Find element in sorted rotated array

Find element in sorted rotated array

To understand how to find element in sorted rotated array, we must understand first, what is a sorted rotated array? An array is called sorted where for all i and j such that i < j, A[i] <= A[j]. A rotation happens when last element of array is push at the start and all elements on array move right by one position. This is called as rotation by 1. If new last element is also pushed to start again all elements are moved to right again, it’s rotation by 2 and so on.

Find element in sorted rotated array
Sorted array
Sorted rotated array

Question which is very commonly asked in Amazon and Microsoft initial hacker round interviews or telephonic interviews : Given a sorted rotated array, find position of an element in that array. For example:

A = [2,3,4,1] Key = 4, Returns 2 which is position of 4 in array

A = [4,5,6,1,2,3] Key = 4 returns 0

Find element in sorted rotated array : Thought process

Before starting with any solution, it’s good to ask some standard questions about an array problem, for example, if duplicate elements are allowed or if negative numbers are allowed in array? It may or may not change the solution, however, it gives an impression that you are concerned about input range and type.

First thing to do in interview is come up with brute force solution, why? There are two reasons : first, it gives you confidence that you have something solved, it may not be optimal way but still you have something. Second, now that you have something written, you can start looking where it takes most of time or space and attack the problem there. It also, helps to identify what properties you are not using which are part of the problem and help your solution.

First thing first, what will be the brute force solution? Simple solution will be to scan through the array and find the key. This algorithm will have O(N) time complexity.

There is no fun in finding an element in sorted array in O(N) 🙂 It would have been the same even if array was not sorted. However, we already know that our array is sorted. It’s also rotated, but let’s forget about that for now. What do we do when we have to find an element in sorted array? Correct, we use binary search.

We split the array in middle and check if element at middle index is the key we are looking for? If yes, bingo! we are done.

If not, if A[mid] is less that or greater than key. If it is less, search in right subarray, and it is greater, search in left subarray. Any which way, our input array is reduced to half. Complexity of binary search algorithm is log (N). We are getting somewhere 🙂

Sorted rotated array

However, our input array in not a plain sorted array, it is rotated too. How does things change with that. First, comparing just middle index and discarding one half of array will not work. Still let’s split the array at middle and see what extra conditions come up?
If A[mid] is equal to key, return middle index.
There are two broad possibilities of rotation of array, either it is rotated more than half of elements or less than half of elements. Can you come up with examples and see how array looks like in both the cases?

If array is rotated by more than half of elements of array, elements from start to mid index will be a sorted.

If array is rotated by less than half of elements of array, elements from mid to end will be sorted.

Next question, how do you identify the case, where array is rotated by more or less than half of elements? Look at examples you come up with and see if there is some condition?

Yes, the condition is that if A[start] < A[mid], array is rotated more than half and if A[start] > A[mid], it is rotated by less than half elements.

Now, that we know, which part of array is sorted and which is not. Can we use that to our advantage?

Case 1 : When array from start to mid is sorted. We will check if key > A[start] and key < A[mid]. If that’s the case, search for key in A[start..mid]. Since, A[start..mid] is sorted, problem reduces to plain binary search. What if key is outside of start and middle bounds, then discard A[start..mid] and look for element in right subarray. Since, A[mid+1..right] is still a sorted rotated array, we follow the same process as we did for the original array.

Case 2 : When array from mid to end is sorted. We will check if key >A[mid] and key < A[end]. If that’s the case, search for key in A[mid+1..end]. Since, A[mid+1..end] is sorted, problem reduces to plain binary search. What if key is outside of mid and end bounds, then discard A[mid..end] and search for element in left subarray. Since, A[start..mid-1] is still a sorted rotated array, we follow the same process as we did for the original array.

Let’s take an example and go through the entire flow and then write concrete algorithm to find element in sorted rotated array.

Below is sorted rotated array given and key to be searched is 6.

We know, A[start] > A[mid], hence check if searched key fall under range A[mid+1..end]. In this case, it does. Hence, we discard A[start..mid].

At this point, we have to options:  either fallback to traditional binary search algorithm or continue with same approach of discarding based on whether key falls in range of sorted array. Both methods work. Let’s continue with same method.

Again find middle of array from middle +1 to end.

A[mid] is still not equal to key. However, A[start] < A[mid]; hence, array from A[start] to A[middle] is sorted. See if our key falls between A[start] and A[mid]? Yes, hence, we discard the right sub array A[mid..End]

Find the middle of remaining array, which is from start to old middle – 1.

Is A[mid] equal to key? No. Since, A[start] is not less than A[mid], see if key falls under A[mid+1..end], it does, hence discard the left subarray.

Now, new middle is equal to key are searching for. Hence return the index.

Similarly, we can find 11 in this array. Can you draw the execution flow that search?

Algorithm to find element in sorted rotated array

  1. Find mid =  (start + end)/ 2
  2. If A[mid] == key; return mid
  3. Else, if A[start] < A[end]
    • We know, left subarray is already sorted.
    • If A[start] < key and A[mid] > key :
      • discard A[mid..end]
      • Continue with new subarray with start and end = mid – 1
    • Else:
      • discard A[start..mid]
      • Continue with new subarray with start = mid + 1 and end
  4. Else
    • We know, right subarray is sorted.
    • If A[mid] < key and A[end] > key :
      • discard A[start..mid]
      • Continue with new subarray with start  = mid + 1 and end
    • Else:
      • discard A[start..mid]
      • Continue with new subarray with start and end = mid – 1

Find element in sorted rotated array : Implementation

package com.company;

/**
 * Created by sangar on 22.3.18.
 */
public class SortedRotatedArray {

    public static int findElementRecursive(int[] input, int start, int end, int key){

        if(start <= end){
            int mid = start + (end - start) / 2;

            if(input[mid] == key) return mid;

            else if(input[start] <= input[mid]){
                /*Left sub array is sorted, check if
                key is with A[start] and A[mid] */
                if(input[start] <= key && input[mid] > key){
                    /*
                      Key lies with left sorted part of array
                     */
                    return findElementRecursive(input, start, mid - 1, key);
                }else{
                    /*
                    Key lies in right subarray
                     */
                    return findElementRecursive(input, mid + 1, end, key);
                }
            }else {
                /*
                 In this case, right subarray is already sorted and
                 check if key falls in range A[mid+1] and A[end]
                 */
                if(input[mid+1] <= key && input[end] > key){
                    /*
                      Key lies with right sorted part of array
                     */
                    return findElementRecursive(input, mid + 1 , end, key);
                }else{
                    /*
                    Key lies in left subarray
                     */
                    return findElementRecursive(input, start, mid - 1, key);
                }
            }
        }
        return -1;
    }

    public static void main(String[] args) {
        int[] input = {10,11,15,17,3,5,6,7,8,9};

        int index = findElementRecursive(input,0, input.length-1, 6);
        System.out.print(index == -1 ? 
              "Element not found" : "Element found at : " + index);

    }
}

Iterative implementation

package com.company;

/**
 * Created by sangar on 22.3.18.
 */
public class SortedRotatedArray {

    public static int findElementIteratve(int[] input, int start, int end, int key) {

        while (start <= end) {
            int mid = start + (end - start) / 2;

            if (input[mid] == key) return mid;

            else if (input[start] <= input[mid]) {
                /*Left sub array is sorted, check if
                key is with A[start] and A[mid] */
                if (input[start] <= key && input[mid] > key) {
                    /*
                      Key lies with left sorted part of array
                     */
                    end = mid - 1;
                } else {
                    /*
                    Key lies in right subarray
                     */
                    start  = mid + 1;
                }
            } else {
                /*
                 In this case, right subarray is already sorted and
                 check if key falls in range A[mid+1] and A[end]
                 */
                if (input[mid + 1] <= key && input[end] > key) {
                    /*
                      Key lies with right sorted part of array
                     */
                    start = mid + 1;
                } else {
                    /*
                    Key lies in left subarray
                     */
                    end  = mid - 1;
                }
            }
        }
        return -1;
    }

    public static void main(String[] args) {
        int[] input = {10,11,15,17,3,5,6,7,8,9};

        int index = findElementIteratve(input,0, input.length-1, 6);
        System.out.print(index == -1 ? "Element not found" : "Element found at : " + index);

    }
}

Complexity of above recursive and iterative algorithm to find an element in a rotated sorted array is O(log n). Recursive implementation has implicit space complexity of O(log n)

What did we learn today? We learned that it’s always better to come up with non-optimized solution first and then try to improve it. Also helps to correlate problem with similar and simpler problem like we understood first what is best way to find an element in sorted array and then extrapolated the solution with additional conditions for our problem.

I hope that this post helped you with this problem and many more similar problems you will see in interviews.

Please share if you have some questions or suggestions. Also, if you want to contribute on this learning process of others, please contact us.

Word break problem

Word break problem

This problem is commonly asked in the Google and Amazon interview. We all know that if you typed string in Google search box does not make sense, Google breaks that into meaningful words and asks us back if we meant those words instead of a single word. This post discusses how can we find if the given string can be broken into meaningful dictionary words. For example, if I typed algorithmsandme and given dictionary is [“algorithms”, “and”, “me”], this string is breakable in meaningful words. but if the string is algorithmsorme this is not breakable into meaningful words. You can find this problem for practice at leetcode.

Word break problem : thoughts

We start with the first character of the string, check if the character itself is a word in the dictionary? If yes, then our problem reduces to the smaller problem, that is to check if substring from index 1 to s.length is breakable or not.
If not, then we check two characters and then three characters and so on till we can check the whole string. As with every character inclusion, the problem reduces in size but remains the same, so ideal case for recursive implementation.

package AlgorithmsAndMe;

import java.util.HashMap;
import java.util.HashSet;
import java.util.List;
import java.util.Set;

public class WordBreak {

    public boolean wordBreak(String s, List<String> wordDict) {
        return wordBreakUtil(s, wordDict, 0, table);
    }

    private boolean wordBreakUtil(String s, 
                                   List<String> wordDict, 
                                   int index) {

        if (index == s.length()) return true;

        boolean isBreakable = false;
        for(int i=index; i<s.length(); i++) {
            isBreakable = isBreakable 
                   || wordDict.contains(s.substring(index, i+1))
                    && wordBreakUtil(s, wordDict, i + 1);
        }

        return isBreakable;
    }
}

If you notice we are solving the same problems again and again in recursive function wordBreakUtil, how can we save that repeated calculations? Best way to save the already solve problems in a cache, that way we can refer to the cache if the problem is already solved or not. If yes, do not solve it again and use the cached value. This approach is called a Top Down approach and uses memoization to avoid repeated subproblems.

package AlgorithmsAndMe;

import java.util.HashMap;
import java.util.HashSet;
import java.util.List;
import java.util.Set;

public class WordBreak {

    public boolean wordBreak(String s, List<String> wordDict) {
        int [] table =  new int[s.length()];
        for(int i=0; i<s.length(); i++){
            table[i] = -1;
        }
        return wordBreakUtilTopDown(s, wordDict, 0, table);
    }

    private boolean wordBreakUtilTopDown(String s, 
                            List<String> wordDict,
                            int index,
                            int[] table) {

        if (index == s.length()) return true;

        if(table[index] < 0) {
            boolean isBreakable = false;
            for (int i = index; i < s.length(); i++) {
                isBreakable = isBreakable 
                        || wordDict.contains(s.substring(index, i + 1))
                        && wordBreakUtilTopDown(s, wordDict, i + 1);
            }
            table[index] = isBreakable ? 1 : 0;
        }
        return table[index] == 1 ? true : false;
    }
  }

If you run the first solution, it will exceed the time limit on leetcode, however, the second implementation should be accepted with 4ms as the time to run. Now you can appreciate the efficiency by memoization.

Word break problem using dynamic programming

In the last two implementations, two things are evident: first, the optimal solution of a subproblem leads to the optimal solution of the original problem. Second, there are overlapping subproblems. These are two must have conditions for applying dynamic programming. We already saw the memoization and top-down approach of DP to avoid repeated solving of subproblems. How can we do it bottom up?

What if store an information if the string till index i is breakable? What will be the base case? The string before index 0 is alway breakable as empty string. So table[0] can be always true. To check if string till index i is breakable or not, we check from index 0 to index i-1 if there is any index j till which string is breakable. If yes, then we just check if substring from index j to i, that will make table[i] as true.

package AlgorithmsAndMe;

import java.util.HashMap;
import java.util.HashSet;
import java.util.List;
import java.util.Set;

public class WordBreak {

    public boolean wordBreak(String s, List<String> wordDict) {
        return wordBreakBottomUp(s, wordDict, 0, table);
    }

    private boolean wordBreakUtilBottomUp(String s, List<String> wordDict){

        if(s == null || s.length() == 0) return false;

        boolean[] table  = new boolean[s.length()+1];

        table[0] = true;
        for (int i = 1; i <= s.length(); i++) {
            for (int j = i - 1; j >= 0; j--) {
                if (table[j] && wordDict.contains(s.substring(j, i))) {
                        table[i] = true;
                    }
                }
            }
        }
        return table[s.length()];
    }
}

The time complexity of the above implementation of the word break problem is O(n2)

If you want to store all the strings which can be generated by breaking a particular word, below is the code.

package AlgorithmsAndMe;

import java.util.*;

public class WordBreak2 {

    public List<String> wordBreak(String s, List<String> wordDict) {
        Map<String, List<String>> map = new HashMap<>();
        return wordBreakUtil2(s, wordDict, map);
    }

    private List<String> wordBreakUtil2(String s,
                                        List<String> wordDict,
                                        Map<String, List<String>> map) {

        if(map.containsKey(s)){
            return map.get(s);
        }

        List<String> result = new ArrayList<String>();
        if (wordDict.contains(s)){
            result.add(s);
        }

        for(int i=1; i<=s.length(); i++) {
            String prefix = s.substring(0, i);
            if(wordDict.contains(prefix)){
                List<String> returnStringsList = wordBreakUtil2(s.substring(i), wordDict, map);

                for(String returnString :returnStringsList ){
                    result.add(prefix + " " + returnString);
                }
            }
        }
        map.put(s,result);

        return result;
    }
}

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