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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;
}
}
How to construct a binary tree using a level order traversal sequence, for example from sequence {1,2,3,#,#,4,#,#,5}, we can construct a binary tree like this:
1
/ \
2 3
/
4
\
5
where '#' signifies a path terminator where no node exists below.
Finally I implement Pham Trung's algorithm by c++
struct TreeNode
{
TreeNode *left;
TreeNode *right;
int val;
TreeNode(int x): left(NULL), right(NULL), val(x) {}
};
TreeNode *build_tree(char nodes[], int n)
{
TreeNode *root = new TreeNode(nodes[0] - '0');
queue<TreeNode*> q;
bool is_left = true;
TreeNode *cur = NULL;
q.push(root);
for (int i = 1; i < n; i++) {
TreeNode *node = NULL;
if (nodes[i] != '#') {
node = new TreeNode(nodes[i] - '0');
q.push(node);
}
if (is_left) {
cur = q.front();
q.pop();
cur->left = node;
is_left = false;
} else {
cur->right = node;
is_left = true;
}
}
return root;
}
Assume using array int[]data with 0-based index, we have a simple function to get children:
Left child
int getLeftChild(int index){
if(index*2 + 1 >= data.length)
return -1;// -1 Means out of bound
return data[(index*2) + 1];
}
Right child
int getRightChild(int index){
if(index*2 + 2 >= data.length)
return -1;// -1 Means out of bound
return data[(index*2) + 2];
}
Edit:
Ok, so by maintaining a queue, we can build this binary tree.
We use a queue to maintain those nodes that are not yet processed.
Using a variable count to keep track of the number of children added for the current node.
First, create a root node, assign it as the current node.
So starting from index 1 (index 0 is the root), as the count is 0, we add this node as left child of the current node.
Increase count. If this node is not '#', add it to the queue.
Moving to the next index, the count is 1, so we add this as right child of current node, reset count to 0 and update current node (by assigning the current node as the first element in the queue). If this node is not '#', add it to the queue.
int count = 0;
Queue q = new Queue();
q.add(new Node(data[0]);
Node cur = null;
for(int i = 1; i < data.length; i++){
Node node = new Node(data[i]);
if(count == 0){
cur = q.dequeue();
}
if(count==0){
count++;
cur.leftChild = node;
}else {
count = 0;
cur.rightChild = node;
}
if(data[i] != '#'){
q.enqueue(node);
}
}
class Node{
int data;
Node leftChild, rightChild;
}
Note: this should only work for a binary tree and not BST.
we can build this binary tree from level order traversal by maintaining a queue. Queue is used to maintain those nodes that are not yet processed.
Using a variable count(index variable) to keep track of the number of children added for the current node.
First, create a root node, assign it as the current node. So starting from index 1,
index value is 1 means, we will add the next value as left node.
index value is 2 means we will add the next value as right node and index value 2 means that we have added left and right node, then do the same for the remaining nodes.
if arr value is -1
3.a. if index value is 1,i.e., there is no left node then change the index variable to add right node.
3.b. if index value is 2, i.e, there is no right node then we have repeat this step for the remaining.
static class Node{
int data;
Node left;
Node right;
Node(int d){
data=d;
left=null;
right=null;
}
}
public static Node constBT(int arr[],int n){
Node root=null;
Node curr=null;
int index=0;
Queue<Node> q=new LinkedList<>();
for(int i=0;i<n;i++){
if(root==null){
root=new Node(arr[i]);
q.add(root);
curr=q.peek();
index=1;
}else{
if(arr[i]==-1){
if(index==1)
index=2;
else{
q.remove();
curr=q.peek();
index=1;
}
}
else if(index==1){
curr.left=new Node(arr[i]);
q.add(curr.left);
index=2;
}else if(index==2){
curr.right=new Node(arr[i]);
q.add(curr.right);
q.remove();
curr=q.peek();
index=1;
}
}
}
return root;
}
My approach is similar to Pham Trung yet intutive. We would maintain an array of Nodes of given data instead of using a queue. We would do reverse engineering on BFS using queue. because BFS for a tree is basically its Level Order Traversal (LOT).
It is important to note that we should have the NULL childs of an node for the LOT to be unique and the reconstruction of Tree from LOT to be possible.
In this case LOT : 1,2,3,-1,-1,4,-1,-1,5
where I have used -1 instead of '#' to represent NULLs
And Tree is
1
/ \
2 3
/ \ /
-1 -1 4
/ \
-1 5
Here, we can easily see that when 1 is popped from the BFS queue, it pushed its left child
(2) and right child (3) in the queue. Similary, for 2 it pushed -1 (NULL) for both of its children. And the process is continued.
So, we can follow the following pseudo code to generate the tree rooted at LOT[0]
j = 1
For every node in LOT:
if n<=j: break
if node != NULL:
make LOT[j] left child of node
if n<=j+1: break
make LOT[j+1] right child of node
j <- j+2
Finally, C++ code for the same
Class Declaration and Preorder traversal
class Node{
public:
int val;
Node* lft, *rgt;
Node(int x ):val(x) {lft=rgt=nullptr;}
};
void preorder(Node* root) {
if(!root) return;
cout<<root->val<<" ";
preorder(root->lft);
preorder(root->rgt);
}
Restoring Tree from LOT Logic
int main(){
int arr[] = {1,2,3,-1,-1,4,-1,-1,5};
int n = sizeof(arr)/sizeof(int);
Node* brr[n];
for(int i=0;i<n;i++) {
if(arr[i]==-1) brr[i] = nullptr;
else brr[i] = new Node(arr[i]);
}
for(int i=0,j=1;j<n;i++) {
if(!brr[i]) continue;
brr[i]->lft = brr[j++];
if(j<n) brr[i]->rgt = brr[j++];
}
preorder(brr[0]);
}
Output: 1 2 3 4 5
I am trying to solve this problem.
The goal is to determine the number of ways a morse string can be interpreted, given a dictionary of word.
What I did is that I first "translated" words from my dictionary into morse. Then, I used a naive algorithm, searching for all the ways it can be interpreted recursively.
#include <iostream>
#include <vector>
#include <map>
#include <string>
#include <iterator>
using namespace std;
string morse_string;
int morse_string_size;
map<char, string> morse_table;
unsigned int sol;
void matches(int i, int factor, vector<string> &dictionary) {
int suffix_length = morse_string_size-i;
if (suffix_length <= 0) {
sol += factor;
return;
}
map<int, int> c;
for (vector<string>::iterator it = dictionary.begin() ; it != dictionary.end() ; it++) {
if (((*it).size() <= suffix_length) && (morse_string.substr(i, (*it).size()) == *it)) {
if (c.find((*it).size()) == c.end())
c[(*it).size()] = 0;
else
c[(*it).size()]++;
}
}
for (map<int, int>::iterator it = c.begin() ; it != c.end() ; it++) {
matches(i+it->first, factor*(it->second), dictionary);
}
}
string encode_morse(string s) {
string ret = "";
for (unsigned int i = 0 ; i < s.length() ; ++i) {
ret += morse_table[s[i]];
}
return ret;
}
int main() {
morse_table['A'] = ".-"; morse_table['B'] = "-..."; morse_table['C'] = "-.-."; morse_table['D'] = "-.."; morse_table['E'] = "."; morse_table['F'] = "..-."; morse_table['G'] = "--."; morse_table['H'] = "...."; morse_table['I'] = ".."; morse_table['J'] = ".---"; morse_table['K'] = "-.-"; morse_table['L'] = ".-.."; morse_table['M'] = "--"; morse_table['N'] = "-."; morse_table['O'] = "---"; morse_table['P'] = ".--."; morse_table['Q'] = "--.-"; morse_table['R'] = ".-."; morse_table['S'] = "..."; morse_table['T'] = "-"; morse_table['U'] = "..-"; morse_table['V'] = "...-"; morse_table['W'] = ".--"; morse_table['X'] = "-..-"; morse_table['Y'] = "-.--"; morse_table['Z'] = "--..";
int T, N;
string tmp;
vector<string> dictionary;
cin >> T;
while (T--) {
morse_string = "";
cin >> morse_string;
morse_string_size = morse_string.size();
cin >> N;
for (int j = 0 ; j < N ; j++) {
cin >> tmp;
dictionary.push_back(encode_morse(tmp));
}
sol = 0;
matches(0, 1, dictionary);
cout << sol;
if (T)
cout << endl << endl;
}
return 0;
}
Now the thing is that I only have 3 seconds of execution time allowed, and my algorithm won't work under this limit of time.
Is this the good way to do this and if so, what am I missing ? Otherwise, can you give some hints about what is a good strategy ?
EDIT :
There can be at most 10 000 words in the dictionary and at most 1000 characters in the morse string.
A solution that combines dynamic programming with a rolling hash should work for this problem.
Let's start with a simple dynamic programming solution. We allocate an vector which we will use to store known counts for prefixes of morse_string. We then iterate through morse_string and at each position we iterate through all words and we look back to see if they can fit into morse_string. If they can fit then we use the dynamic programming vector to determine how many ways we could have build the prefix of morse_string up to i-dictionaryWord.size()
vector<long>dp;
dp.push_back(1);
for (int i=0;i<morse_string.size();i++) {
long count = 0;
for (int j=1;j<dictionary.size();j++) {
if (dictionary[j].size() > i) continue;
if (dictionary[j] == morse_string.substring(i-dictionary[j].size(),i)) {
count += dp[i-dictionary[j].size()];
}
}
dp.push_back(count);
}
result = dp[morse_code.size()]
The problem with this solution is that it is too slow. Let's say that N is the length of morse_string and M is the size of the dictionary and K is the size of the largest word in the dictionary. It will do O(N*M*K) operations. If we assume K=1000 this is about 10^10 operations which is too slow on most machines.
The K cost came from the line dictionary[j] == morse_string.substring(i-dictionary[j].size(),i)
If we could speed up this string matching to constant or log complexity we would be okay. This is where rolling hashing comes in. If you build a rolling hash array of morse_string then the idea is that you can compute the hash of any substring of morse_string in O(1). So you could then do hash(dictionary[j]) == hash(morse_string.substring(i-dictionary[j].size(),i))
This is good but in the presence of imperfect hashing you could have multiple words from the dictionary with the same hash. That would mean that after getting a hash match you would still need to match the strings as well as the hashes. In programming contests, people often assume perfect hashing and skip the string matching. This is often a safe bet especially on a small dictionary. In case it doesn't produce a perfect hashing (which you can check in code) you can always adjust your hash function slightly and maybe the adjusted hash function will produce a perfect hashing.
You have a stack of n boxes, with widths wi, heights hi, and depths
di. The boxes cannot be rotated and can only be stacked on top of one
another if each box in the stack larger than or equal to the box above
it in width, height, and depth. Implement a method to build the
tallest stack possible, where the height of a stack is the sum of the
heights of each box.
I know there are a couple of articles to talk about using dynamic programming to solve it. Since I'd like to do practice in writing recursion code, I wrote the following code:
const int not_possible = 999999;
class box{
public:
int width;
int depth;
int height;
box(int h=not_possible, int d=not_possible, int w=not_possible):
width(w), depth(d), height(h) {}
};
bool check_legal(box lower, box upper){
return (upper.depth<lower.depth) &&
(upper.height<lower.height) &&
(upper.width<lower.width);
}
void highest_stack(const vector<box>& boxes, bool* used, box cur_level, int num_boxes, int height, int& max_height)
{
if(boxes.empty())
return;
bool no_suitable = true;
for(int i = 0; i < num_boxes; ++i){
box cur;
if(!(*(used+i)) && check_legal(cur_level, boxes[i])){
no_suitable = false;
cur = boxes[i];
*(used+i) = true;
highest_stack(boxes, used, cur, num_boxes, height+cur.height, max_height);
*(used+i) = false;
no_suitable = true;
}
}
if(no_suitable){
cout << height << endl; //for debug
if(height > max_height)
max_height = height;
return;
}
}
I've tested it using a lot of examples. For example:
boxes.push_back(box(4,12,32));
boxes.push_back(box(1,2,3));
boxes.push_back(box(2,5,6));
highest_stack(boxes, used, cur, boxes.size(), 0, max_height);
In the function highest_stack, there is one line cout << height << endl; for output. If I comment no_suitable = true;
the output is: 1 2 4; 1 2; 1, 1 4;
if I don't comment no_suitable = true;
the output is: 1 2 4; 2 4; 4; 1 2; 2; 1; 1 4; 0
Both of them can give the correct result which is 7.
My question is:
(1) Can anyone help me verify my solution?
(2) Is there any more elegant recursive code for this problem?
I don't think my code is elegant.
Thanks
I would make a directed graph where the nodes are boxes and the edges go from a box to a box that can be put on top of it. Then I'd use use the longest path algorithm to find the solution.
Design the relation as a Set array of boxes.(Set[]) i.e. each position has a array of boxes.
Initialize each box with an index.
For each box check boxes that can be placed above the current box(box[i]) add it to the set in the set array i.e. set[i].add(box)
Run DFS with the boxes that can be placed above (the role of adjacent)
Maintain a marked[], count[] and boxTo[] arrays of boxes.
Go over the count array and find the largest value.
Traverse the way to the bottom box using the boxTo[] array.
I learned about ternary search from Wikipedia. I'm not sure what they mean by the parameter absolute precision. They didn't elaborate. But here is the pseudocode:
def ternarySearch(f, left, right, absolutePrecision):
#left and right are the current bounds; the maximum is between them
if (right - left) < absolutePrecision:
return (left + right)/2
leftThird = (2*left + right)/3
rightThird = (left + 2*right)/3
if f(leftThird) < f(rightThird):
return ternarySearch(f, leftThird, right, absolutePrecision)
return ternarySearch(f, left, rightThird, absolutePrecision)
I want to find max value from a unimodal function. That means I want to print the border point of the increasing and decreasing sequence. If the sequence is
1 2 3 4 5 -1 -2 -3 -4
then I want to print 5 as output.
Here is my attempt. It isn't giving output. Can you please help or give me link that contains good tutorial on ternary search for self learning?
#include<iostream>
using namespace std;
int ternary_search(int[], int, int, int);
int precval = 1;
int main()
{
int n, arr[100], target;
cout << "\t\t\tTernary Search\n\n" << endl;
//cout << "This program will find max element in an unidomal array." << endl;
cout << "How many integers: ";
cin >> n;
for (int i=0; i<n; i++)
cin >> arr[i];
cout << endl << "The max number in the array is: ";
int res = ternary_search(arr,0,n-1,precval)+0;
cout << res << endl;
return 0;
}
int ternary_search(int arr[], int left, int right, int precval)
{
if (right-left <= precval)
return (arr[right] > arr[left]) ? arr[right] : arr[left];
int first_third = (left * 2 + right) / 3;
int last_third = (left + right * 2) / 3;
if(arr[first_third] < arr[last_third])
return ternary_search(arr, first_third, right, precval);
else
return ternary_search(arr, left, last_third, precval);
}
Thank you in advance.
Absolute precision means the maximum error between the returned result and the true result i.e. max | returned_result - true_result |. In that context, f is a continuous function.
Since you are looking at a discrete function, you can't do much better than get to the point where right - left <= 1. Then, just compare the two resultant values and return the value corresponding to the larger one (since you're looking for max).
EDIT
The first partition point, being mathematically 2/3*left + right/3, should be discretized to ceil(2/3*left + right/3) (so that the relationship is left < first_third <= last_third < right
So first_third = (left*2+right)/3 should be changed to first_third = (left*2 + right + 2)/3.
Try Golden Section search (or Fibonacci search for discrete functions).
It has a smaller number of recursions AND a 50% reduction in evaluations of f, compared to the above ternary search.