How does the Catch2 GENERATE macro work internally? - c++

Recently I learned about the GENERATE macro in Catch2 (from this video). And now I am curious about how it works internally.
Naively one would think that for a test case with k generators (by a generator I mean one GENERATE call site), Catch2 just runs each test case n1 * n2 * ... * nk times, where ni is the number of elements in the i-th generator, each time specifying a different combination of values from those k generators. Indeed, this naive specification seems to hold for a simple test case:
TEST_CASE("Naive") {
auto x = GENERATE(0, 1);
auto y = GENERATE(2, 3);
std::cout << "x = " << x << ", y = " << y << std::endl;
}
As expected, the output is:
x = 0, y = 2
x = 0, y = 3
x = 1, y = 2
x = 1, y = 3
which indicates the test case runs for 2 * 2 == 4 times.
However, it seems that catch isn't implementing it naively, as shown by the following case:
TEST_CASE("Depends on if") {
auto choice = GENERATE(0, 1);
int x = -1, y = -1;
if (choice == 0) {
x = GENERATE(2, 3);
} else {
y = GENERATE(4, 5);
}
std::cout << "choice = " << choice << ", x = " << x << ", y = " << y << std::endl;
}
In the above case, the actual invocation (not callsite) of GENERATE depends on choice. If the logic were implemented naively, one would expect there to be 8 lines of output (since 2 * 2 * 2 == 8):
choice = 0, x = 2, y = -1
choice = 0, x = 2, y = -1
choice = 0, x = 3, y = -1
choice = 0, x = 3, y = -1
choice = 1, x = -1, y = 4
choice = 1, x = -1, y = 4
choice = 1, x = -1, y = 5
choice = 1, x = -1, y = 5
Notice the duplicate lines: the naive permutation still permutes the value of a generator even if it is not actually invoked. For example, y = GENERATE(4, 5) is only invoked if choice == 1, however, even when choice != 1, the implementation still permutes the values 4 and 5, even if those are not used.
The actual output, though, is:
choice = 0, x = 2, y = -1
choice = 0, x = 3, y = -1
choice = 1, x = -1, y = 4
choice = 1, x = -1, y = 5
No duplicate lines. This leads me to suspect that Catch internally uses a stack to track the generators invoked and the order of their latest invocation. Each time a test case finishes one iteration, it traverses the invoked genrators in the reverse order, and advances each generator's value. If such advancement fails (i.e. the sequence of values inside the generator finishes), that generator is reset to its initial state (i.e. ready to emit the first value in sequence); otherwise (the advancement succeeded), the traversal bails out.
In psuedocode it would look like:
for each generator that is invoked in reverse order of latest invocation:
bool success = generator.moveNext();
if success: break;
generator.reset();
This explains the previous cases perfectly. But it does not explain this (rather obscure) one:
TEST_CASE("Non structured generators") {
int x = -1, y = -1;
for (int i = 0; i <= 1; ++i) {
x = GENERATE(0, 1);
if (i == 1) break;
y = GENERATE(2, 3);
}
std::cout << x << "," << y << std::endl;
}
One would expect this to run 4 == 2 * 2 times, and the output being:
x = 0, y = 2
x = 1, y = 2
x = 0, y = 3
x = 1, y = 3
(The x changes before y since x = GENERATE(0, 1) is the last generator invoked)
However, this is not what catch actually does, this is what happens in reality:
x = 0, y = 2
x = 1, y = 2
x = 0, y = 3
x = 1, y = 3
x = 0, y = 2
x = 1, y = 2
x = 0, y = 3
x = 1, y = 3
8 lines of output, which is the first four lines repeated twice.
So my question is, how exactly is GENERATE in Catch2 implemented? I am not looking particularly for detailed code, but a high-level description that could explain what I have seen in the previous examples.

Maybe you can try to see the code generated after the pre-processor using the -E option in GCC.
a.c:
GENERATE(0,1)
gcc -E -CC a.c
How to make G++ preprocessor output a newline in a macro?

Related

In how many different ways can I call a function

In how many ways can I call this function, I found this question in one book and I was like 1 but I'm not sure, I think I can call it just once because, it has 3 integer parameters so that's the only way, I guess. The given answers wear (1, 2, 3 or 4) ways to call the Function.
int Volume (int x, int y=0, int z=3){ return x+y+z;}
What's the correct answer?
Three ways.
Example :
volume(3) //x = 3, y = 0, z = 3
volume(3,4) // x = 3, y = 4, z = 3
volume(3,4,5) // x = 3, y = 4, z = 5

Evenly distribute values into array

I have a fixed size boolean array of size 8. The default value of all elements in the array is false. There will be a number of truth values to fill between 1-8.
I want to distribute the truth values as far away from one another as possible. I also wish to be able to randomize the configuration. In this scenario the array wraps around so position 7 is "next to" position 0 in the array.
here are some examples for fill values. I didn't include all possibilities, but hopefully it gets my point across.
1: [1, 0, 0, 0, 0, 0, 0, 0] or [0, 1, 0, 0, 0, 0, 0, 0]
2: [1, 0, 0, 0, 1, 0, 0, 0] or [0, 1, 0, 0, 0, 1, 0, 0]
3: [1, 0, 0, 1, 0, 0, 1, 0] or [0, 1, 0, 0, 1, 0, 0, 1]
4: [1, 0, 1, 0, 1, 0, 1, 0] or [0, 1, 0, 1, 0, 1, 0, 1]
5: [1, 1, 0, 1, 1, 0, 1, 0]
6: [1, 1, 0, 1, 1, 1, 0, 1]
7: [1, 1, 1, 1, 1, 1, 1, 0]
8: [1, 1, 1, 1, 1, 1, 1, 1]
The closest solution I have come up with so far hasn't quite produced the results I'm looking for...
I seek to write it in c++ but here is a little pseudo-code of my algorithm so far...
not quite working out how I wanted
truths = randBetween(1, 8)
values = [0,0,0,0,0,0,0,0]
startPosition = randBetween(0, 7) //starting index
distance = 4
for(i = 0; i < truths; i++) {
pos = i + startPosition + (i * distance)
values[pos % 8] = 1
}
this is an example output from my current code. those marked with a star are incorrect.
[0, 0, 0, 0, 1, 0, 0, 0]
[0, 1, 0, 0, 1, 0, 0, 0]*
[0, 1, 0, 0, 1, 0, 1, 0]
[0, 1, 0, 1, 1, 0, 1, 0]*
[1, 1, 0, 1, 1, 0, 1, 0]
[1, 1, 0, 1, 1, 1, 1, 0]*
[1, 1, 1, 1, 1, 1, 1, 0]
[1, 1, 1, 1, 1, 1, 1, 1]
I'm looking for a simple way to distribute the truth values evenly throughout the array without having to code for special cases.
Check this out:
#include <cassert>
#include <vector>
#include <iostream>
#include <iomanip>
/**
* Generate an even spaced pattern of ones
* #param arr destination vector of ints
* #param onescnt the requested number of ones
*/
static inline
void gen(std::vector<int>& arr, size_t onescnt) {
const size_t len = arr.size();
const size_t zeroscnt = len - onescnt;
size_t ones = 1;
size_t zeros = 1;
for (size_t i = 0; i < len; ++i) {
if (ones * zeroscnt < zeros * onescnt) {
ones++;
arr[i] = 1;
} else {
zeros++;
arr[i] = 0;
}
}
}
static inline
size_t count(const std::vector<int>& arr, int el) {
size_t cnt = 0;
for (size_t i = 0; i < arr.size(); ++i) {
cnt += arr[i] == el;
}
return cnt;
}
static inline
void gen_print(size_t len, size_t onescnt) {
std::vector<int> arr(len);
gen(arr, onescnt);
std::cout << "gen_printf(" << std::setw(2) << len << ", " << std::setw(2) << onescnt << ") = {";
for (size_t i = 0; i < len; ++i) {
std::cout << arr[i] << ",";
}
std::cout << "}\n";
assert(count(arr, 1) == onescnt);
}
int main() {
for (int i = 0; i <= 8; ++i) {
gen_print(8, i);
}
for (int i = 0; i <= 30; ++i) {
gen_print(30, i);
}
return 0;
}
Generates:
gen_printf( 8, 0) = {0,0,0,0,0,0,0,0,}
gen_printf( 8, 1) = {0,0,0,0,0,0,0,1,}
gen_printf( 8, 2) = {0,0,0,1,0,0,0,1,}
gen_printf( 8, 3) = {0,1,0,0,1,0,0,1,}
gen_printf( 8, 4) = {0,1,0,1,0,1,0,1,}
gen_printf( 8, 5) = {1,0,1,1,0,1,0,1,}
gen_printf( 8, 6) = {1,1,0,1,1,1,0,1,}
gen_printf( 8, 7) = {1,1,1,1,1,1,0,1,}
gen_printf( 8, 8) = {1,1,1,1,1,1,1,1,}
gen_printf(30, 0) = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,}
gen_printf(30, 1) = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,}
gen_printf(30, 2) = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,}
gen_printf(30, 3) = {0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,}
gen_printf(30, 4) = {0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,}
gen_printf(30, 5) = {0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,}
gen_printf(30, 6) = {0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,}
gen_printf(30, 7) = {0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,}
gen_printf(30, 8) = {0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,}
gen_printf(30, 9) = {0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,}
gen_printf(30, 10) = {0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,}
gen_printf(30, 11) = {0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,0,1,}
gen_printf(30, 12) = {0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,}
gen_printf(30, 13) = {0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,}
gen_printf(30, 14) = {0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,}
gen_printf(30, 15) = {0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,}
gen_printf(30, 16) = {1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,}
gen_printf(30, 17) = {1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,}
gen_printf(30, 18) = {1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,}
gen_printf(30, 19) = {1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,}
gen_printf(30, 20) = {1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,}
gen_printf(30, 21) = {1,1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,}
gen_printf(30, 22) = {1,1,0,1,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,0,1,}
gen_printf(30, 23) = {1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,}
gen_printf(30, 24) = {1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,}
gen_printf(30, 25) = {1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,0,1,}
gen_printf(30, 26) = {1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,}
gen_printf(30, 27) = {1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,}
gen_printf(30, 28) = {1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,}
gen_printf(30, 29) = {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,}
gen_printf(30, 30) = {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,}
#edit - better evenly spaced pattern.
Explanation:
So let's take an array of 8 ints and we want to have 5 ones. The ideal ratio of (ones / zeros) in a sequence with 8 elements and 5 ones, well would be (5 / 3). We will never approach such ratio, but we can try.
The idea is to loop through the array and remember the number of ones and zeros we have written in the array. If the ratio of (written ones / written zeros) is lower then the destination ratio (ones / zeros) we want to achieve, we need to put a one to the sequence. Otherwise we put zero in the sequence. The ratio changes and we make the decision next time. The idea is to pursue the ideal ratio of ones per zeros in each slice of the array.
A simple way to do this would be to round the ideal fractional positions.
truths = randBetween(1, 8)
values = [0,0,0,0,0,0,0,0]
offset = randBetween(0, 8 * truths - 1)
for(i = 0; i < truths; i++) {
pos = (offset + (i * 8)) / truths
values[pos % 8] = 1
}
This is an application of Bresenham's line-drawing algorithm. I use it not because it's fast on old hardware, but it places true values exactly.
#include <iostream>
#include <stdexcept>
#include <string>
#include <random>
int main(int argc, char **argv) {
try {
// Read the argument.
if(argc != 2) throw std::invalid_argument("one argument");
int dy = std::stoi(argv[1]);
if(dy < 0 || dy > 8) throw std::out_of_range("[0..8]");
int values[8] = {0};
// https://en.wikipedia.org/wiki/Bresenham%27s_line_algorithm
int dx = 8;
int delta = 2 * dy - dx; // Balance the line. Permute it up later.
for(int x = 0; x < dx; x++) {
if(delta > 0) {
values[x] = 1;
delta -= 2 * dx;
}
delta += 2 * dy;
}
for(int x = 0; x < dx; x++)
std::cout << (x ? ", " : "") << values[x];
std::cout << std::endl;
// Rotate the number by a random amount.
// I'm sure there is an easier way to do this.
// https://stackoverflow.com/questions/7560114/random-number-c-in-some-range
std::random_device rd; // obtain a random number from hardware
std::mt19937 eng(rd()); // seed the generator
std::uniform_int_distribution<> distr(0, dx - 1);
int rotate = distr(eng);
bool first = true;
int x = rotate;
do {
std::cout << (first ? "" : ", ") << values[x];
first = false;
x = (x + 1) % dx;
} while(x != rotate);
std::cout << std::endl;
} catch(const std::exception &e) {
std::cerr << "Something went wrong: " << e.what() << std::endl;
return 1;
}
return 0;
}
Once you have an exact solution, rotate it by a random amount.
0, 1, 0, 0, 1, 0, 1, 0
1, 0, 0, 1, 0, 0, 1, 0
You need to calculate distance dynamically. One element is clear, that can reside at arbitrary location
2 elements is clear, too, distance needs to be 4.
4 elements need a distance of 2
8 elements a distance of 1
More difficult are numbers that don't divide the array:
3 requires a distance of 2.66.
5 requires a distance of 1.6
7 requires a distance of 0.875
Errm... In general, if you have a distance of X.Y, you will have to place some of the elements at distances of X and some at distances of X + 1. X is simple, it will be the result of an integer division: 8 / numberOfElements. The remainder will determine how often you will have to switch to X + 1: 8 % numberOfElements. For 3, this will result in 2, too, so you will have 1x distance of 2 and 2x distance of 3:
[ 1 0 1 0 0 1 0 0 ]
2 3 3 (distance to very first 1)
For 5, you'll get: 8/5 = 1, 8%5 = 3, so: 2x distance of 1, 3x distance of 2
[ 1 1 1 0 1 0 1 0 ]
1 1 2 2 2
For 7 you'll get: 8/7 = 1, 8%7 = 1, so: 7x distance of 1, 1x distance of 2
[ 1 1 1 1 1 1 1 0 ]
1 1 1 1 1 1 2
That will work for arbitrary array length L:
L/n = minimum distance
L%n = number of times to apply minimum distance
L-L%n = number of times to apply minimum distance + 1
Mathematical metrics won't reveal any difference between first applying all smaller distances then all larger ones, human sense for aesthetics, though, might prefer if you alternate between larger and smaller as often as possible – or you apply the algorithm recursively (for larger array length), to get something like 2x2, 3x3, 2x2, 3x3 instead of 4x2 and 6x3.

Largest Area Axis-Aligned Rectangle Inside Convex Polygon [duplicate]

Given an NxN binary matrix (containing only 0's or 1's), how can we go about finding largest rectangle containing all 0's?
Example:
I
0 0 0 0 1 0
0 0 1 0 0 1
II->0 0 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 1 <--IV
0 0 1 0 0 0
IV
For the above example, it is a 6×6 binary matrix. the return value in this case will be Cell 1:(2, 1) and Cell 2:(4, 4). The resulting sub-matrix can be square or rectangular. The return value can also be the size of the largest sub-matrix of all 0's, in this example 3 × 4.
Here's a solution based on the "Largest Rectangle in a Histogram" problem suggested by #j_random_hacker in the comments:
[Algorithm] works by iterating through
rows from top to bottom, for each row
solving this problem, where the
"bars" in the "histogram" consist of
all unbroken upward trails of zeros
that start at the current row (a
column has height 0 if it has a 1 in
the current row).
The input matrix mat may be an arbitrary iterable e.g., a file or a network stream. Only one row is required to be available at a time.
#!/usr/bin/env python
from collections import namedtuple
from operator import mul
Info = namedtuple('Info', 'start height')
def max_size(mat, value=0):
"""Find height, width of the largest rectangle containing all `value`'s."""
it = iter(mat)
hist = [(el==value) for el in next(it, [])]
max_size = max_rectangle_size(hist)
for row in it:
hist = [(1+h) if el == value else 0 for h, el in zip(hist, row)]
max_size = max(max_size, max_rectangle_size(hist), key=area)
return max_size
def max_rectangle_size(histogram):
"""Find height, width of the largest rectangle that fits entirely under
the histogram.
"""
stack = []
top = lambda: stack[-1]
max_size = (0, 0) # height, width of the largest rectangle
pos = 0 # current position in the histogram
for pos, height in enumerate(histogram):
start = pos # position where rectangle starts
while True:
if not stack or height > top().height:
stack.append(Info(start, height)) # push
elif stack and height < top().height:
max_size = max(max_size, (top().height, (pos - top().start)),
key=area)
start, _ = stack.pop()
continue
break # height == top().height goes here
pos += 1
for start, height in stack:
max_size = max(max_size, (height, (pos - start)), key=area)
return max_size
def area(size):
return reduce(mul, size)
The solution is O(N), where N is the number of elements in a matrix. It requires O(ncols) additional memory, where ncols is the number of columns in a matrix.
Latest version with tests is at https://gist.github.com/776423
Please take a look at Maximize the rectangular area under Histogram and then continue reading the solution below.
Traverse the matrix once and store the following;
For x=1 to N and y=1 to N
F[x][y] = 1 + F[x][y-1] if A[x][y] is 0 , else 0
Then for each row for x=N to 1
We have F[x] -> array with heights of the histograms with base at x.
Use O(N) algorithm to find the largest area of rectangle in this histogram = H[x]
From all areas computed, report the largest.
Time complexity is O(N*N) = O(N²) (for an NxN binary matrix)
Example:
Initial array F[x][y] array
0 0 0 0 1 0 1 1 1 1 0 1
0 0 1 0 0 1 2 2 0 2 1 0
0 0 0 0 0 0 3 3 1 3 2 1
1 0 0 0 0 0 0 4 2 4 3 2
0 0 0 0 0 1 1 5 3 5 4 0
0 0 1 0 0 0 2 6 0 6 5 1
For x = N to 1
H[6] = 2 6 0 6 5 1 -> 10 (5*2)
H[5] = 1 5 3 5 4 0 -> 12 (3*4)
H[4] = 0 4 2 4 3 2 -> 10 (2*5)
H[3] = 3 3 1 3 2 1 -> 6 (3*2)
H[2] = 2 2 0 2 1 0 -> 4 (2*2)
H[1] = 1 1 1 1 0 1 -> 4 (1*4)
The largest area is thus H[5] = 12
Here is a Python3 solution, which returns the position in addition to the area of the largest rectangle:
#!/usr/bin/env python3
import numpy
s = '''0 0 0 0 1 0
0 0 1 0 0 1
0 0 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 1
0 0 1 0 0 0'''
nrows = 6
ncols = 6
skip = 1
area_max = (0, [])
a = numpy.fromstring(s, dtype=int, sep=' ').reshape(nrows, ncols)
w = numpy.zeros(dtype=int, shape=a.shape)
h = numpy.zeros(dtype=int, shape=a.shape)
for r in range(nrows):
for c in range(ncols):
if a[r][c] == skip:
continue
if r == 0:
h[r][c] = 1
else:
h[r][c] = h[r-1][c]+1
if c == 0:
w[r][c] = 1
else:
w[r][c] = w[r][c-1]+1
minw = w[r][c]
for dh in range(h[r][c]):
minw = min(minw, w[r-dh][c])
area = (dh+1)*minw
if area > area_max[0]:
area_max = (area, [(r-dh, c-minw+1, r, c)])
print('area', area_max[0])
for t in area_max[1]:
print('Cell 1:({}, {}) and Cell 2:({}, {})'.format(*t))
Output:
area 12
Cell 1:(2, 1) and Cell 2:(4, 4)
Here is J.F. Sebastians method translated into C#:
private Vector2 MaxRectSize(int[] histogram) {
Vector2 maxSize = Vector2.zero;
int maxArea = 0;
Stack<Vector2> stack = new Stack<Vector2>();
int x = 0;
for (x = 0; x < histogram.Length; x++) {
int start = x;
int height = histogram[x];
while (true) {
if (stack.Count == 0 || height > stack.Peek().y) {
stack.Push(new Vector2(start, height));
} else if(height < stack.Peek().y) {
int tempArea = (int)(stack.Peek().y * (x - stack.Peek().x));
if(tempArea > maxArea) {
maxSize = new Vector2(stack.Peek().y, (x - stack.Peek().x));
maxArea = tempArea;
}
Vector2 popped = stack.Pop();
start = (int)popped.x;
continue;
}
break;
}
}
foreach (Vector2 data in stack) {
int tempArea = (int)(data.y * (x - data.x));
if(tempArea > maxArea) {
maxSize = new Vector2(data.y, (x - data.x));
maxArea = tempArea;
}
}
return maxSize;
}
public Vector2 GetMaximumFreeSpace() {
// STEP 1:
// build a seed histogram using the first row of grid points
// example: [true, true, false, true] = [1,1,0,1]
int[] hist = new int[gridSizeY];
for (int y = 0; y < gridSizeY; y++) {
if(!invalidPoints[0, y]) {
hist[y] = 1;
}
}
// STEP 2:
// get a starting max area from the seed histogram we created above.
// using the example from above, this value would be [1, 1], as the only valid area is a single point.
// another example for [0,0,0,1,0,0] would be [1, 3], because the largest area of contiguous free space is 3.
// Note that at this step, the heigh fo the found rectangle will always be 1 because we are operating on
// a single row of data.
Vector2 maxSize = MaxRectSize(hist);
int maxArea = (int)(maxSize.x * maxSize.y);
// STEP 3:
// build histograms for each additional row, re-testing for new possible max rectangluar areas
for (int x = 1; x < gridSizeX; x++) {
// build a new histogram for this row. the values of this row are
// 0 if the current grid point is occupied; otherwise, it is 1 + the value
// of the previously found historgram value for the previous position.
// What this does is effectly keep track of the height of continous avilable spaces.
// EXAMPLE:
// Given the following grid data (where 1 means occupied, and 0 means free; for clairty):
// INPUT: OUTPUT:
// 1.) [0,0,1,0] = [1,1,0,1]
// 2.) [0,0,1,0] = [2,2,0,2]
// 3.) [1,1,0,1] = [0,0,1,0]
//
// As such, you'll notice position 1,0 (row 1, column 0) is 2, because this is the height of contiguous
// free space.
for (int y = 0; y < gridSizeY; y++) {
if(!invalidPoints[x, y]) {
hist[y] = 1 + hist[y];
} else {
hist[y] = 0;
}
}
// find the maximum size of the current histogram. If it happens to be larger
// that the currently recorded max size, then it is the new max size.
Vector2 maxSizeTemp = MaxRectSize(hist);
int tempArea = (int)(maxSizeTemp.x * maxSizeTemp.y);
if (tempArea > maxArea) {
maxSize = maxSizeTemp;
maxArea = tempArea;
}
}
// at this point, we know the max size
return maxSize;
}
A few things to note about this:
This version is meant for use with the Unity API. You can easily make this more generic by replacing instances of Vector2 with KeyValuePair. Vector2 is only used for a convenient way to store two values.
invalidPoints[] is an array of bool, where true means the grid point is "in use", and false means it is not.
Solution with space complexity O(columns) [Can be modified to O(rows) also] and time complexity O(rows*columns)
public int maximalRectangle(char[][] matrix) {
int m = matrix.length;
if (m == 0)
return 0;
int n = matrix[0].length;
int maxArea = 0;
int[] aux = new int[n];
for (int i = 0; i < n; i++) {
aux[i] = 0;
}
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
aux[j] = matrix[i][j] - '0' + aux[j];
maxArea = Math.max(maxArea, maxAreaHist(aux));
}
}
return maxArea;
}
public int maxAreaHist(int[] heights) {
int n = heights.length;
Stack<Integer> stack = new Stack<Integer>();
stack.push(0);
int maxRect = heights[0];
int top = 0;
int leftSideArea = 0;
int rightSideArea = heights[0];
for (int i = 1; i < n; i++) {
if (stack.isEmpty() || heights[i] >= heights[stack.peek()]) {
stack.push(i);
} else {
while (!stack.isEmpty() && heights[stack.peek()] > heights[i]) {
top = stack.pop();
rightSideArea = heights[top] * (i - top);
leftSideArea = 0;
if (!stack.isEmpty()) {
leftSideArea = heights[top] * (top - stack.peek() - 1);
} else {
leftSideArea = heights[top] * top;
}
maxRect = Math.max(maxRect, leftSideArea + rightSideArea);
}
stack.push(i);
}
}
while (!stack.isEmpty()) {
top = stack.pop();
rightSideArea = heights[top] * (n - top);
leftSideArea = 0;
if (!stack.isEmpty()) {
leftSideArea = heights[top] * (top - stack.peek() - 1);
} else {
leftSideArea = heights[top] * top;
}
maxRect = Math.max(maxRect, leftSideArea + rightSideArea);
}
return maxRect;
}
But I get Time Limite exceeded excpetion when I try this on LeetCode. Is there any less complex solution?
I propose a O(nxn) method.
First, you can list all the maximum empty rectangles. Empty means that it covers only 0s. A maximum empty rectangle is such that it cannot be extended in a direction without covering (at least) one 1.
A paper presenting a O(nxn) algorithm to create such a list can be found at www.ulg.ac.be/telecom/rectangles as well as source code (not optimized). There is no need to store the list, it is sufficient to call a callback function each time a rectangle is found by the algorithm, and to store only the largest one (or choose another criterion if you want).
Note that a proof exists (see the paper) that the number of largest empty rectangles is bounded by the number of pixels of the image (nxn in this case).
Therefore, selecting the optimal rectangle can be done in O(nxn), and the overall method is also O(nxn).
In practice, this method is very fast, and is used for realtime video stream analysis.
Here is a version of jfs' solution, which also delivers the position of the largest rectangle:
from collections import namedtuple
from operator import mul
Info = namedtuple('Info', 'start height')
def max_rect(mat, value=0):
"""returns (height, width, left_column, bottom_row) of the largest rectangle
containing all `value`'s.
Example:
[[0, 0, 0, 0, 0, 0, 0, 0, 3, 2],
[0, 4, 0, 2, 4, 0, 0, 1, 0, 0],
[1, 0, 1, 0, 0, 0, 3, 0, 0, 4],
[0, 0, 0, 0, 4, 2, 0, 0, 0, 0],
[0, 0, 0, 2, 0, 0, 0, 0, 0, 0],
[4, 3, 0, 0, 1, 2, 0, 0, 0, 0],
[3, 0, 0, 0, 2, 0, 0, 0, 0, 4],
[0, 0, 0, 1, 0, 3, 2, 4, 3, 2],
[0, 3, 0, 0, 0, 2, 0, 1, 0, 0]]
gives: (3, 4, 6, 5)
"""
it = iter(mat)
hist = [(el==value) for el in next(it, [])]
max_rect = max_rectangle_size(hist) + (0,)
for irow,row in enumerate(it):
hist = [(1+h) if el == value else 0 for h, el in zip(hist, row)]
max_rect = max(max_rect, max_rectangle_size(hist) + (irow+1,), key=area)
# irow+1, because we already used one row for initializing max_rect
return max_rect
def max_rectangle_size(histogram):
stack = []
top = lambda: stack[-1]
max_size = (0, 0, 0) # height, width and start position of the largest rectangle
pos = 0 # current position in the histogram
for pos, height in enumerate(histogram):
start = pos # position where rectangle starts
while True:
if not stack or height > top().height:
stack.append(Info(start, height)) # push
elif stack and height < top().height:
max_size = max(max_size, (top().height, (pos - top().start), top().start), key=area)
start, _ = stack.pop()
continue
break # height == top().height goes here
pos += 1
for start, height in stack:
max_size = max(max_size, (height, (pos - start), start), key=area)
return max_size
def area(size):
return size[0] * size[1]
To be complete, here's the C# version which outputs the rectangle coordinates.
It's based on dmarra's answer but without any other dependencies.
There's only the function bool GetPixel(int x, int y), which returns true when a pixel is set at the coordinates x,y.
public struct INTRECT
{
public int Left, Right, Top, Bottom;
public INTRECT(int aLeft, int aTop, int aRight, int aBottom)
{
Left = aLeft;
Top = aTop;
Right = aRight;
Bottom = aBottom;
}
public int Width { get { return (Right - Left + 1); } }
public int Height { get { return (Bottom - Top + 1); } }
public bool IsEmpty { get { return Left == 0 && Right == 0 && Top == 0 && Bottom == 0; } }
public static bool operator ==(INTRECT lhs, INTRECT rhs)
{
return lhs.Left == rhs.Left && lhs.Top == rhs.Top && lhs.Right == rhs.Right && lhs.Bottom == rhs.Bottom;
}
public static bool operator !=(INTRECT lhs, INTRECT rhs)
{
return !(lhs == rhs);
}
public override bool Equals(Object obj)
{
return obj is INTRECT && this == (INTRECT)obj;
}
public bool Equals(INTRECT obj)
{
return this == obj;
}
public override int GetHashCode()
{
return Left.GetHashCode() ^ Right.GetHashCode() ^ Top.GetHashCode() ^ Bottom.GetHashCode();
}
}
public INTRECT GetMaximumFreeRectangle()
{
int XEnd = 0;
int YStart = 0;
int MaxRectTop = 0;
INTRECT MaxRect = new INTRECT();
// STEP 1:
// build a seed histogram using the first row of grid points
// example: [true, true, false, true] = [1,1,0,1]
int[] hist = new int[Height];
for (int y = 0; y < Height; y++)
{
if (!GetPixel(0, y))
{
hist[y] = 1;
}
}
// STEP 2:
// get a starting max area from the seed histogram we created above.
// using the example from above, this value would be [1, 1], as the only valid area is a single point.
// another example for [0,0,0,1,0,0] would be [1, 3], because the largest area of contiguous free space is 3.
// Note that at this step, the heigh fo the found rectangle will always be 1 because we are operating on
// a single row of data.
Tuple<int, int> maxSize = MaxRectSize(hist, out YStart);
int maxArea = (int)(maxSize.Item1 * maxSize.Item2);
MaxRectTop = YStart;
// STEP 3:
// build histograms for each additional row, re-testing for new possible max rectangluar areas
for (int x = 1; x < Width; x++)
{
// build a new histogram for this row. the values of this row are
// 0 if the current grid point is occupied; otherwise, it is 1 + the value
// of the previously found historgram value for the previous position.
// What this does is effectly keep track of the height of continous avilable spaces.
// EXAMPLE:
// Given the following grid data (where 1 means occupied, and 0 means free; for clairty):
// INPUT: OUTPUT:
// 1.) [0,0,1,0] = [1,1,0,1]
// 2.) [0,0,1,0] = [2,2,0,2]
// 3.) [1,1,0,1] = [0,0,1,0]
//
// As such, you'll notice position 1,0 (row 1, column 0) is 2, because this is the height of contiguous
// free space.
for (int y = 0; y < Height; y++)
{
if (!GetPixel(x, y))
{
hist[y]++;
}
else
{
hist[y] = 0;
}
}
// find the maximum size of the current histogram. If it happens to be larger
// that the currently recorded max size, then it is the new max size.
Tuple<int, int> maxSizeTemp = MaxRectSize(hist, out YStart);
int tempArea = (int)(maxSizeTemp.Item1 * maxSizeTemp.Item2);
if (tempArea > maxArea)
{
maxSize = maxSizeTemp;
maxArea = tempArea;
MaxRectTop = YStart;
XEnd = x;
}
}
MaxRect.Left = XEnd - maxSize.Item1 + 1;
MaxRect.Top = MaxRectTop;
MaxRect.Right = XEnd;
MaxRect.Bottom = MaxRectTop + maxSize.Item2 - 1;
// at this point, we know the max size
return MaxRect;
}
private Tuple<int, int> MaxRectSize(int[] histogram, out int YStart)
{
Tuple<int, int> maxSize = new Tuple<int, int>(0, 0);
int maxArea = 0;
Stack<Tuple<int, int>> stack = new Stack<Tuple<int, int>>();
int x = 0;
YStart = 0;
for (x = 0; x < histogram.Length; x++)
{
int start = x;
int height = histogram[x];
while (true)
{
if (stack.Count == 0 || height > stack.Peek().Item2)
{
stack.Push(new Tuple<int, int>(start, height));
}
else if (height < stack.Peek().Item2)
{
int tempArea = (int)(stack.Peek().Item2 * (x - stack.Peek().Item1));
if (tempArea > maxArea)
{
YStart = stack.Peek().Item1;
maxSize = new Tuple<int, int>(stack.Peek().Item2, (x - stack.Peek().Item1));
maxArea = tempArea;
}
Tuple<int, int> popped = stack.Pop();
start = (int)popped.Item1;
continue;
}
break;
}
}
foreach (Tuple<int, int> data in stack)
{
int tempArea = (int)(data.Item2 * (x - data.Item1));
if (tempArea > maxArea)
{
YStart = data.Item1;
maxSize = new Tuple<int, int>(data.Item2, (x - data.Item1));
maxArea = tempArea;
}
}
return maxSize;
}
An appropriate algorithm can be found within Algorithm for finding the largest inscribed rectangle in polygon (2019).
I implemented it in python:
import largestinteriorrectangle as lir
import numpy as np
grid = np.array([[0, 0, 0, 0, 1, 0],
[0, 0, 1, 0, 0, 1],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1],
[0, 0, 1, 0, 0, 0]],
"bool")
grid = ~grid
lir.lir(grid) # [1, 2, 4, 3]
the result comes as x, y, width, height

All possible combinations of coins

I need to write a program which displays all possible change combinations given an array of denominations [1 , 2, 5, 10, 20, 50, 100, 200] // 1 = 1 cent
Value to make the change from = 300
I'm basing my code on the solution from this site http://www.geeksforgeeks.org/dynamic-programming-set-7-coin-change/
#include<stdio.h>
int count( int S[], int m, int n )
{
int i, j, x, y;
// We need n+1 rows as the table is consturcted in bottom up manner using
// the base case 0 value case (n = 0)
int table[n+1][m];
// Fill the enteries for 0 value case (n = 0)
for (i=0; i<m; i++)
table[0][i] = 1;
// Fill rest of the table enteries in bottom up manner
for (i = 1; i < n+1; i++)
{
for (j = 0; j < m; j++)
{
// Count of solutions including S[j]
x = (i-S[j] >= 0)? table[i - S[j]][j]: 0;
// Count of solutions excluding S[j]
y = (j >= 1)? table[i][j-1]: 0;
// total count
table[i][j] = x + y;
}
}
return table[n][m-1];
}
// Driver program to test above function
int main()
{
int arr[] = {1, 2, 5, 10, 20, 50, 100, 200}; //coins array
int m = sizeof(arr)/sizeof(arr[0]);
int n = 300; //value to make change from
printf(" %d ", count(arr, m, n));
return 0;
}
The program runs fine. It displays the number of all possible combinations, but I need it to be more advanced. The way I need it to work is to display the result in following fashion:
1 cent: n number of possible combinations.
2 cents:
5 cents:
and so on...
How can I modify the code to achieve that ?
Greedy Algorithm Approach
Have this denominations in an int array say, int den[] = [1 , 2, 5, 10, 20, 50, 100, 200]
Iterate over this array
For each iteration do the following
Take the element in the denominations array
Divide the change to be allotted number by the element in denominations array number
If the change allotted number is perfectly divisible by the number in denomination array then you are done with the change for that number.
If the number is not perfectly divisible then check for the remainder and do the same iteration with other number
Exit the inner iteration once you get the value equal to the change number
Do the same for the next denomination available in our denomination array.
Explained with example
den = [1 , 2, 5, 10, 20, 50, 100, 200]
Change to be alloted : 270, let take this as x
and y be the temporary variable
Change map z[coin denomination, count of coins]
int y, z[];
First iteration :
den = 1
x = 270
y = 270/1;
if x is equal to y*den
then z[den, y] // z[1, 270]
Iteration completed
Second Iteration:
den = 2
x = 270
y = 270/2;
if x is equal to y*den
then z[den , y] // [2, 135]
Iteration completed
Lets take a odd number
x = 217 and den = 20
y= 217/20;
now x is not equal to y*den
then update z[den, y] // [20, 10]
find new x = x - den*y = 17
x=17 and identify the next change value by greedy it would be 10
den = 10
y = 17/10
now x is not equal to y*den
then update z[den, y] // [10, 1]
find new x = x - den*y = 7
then do the same and your map would be having following entries
[20, 10]
[10, 1]
[5, 1]
[2, 1]

How should I go about solving this recursion without trial and error

int sum_down(int x)
{
if (x >= 0)
{
x = x - 1;
int y = x + sum_down(x);
return y + sum_down(x);
}
else
{
return 1;
}
}
What is this smallest integer value of the parameter x, so that the returned value is greater than 1.000.000 ?
Right now I am just doing it by trial and error and since this question is asked via a paper format. I don't think I will have enough time to do trial and error. Question is, how do you guys visualise this quickly such that it can be solved easily. Thanks guys and I am new to programming so thanks in advance!
The recursion logic:
x = x - 1;
int y = x + sum_down(x);
return y + sum_down(x);
can be simplified to:
x = x - 1;
int y = x + sum_down(x) + sum_down(x);
return y;
which can be simplified to:
int y = (x-1) + sum_down(x-1) + sum_down(x-1);
return y;
which can be simplified to:
return (x-1) + 2*sum_down(x-1);
Put in mathematical form,
f(N) = (N-1) + 2*f(N-1)
with the recursion terminating when N is -1. f(-1) = 1.
Hence,
f(0) = -1 + 2*1 = 1
f(1) = 0 + 2*1 = 2
f(2) = 1 + 2*2 = 5
...
f(18) = 17 + 2*f(17) = 524269
f(19) = 18 + 2*524269 = 1048556
Your program can be written this way (sorry about c#):
public static void Main()
{
int i = 0;
int j = 0;
do
{
i++;
j = sum_down(i);
Console.Out.WriteLine("j:" + j);
} while (j < 1000000);
Console.Out.WriteLine("i:" + i);
}
static int sum_down(int x)
{
if (x >= 0)
{
return x - 1 + 2 * sum_down(x - 1);
}
else
{
return 1;
}
}
So at first iteration you'll get 2, then 5, then 12... So you can neglect the x-1 part since it'll stay little compared to the multiplication.
So we have:
i = 1 => sum_down ~= 4 (real is 2)
i = 2 => sum_down ~= 8 (real is 5)
i = 3 => sum_down ~= 16 (real is 12)
i = 4 => sum_down ~= 32 (real is 27)
i = 5 => sum_down ~= 64 (real is 58)
So we can say that sum_down(x) ~= 2^x+1. Then it's just basic math with 2^x+1 < 1 000 000 which is 19.
A bit late, but it's not that hard to get an exact non-recursive formula.
Write it up mathematically, as explained in other answers already:
f(-1) = 1
f(x) = 2*f(x-1) + x-1
This is the same as
f(-1) = 1
f(x+1) = 2*f(x) + x
(just switched from x and x-1 to x+1 and x, difference 1 in both cases)
The first few x and f(x) are:
x: -1 0 1 2 3 4
f(x): 1 1 2 5 12 27
And while there are many arbitrary complicated ways to transform this into a non-recursive formula, with easy ones it often helps to write up what the difference is between each two elements:
x: -1 0 1 2 3 4
f(x): 1 1 2 5 12 27
0 1 3 7 15
So, for some x
f(x+1) - f(x) = 2^(x+1) - 1
f(x+2) - f(x) = (f(x+2) - f(x+1)) + (f(x+1) - f(x)) = 2^(x+2) + 2^(x+1) - 2
f(x+n) - f(x) = sum[0<=i<n](2^(x+1+i)) - n
With eg. a x=0 inserted, to make f(x+n) to f(n):
f(x+n) - f(x) = sum[0<=i<n](2^(x+1+i)) - n
f(0+n) - f(0) = sum[0<=i<n](2^(0+1+i)) - n
f(n) - 1 = sum[0<=i<n](2^(i+1)) - n
f(n) = sum[0<=i<n](2^(i+1)) - n + 1
f(n) = sum[0<i<=n](2^i) - n + 1
f(n) = (2^(n+1) - 2) - n + 1
f(n) = 2^(n+1) - n - 1
No recursion anymore.
How about this :
int x = 0;
while (sum_down(x) <= 1000000)
{
x++;
}
The loop increments x until the result of sum_down(x) is superior to 1.000.000.
Edit : The result would be 19.
While trying to understand and simplify the recursion logic behind the sum_down() function is enlightening and informative, this snippet tend to be logical and pragmatic in that it does not try and solve the problem in terms of context, but in terms of results.
Two lines of Python code to answer your question:
>>> from itertools import * # no code but needed for dropwhile() and count()
Define the recursive function (See R Sahu's answer)
>>> f = lambda x: 1 if x<0 else (x-1) + 2*f(x-1)
Then use the dropwhile() function to remove elements from the list [0, 1, 2, 3, ....] for which f(x)<=1000000, resulting in a list of integers for which f(x) > 1000000. Note: count() returns an infinite "list" of [0, 1, 2, ....]
The dropwhile() function returns a Python generator so we use next() to get the first value of the list:
>>> next(dropwhile(lambda x: f(x)<=1000000, count()))
19