I have a non-linear system I'm using Ceres to solve. It's a sparse system with a sparse block structure. Since I'm also working on image data, I've based my code off of the 'denoising.cc' example.
The issue I'm encountering is that my code fails with "Terminating: Residual and Jacobian evaluation failed.". I'm able to fix the issue by hard-coding the variable 'num_weights' in Evaluate.
The issue persists when I call this function on one or on many pixels. For each pixel, my weights are different.
Any insight as to why this is will help.
Thanks!
Cost::Cost(const std::vector<double> &weights) : _weights(weights)
{
set_num_residuals(1);
mutable_parameter_block_sizes()->push_back(1); //has more parameters than weights
for (int i = 0; i < _weights.size(); ++i)
mutable_parameter_block_sizes()->push_back(1);
}
bool Cost::Evaluate(double const* const* parameters,
double *residuals,
double **jacobians) const
{
int num_weights = (int)_weights.size();
float d0 = parameters[0][0];
residuals[0] = d0;
for (int i = 0; i < num_weights; ++i)
{
residuals[0] += parameters[i+1][0];
}
if (jacobians != NULL)
{
for (int i = 0; i < num_weights+1; ++i)
{
if (jacobians[i] != NULL)
{
jacobians[i][0] = 0;
}
}
}
return true;
}
Just figured it out.
Basically, my vector "std::vector weights" went out of scope. The ceres cost function does not retain ownership of the vector, so "num_weights" evaluated to 0.
It is a interview question. Given an array, e.g., [3,2,1,2,7], we want to make all elements in this array unique by incrementing duplicate elements and we require the sum of the refined array is minimal. For example the answer for [3,2,1,2,7] is [3,2,1,4,7] and its sum is 17. Any ideas?
It's not quite as simple as my earlier comment suggested, but it's not terrifically complicated.
First, sort the input array. If it matters to be able to recover the original order of the elements then record the permutation used for the sort.
Second, scan the sorted array from left to right (ie from low to high). If an element is less than or equal to the element to its left, set it to be one greater than that element.
Pseudocode
sar = sort(input_array)
for index = 2:size(sar) ! I count from 1
if sar(index)<=sar(index-1) sar(index) = sar(index-1)+1
forend
Is the sum of the result minimal ? I've convinced myself that it is through some head-scratching and trials but I haven't got a formal proof.
If you only need to find ONE of the best solution, here's the algorythm with some explainations.
The idea of this problem is to find an optimal solution, which can be found only by testing all existing solutions (well, they're infinite, let's stick with the reasonable ones).
I wrote a program in C, because I'm familiar with it, but you can port it to any language you want.
The program does this: it tries to increment one value to the max possible (I'll explain how to find it in the comments under the code sections), than if the solution is not found, decreases this value and goes on with the next one and so on.
It's an exponential algorythm, so it will be very slow on large values of duplicated data (yet, it assures you the best solution is found).
I tested this code with your example, and it worked; not sure if there's any bug left, but the code (in C) is this.
#include <stdio.h>
#include <stdlib.h>
#include <limits.h>
typedef int BOOL; //just to ease meanings of values
#define TRUE 1
#define FALSE 0
Just to ease comprehension, I did some typedefs. Don't worry.
typedef struct duplicate { //used to fasten the algorythm; it uses some more memory just to assure it's ok
int value;
BOOL duplicate;
} duplicate_t;
int maxInArrayExcept(int *array, int arraySize, int index); //find the max value in array except the value at the index given
//the result is the max value in the array, not counting th index
int *findDuplicateSum(int *array, int arraySize);
BOOL findDuplicateSum_R(duplicate_t *array, int arraySize, int *tempSolution, int *solution, int *totalSum, int currentSum); //resursive function used to find solution
BOOL check(int *array, int arraySize); //checks if there's any repeated value in the solution
These are all the functions we'll need. All split up for comprehension purpose.
First, we have a struct. This struct is used to avoid checking, for every iteration, if the value on a given index was originally duplicated. We don't want to modify any value not duplicated originally.
Then, we have a couple functions: first, we need to see the worst case scenario: every value after the duplicated ones is already occupied: then we need to increment the duplicated value up to the maximum value reached + 1.
Then, there are the main Function we'll discute later about.
The check Function only checks if there's any duplicated value in a temporary solution.
int main() { //testing purpose
int i;
int testArray[] = { 3,2,1,2,7 }; //test array
int nTestArraySize = 5; //test array size
int *solutionArray; //needed if you want to use the solution later
solutionArray = findDuplicateSum(testArray, nTestArraySize);
for (i = 0; i < nTestArraySize; ++i) {
printf("%d ", solutionArray[i]);
}
return 0;
}
This is the main Function: I used it to test everything.
int * findDuplicateSum(int * array, int arraySize)
{
int *solution = malloc(sizeof(int) * arraySize);
int *tempSolution = malloc(sizeof(int) * arraySize);
duplicate_t *duplicate = calloc(arraySize, sizeof(duplicate_t));
int i, j, currentSum = 0, totalSum = INT_MAX;
for (i = 0; i < arraySize; ++i) {
tempSolution[i] = solution[i] = duplicate[i].value = array[i];
currentSum += array[i];
for (j = 0; j < i; ++j) { //to find ALL the best solutions, we should also put the first found value as true; it's just a line more
//yet, it saves the algorythm half of the duplicated numbers (best/this case scenario)
if (array[j] == duplicate[i].value) {
duplicate[i].duplicate = TRUE;
}
}
}
if (findDuplicateSum_R(duplicate, arraySize, tempSolution, solution, &totalSum, currentSum));
else {
printf("No solution found\n");
}
free(tempSolution);
free(duplicate);
return solution;
}
This Function does a lot of things: first, it sets up the solution array, then it initializes both the solution values and the duplicate array, that is the one used to check for duplicated values at startup. Then, we find the current sum and we set the maximum available sum to the maximum integer possible.
Then, the recursive Function is called; this one gives us the info about having found the solution (that should be Always), then we return the solution as an array.
int findDuplicateSum_R(duplicate_t * array, int arraySize, int * tempSolution, int * solution, int * totalSum, int currentSum)
{
int i;
if (check(tempSolution, arraySize)) {
if (currentSum < *totalSum) { //optimal solution checking
for (i = 0; i < arraySize; ++i) {
solution[i] = tempSolution[i];
}
*totalSum = currentSum;
}
return TRUE; //just to ensure a solution is found
}
for (i = 0; i < arraySize; ++i) {
if (array[i].duplicate == TRUE) {
if (array[i].duplicate <= maxInArrayExcept(solution, arraySize, i)) { //worst case scenario, you need it to stop the recursion on that value
tempSolution[i]++;
return findDuplicateSum_R(array, arraySize, tempSolution, solution, totalSum, currentSum + 1);
tempSolution[i]--; //backtracking
}
}
}
return FALSE; //just in case the solution is not found, but we won't need it
}
This is the recursive Function. It first checks if the solution is ok and if it is the best one found until now. Then, if everything is correct, it updates the actual solution with the temporary values, and updates the optimal condition.
Then, we iterate on every repeated value (the if excludes other indexes) and we progress in the recursion until (if unlucky) we reach the worst case scenario: the check condition not satisfied above the maximum value.
Then we have to backtrack and continue with the iteration, that will go on with other values.
PS: an optimization is possible here, if we move the optimal condition from the check into the for: if the solution is already not optimal, we can't expect to find a better one just adding things.
The hard code has ended, and there are the supporting functions:
int maxInArrayExcept(int *array, int arraySize, int index) {
int i, max = 0;
for (i = 0; i < arraySize; ++i) {
if (i != index) {
if (array[i] > max) {
max = array[i];
}
}
}
return max;
}
BOOL check(int *array, int arraySize) {
int i, j;
for (i = 0; i < arraySize; ++i) {
for (j = 0; j < i; ++j) {
if (array[i] == array[j]) return FALSE;
}
}
return TRUE;
}
I hope this was useful.
Write if anything is unclear.
Well, I got the same question in one of my interviews.
Not sure if you still need it. But here's how I did it. And it worked well.
num_list1 = [2,8,3,6,3,5,3,5,9,4]
def UniqueMinSumArray(num_list):
max=min(num_list)
for i,V in enumerate(num_list):
while (num_list.count(num_list[i])>1):
if (max > num_list[i]+1) :
num_list[i] = max + 1
else:
num_list[i]+=1
max = num_list[i]
i+=1
return num_list
print (sum(UniqueMinSumArray(num_list1)))
You can try with your list of numbers and I am sure it will give you the correct unique minimum sum.
I got the same interview question too. But my answer is in JS in case anyone is interested.
For sure it can be improved to get rid of for loop.
function getMinimumUniqueSum(arr) {
// [1,1,2] => [1,2,3] = 6
// [1,2,2,3,3] = [1,2,3,4,5] = 15
if (arr.length > 1) {
var sortedArr = [...arr].sort((a, b) => a - b);
var current = sortedArr[0];
var res = [current];
for (var i = 1; i + 1 <= arr.length; i++) {
// check current equals to the rest array starting from index 1.
if (sortedArr[i] > current) {
res.push(sortedArr[i]);
current = sortedArr[i];
} else if (sortedArr[i] == current) {
current = sortedArr[i] + 1;
// sortedArr[i]++;
res.push(current);
} else {
current++;
res.push(current);
}
}
return res.reduce((a,b) => a + b, 0);
} else {
return 0;
}
}
I´m trying to find the maximum number of point that I can place in a given surface considering that the points have to present a certain distance between themselves.
I had thought to discretize the surface with a small resolution and try all combinations. Unfortunately, this method is very slow.Another solution could be the search of valid points and from each valid position search new valid points using a recursive function. Altough this method is faster, its computation time grows exponentially with the surface.
So, I wonder if it exists other algorithm that permits a faster solution.
My code in C is as follows:
int locatePoints(int p_valids, int achieved, int points[][2]){
int row_ini;
int row_end;
int points_max[N][2];
int points_act[N][2];
int valids_act;
int quantity_max;
int valid;
int point_to5;
int SquaredDistance;
int i,j,k,l;
if (p_valids==0){
row_ini = 0;
}
else{
row_ini = points[p_valids-1][0];
}
row_end = row_ini+6;
if (row_end>ROW_LIMIT){
row_end = ROW_LIMIT;
}
for(i=0;i<p_valids;i++){
points_act[i][0] = points[i][0];
points_act[i][1] = points[i][1];
points_max[i][0]=points[i][0];
points_max[i][1]=points[i][1];
}
quantity_max = p_valids;
for(i=row_ini;i<row_end;i++){
for(j=0;j<COL_LIMIT;j++){
valid = 1;
point_to5 = 0;
for (k=0;k<p_valids && valid;k++){
SquaredDistance = (points[k][0]-i)*(points[k][0]-i)+(points[k][1]-j)*(points[k][1]-j);
if (SquaredDistance<25){
if (SquaredDistance<16){
valid = 0;
}
else{
point_to5++;
if (point_to5>1){
valid = 0;
}
}
}
}
if (valid){
points_act[p_valids][0] = i;
points_act[p_valids][1] = j;
valids_act = locatePoints(p_valids+1, quantity_max, points_act);
if (valids_act>quantity_max){
quantity_max = valids_act;
for(k=0;k<quantity_max;k++){
points_max[k][0]=points_act[k][0];
points_max[k][1]=points_act[k][1];
}
}
}
}
}
for(k=0;k<quantity_max;k++){
points[k][0]= points_max[k][0];
points[k][1]= points_max[k][1];
}
return quantity_max;
}
I'm using GA, so I took example from this page (http://www.ai-junkie.com/ga/intro/gat3.html) and tried to do on my own.
The problem is, it doesn't work. For example, maximum fitness does not always grow in the next generation, but becomes smallest. Also, after some number of generations, it just stops getting better. For example, in first 100 generations, it found the largest circle with radius 104. And in next 900 largest radius is 107. And after drawing it, I see that it can grow much more.
Here is my code connected with GA. I leave out generating random circles, decoding and drawing.
private Genome ChooseParent(Genome[] population, Random r)
{
double sumFitness = 0;
double maxFitness = 0;
for (int i = 0; i < population.Length; i++)
{
sumFitness += population[i].fitness;
if (i == 0 || maxFitness < population[i].fitness)
{
maxFitness = population[i].fitness;
}
}
sumFitness = population.Length * maxFitness - sumFitness;
double randNum = r.NextDouble() *sumFitness;
double acumulatedSum = 0;
for(int i=0;i<population.Length;i++)
{
acumulatedSum += population[i].fitness;
if(randNum<acumulatedSum)
{
return population[i];
}
}
return population[0];
}
private void Crossover(Genome parent1, Genome parent2, Genome child1, Genome child2, Random r)
{
double d=r.NextDouble();
if(d>this.crossoverRate || child1.Equals(child2))
{
for (int i = 0; i < parent1.bitNum; i++)
{
child1.bit[i] = parent1.bit[i];
child2.bit[i] = parent2.bit[i];
}
}
else
{
int cp = r.Next(parent1.bitNum - 1);
for (int i = 0; i < cp; i++)
{
child1.bit[i] = parent1.bit[i];
child2.bit[i] = parent2.bit[i];
}
for (int i = cp; i < parent1.bitNum; i++)
{
child1.bit[i] = parent2.bit[i];
child2.bit[i] = parent1.bit[i];
}
}
}
private void Mutation(Genome child, Random r)
{
for(int i=0;i<child.bitNum;i++)
{
if(r.NextDouble()<=this.mutationRate)
{
child.bit[i] = (byte)(1 - child.bit[i]);
}
}
}
public void Run()
{
for(int generation=0;generation<1000;generation++)
{
CalculateFitness(population);
System.Diagnostics.Debug.WriteLine(maxFitness);
population = population.OrderByDescending(x => x).ToArray();
//ELITIZM
Copy(population[0], newpopulation[0]);
Copy(population[1], newpopulation[1]);
for(int i=1;i<this.populationSize/2;i++)
{
Genome parent1 = ChooseParent(population, r);
Genome parent2 = ChooseParent(population, r);
Genome child1 = newpopulation[2 * i];
Genome child2 = newpopulation[2 * i + 1];
Crossover(parent1, parent2, child1, child2, r);
Mutation(child1, r);
Mutation(child2, r);
}
Genome[] tmp = population;
population = newpopulation;
newpopulation = tmp;
DekodePopulation(population); //decoding and fitness calculation for each member of population
}
}
If someone can point on potential problem that caused such behaviour and ways to fix it, I'll be grateful.
Welcome to the world of genetic algorithms!
I'll go through your issues and suggest a potential problem. Here we go:
maximum fitness does not always grow in the next generation, but becomes smallest - You probably meant smaller. This is weird since you employed elitism, so each generation's best individual should be at least as good as in the previous one. I suggest you check your code for mistakes because this really should not happen. However, the fitness does not need to always grow. It is impossible to achieve this in GA - it's a stochastic algorithm, working with randomness - suppose that, by chance, no mutation nor crossover happens in a generation - then the fitness cannot improve to the next generation since there is no change.
after some number of generations, it just stops getting better. For example, in first 100 generations, it found the largest circle with radius 104. And in next 900 largest radius is 107. And after drawing it, I see that it can grow much more. - this is (probably) a sign of a phenomenon called premature convergence and it's, unfortunately, a "normal" thing in genetic algorithm. Premature convergence is a situation when the whole population converges to a single solution or to a set of solutions which are near each other and which is/are sub-optimal (i.e. it is not the best possible soluion). When this happens, the GA has a very hard time escaping this local optimum. You can try to tweak the parameters, especially the mutation probability, to force more exploration.
Also, another very important thing that can cause problems is the encoding, i.e. how is the bit string mapped to the circle. If the encoding is much too indirect, it can lead to poor performance of the GA. GAs work when there are some building blocks in the genotype which can be exchanged between among the population. If there are no such blocks, the performance of a GA is usually going to be poor.
I have implemented this exercise and achieved good results. Here is the link:
https://github.com/ManhTruongDang/ai-junkie
Hope this can be of use to you.
The environment: I am working in a proprietary scripting language where there is no such thing as a user-defined function. I have various loops and local variables of primitive types that I can create and use.
I have two related arrays, "times" and "values". They both contain floating point values. I want to numerically sort the "times" array but have to be sure that the same operations are applied on the "values" array. What's the most efficient way I can do this without the benefit of things like recursion?
You could maintain an index table and sort the index table instead.
This way you will not have to worry about times and values being consistent.
And whenever you need a sorted value, you can lookup on the sorted index.
And if in the future you decided there was going to be a third value, the sorting code will not need any changes.
Here's a sample in C#, but it shouldn't be hard to adapt to your scripting language:
static void Main() {
var r = new Random();
// initialize random data
var index = new int[10]; // the index table
var times = new double[10]; // times
var values = new double[10]; // values
for (int i = 0; i < 10; i++) {
index[i] = i;
times[i] = r.NextDouble();
values[i] = r.NextDouble();
}
// a naive bubble sort
for (int i = 0; i < 10; i++)
for (int j = 0; j < 10; j++)
// compare time value at current index
if (times[index[i]] < times[index[j]]) {
// swap index value (times and values remain unchanged)
var temp = index[i];
index[i] = index[j];
index[j] = temp;
}
// check if the result is correct
for (int i = 0; i < 10; i++)
Console.WriteLine(times[index[i]]);
Console.ReadKey();
}
Note: I used a naive bubble sort there, watchout. In your case, an insertion sort is probably a good candidate. Since you don't want complex recursions.
Just take your favourite sorting algorithm (e.g. Quicksort or Mergesort) and use it to sort the "values" array. Whenever two values are swapped in "values", also swap the values with the same indices in the "times" array.
So basically you can take any fast sorting algorithm and modify the swap() operation so that elements in both arrays are swapped.
Take a look at the Bottom-Up mergesort at Algorithmist. It's a non-recursive way of performing a mergesort. The version presented there uses function calls, but that can be inlined easily enough.
Like martinus said, every time you change a value in one array, do the exact same thing in the parallel array.
Here's a C-like version of a stable-non-recursive mergesort that makes no function calls, and uses no recursion.
const int arrayLength = 40;
float times_array[arrayLength];
float values_array[arrayLength];
// Fill the two arrays....
// Allocate two buffers
float times_buffer[arrayLength];
float values_buffer[arrayLength];
int blockSize = 1;
while (blockSize <= arrayLength)
{
int i = 0;
while (i < arrayLength-blockSize)
{
int begin1 = i;
int end1 = begin1 + blockSize;
int begin2 = end1;
int end2 = begin2 + blockSize;
int bufferIndex = begin1;
while (begin1 < end1 && begin2 < end2)
{
if ( values_array[begin1] > times_array[begin2] )
{
times_buffer[bufferIndex] = times_array[begin2];
values_buffer[bufferIndex++] = values_array[begin2++];
}
else
{
times_buffer[bufferIndex] = times_array[begin1];
values_buffer[bufferIndex++] = values_array[begin1++];
}
}
while ( begin1 < end1 )
{
times_buffer[bufferIndex] = times_array[begin1];
values_buffer[bufferIndex++] = values_array[begin1++];
}
while ( begin2 < end2 )
{
times_buffer[bufferIndex] = times_array[begin2];
values_buffer[bufferIndex++] = values_array[begin2++];
}
for (int k = i; k < i + 2 * blockSize; ++k)
{
times_array[k] = times_buffer[k];
values_array[k] = values_buffer[k];
}
i += 2 * blockSize;
}
blockSize *= 2;
}
I wouldn't suggest writing your own sorting routine, as the sorting routines provided as part of the Java language are well optimized.
The way I'd solve this is to copy the code in the java.util.Arrays class into your own class i.e. org.mydomain.util.Arrays. And add some comments telling yourself not to use the class except when you must have the additional functionality that you're going to add. The Arrays class is quite stable so this is less, less ideal than it would seem, but it's still less than ideal. However, the methods you need to change are private, so you've no real choice.
You then want to create an interface along the lines of:
public static interface SwapHook {
void swap(int a, int b);
}
You then need to add this to the sort method you're going to use, and to every subordinate method called in the sorting procedure, which swaps elements in your primary array. You arrange for the hook to get called by your modified sorting routine, and you can then implement the SortHook interface to achieve the behaviour you want in any secondary (e.g. parallel) arrays.
HTH.