Algorithm on hexagonal grid - c++

Hexagonal grid is represented by a two-dimensional array with R rows and C columns. First row always comes "before" second in hexagonal grid construction (see image below). Let k be the number of turns. Each turn, an element of the grid is 1 if and only if the number of neighbours of that element that were 1 the turn before is an odd number. Write C++ code that outputs the grid after k turns.
Limitations:
1 <= R <= 10, 1 <= C <= 10, 1 <= k <= 2^(63) - 1
An example with input (in the first row are R, C and k, then comes the starting grid):
4 4 3
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 0
Simulation: image, yellow elements represent '1' and blank represent '0'.
This problem is easy to solve if I simulate and produce a grid each turn, but with big enough k it becomes too slow. What is the faster solution?
EDIT: code (n and m are used instead R and C) :
#include <cstdio>
#include <cstring>
using namespace std;
int old[11][11];
int _new[11][11];
int n, m;
long long int k;
int main() {
scanf ("%d %d %lld", &n, &m, &k);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) scanf ("%d", &old[i][j]);
}
printf ("\n");
while (k) {
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
int count = 0;
if (i % 2 == 0) {
if (i) {
if (j) count += old[i-1][j-1];
count += old[i-1][j];
}
if (j) count += (old[i][j-1]);
if (j < m-1) count += (old[i][j+1]);
if (i < n-1) {
if (j) count += old[i+1][j-1];
count += old[i+1][j];
}
}
else {
if (i) {
if (j < m-1) count += old[i-1][j+1];
count += old[i-1][j];
}
if (j) count += old[i][j-1];
if (j < m-1) count += old[i][j+1];
if (i < n-1) {
if (j < m-1) count += old[i+1][j+1];
count += old[i+1][j];
}
}
if (count % 2) _new[i][j] = 1;
else _new[i][j] = 0;
}
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) old[i][j] = _new[i][j];
}
k--;
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
printf ("%d", old[i][j]);
}
printf ("\n");
}
return 0;
}

For a given R and C, you have N=R*C cells.
If you represent those cells as a vector of elements in GF(2), i.e, 0s and 1s where arithmetic is performed mod 2 (addition is XOR and multiplication is AND), then the transformation from one turn to the next can be represented by an N*N matrix M, so that:
turn[i+1] = M*turn[i]
You can exponentiate the matrix to determine how the cells transform over k turns:
turn[i+k] = (M^k)*turn[i]
Even if k is very large, like 2^63-1, you can calculate M^k quickly using exponentiation by squaring: https://en.wikipedia.org/wiki/Exponentiation_by_squaring This only takes O(log(k)) matrix multiplications.
Then you can multiply your initial state by the matrix to get the output state.
From the limits on R, C, k, and time given in your question, it's clear that this is the solution you're supposed to come up with.

There are several ways to speed up your algorithm.
You do the neighbour-calculation with the out-of bounds checking in every turn. Do some preprocessing and calculate the neighbours of each cell once at the beginning. (Aziuth has already proposed that.)
Then you don't need to count the neighbours of all cells. Each cell is on if an odd number of neighbouring cells were on in the last turn and it is off otherwise.
You can think of this differently: Start with a clean board. For each active cell of the previous move, toggle the state of all surrounding cells. When an even number of neighbours cause a toggle, the cell is on, otherwise the toggles cancel each other out. Look at the first step of your example. It's like playing Lights Out, really.
This method is faster than counting the neighbours if the board has only few active cells and its worst case is a board whose cells are all on, in which case it is as good as neighbour-counting, because you have to touch each neighbours for each cell.
The next logical step is to represent the board as a sequence of bits, because bits already have a natural way of toggling, the exclusive or or xor oerator, ^. If you keep the list of neigbours for each cell as a bit mask m, you can then toggle the board b via b ^= m.
These are the improvements that can be made to the algorithm. The big improvement is to notice that the patterns will eventually repeat. (The toggling bears resemblance with Conway's Game of Life, where there are also repeating patterns.) Also, the given maximum number of possible iterations, 2⁶³ is suspiciously large.
The playing board is small. The example in your question will repeat at least after 2¹⁶ turns, because the 4×4 board can have at most 2¹⁶ layouts. In practice, turn 127 reaches the ring pattern of the first move after the original and it loops with a period of 126 from then.
The bigger boards may have up to 2¹⁰⁰ layouts, so they may not repeat within 2⁶³ turns. A 10×10 board with a single active cell near the middle has ar period of 2,162,622. This may indeed be a topic for a maths study, as Aziuth suggests, but we'll tacke it with profane means: Keep a hash map of all previous states and the turns where they occurred, then check whether the pattern has occurred before in each turn.
We now have:
a simple algorithm for toggling the cells' state and
a compact bitwise representation of the board, which allows us to create a hash map of the previous states.
Here's my attempt:
#include <iostream>
#include <map>
/*
* Bit representation of a playing board, at most 10 x 10
*/
struct Grid {
unsigned char data[16];
Grid() : data() {
}
void add(size_t i, size_t j) {
size_t k = 10 * i + j;
data[k / 8] |= 1u << (k % 8);
}
void flip(const Grid &mask) {
size_t n = 13;
while (n--) data[n] ^= mask.data[n];
}
bool ison(size_t i, size_t j) const {
size_t k = 10 * i + j;
return ((data[k / 8] & (1u << (k % 8))) != 0);
}
bool operator<(const Grid &other) const {
size_t n = 13;
while (n--) {
if (data[n] > other.data[n]) return true;
if (data[n] < other.data[n]) return false;
}
return false;
}
void dump(size_t n, size_t m) const {
for (size_t i = 0; i < n; i++) {
for (size_t j = 0; j < m; j++) {
std::cout << (ison(i, j) ? 1 : 0);
}
std::cout << '\n';
}
std::cout << '\n';
}
};
int main()
{
size_t n, m, k;
std::cin >> n >> m >> k;
Grid grid;
Grid mask[10][10];
for (size_t i = 0; i < n; i++) {
for (size_t j = 0; j < m; j++) {
int x;
std::cin >> x;
if (x) grid.add(i, j);
}
}
for (size_t i = 0; i < n; i++) {
for (size_t j = 0; j < m; j++) {
Grid &mm = mask[i][j];
if (i % 2 == 0) {
if (i) {
if (j) mm.add(i - 1, j - 1);
mm.add(i - 1, j);
}
if (j) mm.add(i, j - 1);
if (j < m - 1) mm.add(i, j + 1);
if (i < n - 1) {
if (j) mm.add(i + 1, j - 1);
mm.add(i + 1, j);
}
} else {
if (i) {
if (j < m - 1) mm.add(i - 1, j + 1);
mm.add(i - 1, j);
}
if (j) mm.add(i, j - 1);
if (j < m - 1) mm.add(i, j + 1);
if (i < n - 1) {
if (j < m - 1) mm.add(i + 1, j + 1);
mm.add(i + 1, j);
}
}
}
}
std::map<Grid, size_t> prev;
std::map<size_t, Grid> pattern;
for (size_t turn = 0; turn < k; turn++) {
Grid next;
std::map<Grid, size_t>::const_iterator it = prev.find(grid);
if (1 && it != prev.end()) {
size_t start = it->second;
size_t period = turn - start;
size_t index = (k - turn) % period;
grid = pattern[start + index];
break;
}
prev[grid] = turn;
pattern[turn] = grid;
for (size_t i = 0; i < n; i++) {
for (size_t j = 0; j < m; j++) {
if (grid.ison(i, j)) next.flip(mask[i][j]);
}
}
grid = next;
}
for (size_t i = 0; i < n; i++) {
for (size_t j = 0; j < m; j++) {
std::cout << (grid.ison(i, j) ? 1 : 0);
}
std::cout << '\n';
}
return 0;
}
There is probably room for improvement. Especially, I'm not so sure how it fares for big boards. (The code above uses an ordered map. We don't need the order, so using an unordered map will yield faster code. The example above with a single active cell on a 10×10 board took significantly longer than a second with an ordered map.)

Not sure about how you did it - and you should really always post code here - but let's try to optimize things here.
First of all, there is not really a difference between that and a quadratic grid. Different neighbor relationships, but I mean, that is just a small translation function. If you have a problem there, we should treat this separately, maybe on CodeReview.
Now, the naive solution is:
for all fields
count neighbors
if odd: add a marker to update to one, else to zero
for all fields
update all fields by marker of former step
this is obviously in O(N). Iterating twice is somewhat twice the actual run time, but should not be that bad. Try not to allocate space every time that you do that but reuse existing structures.
I'd propose this solution:
at the start:
create a std::vector or std::list "activated" of pointers to all fields that are activated
each iteration:
create a vector "new_activated"
for all items in activated
count neighbors, if odd add to new_activated
for all items in activated
set to inactive
replace activated by new_activated*
for all items in activated
set to active
*this can be done efficiently by putting them in a smart pointer and use move semantics
This code only works on the activated fields. As long as they stay within some smaller area, this is far more efficient. However, I have no idea when this changes - if there are activated fields all over the place, this might be less efficient. In that case, the naive solution might be the best one.
EDIT: after you now posted your code... your code is quite procedural. This is C++, use classes and use representation of things. Probably you do the search for neighbors right, but you can easily make mistakes there and therefore should isolate that part in a function, or better method. Raw arrays are bad and variables like n or k are bad. But before I start tearing your code apart, I instead repeat my recommendation, put the code on CodeReview, having people tear it apart until it is perfect.

This started off as a comment, but I think it could be helpful as an answer in addition to what has already been stated.
You stated the following limitations:
1 <= R <= 10, 1 <= C <= 10
Given these restrictions, I'll take the liberty to can represent the grid/matrix M of R rows and C columns in constant space (i.e. O(1)), and also check its elements in O(1) instead of O(R*C) time, thus removing this part from our time-complexity analysis.
That is, the grid can simply be declared as bool grid[10][10];.
The key input is the large number of turns k, stated to be in the range:
1 <= k <= 2^(63) - 1
The problem is that, AFAIK, you're required to perform k turns. This makes the algorithm be in O(k). Thus, no proposed solution can do better than O(k)[1].
To improve the speed in a meaningful way, this upper-bound must be lowered in some way[1], but it looks like this cannot be done without altering the problem constraints.
Thus, no proposed solution can do better than O(k)[1].
The fact that k can be so large is the main issue. The most anyone can do is improve the rest of the implementation, but this will only improve by a constant factor; you'll have to go through k turns regardless of how you look at it.
Therefore, unless some clever fact and/or detail is found that allows this bound to be lowered, there's no other choice.
[1] For example, it's not like trying to determine if some number n is prime, where you can check all numbers in the range(2, n) to see if they divide n, making it a O(n) process, or notice that some improvements include only looking at odd numbers after checking n is not even (constant factor; still O(n)), and then checking odd numbers only up to √n, i.e., in the range(3, √n, 2), which meaningfully lowers the upper-bound down to O(√n).

Related

Reversing a sequence of cyclic shifts

Here's the problem I have a hard time solving.
You are given a ciphertext Y and a sequence of cyclic shifts that had produced Y from string Z, the shift with parameters (i, j, k) applies to the substring Z[i..j] (from i-th to j-th character, inclusive) and cyclicly rotates it to the right k times. String characters are numbered starting from one. Given the above information, your task is to guess the initial cleartext X.
Input:
The first line contains the ciphertext, which is a nonempty string consisting of N lowercase English letters (1 ≤ N ≤ 50000). The second line contains the number of shifts M (1 ≤ M ≤ 50000).
The following M lines describe the sequence of cyclic shifts (in the order of their application to the cleartext). Each shift is described by three parameters i, j, k (1 ≤ i < j ≤ N, 1 ≤ k ≤ j − i).
Example of input:
logoduck
3
1 3 1
4 5 1
1 4 1
As output you should provide the deciphered text (``goodluck'' for example).
The obvious approach is to try and reverse each shift starting from the last one. It seems that this approach is not time-efficient. However, I can't come up with any ideas how to do it any other way, so any help is appreciated.
I attach my code:
#include <iostream>
#include <vector>
#include <string>
int main() {
std::string message;
std::cin >> message;
int number_of_elements = message.size();
int elements[number_of_elements];
for (int i = 0; i < number_of_elements; ++i) {
elements[i] = i;
}
int number_of_shifts;
std::cin >> number_of_shifts;
std::vector<std::vector<int>> shifts(number_of_shifts);
for (int iterator = 0; iterator < number_of_shifts; ++iterator) {
int left, right, by;
std::cin >> left >> right >> by;
--left;
--right;
shifts[iterator].push_back(left);
shifts[iterator].push_back(right);
shifts[iterator].push_back(by);
}
for (int iterator = number_of_shifts - 1; -1 < iterator; --iterator) {
int current[number_of_elements];
int left, right, by;
left = shifts[iterator][0];
right = shifts[iterator][1];
by = shifts[iterator][2];
for (int j = right; left - 1 < j; --j) {
if (j - by < left) {
current[right + 1 - (left - (j - by))] = elements[j];
} else {
current[j - by] = elements[j];
}
}
for (int j = left; j < right + 1; ++j) {
elements[j] = current[j];
}
}
for (int i = 0; i < number_of_elements; ++i) {
std::cout << message.substr(elements[i], 1);
}
return 0;
}
You can do this in O(M log N) time using a data structure called a "rope", which is like a string that supports splitting and concatenation in O(log N) time: https://en.wikipedia.org/wiki/Rope_(data_structure)
Rotations can be build from these operations, of course.
The implementation is a binary tree, B-tree, or similar with limited-size strings in the leaves.
It's not hard to find C++ implementations, but unfortunately, the STL doesn't have one. If you have to implement it yourself, it's a little tricky.

Error trying to find all the prime numbers from 2 to n using Sieve of Eratosthenes in C++

I need to find all the prime numbers from 2 to n using the Sieve of Eratosthenes. I looked on Wikipedia(Sieve of Eratosthenes) to find out what the Sieve of Eratosthenes was, and it gave me this pseudocode:
Input: an integer n > 1
Let A be an array of Boolean values, indexed by integers 2 to n,
initially all set to true.
for i = 2, 3, 4, ..., not exceeding √n:
if A[i] is true:
for j = i2, i2+i, i2+2i, i2+3i, ..., not exceeding n :
A[j] := false
Output: all i such that A[i] is true.
So I used this and translated it to C++. It looks fine to me, but I have a couple errors. Firstly, if I input 2 or 3 into n, it says:
terminate called after throwing an instance of 'Range_error'
what(): Range_error: 2
Also, whenever I enter a 100 or anything else (4, 234, 149, 22, anything), it accepts the input for n, and doesn't do anything. Here is my C++ translation:
#include "std_lib_facilities.h"
int main()
{
/* this program will take in an input 'n' as the maximum value. Then it will calculate
all the prime numbers between 2 and n. It follows the Sieve of Eratosthenes with
the algorithms from Wikipedia's pseudocode translated by me into C++*/
int n;
cin >> n;
vector<string>A;
for(int i = 2; i <= n; ++i) // fills the whole table with "true" from 0 to n-2
A.push_back("true");
for(int i = 2; i <= sqrt(n); ++i)
{
i -= 2; // because I built the vector from 0 to n-2, i need to reflect that here.
if(A[i] == "true")
{
for(int j = pow(i, 2); j <= n; j += i)
{
A[j] = "false";
}
}
}
//print the prime numbers
for(int i = 2; i <= n; ++i)
{
if(A[i] == "true")
cout << i << '\n';
}
return 0;
}
The issue is coming from the fact that the indexes are not in line with the value they are representing, i.e., they are moved down by 2. By doing this operation, they no longer have the same mathematical properties.
Basically, the value 3 is at position 1 and the value 4 is at position 2. When you are testing for division, you are using the positions as they were values. So instead of testing if 4%3==0, you are testing that 2%1=0.
In order to make your program works, you have to remove the -2 shifting of the indexes:
int main()
{
int n;
cin >> n;
vector<string>A;
for(int i = 0; i <= n; ++i) // fills the whole table with "true" from 0 to n-2
A.push_back("true");
for(int i = 2; i <= sqrt(n); ++i)
{
if(A[i] == "true")
{
for(int j = pow(i, 2); j <= n; j += i)
{
A[j] = "false";
}
}
}
//print the prime numbers
for(int i = 2; i <= n; ++i)
{
if(A[i] == "true")
cout << i << '\n';
}
return 0;
}
I agree with other comments, you could use a vector of bools. And directly initialize them with the right size and value:
std::vector<bool> A(n, false);
Here you push back n-1 elements
vector<string>A;
for(int i = 2; i <= n; ++i) // fills the whole table with "true" from 0 to n-2
A.push_back("true");
but here you access your vector from A[2] to A[n].
//print the prime numbers
for(int i = 2; i <= n; ++i)
{
if(A[i] == "true")
cout << i << '\n';
}
A has elements at positions A[0] to A[n-2]. You might correct this defect by initializing your vector differently. For example as
vector<string> A(n+1, "true");
This creates a vector A with n+1 strings with default values "true" which can be accessed through A[0] to A[n]. With this your code should run, even if it has more deficits. But I think you learn most if you just try to successfully implement the sieve and then look for (good) alternatives in the internet.
This is painful. Why are you using a string array to store boolean values, and not, let's say, an array of boolean values? Why are you leaving out the first two array elements, forcing you to do some adjustment of all indices? Which you then forget half the time, totally breaking your code? At least you should change this line:
i -= 2; // because I built the vector from 0 to n-2, i need to reflect that here.
to:
i -= 2; // because I left the first two elements out, I that here.
// But only here, doing it everywhere is too annoying.
As a result of that design decision, when you execute this line:
for(int j = pow(i, 2); j <= n; j += i)
i is actually zero which means j will stay zero forever.

Find which two values in an array maximize a given expression?

I met a very simple interview question, but my solution is incorrect. Any helps on this? 1)any bugs in my solution? 2)any good idea for time complexity O(n)?
Question:
Given an int array A[], define X=A[i]+A[j]+(j-i), j>=i. Find max value of X?
My solution is:
int solution(vector<int> &A){
if(A.empty())
return -1;
long long max_dis=-2000000000, cur_dis;
int size = A.size();
for(int i=0;i<size;i++){
for(int j=i;j<size;j++){
cur_dis=A[j]+A[i]+(j-i);
if(cur_dis > max_dis)
max_dis=cur_dis;
}
}
return max_dis;
}
The crucial insight is that it can be done in O(n) only if you track where potentially useful values are even before you're certain they'll prove usable.
Start with best_i = best_j = max_i = 0. The first two track the i and j values to use in the solution. The next one will record the index with the highest contributing factor for i, i.e. where A[i] - i is highest.
Let's call the value of X for some values of i and j "Xi,j", and start by recording our best solution so far ala Xbest = X0,0
Increment n along the array...
whenever the value at [n] gives a better "i" contribution for A[i] - i than max_i, update max_i.
whenever using n as the "j" index yields Xmax_i,n greater than Xbest, best_i = max_i, best_j = n.
Discussion - why/how it works
j_random_hacker's comment suggests I sketch a proof, but honestly I've no idea where to start. I'll try to explain as best I can - if someone else has a better explanation please chip in....
Restating the problem: greatest Xi,j where j >= i. Given we can set an initial Xbest of X0,0, the problem is knowing when to update it and to what. As we contemplate successive indices in the array as potential values for j, we want to generate Xi,j=n for some i (discussed next) to compare with Xbest. But, what i value to use? Well, given any index from 0 to n is <= j, the j >= i constraint isn't relevant if we pick the best i value from the indices we've already visited. We work out the best i value by separating the i-related contribution to X from the j-related contribution - A[i] - i - so in preparation for considering whether we've a new best solution with j=n we must maintain the best_i variable too as we go.
A way to approach the problem
For whatever it's worth - when I was groping around for a solution, I wrote down on paper some imaginary i and j contributions that I could see covered the interesting cases... where Ci and Cj are the contributions related to n's use as i and j respectively, something like
n 0 1 2 3 4
Ci 4 2 8 3 1
Cj 12 4 3 5 9
You'll notice I didn't bother picking values where Ci could be A[i] - i while Cj was A[j] + j... I could see the emerging solution should work for any formulas, and that would have just made it harder to capture the interesting cases. So - what's the interesting case? When n = 2 the Ci value is higher than anything we've seen in earlier elements, but given only knowledge of those earlier elements we can't yet see a way to use it. That scenario is the single "great" complication of the problem. What's needed is a Cj value of at least 9 so Xbest is improved, which happens to come along when n = 4. If we'd found an even better Ci at [3] then we'd of course want to use that. best_i tracks where that waiting-on-a-good-enough-Cj value index is.
Longer version of my comment: what about iterating the array from both ends, trying to find the highest number, while decreasing it by the distance from the appripriate end. Would that find the correct indexes (and thus the correct X)?
#include <vector>
#include <algorithm>
#include <iostream>
#include <random>
#include <climits>
long long brutal(const std::vector<int>& a) {
long long x = LLONG_MIN;
for(int i=0; i < a.size(); i++)
for(int j=i; j < a.size(); j++)
x = std::max(x, (long long)a[i] + a[j] + j-i);
return x;
}
long long smart(const std::vector<int>& a) {
if(a.size() == 0) return LLONG_MIN;
long long x = LLONG_MIN, y = x;
for(int i = 0; i < a.size(); i++)
x = std::max(x, (long long)a[i]-i);
for(int j = 0; j < a.size(); j++)
y = std::max(y, (long long)a[j]+j);
return x + y;
}
int main() {
std::random_device rd;
std::uniform_int_distribution<int> rlen(0, 1000);
std::uniform_int_distribution<int> rnum(INT_MIN,INT_MAX);
std::vector<int> v;
for(int loop = 0; loop < 10000; loop++) {
v.resize(rlen(rd));
for(int i = 0; i < v.size(); i++)
v[i] = rnum(rd);
if(brutal(v) != smart(v)) {
std::cout << "bad" << std::endl;
return -1;
}
}
std::cout << "good" << std::endl;
}
I'll write in pseudo code because I don't have much time, but this should be the most performing way using recursion
compare(array, left, right)
val = array[left] + array[right] + (right - left);
if (right - left) > 1
val1 = compare(array, left, right-1);
val2 = compare(array, left+1, right);
val = Max(Max(val1,val2),val);
end if
return val
and than you call simply
compare(array,0,array.length);
I think I found a incredibly faster solution but you need to check it:
you need to rewrite your array as follow
Array[i] = array[i] + (MOD((array.lenght / 2) - i));
Then you just find the 2 highest value of the array and sum them, that should be your solution, almost O(n)
wait maybe I'm missing something... I have to check.
Ok you get the 2 highest value from this New Array, and save the positions i, and j. Then you need to calculate from the original array your result.
------------ EDIT
This should be an implementation of the method suggested by Tony D (in c#) that I tested.
int best_i, best_j, max_i, currentMax;
best_i = 0;
best_j = 0;
max_i = 0;
currentMax = 0;
for (int n = 0; n < array.Count; n++)
{
if (array[n] - n > array[max_i] - max_i) max_i = n;
if (array[n] + array[max_i] - (n - max_i) > currentMax)
{
best_i = max_i;
best_j = n;
currentMax = array[n] + array[max_i] - (n - max_i);
}
}
return currentMax;
Question:
Given an int array A[], define X=A[i]+A[j]+(j-i), j>=i. Find max value of X?
Answer O(n):
lets rewrite the formula: X = A[i]-i + A[j]+j
we can track the highest A[i]-i we got and the highest A[j]+j we got. We loop over the array once and update both of our max values. After looping once we return the sum of A[i]-i + A[j]+j, which equals X.
We absolutely don't care about the j>=i constraint, because it is always true when we maximize both A[i]-i and A[j]+j
Code:
int solution(vector<int> &A){
if(A.empty()) return -1;
long long max_Ai_part =-2000000000;
long long max_Aj_part =-2000000000;
int size = A.size();
for(int i=0;i<size;i++){
if(max_Ai_part < A[i] - i)
max_Ai_part = A[i] - i;
if(max_Aj_part < A[j] + j)
max_Ai_part = A[j] - j;
}
return max_Ai_part + max_Aj_part;
}
Bonus:
most people get confused with the j>=i constraint. If you have a feeling for numbers, you should be able to see that i should tend to be lower than j.
Assume we have our formula, it is maximized and i > j. (this is impossible, but lets check it out)
we define x1 := j-i and x2 = i-j
A[i]+A[j]+j-i = A[i]+A[j] + x1, x1 < 0
we could then swap i with j and end up with this:
A[j]+A[i]+i-j = A[i]+A[j] + x2, x2 > 0
it is basically the same formula, but now because i > j the second formula will be greater than the first. In other words we could increase the maximum by swapping i and j which can't be true if we already had the maximum.
If we ever find a maximum, i cannot be greater than j.

Dynamic Programming for analyzing a Vector of Vector of Bools

Here's the problem I'm trying to solve.
Given a square of bools, I want to find the size of largest subsquare entirely full of trues (1's). Also, I am allowed O(n^2) memory requirement as well as the run time must be O(n^2). The header to the function will look like the following
unsigned int largestCluster(const vector<vector<bool>> &map);
Some other things to note will be there always be at least one 1 (a 1 x 1 subsquare) and the input will also always be a square.
Now for my attempts at the problem:
Given this is based on the concept of dynamic programming, which to my limited understanding, helps store information that is previously found for later use. So if my understanding is correcting, Prim's algorithm would be an example of a dynamic algorithm because it remembers what vertices we've visited, the smallest distance to a vertice, and the parent that enables that smallest distance.
I tried analyzing the map and keeping track of the number of true neighbors, a true location location has. I was thinking if a spot had 4 true neighbors than that is a potential subsquare. However, this didn't help with subsquares of size 4 or less..
I tried to include a lot of detail in this question for help as I'm trying to game plan a way to tackle this problem because I don't believe it's going to require writing a lengthy function. Thanks for any help
Here's my nomination. Dynamic programming, O(n^2) complexity. I realize that I probably just did somebody's homework, but it looked like an intriguing little problem.
int largestCluster(const std::vector<std::vector<bool> > a)
{
const int n = a.size();
std::vector<std::vector<short> > s;
s.resize(n);
for (int i = 0; i < n; ++i)
{
s[i].resize(n);
}
s[0][0] = a[0][0] ? 1 : 0;
int maxSize = s[0][0];
for (int k = 1; k < n; ++k)
{
s[k][0] = a[k][0] ? 1 : 0;
for (int j = 1; j < k; ++j)
{
if (a[k][j])
{
int m = s[k - 1][j - 1];
if (s[k][j - 1] < m)
{
m = s[k][j - 1];
}
if (s[k - 1][j] < m)
{
m = s[k - 1][j];
}
s[k][j] = ++m;
if (m > maxSize)
{
maxSize = m;
}
}
else
{
s[k][j] = 0;
}
}
s[0][k] = a[0][k] ? 1 : 0;
for (int i = 1; i <= k; ++i)
{
if (a[i][k])
{
int m = s[i - 1][k - 1];
if (s[i - 1][k] < m)
{
m = s[i - 1][k];
}
if (s[i][k - 1] < m)
{
m = s[i][k - 1];
}
s[i][k] = ++m;
if (m > maxSize)
{
maxSize = m;
}
}
else
{
s[i][k] = 0;
}
}
}
return maxSize;
}
If you want a dynamic programming approach one strategy I could think of would be to consider a box (base case 1 entry) as a potential upper left corner of a larger box and start by the bottom right corner of your large square, you then need to evaluate only the "boxes" (using information previously stored to only consider the largest cluster so far) that are to the right, bottom, and diagonally right-bottom of that we are now evaluating.
By saving information about each edge we would be respecting the O(n^2) (though not o(n^2)) however for the run-time you need to work on the details of the approach to get to O(n^2)
This is just a rough draft idea as I don't have much time, and I would appreciate any more hints/comments about this myself.

Backtracking - Filling a grid with coins

I was trying to do this question i came across while looking up interview questions. We are asked the number of ways of placing r coins on a n*m grid such that each row and col contain at least one coin.
I thought of a backtracking solution, processing each cell in the grid in a row major order, I have set up my recursion in this way. Seems my approach is faulty because it outputs 0 every time. Could someone please help me find the error in my approach. ? Thanks.
constraints. n , m < 200 and r < n*m;
Here is the code i came up with.
#include<cstdio>
#define N 201
int n, m , r;
int used[N][N];
int grid[N][N] ; // 1 is coin is placed . 0 otherwise. // -1 undecided.
bool isOk()
{
int rows[N];
int cols[N];
for(int i = 0 ; i < n ; i++) rows[i] = 0;
for(int i = 0 ; i < m ; i++) cols[i] = 0;
int sum = 0;
for(int i = 0 ; i < n ; i++)for(int j = 0; j < m ; j++)
{
if(grid[i][j]==1)
{
rows[i]++;
cols[j]++;
sum++;
}
}
for(int i = 0 ; i < n ; i++)
{
if(rows[i]==0) return false;
}
for(int j = 0 ; j < n ; j++)
{
if(cols[j]==0) return false;
}
if(sum==r) return true;
else return false;
}
int calc_ways(int row , int col, int coins)
{
if(row >= n) return 0;
if(col >= m) return 0;
if(coins > r) return 0;
if(coins == r)
{
bool res = isOk();
if(res) return 1;
else 0;
}
if(row == n - 1 and col== m- 1)
{
bool res = isOk();
if(res) return 1;
else return 0;
}
int nrow, ncol;
if(col + 1 >= m)
{
nrow = row + 1;
ncol = 0;
}
else
{
nrow = row;
ncol = col + 1;
}
if(used[row][col]) return calc_ways(nrow, ncol, coins);
int ans = 0;
used[row][col] = 1;
grid[row][col] = 0;
ans += calc_ways(nrow , ncol , coins);
grid[row][col] = 1;
ans += calc_ways(nrow , ncol , coins + 1);
return ans;
}
int main()
{
int t;
scanf("%d" , &t);
while(t--)
{
scanf("%d %d %d" , &n , &m , &r);
for(int i = 0 ; i <= n ; i++)
{
for(int j = 0; j <= m ; j++)
{
used[i][j] = 0;
grid[i][j] = -1;
}
}
printf("%d\n" , calc_ways(0 , 0 , 0 ));
}
return 0;
}
You barely need a program to solve this at all.
Without loss of generality, let m <= n.
To begin with, we must have n <= r, otherwise no solution is possible.
Then, we subdivide the problem into a square of size m x m, on to which we will place m coins along the major diagonal, and a remainder, on to which we will place n - m coins so as to fulfil the remaining condition.
There is one way to place the coins along the major diagonal of the square.
There are m^(n - m) possibilities for the remainder.
We can permute the total so far in n! ways, although some of those will be duplicates (how many is left as an exercise for the student).
Furthermore, there are r - n coins left to place and (m - 1)n places left to put them.
Putting these all together we have an upper bound of
1 x m^(n - m) x n! x C((m - 1)n, r - n)
solutions to the problem. Divide this number by the number of duplicate permutations and you're done.
Problem 1
The code will start by placing a coin on each square and marking each square as used.
It will then test the final position and decide that the final position does not meet the goal of r coins.
Next it will start backtracking, but will never actually try another choice because used[row][col] is set to 1 and this shortcircuits the code to place coins.
In other words, one problem is that entries in "used" are set, but never cleared during the recursion.
Problem 2
Another problem with the code is that if n,m are of size 200, then it will never complete.
The issue is that this backtracking code has complexity O(2^(n*m)) as it will try all possible combinations of placing coins (many universe lifetimes for n=m=200...).
I would recommend you look at a different approach. For example, you might want to consider dynamic programming to compute how many ways there are of placing "k" coins on the remaining "a" columns of the board such that we make sure that we place coins on the "b" rows of the board that currently have no coins.
It can be treated as total ways in which d grid can b filled with r coins -(total ways leaving a single row nd filling in d rest -total ways leaving a single column nd filling in d rest- total ways leaving a row nd column together nd filling d rest) which implies
p(n*m ,r) -( (p((n-1)*m , r) * c(n,1)) +(p((m-1)*n , r) * c(m,1))+(p((n-1)*(m-1) , r) * c(n,1)*c(m,1)) )
I just think so but not sure of it!