Find duplicate element in array in time O(n) - java

I have been asked this question in a job interview and I have been wondering about the right answer.
You have an array of numbers from 0 to n-1, one of the numbers is removed, and replaced with a number already in the array which makes a duplicate of that number. How can we detect this duplicate in time O(n)?
For example, an array of 4,1,2,3 would become 4,1,2,2.
The easy solution of time O(n2) is to use a nested loop to look for the duplicate of each element.


This can be done in O(n) time and O(1) space.
(The algorithm only works because the numbers are consecutive integers in a known range):
In a single pass through the vector, compute the sum of all the numbers, and the sum of the squares of all the numbers.
Subtract the sum of all the numbers from N(N-1)/2. Call this A.
Subtract the sum of the squares from N(N-1)(2N-1)/6. Divide this by A. Call the result B.
The number which was removed is (B + A)/2 and the number it was replaced with is (B - A)/2.
Example:
The vector is [0, 1, 1, 2, 3, 5]:
N = 6
Sum of the vector is 0 + 1 + 1 + 2 + 3 + 5 = 12. N(N-1)/2 is 15. A = 3.
Sum of the squares is 0 + 1 + 1 + 4 + 9 + 25 = 40. N(N-1)(2N-1)/6 is 55. B = (55 - 40)/A = 5.
The number which was removed is (5 + 3) / 2 = 4.
The number it was replaced by is (5 - 3) / 2 = 1.
Why it works:
The sum of the original vector [0, ..., N-1] is N(N-1)/2. Suppose the value a was removed and replaced by b. Now the sum of the modified vector will be N(N-1)/2 + b - a. If we subtract the sum of the modified vector from N(N-1)/2 we get a - b. So A = a - b.
Similarly, the sum of the squares of the original vector is N(N-1)(2N-1)/6. The sum of the squares of the modified vector is N(N-1)(2N-1)/6 + b2 - a2. Subtracting the sum of the squares of the modified vector from the original sum gives a2 - b2, which is the same as (a+b)(a-b). So if we divide it by a - b (i.e., A), we get B = a + b.
Now B + A = a + b + a - b = 2a and B - A = a + b - (a - b) = 2b.


We have the original array int A[N]; Create a second array bool B[N] too, of type bool=false. Iterate the first array and set B[A[i]]=true if was false, else bing!


You can do it in O(N) time without any extra space. Here is how the algorithm works :
Iterate through array in the following manner :
For each element encountered, set its corresponding index value to negative.
Eg : if you find a[0] = 2. Got to a[2] and negate the value.
By doing this you flag it to be encountered. Since you know you cannot have negative numbers, you also know that you are the one who negated it.
Check if index corresponding to the value is already flagged negative, if yes you get the duplicated element. Eg : if a[0]=2 , go to a[2] and check if it is negative.
Lets say you have following array :
int a[] = {2,1,2,3,4};
After first element your array will be :
int a[] = {2,1,-2,3,4};
After second element your array will be :
int a[] = {2,-1,-2,3,4};
When you reach third element you go to a[2] and see its already negative. You get the duplicate.


Scan the array 3 times:
XOR together all the array elements -> A. XOR together all the numbers from 0 to N-1 -> B. Now A XOR B = X XOR D, where X is the removed element, and D is the duplicate element.
Choose any non-zero bit in A XOR B. XOR together all the array elements where this bit is set -> A1. XOR together all the numbers from 0 to N-1 where this bit is set -> B1. Now either A1 XOR B1 = X or A1 XOR B1 = D.
Scan the array once more and try to find A1 XOR B1. If it is found, this is the duplicate element. If not, the duplicate element is A XOR B XOR A1 XOR B1.


Use a HashSet to hold all numbers already seen. It operates in (amortized) O(1) time, so the total is O(N).


I suggest using a BitSet. We know N is small enough for array indexing, so the BitSet will be of reasonable size.
For each element of the array, check the bit corresponding to its value. If it is already set, that is the duplicate. If not, set the bit.


#rici is right about the time and space usage: "This can be done in O(n) time and O(1) space."
However, the question can be expanded to broader requirement: it's not necessary that there is only one duplicate number, and numbers might not be consecutive.
OJ puts it this way here:
(note 3 apparently can be narrowed)
Given an array nums containing n + 1 integers where each integer is between 1 and n (inclusive), prove that at least one duplicate number must exist. Assume that there is only one duplicate number, find the duplicate one.
Note:
You must not modify the array (assume the array is read only).
You must use only constant, O(1) extra space.
Your runtime complexity should be less than O(n2).
There is only one duplicate number in the array, but it could be repeated more than once.
The question is very well explained and answered here by Keith Schwarz, using Floyd's cycle-finding algorithm:
The main trick we need to use to solve this problem is to notice that because we have an array of n elements ranging from 0 to n - 2, we can think of the array as defining a function f from the set {0, 1, ..., n - 1} onto itself. This function is defined by f(i) = A[i]. Given this setup, a duplicated value corresponds to a pair of indices i != j such that f(i) = f(j). Our challenge, therefore, is to find this pair (i, j). Once we have it, we can easily find the duplicated value by just picking f(i) = A[i].
But how are we to find this repeated value? It turns out that this is a well-studied problem in computer science called cycle detection. The general form of the problem is as follows. We are given a function f. Define the sequence x_i as
x_0 = k (for some k)
x_1 = f(x_0)
x_2 = f(f(x_0))
...
x_{n+1} = f(x_n)
Assuming that f maps from a domain into itself, this function will have one of three forms. First, if the domain is infinite, then the sequence could be infinitely long and nonrepeating. For example, the function f(n) = n + 1 on the integers has this property - no number is ever duplicated. Second, the sequence could be a closed loop, which means that there is some i so that x_0 = x_i. In this case, the sequence cycles through some fixed set of values indefinitely. Finally, the sequence could be "rho-shaped." In this case, the sequence looks something like this:
x_0 -> x_1 -> ... x_k -> x_{k+1} ... -> x_{k+j}
^ |
| |
+-----------------------+
That is, the sequence begins with a chain of elements that enters a cycle, then cycles around indefinitely. We'll denote the first element of the cycle that is reached in the sequence the "entry" of the cycle.
An python implementation can also be found here:
def findDuplicate(self, nums):
# The "tortoise and hare" step. We start at the end of the array and try
# to find an intersection point in the cycle.
slow = 0
fast = 0
# Keep advancing 'slow' by one step and 'fast' by two steps until they
# meet inside the loop.
while True:
slow = nums[slow]
fast = nums[nums[fast]]
if slow == fast:
break
# Start up another pointer from the end of the array and march it forward
# until it hits the pointer inside the array.
finder = 0
while True:
slow = nums[slow]
finder = nums[finder]
# If the two hit, the intersection index is the duplicate element.
if slow == finder:
return slow


Use hashtable. Including an element in a hashtable is O(1).


One working solution:
asume number are integers
create an array of [0 .. N]
int[] counter = new int[N];
Then iterate read and increment the counter:
if (counter[val] >0) {
// duplicate
} else {
counter[val]++;
}


This can be done in O(n) time and O(1) space.
Without modifying the input array
The idea is similar to finding the starting node of a loop in a linked list.
Maintain two pointers: fast and slow
slow = a[0]
fast = a[a[0]]
loop till slow != fast
Once we find the loop (slow == fast)
Reset slow back to zero
slow = 0
find the starting node
while(slow != fast){
slow = a[slow];
fast = a[fast];
}
slow is your duplicate number.
Here's a Java implementation:
class Solution {
public int findDuplicate(int[] nums) {
if(nums.length <= 1) return -1;
int slow = nums[0], fast = nums[nums[0]]; //slow = head.next, fast = head.next.next
while(slow != fast){ //check for loop
slow = nums[slow];
fast = nums[nums[fast]];
}
if(slow != fast) return -1;
slow = 0; //reset one pointer
while(slow != fast){ //find starting point of loop
slow = nums[slow];
fast = nums[fast];
}
return slow;
}
}


This is an alternative solution in O(n) time and O(1) space. It is similar to rici's. I find it a bit easier to understand but, in practice, it will overflow faster.
Let X be the missing number and R be the repeated number.
We can assume the numbers are from [1..n], i.e. zero does not appear. In fact, while looping through the array, we can test if zero was found and return immediately if not.
Now consider:
sum(A) = n (n + 1) / 2 - X + R
product(A) = n! R / X
where product(A) is the product of all element in A skipping the zero. We have two equations in two unknowns from which X and R can be derived algebraically.
Edit: by popular demand, here is a worked-out example:
Let's set:
S = sum(A) - n (n + 1) / 2
P = n! / product(A)
Then our equations become:
R - X = S
X = R P
which can be solved to:
R = S / (1 - P)
X = P R = P S / (1 - P)
Example:
A = [0 1 2 2 4]
n = A.length - 1 = 4
S = (1 + 2 + 2 + 4) - 4 * 5 / 2 = -1
P = 4! / (1 * 2 * 2 * 4) = 3 / 2
R = -1 / (1 - 3/2) = -1 / -1/2 = 2
X = 3/2 * 2 = 3


You could proceed as follows:
sort your array by using a Linear-time sorting algorithm (e.g. Counting sort) - O(N)
scan the sorted array and stop as soon as two consecutive elements are equal - O(N)


public class FindDuplicate {
public static void main(String[] args) {
// assume the array is sorted, otherwise first we have to sort it.
// time efficiency is o(n)
int elementData[] = new int[] { 1, 2, 3, 3, 4, 5, 6, 8, 8 };
int count = 1;
int element1;
int element2;
for (int i = 0; i < elementData.length - 1; i++) {
element1 = elementData[i];
element2 = elementData[count];
count++;
if (element1 == element2) {
System.out.println(element2);
}
}
}
}


public void duplicateNumberInArray {
int a[] = new int[10];
Scanner inp = new Scanner(System.in);
for(int i=1;i<=5;i++){
System.out.println("enter no. ");
a[i] = inp.nextInt();
}
Set<Integer> st = new HashSet<Integer>();
Set<Integer> s = new HashSet<Integer>();
for(int i=1;i<=5;i++){
if(!st.add(a[i])){
s.add(a[i]);
}
}
Iterator<Integer> itr = s.iterator();
System.out.println("Duplicate numbers are");
while(itr.hasNext()){
System.out.println(itr.next());
}
}
First of all creating an array of integer using Scanner class. Then iterating a loop through the numbers and checking if the number can be added to set (Numbers can be added to set only when that particular number should not be in set already, means set does not allow duplicate no. to add and return a boolean vale FALSE on adding duplicate value).If no. cannot be added means it is duplicate so add that duplicate number into another set, so that we can print later. Please note onething that we are adding the duplicate number into a set because it might be possible that duplicate number might be repeated several times, hence add it only once.At last we are printing set using Iterator.


//This is similar to the HashSet approach but uses only one data structure:
int[] a = { 1, 4, 6, 7, 4, 6, 5, 22, 33, 44, 11, 5 };
LinkedHashMap<Integer, Integer> map = new LinkedHashMap<Integer, Integer>();
for (int i : a) {
map.put(i, map.containsKey(i) ? (map.get(i)) + 1 : 1);
}
Set<Entry<Integer, Integer>> es = map.entrySet();
Iterator<Entry<Integer, Integer>> it = es.iterator();
while (it.hasNext()) {
Entry<Integer, Integer> e = it.next();
if (e.getValue() > 1) {
System.out.println("Dupe " + e.getKey());
}
}


We can do using hashMap efficiently:
Integer[] a = {1,2,3,4,0,1,5,2,1,1,1,};
HashMap<Integer,Integer> map = new HashMap<Integer,Integer>();
for(int x : a)
{
if (map.containsKey(x)) map.put(x,map.get(x)+1);
else map.put(x,1);
}
Integer [] keys = map.keySet().toArray(new Integer[map.size()]);
for(int x : keys)
{
if(map.get(x)!=1)
{
System.out.println(x+" repeats : "+map.get(x));
}
}


This program is based on c# and if you want to do this program using another programming language you have to firstly change an array in accending order and compare the first element to the second element.If it is equal then repeated number found.Program is
int[] array=new int[]{1,2,3,4,5,6,7,8,9,4};
Array.Sort(array);
for(int a=0;a<array.Length-1;a++)
{
if(array[a]==array[a+1]
{
Console.WriteLine("This {0} element is repeated",array[a]);
}
}
Console.WriteLine("Not repeated number in array");


sort the array O(n ln n)
using the sliding window trick to traverse the array O(n)
Space is O(1)
Arrays.sort(input);
for(int i = 0, j = 1; j < input.length ; j++, i++){
if( input[i] == input[j]){
System.out.println(input[i]);
while(j < input.length && input[i] == input[j]) j++;
i = j - 1;
}
}
Test case int[] { 1, 2, 3, 7, 7, 8, 3, 5, 7, 1, 2, 7 }
output 1, 2, 3, 7


Traverse through the array and check the sign of array[abs(array[i])], if positive make it as negative and if it is negative then print it, as follows:
import static java.lang.Math.abs;
public class FindRepeatedNumber {
private static void findRepeatedNumber(int arr[]) {
int i;
for (i = 0; i < arr.length; i++) {
if (arr[abs(arr[i])] > 0)
arr[abs(arr[i])] = -arr[abs(arr[i])];
else {
System.out.print(abs(arr[i]) + ",");
}
}
}
public static void main(String[] args) {
int arr[] = { 4, 2, 4, 5, 2, 3, 1 };
findRepeatedNumber(arr);
}
}
Reference: http://www.geeksforgeeks.org/find-duplicates-in-on-time-and-constant-extra-space/


As described,
You have an array of numbers from 0 to n-1, one of the numbers is
removed, and replaced with a number already in the array which makes a
duplicate of that number.
I'm assuming elements in the array are sorted except the duplicate entry. If this is the scenario , we can achieve the goal easily as below :
public static void main(String[] args) {
//int arr[] = { 0, 1, 2, 2, 3 };
int arr[] = { 1, 2, 3, 4, 3, 6 };
int len = arr.length;
int iMax = arr[0];
for (int i = 1; i < len; i++) {
iMax = Math.max(iMax, arr[i]);
if (arr[i] < iMax) {
System.out.println(arr[i]);
break;
}else if(arr[i+1] <= iMax) {
System.out.println(arr[i+1]);
break;
}
}
}
O(n) time and O(1) space ;please share your thoughts.


Here is the simple solution with hashmap in O(n) time.
#include<iostream>
#include<map>
using namespace std;
int main()
{
int a[]={1,3,2,7,5,1,8,3,6,10};
map<int,int> mp;
for(int i=0;i<10;i++){
if(mp.find(a[i]) == mp.end())
mp.insert({a[i],1});
else
mp[a[i]]++;
}
for(auto i=mp.begin();i!=mp.end();++i){
if(i->second > 1)
cout<<i->first<<" ";
}
}


int[] a = {5, 6, 8, 9, 3, 4, 2, 9 };
int[] b = {5, 6, 8, 9, 3, 6, 1, 9 };
for (int i = 0; i < a.Length; i++)
{
if (a[i] != b[i])
{
Console.Write("Original Array manipulated at position {0} + "\t\n"
+ "and the element is {1} replaced by {2} ", i,
a[i],b[i] + "\t\n" );
break;
}
}
Console.Read();
///use break if want to check only one manipulation in original array.
///If want to check more then one manipulation in original array, remove break


This video If Programming Was An Anime is too fun not to share. It is the same problem and the video has the answers:
Sorting
Creating a hashmap/dictionary.
Creating an array. (Though this is partially skipped over.)
Using the Tortoise and Hare Algorithm.
Note: This problem is more of a trivia problem than it is real world. Any solution beyond a hashmap is premature optimization, except in rare limited ram situations, like embedded programming.
Furthermore, when is the last time you've seen in the real world an array where all of the variables within the array fit within the size of the array? Eg, if the data in the array is bytes (0-255) when do you have an array 256 elements or larger without nulls or inf within it, and you need to find a duplicate number? This scenario is so rare you will probably never get to use this trick in your entire career.
Because it is a trivia problem and is not real world the question, I'd be cautious accepting an offer from a company that asks trivia questions like this, because people will pass the interview by sheer luck instead of skill. This implies the devs there are not guaranteed to be skilled, which unless you're okay teaching your seniors skills, you might have a bad time.


int a[] = {2,1,2,3,4};
int b[] = {0};
for(int i = 0; i < a.size; i++)
{
if(a[i] == a[i+1])
{
//duplicate found
//copy it to second array
b[i] = a[i];
}
}

Related

How to reverse slice of array java

I've been attempting to reverse a slice of a list in java.
The equivalent in python (though perhaps not the best way - I don't care about this, just want to get my point across) would be:
myList = [0,1,2,3,4,5,6,7,8,9,10]
reverseSlice = myList[2:6]
reverseSlice.reverse()
myList[2:6] = reverseSlice
When I tried to manually implement this in Java, I tried this:
public static int[] reverse(int[] x,int a1, int a2) {
a1--;
int[] rArr = new int[a2-a1+1];
for (int ind = a1; ind<a2; ind++) {
rArr[a2-ind-1] = x[ind];
}
for (int ind = a1; ind<a2; ind++) {
x[ind] = rArr[ind];
}
return x;
}
However, when I run this:
int[] cows1 = new int[]{0,1,2,3,4,5,6,7,8,9,10};
cows1 = reverse(cows1,2,6);
for (int i : cows1) {
System.out.print(i + " ");
}
I get 0 4 3 2 1 0 6 7 8 9 10 .
I'm really confused how I got this, as in my function, I don't introduce new values, so the "0" shouldn't have appeared.
My Question: Why is my code returning values that are not in the list?

The 0 value which is coming might be because of accessing an uninitialized value from rArr array. For the purpose of reversing, an extra array is not required. You can simply swap the values from the starting index to the end index. I have tested the following code and it gives correct output
public class Solution2 {
public static void main(String args[])
{
int[] cows1 = new int[]{0,1,2,3,4,5,6,7,8,9,10};
cows1 = reverse(cows1,2,6);
// both indices are inclusive
for (int i : cows1) {
System.out.print(i + " ");
}
}
public static int[] reverse(int[] x,int a1, int a2) {
for (int i = 0; i < (a2-a1+1)/2; i++) {
swap(x,a1+i,a2-i);
}
return x;
}
private static void swap(int[] x, int i, int j) {
// System.out.println("values swappeed are "+ x[i] + " " + x[j]);
int temp = x[i];
x[i] = x[j];
x[j] = temp;
}
}
and the output comes as
0 1 6 5 4 3 2 7 8 9 10

The problem seems to be the first line of your reverse function: a1--; not preserving the original value for your start index.
The method call
The purpose of the reverse function is to reverse only the portion of the array bound by variable values a1 and a2. On a side note, never use meaningless names like that. Better names for these variables could've been startIndex and endIndex or similar naming (a1 and a2 don't really mean anything).
For values (2, 6), the reverse function should extract the portion of the array starting at index 2 and ending on index 6. I am assuming the end index is not inclusive, so it should only grab values at index 2, 3, 4, and 5. However, the first thing the method does is decrement the start index, so it actually starts at index 1.
From there, the function successfully reverse the values located at these indices [4,3,2,1]. The question is now, what happens to the value at index 5? Let's see what this part of the code does new int[a2-a1+1]. The value of a2 is 6 and the new value of a1 is 1 because it was decremented in the first line. That means that your new array is of size [6-1+1] which is 6. That seems to be incorrect. Only 4 values are required to be reversed and the array is one element too big. The last index of this array is defaulted to integer value of 0.
The swap
First loop:
for (int ind = a1; ind<a2; ind++) { // starts with a1 = 1 and a2 = 6
rArr[a2-ind-1] = x[ind]; // rArr[4] = x[1]: first iteration is off by 1
}
Second loop:
for (int ind = a1; ind<a2; ind++) { // a1 = 1, a2 = 6 (starts at 1 and ends at 5 (rArr doesn't have an index 5)
x[ind] = rArr[ind]; // first iteration: x[1] = rArr[1], last iteration x[5] = rArr[5] -> this last index is accessible because array was too big and was initialized to a primitive `int` default value (0).
}
The solution
Make sure your start and end indices are preserved.
Allocate the temp array (if needed) using the original start and end indices for your calculation.
Inside the reverse() method, your first line should've been:
int[] rArr = new int[a2-a1]; // allocates an array of size 4 (6-2)
From here, if the rest of the method is wrong, you will not see a zero anywhere in the reversed array. At worst, you would've seen an offset problem if your start index was erroneously calculated. And that, I think, would've been easy to spot and fix.
If you are new at programming, and even if you are more of an "expert", you should always work out these problems on paper (or on a board) before you attempt to code and walk through your logic. You will be amazed as to how many logic errors you will catch before you code.

Java merge adjacent array elements to yield maximum min value

excuse me for the confusing title, I need to implement an algorithm which can be simplified as the following:
given an array of integers, and the number of merges needed (denoted as k), return the maximum min value of the merged array, a merge can only happen with adjacent elements.
E.g. array = [3,2,8,2,9], k = 2
The maximum min value of the array after two merges is 5, and the merged array is [5, 10, 9]. In this example, we need to merge elements 3 & 2 and 8 & 2.
Any other merge strategies will yield min val that is smaller or equal to 5, e.g.:
[5,8,11], [3,10,11], [13,2,9](merged elements can be merged again)
What is the best data structure to represent the data and what is an efficient algorithm to address the problem? As far as I can think of, a greedy algorithm needs to be applied where a merge needs to happen with the current min value of the array and one of its smaller neighboring element.
Edit: I just realized that greedy algorithm might not apply, sorry for the misleading comment, if it doesn't distinguish between merging with left or right elements, this will generate the wrong answer. Take this as an example, given an array = [4,5,3,5], and we need to remove 2 elements.
With greedy, [4,5,3,5] -> [4,8,5] -> [12,5], so the answer is 5; however the correct answer should be 8 with the following merge sequence:
[4,5,3,5] -> [4,5,8] -> [9,8]

ValPosFrom is a simple class that stores those things, from being the place to merge from . you can get non deterministic results from things like List = 3,2,6,3,2 and k=1 it will merge one of the 2 mins to 5 but it doesn't matter which one. it converges when all of any positions neighbors values are unique.
private static List<Integer> merge(List<Integer> things, int merges) {
List<Integer> result = new ArrayList<>(things);
for (int i = 0; i < merges; i++) {
int min = Integer.MAX_VALUE;
List<Integer> positions = new ArrayList<>();
for (int j = 0; j < result.size(); j++) {
if (result.get(j) < min) {
positions.clear();
positions.add(j);
min = result.get(j);
} else if (result.get(j) == min) {
positions.add(j);
}
}
List<ValPosFrom> neighbors = new ArrayList<>();
positions.forEach(p -> {
if (p - 1 >= 0) {
neighbors.add(new ValPosFrom(result.get(p - 1), p - 1, p));
}
if (p + 1 < result.size()) {
neighbors.add(new ValPosFrom(result.get(p + 1), p + 1, p));
}
});
ValPosFrom vpf = Collections.min(neighbors, Comparator.comparingInt(v -> v.val));
result.set(vpf.pos, result.get(vpf.pos) + result.get(vpf.from));
result.remove(vpf.from);
}
return result;
}

Find what number it used to be before element reset

Given an array with any size, in my case the array size is 5.
This array contains ALL numbers from 1 to 5 (must contain all of them)
[1 | 2 | 3 | 4 | 5]
0 1 2 3 4
And now, one element was reset and was set to 0, and the mission is to find what number it used to be before it turned 0.
So I have this simple solution:
Explained: First, loop from 1 to 5, create an inner loop to check if the i from the first loop exists in the whole array, if it doesn't exist, that means that it is the value it used to be before 0, because the array contained all numbers from 1 to 5 or 1 to 100 (doesn't matter) and there's on'y one rested element.
Code:
int[] numbers = new int[]{1, 2, 3, 4, 5};
numbers[1] = 0;
int lost = -1;
loop:
for (int i = 1; i <= numbers.length; i++) {
for (int j = 0; j < numbers.length; j++) {
if (numbers[j] == i) {
continue loop;
}
}
lost = i;
break loop;
}
System.out.println(lost);
That solution is not bad, but I think there's a better solution, something more stable.
I have thought about it mathematically, in our example:
1 + x + 3 + 4 + 5 = 15
x = 2
Mathematically, it's really easy. Is there a way this can be done in a programming language as easy as it is mathematically?
Any better algorithms you can think of to solve this question?

This works for ONE element being reset. Just subtract each remaining element from the sum and what ever is left over would have been the previous number the element was before it was reset.
public static void main(String[] args) throws Exception {
int sum = 15;
int[] numbers = new int[] { 1, 2, 3, 4, 5 };
numbers[4] = 0;
for (int i = 0; i < numbers.length; i++) {
sum -= numbers[i];
}
System.out.println(sum);
}
Results:
5

There is one more possibility, you can use HashMaps!
you don't have to traverse through any "for loops" then.
You can use Hashmaps to check whether is there any value for the key of "0", if yes then that is the case where some number is reset to 0.
Then you can traverse the Hashmap and compare which value is missing.
Everything is done With O(1) complexity and worst case of O(n) complexity.

How to find minimum number of jumps to reach the end of the array in O(n) time

Question
Given an array of integers where each element represents the max number of steps that can be made forward from that element.
Write a function to return the minimum number of jumps to reach the
end of the array (starting from the first element). If an element is
0, then cannot move through that element.
Example
Input: arr[] = {1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9}
Output: 3 (1-> 3 -> 8 ->9)
Found multiple ways from Dynamic Programming approach to other linear approaches. I am not able to understand the approach which is said to linear in time. HERE is the link where a linear approach is proposed.
I am not able to understand it at all. What I could understand is that author is suggesting to do a greedy approach and see if we reach end .. if not then backtrack ?

The time complexity of the solution proposed on the site is linear because you only iterate over the array once. The algorithm avoids the inner iteration of my proposed solution by using some clever tricks.
The variable maxReach stores at all time the maximal reachable position in the array. jump stores the amount of jumps necessary to reach that position. step stores the amount of steps we can still take (and is initialized with the amount of steps at the first array position)
During the iteration, the above values are updated as follows:
First we test whether we have reached the end of the array, in which case we just need to return the jump variable.
Next we update the maximal reachable position. This is equal to the maximum of maxReach and i+A[i] (the number of steps we can take from the current position).
We used up a step to get to the current index, so steps has to be decreased.
If no more steps are remaining (i.e. steps=0, then we must have used a jump. Therefore increase jump. Since we know that it is possible somehow to reach maxReach, we initialize the steps to the amount of steps to reach maxReach from position i.
public class Solution {
public int jump(int[] A) {
if (A.length <= 1)
return 0;
int maxReach = A[0];
int step = A[0];
int jump = 1;
for (int i = 1; i < A.length; i++) {
if (i == A.length - 1)
return jump;
if (i + A[i] > maxReach)
maxReach = i + A[i];
step--;
if (step == 0) {
jump++;
step = maxReach - i;
}
}
return jump;
}
}
Example:
int A[] = {1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9}
int maxReach = A[0]; // A[0]=1, so the maximum index we can reach at the moment is 1.
int step = A[0]; // A[0] = 1, the amount of steps we can still take is also 1.
int jump = 1; // we will always need to take at least one jump.
/*************************************
* First iteration (i=1)
************************************/
if (i + A[i] > maxReach) // 1+3 > 1, we can reach further now!
maxReach = i + A[i] // maxReach = 4, we now know that index 4 is the largest index we can reach.
step-- // we used a step to get to this index position, so we decrease it
if (step == 0) {
++jump; // we ran out of steps, this means that we have made a jump
// this is indeed the case, we ran out of the 1 step we started from. jump is now equal to 2.
// but we can continue with the 3 steps received at array position 2.
steps = maxReach-i // we know that by some combination of 2 jumps, we can reach position 4.
// therefore in the current situation, we can minimaly take 3
// more steps to reach position 4 => step = 3
}
/*************************************
* Second iteration (i=2)
************************************/
if (i + A[i] > maxReach) // 2+5 > 4, we can reach further now!
maxReach = i + A[i] // maxReach = 7, we now know that index 7 is the largest index we can reach.
step-- // we used a step so now step = 2
if (step==0){
// step
}
/*************************************
* Second iteration (i=3)
************************************/
if (i + A[i] > maxReach) // 3+8 > 7, we can reach further now!
maxReach = i + A[i] // maxReach = 11, we now know that index 11 is the largest index we can reach.
step-- // we used a step so now step = 1
if (step==0){
// step
}
/*************************************
* Third iteration (i=4)
************************************/
if (i + A[i] > maxReach) // 4+9 > 11, we can reach further now!
maxReach = i + A[i] // maxReach = 13, we now know that index 13 is the largest index we can reach.
step-- // we used a step so now step = 0
if (step == 0) {
++jump; // we ran out of steps, this means that we have made a jump.
// jump is now equal to 3.
steps = maxReach-i // there exists a combination of jumps to reach index 13, so
// we still have a budget of 9 steps
}
/************************************
* remaining iterations
***********************************
// nothing much changes now until we reach the end of the array.
My suboptimal algorithm which works in O(nk) time with n the number of elements in the array and k the largest element in the array and uses an internal loop over array[i]. This loop is avoided by the below algorithm.
Code
public static int minimum_steps(int[] array) {
int[] min_to_end = new int[array.length];
for (int i = array.length - 2; i >= 0; --i) {
if (array[i] <= 0)
min_to_end[i] = Integer.MAX_VALUE;
else {
int minimum = Integer.MAX_VALUE;
for (int k = 1; k <= array[i]; ++k) {
if (i + k < array.length)
minimum = Math.min(min_to_end[i+k], minimum);
else
break;
}
min_to_end[i] = minimum + 1;
}
}
return min_to_end[0];
}

Here is the basic intuition regarding the above problem's greedy approach and rest are the code requirements.
Given array is Input: a[] = {1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9}.
Now we start from the 1st element i.e i=0 and a[i] = 1. So seeing this we can take at most a jump of size 1, so since we don't have any other choice so we make this step happen.
Currently we are at i=1 and a[i]=3. So we currently can make a jump of size 3, but instead we consider all possible jumps we can make from the current location and attain the maximum distance which is within bounds(of the array). So what are our choices? we can make a jump of 1 step, or 2 steps or 3 steps. So we investigate from current location for each size jumps and choose the one which can take us maximum further into the array.
Once we have decided, which one we stick to, we take that jump size and update our number of jumps made so far and also where we can reach at most and how many steps we have now to decide our next move. And that's it. This is how finally we select the best option linearly traversing the array.
So this is the basic idea of the algo you might be looking for, next is to code it for the algorithm to work. Cheers!
Hope somebody time travels and finds the intuition helpful !! :) :P
"Years late to the party" #Vasilescu Andrei - well said. Sometimes it feels to me that we are time travelers.

Many of the answers here so far are great, but I feel I can help explain why the algorithm is correct and the intuition behind it.
I like this problem because it's one where the intuitive dynamic programming approach runs in O(n^2) worst-case, and a greedy approach (the one that motivated this question) runs in O(n) worst-case (it actually only visits each element of the array once). This algorithm is also for me somewhat reminiscent of Dijkstra's algorithm which solves another single-source shortest-path problem and that is also greedy.
To start, remember from the problem statement that A[i] holds the maximum position you can jump to from that index, but you can take a shorter jump from i if A[i]>1, so a shortest sequence of jumps from i=0 could be one with shorter jumps than what's allowed on each index. This is important, since you will see that the algorithm never considers those smaller jumps or their locations explicitly.
Second, it helps to think of the algorithm that you mentioned as one that gives itself "strands of rope" (steps = maxReach - i;) to reach the end, and that it consumes this rope (steps--;) as it tries to advance through the array.
Third, note that the algorithm is not keeping track of the specific indices i that may be part of a shortest sequence from the beginning to the end of the input array A. In fact, the algorithm only increases the variable jump (it gives itself a new strand of rope) when it "runs out of rope" (from the previous strand), so that it can keep iterating in the main loop to "try" to reach the end.
More specifically for the algorithm to be correct it needs to:
Keep track of "how far it can reach" (maxReach) from each location i as it moves forward through the array. Note that this quantity is updated for each location even if it's clear already at that moment that reaching that new location will require it to take more "jumps" as you exceed the number of steps (i.e. you run out of rope) that you gave yourself earlier, even if no shortest path would actually visit that element. The goal of these updates is to identify how far the next jump could reach so that it can give itself that much rope once it exhausted the current one.
Account for the minimum number of jumps (jumps) you must take if you want to continue iterating through the array to reach the end, as you run out of rope (steps) from the previous strand.
The algorithm that you linked to, for reference:
public class Solution {
public int jump(int[] A) {
if (A.length <= 1)
return 0;
int maxReach = A[0];
int step = A[0];
int jump = 1;
for (int i = 1; i < A.length; i++) {
if (i == A.length - 1)
return jump;
if (i + A[i] > maxReach)
maxReach = i + A[i];
step--;
if (step == 0) {
jump++;
step = maxReach - i;
}
}
return jump;
}
}

Years late to the party , but here is another O(n) solution that made sense for me.
/// <summary>
///
/// The actual problem is if it's worth not to jump to the rightmost in order to land on a value that pushes us further than if we jumped on the rightmost.
///
/// However , if we approach the problem from the end, we go end to start,always jumping to the leftmost
///
/// with this approach , these is no point in not jumping to the leftmost from end to start , because leftmost will always be the index that has the leftmost leftmost :) , so always choosing leftmost is the fastest way to reach start
///
/// </summary>
/// <param name="arr"></param>
static void Jumps (int[] arr)
{
var LeftMostReacher = new int[arr.Length];
//let's see , for each element , how far back can it be reached from
LeftMostReacher[0] = -1; //the leftmost reacher of 0 is -1
var unReachableIndex = 1; //this is the first index that hasn't been reached by anyone yet
//we use this unReachableIndex var so each index's leftmost reacher is the first that was able to jump to it . Once flagged by the first reacher , new reachers can't be the leftmost anymore so they check starting from unReachableIndex
// this design insures that the inner loop never flags the same index twice , so the runtime of these two loops together is O(n)
for (int i = 0; i < arr.Length; i++)
{
int maxReach = i + arr[i];
for (; unReachableIndex <= maxReach && unReachableIndex < arr.Length; unReachableIndex++)
{
LeftMostReacher[unReachableIndex] = i;
}
}
// we just go back from the end and then reverse the path
int index = LeftMostReacher.Length - 1;
var st = new Stack<int>();
while (index != -1)
{
st.Push(index);
index = LeftMostReacher[index];
}
while (st.Count != 0)
{
Console.Write(arr[st.Pop()] + " ");
}
Console.WriteLine();
}
static void Main ()
{
var nrs = new[] { 1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9 };
Jumps(nrs);
}

Simple python code for the Minimum number of jumps to reach end problem.
ar=[1, 3, 6, 3, 2, 3, 6, 8, 9, 5]
minJumpIdx=0
res=[0]*len(ar)
i=1
while(i<len(ar) and i>minJumpIdx):
if minJumpIdx+ar[minJumpIdx]>=i:
res[i]=res[minJumpIdx]+1
i+=1
else:
minJumpIdx+=1
if res[-1]==0:
print(-1)
else:
print(res[-1])

Here is another linear solution. The code is longer than the one suggested in the leet code link, but I think it is easier to understand. It is based on a two observations: the number of steps required to reach the i + 1 position is never less than the number of steps required to reach the i position and each element each element assigns its value + 1 to i + 1 ... i + a[i] segment.
public class Solution {
public int jump(int[] a) {
int n = a.length;
// count[i] is the number of "open" segments with value i
int[] count = new int[n];
// the number of steps to reach the i-th position
int[] dp = new int[n];
Arrays.fill(dp, n);
// toDelete[i] is the list of values of segments
// that close in the i-th position
ArrayList<Integer>[] toDelete = new ArrayList[n];
for (int i = 0; i < n; i++)
toDelete[i] = new ArrayList<>();
// Initially, the value is 0(for the first element).
toDelete[0].add(0);
int min = 0;
count[0]++;
for (int i = 0; i < n; i++) {
// Finds the new minimum. It uses the fact that it cannot decrease.
while (min < n && count[min] == 0)
min++;
// If min == n, then there is no path. So we can stop.
if (min == n)
break;
dp[i] = min;
if (dp[i] + 1 < n) {
// Creates a new segment from i + 1 to i + a[i] with dp[i] + 1 value
count[dp[i] + 1]++;
if (i + a[i] < n)
toDelete[i + a[i]].add(dp[i] + 1);
}
// Processes closing segments in this position.
for (int deleted : toDelete[i])
count[deleted]--;
}
return dp[n - 1];
}
}
Complexity analysis:
The total number of elements in toDelete lists is O(n). It is the case because at each position i at most one element is added. That's why processing all elements in all toDelete lists requires linear time.
The min value can only increase. That's why the inner while loop makes at most n iterations in total.
The outer for loop obviously makes n iterations. Thus, the time complexity is linear.

Okay, it took me good amount of time to wrap my head around the O(n) algo, I will try to explain the logic to my best simplest possible:
At each "i" in the array, you know with that value what is the currentFarthest value, you can reach up to, & also the currentEnd value, whenever you hit the currentEnd value, you know its time to make a jump & update currentEnd with currentFarthest.
The picture below might help :

I have done this with Python.
Less complex code with simple terms. This might help you.
def minJump(a):
end=len(a)
count=0
i=a[0]
tempList1=a
while(i<=end):
if(i==0):
return 0
tempList1=a[count+1:count+i+1]
max_index=a.index(max(tempList1))
count+=1
i=a[max_index]
end=end-max_index
return count+1

Another O(n) Solution With the best explanation
The following solution provides with o(n) time complexity
For solving minimum jumps to reach the end of the array,
For every jump index, we consider need to evaluate the corresponding step values in the index and using the index value divides the array into sub-parts and find out the maximum steps covered index.
Following code and explanation will give you a clear idea:
In each sub-array find out the max distance covered index as the first part of the array, and the second array
Input Array : {1, 3, 5, 9, 6, 2, 6, 7, 6, 8, 9} -> index position starts with 0
Steps :
Initial step is considering the first index and incrementing the jump
Jump = 1
1, { 3, 5, 9, 6, 2, 6, 7, 6, 8, 9} -> 1 is considered as a first jump
next step
From the initial step there is only one step to move so
Jump = 2
1,3, { 5, 9, 6,2, 6, 7, 6, 8, 9} -> 1 is considered as a first jump
next step
Now we have a flexibility to choose any of {5,9,6} because of last step says we can move upto 3 steps
Consider it as a subarray, evaluate the max distance covers with each index position
As {5,9,6} index positions are {2,3,4}
so the total farther steps we can cover:
{7,12,10} -> we can assume it as {7,12} & {10} are 2 sub arrays where left part of arrays says max distance covered with 2 steps and right side array says max steps cover with remaining values
next step:
Considering the maximum distanc covered in first array we iterate the remaining next elements
1,3,9 {6,2, 6, 7, 6, 8, 9}
From above step ww already visited the 4th index we continue with next 5th index as explained above
{6,2, 6, 7, 6, 8, 9} index positions {4,5,6,7,8,9,10}
{10,7,12,14,14,17,19}
Max step covers here is 19 which corresponding index is 10
Code
//
// Created by Praveen Kanike on 07/12/20.
//
#include <iostream>
using namespace std;
// Returns minimum number of jumps
// to reach arr[n-1] from arr[0]
int minJumps(int arr[], int n)
{
// The number of jumps needed to
// reach the starting index is 0
if (n <= 1)
return 0;
// Return -1 if not possible to jump
if (arr[0] == 0)
return -1;
// stores the number of jumps
// necessary to reach that maximal
// reachable position.
int jump = 1;
// stores the subarray last index
int subArrEndIndex = arr[0];
int i = 1;
//maximum steps covers in first half of sub array
int subArrFistHalfMaxSteps = 0;
//maximum steps covers in second half of sub array
int subArrSecondHalfMaxSteps =0;
// Start traversing array
for (i = 1; i < n;) {
subArrEndIndex = i+subArrEndIndex;
// Check if we have reached the end of the array
if(subArrEndIndex >= n)
return jump;
int firstHalfMaxStepIndex = 0;
//iterate the sub array and find out the maxsteps cover index
for(;i<subArrEndIndex;i++)
{
int stepsCanCover = arr[i]+i;
if(subArrFistHalfMaxSteps < stepsCanCover)
{
subArrFistHalfMaxSteps = stepsCanCover;
subArrSecondHalfMaxSteps = 0;
firstHalfMaxStepIndex = i;
}
else if(subArrSecondHalfMaxSteps < stepsCanCover)
{
subArrSecondHalfMaxSteps = stepsCanCover;
}
}
if(i > subArrFistHalfMaxSteps)
return -1;
jump++;
//next subarray end index and so far calculated sub array max step cover value
subArrEndIndex = arr[firstHalfMaxStepIndex];
subArrFistHalfMaxSteps = subArrSecondHalfMaxSteps;
}
return -1;
}
// Driver program to test above function
int main()
{
int arr[] = {100, 3, 5, 9, 6, 2, 6, 7, 6, 8, 9};
int size = sizeof(arr) / sizeof(int);
// Calling the minJumps function
cout << ("Minimum number of jumps to reach end is %d ",
minJumps(arr, size));
return 0;
}

Just in case you need to write a python solution for the greedy approach this code will get you covered for the above problem :)
def minJumps(self, arr, n):
#code here
if(n <= 1):
return(0)
if(arr[0] == 0):
return -1
maxrange, step = arr[0], arr[0]
jumps = 1
for i in range(1,n):
if (i == len(arr) - 1):
return jumps
maxrange = max(maxrange, i+arr[i])
step -= 1
if(step == 0):
jumps += 1
if(i>=maxrange):
return -1
step = maxrange - i
return(jumps)

Here is another solution. In this solution, the worst-case complexity is O(n) while the average-case complexity is less than O(n). I don't know how to do average-case complexity analysis. So, I can't tell the exact average-case complexity. But yeah it is faster than 99.22% of submissions on leet code.
def minJumps(self, arr, n):
current_w=0 # current_index
last_w=n-1 # last_index
max_reach=0 # value of index upto which we have analysed array for optimum solution
max_jumps=arr[0] # maximum jumps that can be taken from a current_index
hop=0 # total jumps
while current_w<last_w:
max_jumps=arr[current_w]
if max_jumps==0:
return -1
if max_jumps==1:
max_reach=max_jumps+current_w
current_w+=1
elif max_jumps<last_w-current_w: # if maximum steps does not reach to last index
can_jump_to=arr[max_reach+1:max_jumps+current_w+1] # subarray in which we have to search for a wall,jumping to which can take us to required solution
jump_to=max(range(len(can_jump_to)),key=lambda x: x+can_jump_to[x])+max_reach+1 # finding index of wall whoose definition mentioned in above comment
max_reach=max_jumps+current_w #updating max_reach
current_w=jump_to #updating current position
else:
current_w=last_w
hop+=1
return hop

static void minJumps(int a[] , int n)
{
int dp[] = new int[n];
dp[0] = 0; //As the min jumps needed to get to first index is zero only.
//Fill the rest of the array with INT_MAX val so we can make math.min comparisions.
for(int i=1;i<n;i++)
dp[i] = Integer.MAX_VALUE;
for(int i=1;i<n;i++)
{
for(int j=0;j<i;j++)
{ //If we have enough jumps from the position j to reach i.
if(j+a[j]>=i)
{ //Take the min of current stored value & jumps req to
//reach i from j by getting jumps req to reach j plus 1.
//(Plus 1 because since we have enough jumps to reach 1 from j we
//simply add 1 by taking the jumps required to reach j.)
dp[i] = Math.min(dp[i],dp[j]+1);
}
}
}
//If first element has zero jumps in store or if the final jumps value
//becomes MAX value because there's an element in between which gives zero
//jumps.
if(a[0]==0 || dp[n-1] == Integer.MAX_VALUE )
System.out.println("-1");
else System.out.println(dp[n-1]);
}

Modifying the greatest elements of an array without changing their position?

I'm trying to figure out how to modify the n greatest elements of an array without modifying their position. For example, suppose I have an array of ints {5, 2, 3, 4, 8, 9, 1, 3};
I want to add 1 to the two greatest elements, making the array {5, 2, 3, 4, 9, 10, 1, 3}.
All of the methods I can think of to go about doing this end up feeling clunky and unintuitive when I try to implement them, signaling to me that I'm not thinking about it correctly. For example, I could use a TreeMap with the values of the array as keys and their indices as values to find the greatest values, modify them, and then throw them back into the array, but then I would have have to implement my own Comparator to sort the TreeMap in reverse order(unless there's an easier way I'm not aware of?). I was also considering copying the contents of the array into a list, iterating through n times, each time finding the greatest element and its index, putting the modified greatest element back into the array at that index, removing the element from the list, and repeat, but that feels sloppy and inefficient to me.
Any suggestions as to how to approach this type of problem?

The simplest thing would be to scan your array, and store the indices of the n highest values. Increment the values of those elements.
This is going to be O(n) performance, and I don't think any fancier methods can beat that.
edit to add: you can sort the array in place in O(n) at best, in which case you can get the n highest values very quickly, but the requirement is to not change position of the elements, so you'd have to start with a copy of the array if you wanted to do that (or preserve ordering information so you could put everything back afterward).

You might be over engineering the solution to this problem: scan the array, from beginning to end, and mark the two largest elements. Return to the two greatest elements and add 1 to it. The solution shouldn't be longer than 10 lines.

Loop over the array and keep track of the indices and values of the two largest items
a. Initialize the tracker with -1 for an index and MIN_INT for a value or the first two values of the array
b. At each step of the loop compare the current value against the two tracker values and update if necessary
Increment the two items
Any algorithm you choose should be O(n) for this. Sorting and n passes are way overkill.

Find the nth largest element (call it K) using techniques here and here (can be done in linear time), then go through the array modifying all elements >= K.

i would do something like this
int[] indices = new int[2];
int[] maximas = new int[] { 0, 0 };
int[] data = new int[] { 3, 4, 5, 1, 9 };
for (int i = 0; i < 5; ++i)
{
if (data[i] > maximas[1])
{
maximas[0] = maximas[1];
maximas[1] = data[i];
indices[0] = indices[1];
indices[1] = i;
}
else if (data[i] > maximas[0])
{
maximas[0] = data[i];
indices[0] = i;
}
}
didn't test it, but I think it should work :)

I have tought a bit about this but I cannot achieve more than worstcase:
O( n + (m-n) * n ) : (m > n)
best case:
O(m) : (m <= n)
where m = number of values, n = number of greatest value to search
This is the implementation in C#, but you can easily adapt to java:
int n = 3;
List<int> values = new List<int> {1,1,1,8,7,6,5};
List<int> greatestIndexes = new List<int>();
for (int i = 0; i < values.Count; i++) {
if (greatestIndexes.Count < n)
{
greatestIndexes.Add(i);
}
else {
int minIndex = -1, minValue = int.MaxValue;
for (int j = 0; j < n; j++)
{
if (values[greatestIndexes[j]] < values[i]) {
if (minValue > values[greatestIndexes[j]])
{
minValue = values[greatestIndexes[j]];
minIndex = j;
}
}
}
if (minIndex != -1)
{
greatestIndexes.RemoveAt(minIndex);
greatestIndexes.Add(i);
}
}
}
foreach (var i in greatestIndexes) {
Console.WriteLine(values[i]);
}
Output:
8
7
6

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