1435, 1530, 1827, 2187, 6880,
102510, 104260, 105210, 105264, 105750, 108135,
110758, 115672, 116725, 117067, 118440,
120600, 123354, 124483, 125248, 125433, 125460, 125500,
13078260,
16758243290880,
24959017348650]
];
tests.forEach(function (vampires, shouldBeVampire) {
vampires.forEach(function (vampire) {
var isVampire = vampireFangs(vampire);
if (!isVampire !== !shouldBeVampire) {
output.textContent = 'Unexpected: vampireFangs('
+ vampire + ') returns ' + JSON.stringify(isVampire);
throw 'Test failed';
}
});
});
output.textContent = 'All tests passed.';
N: <input value="1047527295416280"><button>Vampire Check</button>
<pre></pre>
As JavaScript uses 64 bit floating point representation, the above snippet only accepts to numbers up to 253-1. Above that limit there would be loss of precision and consequently unreliable results.
As Python does not have such limitation, I also put a Python implementation on eval.in. That site has a limitation on execution times, so you'd have to run it elsewhere if that becomes an issue.
In pseudocode:
if digitcount is odd return false
if digitcount is 2 return false
for A = each permutation of length digitcount/2 selected from all the digits,
for B = each permutation of the remaining digits,
if either A or B starts with a zero, continue
if both A and B end in a zero, continue
if A*B == the number, return true
There are a number of optimizations that could still be performed here, mostly in terms of ensuring that each possible pair of factors is tried only once. In other words, how to best check for repeating digits when selecting permutations?
But that's the gist of the algorithm I would use.
P.S.: You're not looking for primes, so why use a primality test? You just care about whether these are vampire numbers; there are only a very few possible factors. No need to check all the numbers up to sqrt(number).
Here are some suggestions:
First a simple improvement: if the number of digits is < 4 or odd return false (or if v is negative too).
You don't need to sort v, it is enough to count how many times each digit occurs O(n).
You don't have to check each number, only the combinations that are possible with the digits. This could be done by backtracking and significantly reduces the amount of numbers that have to be checked.
The final sort to check if all digits were used isn't needed either, just add up the used digits of both numbers and compare with the occurences in v.
Here is the code for a JS-like language with integers that never overflow, the V parameter is an integer string without leading 0s:
Edit: As it turns out the code is not only JS-like, but valid JS code and it had no problem to decide that 1047527295416280 is indeed a vampire number (jsfiddle).
var V, v, isVmp, digits, len;
function isVampire(numberString) {
V = numberString;
if (V.length < 4 || V.length % 2 == 1 )
return false;
v = parseInt(V);
if (v < 0)
return false;
digits = countDigits(V);
len = V.length / 2;
isVmp = false;
checkNumbers();
return isVmp;
}
function countDigits(s) {
var offset = "0".charCodeAt(0);
var ret = [0,0,0,0,0,0,0,0,0,0];
for (var i = 0; i < s.length; i++)
ret[s.charCodeAt(i) - offset]++;
return ret;
}
function checkNumbers(number, depth) {
if (isVmp)
return;
if (typeof number == 'undefined') {
for (var i = 1; i < 10; i++) {
if (digits[i] > 0) {
digits[i]--;
checkNumbers(i, len - 1);
digits[i]++;
}
}
} else if (depth == 0) {
if (v % number == 0) {
var b = v / number;
if (number % 10 != 0 || b % 10 != 0) {
var d = countDigits('' + b);
if (d[0] == digits[0] && d[1] == digits[1] && d[2] == digits[2] &&
d[3] == digits[3] && d[4] == digits[4] && d[5] == digits[5] &&
d[6] == digits[6] && d[7] == digits[7] && d[8] == digits[8] &&
d[9] == digits[9])
isVmp = true;
}
}
} else {
for (var i = 0; i < 10; i++) {
if (digits[i] > 0) {
digits[i]--;
checkNumbers(number * 10 + i, depth - 1);
digits[i]++;
}
}
}
}
I have a very large set (billions or more, it's expected to grow exponentially to some level), and I want to generate seemingly random elements from it without repeating. I know I can pick a random number and repeat and record the elements I have generated, but that takes more and more memory as numbers are generated, and wouldn't be practical after couple millions elements out.
I mean, I could say 1, 2, 3 up to billions and each would be constant time without remembering all the previous, or I can say 1,3,5,7,9 and on then 2,4,6,8,10, but is there a more sophisticated way to do that and eventually get a seemingly random permutation of that set?
Update
1, The set does not change size in the generation process. I meant when the user's input increases linearly, the size of the set increases exponentially.
2, In short, the set is like the set of every integer from 1 to 10 billions or more.
3, In long, it goes up to 10 billion because each element carries the information of many independent choices, for example. Imagine an RPG character that have 10 attributes, each can go from 1 to 100 (for my problem different choices can have different ranges), thus there's 10^20 possible characters, number "10873456879326587345" would correspond to a character that have "11, 88, 35...", and I would like an algorithm to generate them one by one without repeating, but makes it looks random.
Thanks for the interesting question. You can create a "pseudorandom"* (cyclic) permutation with a few bytes using modular exponentiation. Say we have n elements. Search for a prime p that's bigger than n+1. Then find a primitive root g modulo p. Basically by definition of primitive root, the action x --> (g * x) % p is a cyclic permutation of {1, ..., p-1}. And so x --> ((g * (x+1))%p) - 1 is a cyclic permutation of {0, ..., p-2}. We can get a cyclic permutation of {0, ..., n-1} by repeating the previous permutation if it gives a value bigger (or equal) n.
I implemented this idea as a Go package. https://github.com/bwesterb/powercycle
package main
import (
"fmt"
"github.com/bwesterb/powercycle"
)
func main() {
var x uint64
cycle := powercycle.New(10)
for i := 0; i < 10; i++ {
fmt.Println(x)
x = cycle.Apply(x)
}
}
This outputs something like
0
6
4
1
2
9
3
5
8
7
but that might vary off course depending on the generator chosen.
It's fast, but not super-fast: on my five year old i7 it takes less than 210ns to compute one application of a cycle on 1000000000000000 elements. More details:
BenchmarkNew10-8 1000000 1328 ns/op
BenchmarkNew1000-8 500000 2566 ns/op
BenchmarkNew1000000-8 50000 25893 ns/op
BenchmarkNew1000000000-8 200000 7589 ns/op
BenchmarkNew1000000000000-8 2000 648785 ns/op
BenchmarkApply10-8 10000000 170 ns/op
BenchmarkApply1000-8 10000000 173 ns/op
BenchmarkApply1000000-8 10000000 172 ns/op
BenchmarkApply1000000000-8 10000000 169 ns/op
BenchmarkApply1000000000000-8 10000000 201 ns/op
BenchmarkApply1000000000000000-8 10000000 204 ns/op
Why did I say "pseudorandom"? Well, we are always creating a very specific kind of cycle: namely one that uses modular exponentiation. It looks pretty pseudorandom though.
I would use a random number and swap it with an element at the beginning of the set.
Here's some pseudo code
set = [1, 2, 3, 4, 5, 6]
picked = 0
Function PickNext(set, picked)
If picked > Len(set) - 1 Then
Return Nothing
End If
// random number between picked (inclusive) and length (exclusive)
r = RandomInt(picked, Len(set))
// swap the picked element to the beginning of the set
result = set[r]
set[r] = set[picked]
set[picked] = result
// update picked
picked++
// return your next random element
Return temp
End Function
Every time you pick an element there is one swap and the only extra memory being used is the picked variable. The swap can happen if the elements are in a database or in memory.
EDIT Here's a jsfiddle of a working implementation http://jsfiddle.net/sun8rw4d/
JavaScript
var set = [];
set.picked = 0;
function pickNext(set) {
if(set.picked > set.length - 1) { return null; }
var r = set.picked + Math.floor(Math.random() * (set.length - set.picked));
var result = set[r];
set[r] = set[set.picked];
set[set.picked] = result;
set.picked++;
return result;
}
// testing
for(var i=0; i<100; i++) {
set.push(i);
}
while(pickNext(set) !== null) { }
document.body.innerHTML += set.toString();
EDIT 2 Finally, a random binary walk of the set. This can be accomplished with O(Log2(N)) stack space (memory) which for 10billion is only 33. There's no shuffling or swapping involved. Using trinary instead of binary might yield even better pseudo random results.
// on the fly set generator
var count = 0;
var maxValue = 64;
function nextElement() {
// restart the generation
if(count == maxValue) {
count = 0;
}
return count++;
}
// code to pseudo randomly select elements
var current = 0;
var stack = [0, maxValue - 1];
function randomBinaryWalk() {
if(stack.length == 0) { return null; }
var high = stack.pop();
var low = stack.pop();
var mid = ((high + low) / 2) | 0;
// pseudo randomly choose the next path
if(Math.random() > 0.5) {
if(low <= mid - 1) {
stack.push(low);
stack.push(mid - 1);
}
if(mid + 1 <= high) {
stack.push(mid + 1);
stack.push(high);
}
} else {
if(mid + 1 <= high) {
stack.push(mid + 1);
stack.push(high);
}
if(low <= mid - 1) {
stack.push(low);
stack.push(mid - 1);
}
}
// how many elements to skip
var toMid = (current < mid ? mid - current : (maxValue - current) + mid);
// skip elements
for(var i = 0; i < toMid - 1; i++) {
nextElement();
}
current = mid;
// get result
return nextElement();
}
// test
var result;
var list = [];
do {
result = randomBinaryWalk();
list.push(result);
} while(result !== null);
document.body.innerHTML += '<br/>' + list.toString();
Here's the results from a couple of runs with a small set of 64 elements. JSFiddle http://jsfiddle.net/yooLjtgu/
30,46,38,34,36,35,37,32,33,31,42,40,41,39,44,45,43,54,50,52,53,51,48,47,49,58,60,59,61,62,56,57,55,14,22,18,20,19,21,16,15,17,26,28,29,27,24,25,23,6,2,4,5,3,0,1,63,10,8,7,9,12,11,13
30,14,22,18,16,15,17,20,19,21,26,28,29,27,24,23,25,6,10,8,7,9,12,13,11,2,0,63,1,4,5,3,46,38,42,44,45,43,40,41,39,34,36,35,37,32,31,33,54,58,56,55,57,60,59,61,62,50,48,49,47,52,51,53
As I mentioned in my comment, unless you have an efficient way to skip to a specific point in your "on the fly" generation of the set this will not be very efficient.
if it is enumerable then use a pseudo-random integer generator adjusted to the period 0 .. 2^n - 1 where the upper bound is just greater than the size of your set and generate pseudo-random integers discarding those more than the size of your set. Use those integers to index items from your set.
Pre- compute yourself a series of indices (e.g. in a file), which has the properties you need and then randomly choose a start index for your enumeration and use the series in a round-robin manner.
The length of your pre-computed series should be > the maximum size of the set.
If you combine this (depending on your programming language etc.) with file mappings, your final nextIndex(INOUT state) function is (nearly) as simple as return mappedIndices[state++ % PERIOD];, if you have a fixed size of each entry (e.g. 8 bytes -> uint64_t).
Of course, the returned value could be > your current set size. Simply draw indices until you get one which is <= your sets current size.
Update (In response to question-update):
There is another option to achieve your goal if it is about creating 10Billion unique characters in your RPG: Generate a GUID and write yourself a function which computes your number from the GUID. man uuid if you are are on a unix system. Else google it. Some parts of the uuid are not random but contain meta-info, some parts are either systematic (such as your network cards MAC address) or random, depending on generator algorithm. But they are very very most likely unique. So, whenever you need a new unique number, generate a uuid and transform it to your number by means of some algorithm which basically maps the uuid bytes to your number in a non-trivial way (e.g. use hash functions).
Imagine you have 3 buckets, but each of them has a hole in it. I'm trying to fill a bath tub. The bath tub has a minimum level of water it needs and a maximum level of water it can contain. By the time you reach the tub with the bucket it is not clear how much water will be in the bucket, but you have a range of possible values.
Is it possible to adequately fill the tub with water?
Pretty much you have 3 ranges (min,max), is there some sum of them that will fall within a 4th range?
For example:
Bucket 1 : 5-10L
Bucket 2 : 15-25L
Bucket 3 : 10-50L
Bathtub 100-150L
Is there some guaranteed combination of 1 2 and 3 that will fill the bathtub within the requisite range? Multiples of each bucket can be used.
EDIT: Now imagine there are 50 different buckets?
If the capacity of the tub is not very large ( not greater than 10^6 for an example), we can solve it using dynamic programming.
Approach:
Initialization: memo[X][Y] is an array to memorize the result. X = number of buckets, Y = maximum capacity of the tub. Initialize memo[][] with -1.
Code:
bool dp(int bucketNum, int curVolume){
if(curVolume > maxCap)return false; // pruning extra branches
if(curVolume>=minCap && curVolume<=maxCap){ // base case on success
return true;
}
int &ret = memo[bucketNum][curVolume];
if(ret != -1){ // this state has been visited earlier
return false;
}
ret = false;
for(int i = minC[bucketNum]; i < = maxC[bucketNum]; i++){
int newVolume = curVolume + i;
for(int j = bucketNum; j <= 3; j++){
ret|=dp(j,newVolume);
if(ret == true)return ret;
}
}
return ret;
}
Warning: Code not tested
Here's a naïve recursive solution in python that works just fine (although it doesn't find an optimal solution):
def match_helper(lower, upper, units, least_difference, fail = dict()):
if upper < lower + least_difference:
return None
if fail.get((lower,upper)):
return None
exact_match = [ u for u in units if u['lower'] >= lower and u['upper'] <= upper ]
if exact_match:
return [ exact_match[0] ]
for unit in units:
if unit['upper'] > upper:
continue
recursive_match = match_helper(lower - unit['lower'], upper - unit['upper'], units, least_difference)
if recursive_match:
return [unit] + recursive_match
else:
fail[(lower,upper)] = 1
return None
def match(lower, upper):
units = [
{ 'name': 'Bucket 1', 'lower': 5, 'upper': 10 },
{ 'name': 'Bucket 2', 'lower': 15, 'upper': 25 },
{ 'name': 'Bucket 3', 'lower': 10, 'upper': 50 }
]
least_difference = min([ u['upper'] - u['lower'] for u in units ])
return match_helper(
lower = lower,
upper = upper,
units = sorted(units, key = lambda u: u['upper']),
least_difference = min([ u['upper'] - u['lower'] for u in units ]),
)
result = match(100, 175)
if result:
lower = sum([ u['lower'] for u in result ])
upper = sum([ u['upper'] for u in result ])
names = [ u['name'] for u in result ]
print lower, "-", upper
print names
else:
print "No solution"
It prints "No solution" for 100-150, but for 100-175 it comes up with a solution of 5x bucket 1, 5x bucket 2.
Assuming you are saying that the "range" for each bucket is the amount of water that it may have when it reaches the tub, and all you care about is if they could possibly fill the tub...
Just take the "max" of each bucket and sum them. If that is in the range of what you consider the tub to be "filled" then it can.
Updated:
Given that buckets can be used multiple times, this seems to me like we're looking for solutions to a pair of equations.
Given buckets x, y and z we want to find a, b and c:
a*x.min + b*y.min + c*z.min >= bathtub.min
and
a*x.max + b*y.max + c*z.max <= bathtub.max
Re: http://en.wikipedia.org/wiki/Diophantine_equation
If bathtub.min and bathtub.max are both multiples of the greatest common divisor of a,b and c, then there are infinitely many solutions (i.e. we can fill the tub), otherwise there are no solutions (i.e. we can never fill the tub).
This can be solved with multiple applications of the change making problem.
Each Bucket.Min value is a currency denomination, and Bathtub.Min is the target value.
When you find a solution via a change-making algorithm, then apply one more constraint:
sum(each Bucket.Max in your solution) <= Bathtub.max
If this constraint is not met, throw out this solution and look for another. This will probably require a change to a standard change-making algorithm that allows you to try other solutions when one is found to not be suitable.
Initially, your target range is Bathtub.Range.
Each time you add an instance of a bucket to the solution, you reduce the target range for the remaining buckets.
For example, using your example buckets and tub:
Target Range = 100..150
Let's say we want to add a Bucket1 to the candidate solution. That then gives us
Target Range = 95..140
because if the rest of the buckets in the solution total < 95, then this Bucket1 might not be sufficient to fill the tub to 100, and if the rest of the buckets in the solution total > 140, then this Bucket1 might fill the tub over 150.
So, this gives you a quick way to check if a candidate solution is valid:
TargetRange = Bathtub.Range
foreach Bucket in CandidateSolution
TargetRange.Min -= Bucket.Min
TargetRange.Max -= Bucket.Max
if TargetRange.Min == 0 AND TargetRange.Max >= 0 then solution found
if TargetRange.Min < 0 or TargetRange.Max < 0 then solution is invalid
This still leaves the question - How do you come up with the set of candidate solutions?
Brute force would try all possible combinations of buckets.
Here is my solution for finding the optimal solution (least number of buckets). It compares the ratio of the maximums to the ratio of the minimums, to figure out the optimal number of buckets to fill the tub.
private static void BucketProblem()
{
Range bathTub = new Range(100, 175);
List<Range> buckets = new List<Range> {new Range(5, 10), new Range(15, 25), new Range(10, 50)};
Dictionary<Range, int> result;
bool canBeFilled = SolveBuckets(bathTub, buckets, out result);
}
private static bool BucketHelper(Range tub, List<Range> buckets, Dictionary<Range, int> results)
{
Range bucket;
int startBucket = -1;
int fills = -1;
for (int i = buckets.Count - 1; i >=0 ; i--)
{
bucket = buckets[i];
double maxRatio = (double)tub.Maximum / bucket.Maximum;
double minRatio = (double)tub.Minimum / bucket.Minimum;
if (maxRatio >= minRatio)
{
startBucket = i;
if (maxRatio - minRatio > 1)
fills = (int) minRatio + 1;
else
fills = (int) maxRatio;
break;
}
}
if (startBucket < 0)
return false;
bucket = buckets[startBucket];
tub.Maximum -= bucket.Maximum * fills;
tub.Minimum -= bucket.Minimum * fills;
results.Add(bucket, fills);
return tub.Maximum == 0 || tub.Minimum <= 0 || startBucket == 0 || BucketHelper(tub, buckets.GetRange(0, startBucket), results);
}
public static bool SolveBuckets(Range tub, List<Range> buckets, out Dictionary<Range, int> results)
{
results = new Dictionary<Range, int>();
buckets = buckets.OrderBy(b => b.Minimum).ToList();
return BucketHelper(new Range(tub.Minimum, tub.Maximum), buckets, results);
}
I need to make a chart with an optimized y axis maximum value.
The current method I have of making charts simply uses the maximum value of all the graphs, then divides it by ten, and uses that as grid lines. I didn't write it.
Update Note: These graphs have been changed. As soon as I fixed the code, my dynamic graphs started working, making this question nonsensical (because the examples no longer had any errors in them). I've updated these with static images, but some of the answers refrence different values. Keep that in mind.
There were between 12003 and 14003 inbound calls so far in February. Informative, but ugly.
I'd like to avoid charts that look like a monkey came up with the y-axis numbers.
Using the Google charts API helps a little bit, but it's still not quite what I want.
The numbers are clean, but the top of the y value is always the same as the maximum value on the chart. This chart scales from 0 to 1357. I need to have calculated the proper value of 1400, problematically.
I'm throwing in rbobby's defanition of a 'nice' number here because it explains it so well.
A "nice" number is one that has 3 or fewer non-zero digits (eg. 1230000)
A "nice" number has the same or few non-zero digits than zero digits (eg 1230 is not nice, 1200 is nice)
The nicest numbers are ones with multiples of 3 zeros (eg. "1,000", "1,000,000")
The second nicest numbers are onces with multples of 3 zeros plus 2 zeros (eg. "1,500,000", "1,200")
Solution
I found the way to get the results that I want using a modified version of Mark Ransom's idea.
Fist, Mark Ransom's code determines the optimum spacing between ticks, when given the number of ticks. Sometimes this number ends up being more than twice what the highest value on the chart is, depending on how many grid lines you want.
What I'm doing is I'm running Mark's code with 5, 6, 7, 8, 9, and 10 grid lines (ticks) to find which of those is the lowest. With a value of 23, the height of the chart goes to 25, with a grid line at 5, 10, 15, 20, and 25. With a value of 26, the chart's height is 30, with grid lines at 5, 10, 15, 20, 25, and 30. It has the same spacing between grid lines, but there are more of them.
So here's the steps to just-about copy what Excel does to make charts all fancy.
Temporarily bump up the chart's highest value by about 5% (so that there is always some space between the chart's highest point and the top of the chart area. We want 99.9 to round up to 120)
Find the optimum grid line placement
for 5, 6, 7, 8, 9, and 10 grid
lines.
Pick out the lowest of those numbers. Remember the number of grid lines it took to get that value.
Now you have the optimum chart height. The lines/bar will never butt up against the top of the chart and you have the optimum number of ticks.
PHP:
function roundUp($maxValue){
$optiMax = $maxValue * 2;
for ($i = 5; $i <= 10; $i++){
$tmpMaxValue = bestTick($maxValue,$i);
if (($optiMax > $tmpMaxValue) and ($tmpMaxValue > ($maxValue + $maxValue * 0.05))){
$optiMax = $tmpMaxValue;
$optiTicks = $i;
}
}
return $optiMax;
}
function bestTick($maxValue, $mostTicks){
$minimum = $maxValue / $mostTicks;
$magnitude = pow(10,floor(log($minimum) / log(10)));
$residual = $minimum / $magnitude;
if ($residual > 5){
$tick = 10 * $magnitude;
} elseif ($residual > 2) {
$tick = 5 * $magnitude;
} elseif ($residual > 1){
$tick = 2 * $magnitude;
} else {
$tick = $magnitude;
}
return ($tick * $mostTicks);
}
Python:
import math
def BestTick(largest, mostticks):
minimum = largest / mostticks
magnitude = 10 ** math.floor(math.log(minimum) / math.log(10))
residual = minimum / magnitude
if residual > 5:
tick = 10 * magnitude
elif residual > 2:
tick = 5 * magnitude
elif residual > 1:
tick = 2 * magnitude
else:
tick = magnitude
return tick
value = int(input(""))
optMax = value * 2
for i in range(5,11):
maxValue = BestTick(value,i) * i
print maxValue
if (optMax > maxValue) and (maxValue > value + (value*.05)):
optMax = maxValue
optTicks = i
print "\nTest Value: " + str(value + (value * .05)) + "\n\nChart Height: " + str(optMax) + " Ticks: " + str(optTicks)
This is from a previous similar question:
Algorithm for "nice" grid line intervals on a graph
I've done this with kind of a brute
force method. First, figure out the
maximum number of tick marks you can
fit into the space. Divide the total
range of values by the number of
ticks; this is the minimum
spacing of the tick. Now calculate
the floor of the logarithm base 10 to
get the magnitude of the tick, and
divide by this value. You should end
up with something in the range of 1 to
10. Simply choose the round number greater than or equal to the value and
multiply it by the logarithm
calculated earlier. This is your
final tick spacing.
Example in Python:
import math
def BestTick(largest, mostticks):
minimum = largest / mostticks
magnitude = 10 ** math.floor(math.log(minimum) / math.log(10))
residual = minimum / magnitude
if residual > 5:
tick = 10 * magnitude
elif residual > 2:
tick = 5 * magnitude
elif residual > 1:
tick = 2 * magnitude
else:
tick = magnitude
return tick
You could round up to two significant figures. The following pseudocode should work:
// maxValue is the largest value in your chart
magnitude = floor(log10(maxValue))
base = 10^(magnitude - 1)
chartHeight = ceiling(maxValue / base) * base
For example, if maxValue is 1357, then magnitude is 3 and base is 100. Dividing by 100, rounding up, and multiplying by 100 has the result of rounding up to the next multiple of 100, i.e. rounding up to two significant figures. In this case, the result if 1400 (1357 ⇒ 13.57 ⇒ 14 ⇒ 1400).
In the past I've done this in a brute force-ish sort of way. Here's a chunk of C++ code that works well... but for a hardcoded lower and upper limits (0 and 5000):
int PickYUnits()
{
int MinSize[8] = {20, 20, 20, 20, 20, 20, 20, 20};
int ItemsPerUnit[8] = {5, 10, 20, 25, 50, 100, 250, 500};
int ItemLimits[8] = {20, 50, 100, 250, 500, 1000, 2500, 5000};
int MaxNumUnits = 8;
double PixelsPerY;
int PixelsPerAxis;
int Units;
//
// Figure out the max from the dataset
// - Min is always 0 for a bar chart
//
m_MinY = 0;
m_MaxY = -9999999;
m_TotalY = 0;
for (int j = 0; j < m_DataPoints.GetSize(); j++) {
if (m_DataPoints[j].m_y > m_MaxY) {
m_MaxY = m_DataPoints[j].m_y;
}
m_TotalY += m_DataPoints[j].m_y;
}
//
// Give some space at the top
//
m_MaxY = m_MaxY + 1;
//
// Figure out the size of the range
//
double yRange = (m_MaxY - m_MinY);
//
// Pick the initial size
//
Units = MaxNumUnits;
for (int k = 0; k < MaxNumUnits; k++)
{
if (yRange < ItemLimits[k])
{
Units = k;
break;
}
}
//
// Adjust it upwards based on the space available
//
PixelsPerY = m_rcGraph.Height() / yRange;
PixelsPerAxis = (int)(PixelsPerY * ItemsPerUnit[Units]);
while (PixelsPerAxis < MinSize[Units]){
Units += 1;
PixelsPerAxis = (int)(PixelsPerY * ItemsPerUnit[Units]);
if (Units == 5)
break;
}
return ItemsPerUnit[Units];
}
However something in what you've said tweaked me. To pick nice axis numbers a definition of "nice number" would help:
A "nice" number is one that has 3 or fewer non-zero digits (eg. 1230000)
A "nice" number has the same or few non-zero digits than zero digits (eg 1230 is not nice, 1200 is nice)
The nicest numbers are ones with multiples of 3 zeros (eg. "1,000", "1,000,000")
The second nicest numbers are onces with multples of 3 zeros plus 2 zeros (eg. "1,500,000", "1,200")
Not sure if the above definition is "right" or actually helpful (but with the definition in hand it then becomes a simpler task to devise an algorithm).
A slight refinement and tested... (works for fractions of units and not just integers)
public void testNumbers() {
double test = 0.20000;
double multiple = 1;
int scale = 0;
String[] prefix = new String[]{"", "m", "u", "n"};
while (Math.log10(test) < 0) {
multiple = multiple * 1000;
test = test * 1000;
scale++;
}
double tick;
double minimum = test / 10;
double magnitude = 100000000;
while (minimum <= magnitude){
magnitude = magnitude / 10;
}
double residual = test / (magnitude * 10);
if (residual > 5) {
tick = 10 * magnitude;
} else if (residual > 2) {
tick = 5 * magnitude;
} else if (residual > 1) {
tick = 2 * magnitude;
} else {
tick = magnitude;
}
double curAmt = 0;
int ticks = (int) Math.ceil(test / tick);
for (int ix = 0; ix < ticks; ix++) {
curAmt += tick;
BigDecimal bigDecimal = new BigDecimal(curAmt);
bigDecimal.setScale(2, BigDecimal.ROUND_HALF_UP);
System.out.println(bigDecimal.stripTrailingZeros().toPlainString() + prefix[scale] + "s");
}
System.out.println("Value = " + test + prefix[scale] + "s");
System.out.println("Tick = " + tick + prefix[scale] + "s");
System.out.println("Ticks = " + ticks);
System.out.println("Scale = " + multiple + " : " + scale);
}
If you want 1400 at the top, how about adjusting the last two parameters to 1400 instead of 1357:
You could use div and mod. For example.
Let's say you want your chart to round up by increments of 20 (just to make it more a more arbitrary number than your typical "10" value).
So I would assume that 1, 11, 18 would all round up to 20. But 21, 33, 38 would round to 40.
To come up with the right value do the following:
Where divisor = your rounding increment.
divisor = 20
multiple = maxValue / divisor; // Do an integer divide here.
if (maxValue modulus divisor > 0)
multiple++;
graphMax = multiple * maxValue;
So now let's plugin real numbers:
divisor = 20;
multiple = 33 / 20; (integer divide)
so multiple = 1
if (33 modulus 20 > 0) (it is.. it equals 13)
multiple++;
so multiple = 2;
graphMax = multiple (2) * maxValue (20);
graphMax = 40;