Searching for an element in a circular sorted array

We want to search for a given element in a circular sorted array in complexity not greater than 开发者_开发问答O(log n).

Example: Search for 13 in {5,9,13,1,3}.

My idea was to convert the circular array into a regular sorted array then do a binary search on the resulting array, but my problem was the algorithm I came up was stupid that it takes O(n) in the worst case:

for(i = 1; i < a.length; i++){
    if (a[i] < a[i-1]){
        minIndex = i; break;
    }
}

then the corresponding index of ith element will be determined from the following relation:

(i + minInex - 1) % a.length

it is clear that my conversion (from circular to regular) algorithm may take O(n), so we need a better one.

According to ire_and_curses idea, here is the solution in Java:

public int circularArraySearch(int[] a, int low, int high, int x){
    //instead of using the division op. (which surprisingly fails on big numbers)
    //we will use the unsigned right shift to get the average
    int mid = (low + high) >>> 1;
    if(a[mid] == x){
        return mid;
    }
    //a variable to indicate which half is sorted
    //1 for left, 2 for right
    int sortedHalf = 0;
    if(a[low] <= a[mid]){
        //the left half is sorted
        sortedHalf = 1;
        if(x <= a[mid] && x >= a[low]){
            //the element is in this half
            return binarySearch(a, low, mid, x);
        }
    }
    if(a[mid] <= a[high]){
        //the right half is sorted
        sortedHalf = 2;
        if(x >= a[mid] && x<= a[high] ){
            return binarySearch(a, mid, high, x);
        }
    }
    // repeat the process on the unsorted half
    if(sortedHalf == 1){
        //left is sorted, repeat the process on the right one
        return circularArraySearch(a, mid, high, x);
    }else{
        //right is sorted, repeat the process on the left
        return circularArraySearch(a, low, mid, x);
    }
}

Hopefully this will work.

You can do this by taking advantage of the fact that the array is sorted, except for the special case of the pivot value and one of its neighbours.

• Find the middle value of the array a.
• If a[0] < a[mid], then all values in the first half of the array are sorted.
• If a[mid] < a[last], then all values in the second half of the array are sorted.
• Take the sorted half, and check whether your value lies within it (compare to the maximum idx in that half).
• If so, just binary search that half.
• If it doesn't, it must be in the unsorted half. Take that half and repeat this process, determining which half of that half is sorted, etc.

Not very elegant, but of the top off my head - just use binary search to find the pivot of the rotated array, and then perform binary search again, compensating for the offset of the pivot. Kind of silly to perform two full searches, but it does fulfill the condition, since O(log n) + O(log n) == O(log n). Keep it simple and stupid(tm)!

This is an example that works in Java. Since this is a sorted array you take advantage of this and run a Binary Search, however it needs to be slightly modified to cater for the position of the pivot.

The method looks like this:

private static int circularBinSearch ( int key, int low, int high )
{
    if (low > high)
    {
        return -1; // not found
    }

    int mid = (low + high) / 2;
    steps++;

    if (A[mid] == key)
    {
        return mid;
    }
    else if (key < A[mid])
    {
        return ((A[low] <= A[mid]) && (A[low] > key)) ?
               circularBinSearch(key, mid + 1, high) :
               circularBinSearch(key, low, mid - 1);
    }
    else // key > A[mid]
    {
        return ((A[mid] <= A[high]) && (key > A[high])) ?
               circularBinSearch(key, low, mid - 1) :
               circularBinSearch(key, mid + 1, high);
    }
}

Now to ease any worries, here's a nice little class that verifies the algorithm:

public class CircularSortedArray
{
    public static final int[] A = {23, 27, 29, 31, 37, 43, 49, 56, 64, 78, 
                                   91, 99, 1, 4, 11, 14, 15, 17, 19};
    static int steps;

    // ---- Private methods ------------------------------------------

    private static int circularBinSearch ( int key, int low, int high )
    {
        ... copy from above ...
    }

    private static void find ( int key )
    {
        steps = 0;
        int index = circularBinSearch(key, 0, A.length-1);
        System.out.printf("key %4d found at index %2d in %d steps\n",
                          key, index, steps);
    }

    // ---- Static main -----------------------------------------------

    public static void main ( String[] args )
    {
        System.out.println("A = " + Arrays.toString(A));
        find(44);   // should not be found
        find(230);
        find(-123);

        for (int key: A)  // should be found at pos 0..18
        {
            find(key);
        }
    }
}

That give you an output of:

A = [23, 27, 29, 31, 37, 43, 49, 56, 64, 78, 91, 99, 1, 4, 11, 14, 15, 17, 19]
key   44 found at index -1 in 4 steps
key  230 found at index -1 in 4 steps
key -123 found at index -1 in 5 steps
key   23 found at index  0 in 4 steps
key   27 found at index  1 in 3 steps
key   29 found at index  2 in 4 steps
key   31 found at index  3 in 5 steps
key   37 found at index  4 in 2 steps
key   43 found at index  5 in 4 steps
key   49 found at index  6 in 3 steps
key   56 found at index  7 in 4 steps
key   64 found at index  8 in 5 steps
key   78 found at index  9 in 1 steps
key   91 found at index 10 in 4 steps
key   99 found at index 11 in 3 steps
key    1 found at index 12 in 4 steps
key    4 found at index 13 in 5 steps
key   11 found at index 14 in 2 steps
key   14 found at index 15 in 4 steps
key   15 found at index 16 in 3 steps
key   17 found at index 17 in 4 steps
key   19 found at index 18 in 5 steps

You have three values, l,m,h for the values at the low, mid and high indexes of your search. If you think were you would continue searching for each possibility:

// normal binary search
l < t < m    - search(t,l,m)
m < t < h    - search(t,m,h)

// search over a boundary
l > m, t < m - search(t,l,m)
l > m, t > l - search(t,l,m)
m > h, t > m - search(t,m,h)  
m > h, t < h - search(t,m,h)  

It's a question of considering where the target value could be, and searching that half of the space. At most one half of the space will have the wrap-over in it, and it is easy to determine whether or not the target value is in that half or the other.

It's sort of a meta question - do you think of binary search it terms of how it is often presented - finding a value between two points, or more generally as a repeated division of an abstract search space.

You can use binary search to find the location of smallest element and reduce it to O(Log n).

You can find the location by (this is just a sketch of algorithm, it's inaccurate but you can get the idea from it):
1. i <- 1
2. j <- n
3. while i < j
3.1. k <- (j-i) / 2
3.2. if arr[k] < arr[i] then j <- k 3.3. else i <- k

After finding the location of the smallest element you can treat the array as two sorted arrays.

You just use a simple binary search as if it were a regular sorted array. The only trick is you need to rotate the array indexes:

(index + start-index) mod array-size

where the start-index is the offset of the first element in the circular array.

public static int _search(int[] buff, int query){
    int s = 0;
    int e = buff.length;
    int m = 0; 

    while(e-s>1){
        m = (s+e)/2;
        if(buff[offset(m)] == query){
            return offset(m);
        } else if(query < buff[offset(m)]){
            e = m;
        } else{
            s = m;
        }
    }

    if(buff[offset(end)]==query) return end;
    if(buff[offset(start)]==query) return start;
    return -1;
}

public static int offset(int j){
    return (dip+j) % N;
}

Check this coe,

    def findkey():
    key = 3

    A=[10,11,12,13,14,1,2,3]
    l=0
    h=len(A)-1
    while True:
        mid = l + (h-l)/2
        if A[mid] == key:
            return mid
        if A[l] == key:
            return l
        if A[h] == key:
            return h
        if A[l] < A[mid]:
            if key < A[mid] and key > A[l]:
                h = mid - 1
            else:
                l = mid + 1
        elif A[mid] < A[h]:
            if key > A[mid] and key < A[h]:
                l = mid + 1
            else:
                h = mid - 1

if __name__ == '__main__':
    print findkey()

Here is an idea, related to binary search. Just keep backing up your index for the right array-index bound, the left index bound is stored in the step size:

step = n
pos = n
while( step > 0 ):
    test_idx = pos - step   #back up your current position
    if arr[test_idx-1] < arr[pos-1]:
        pos = test_idx
    if (pos == 1) break
    step /= 2 #floor integer division
return arr[pos]

To avoid the (pos==1) thing, we could back up circularly (go into negative numbers) and take (pos-1) mod n.

I think you can find the offset using this code:

public static int findOffset(int [] arr){
        return findOffset(arr,0,arr.length-1);
    }
private static int findOffset(int[] arr, int start, int end) {

    if(arr[start]<arr[end]){
        return -1;
    }
    if(end-start==1){
        return end;
    }
    int mid = start + ((end-start)/2);
    if(arr[mid]<arr[start]){
        return findOffset(arr,start,mid);
    }else return findOffset(arr,mid,end);
}

Below is a implementation in C using binary search.

int rotated_sorted_array_search(int arr[], int low, int high, int target)
{
    while(low<=high)
    {
        int mid = (low+high)/2;

        if(target == arr[mid])
            return mid;

        if(arr[low] <= arr[mid])
        {
            if(arr[low]<=target && target < arr[mid])
            {
                high = mid-1;
            }
            else
                low = mid+1;
        }
        else
        {
            if(arr[mid]< target && target <=arr[high])
            {
                low = mid+1;
            }
            else
                high = mid-1;
        }
    }
    return -1;
}

Here's a solution in javascript. Tested it with a few different arrays and it seems to work. It basically uses the same method described by ire_and_curses:

function search(array, query, left, right) {
  if (left > right) {
    return -1;
  }

  var midpoint = Math.floor((left + right) / 2);
  var val = array[midpoint];
  if(val == query) {
    return midpoint;
  }

  // Look in left half if it is sorted and value is in that 
  // range, or if right side is sorted and it isn't in that range.
  if((array[left] < array[midpoint] && query >= array[left] && query <= array[midpoint])
    || (array[midpoint] < array[right] 
        && !(query >= array[midpoint] && query <= array[right]))) {
    return search(array, query, left, midpoint - 1);
  } else {
    return search(array, query, midpoint + 1, right);
  }
}

Simple binary search with a little change.

Index of rotating array= (i+pivot)%size

pivot is the index i+1 where a[i]>a[i+1].

#include <stdio.h>
#define size 5
#define k 3
#define value 13
int binary_search(int l,int h,int arr[]){

int mid=(l+h)/2;

if(arr[(mid+k)%size]==value)
    return (mid+k)%size;

if(arr[(mid+k)%size]<value)
    binary_search(mid+1,h,arr);
else
    binary_search(l,mid,arr);
}

int main() {
    int arr[]={5,9,13,1,3};
    printf("found at: %d\n", binary_search(0,4,arr));
    return 0;
}

A simple method in Ruby

def CircularArraySearch(a, x)
  low = 0
  high = (a.size) -1

  while low <= high
    mid = (low+high)/2
    if a[mid] == x
      return mid
    end
    if a[mid] <= a[high]
      if (x > a[mid]) && (x <= a[high])
        low = mid + 1
      elsif high = mid -1
      end
    else
      if (a[low] <= x) && (x < a[mid])
        high = mid -1
      else
        low = mid +1
      end
    end
  end
  return -1
end

a = [12, 14, 18, 2, 3, 6, 8, 9]
x = gets.to_i
p CircularArraySearch(a, x)

Though approved answer is optimal but we can also do with a similar and cleaner algorithm.

1. run a binary search to find pivot element (where the array is rotated). O(logn)
2. left half of the pivot will be sorted in decreasing order, run a backward binary search here for the key. O(logn)
3. right half of the pivot will be sorted in increasing order, run a forward binary search in this half for the key. O(logn)
4. return found key index from step 2 and 3.

Total Time Complexity: O(logn)

Thoughts welcome.

guirgis: It is lame to post an interview question, guess you didn't get the job :-(

Use a special cmp function and you need only one pass with regular binary search. Something like:

def rotatedcmp(x, y):
  if x and y < a[0]:
    return cmp(x, y)
  elif x and y >= a[0]:
    return cmp(x, y)
  elif x < a[0]:
    return x is greater
  else:
    return y is greater

If you can depend on int underflow subtract a[0] - MIN_INT from each element as it is accessed and use regular compare.