Finding multiple entries with binary search

Question

I use standard binary search to quickly return a single object in a sorted list (with respect to a sortable property).

Now I need to modify the search so that ALL matching list entries are returned. How should I best do this?

Answer 1

Well, as the list is sorted, all the entries you are interested in are contiguous. This means you need to find the first item equal to the found item, looking backwards from the index which was produced by the binary search. And the same about last item.

You can simply go backwards from the found index, but this way the solution may be as slow as O(n) if there are a lot of items equal to the found one. So you should better use exponential search: double your jumps as you find more equal items. This way your whole search is still O(log n).

Answer 2

First let's recall the naive binary search code snippet:

int bin_search(int arr[], int key, int low, int high) {     if (low > high)         return -1;      int mid = low + ((high - low) >> 1);      if (arr[mid] == key) return mid;     if (arr[mid] > key)         return bin_search(arr, key, low, mid - 1);     else         return bin_search(arr, key, mid + 1, high); } 

Quoted from Prof.Skiena: Suppose we delete the equality test if (s[middle] == key) return(middle); from the implementation above and return the index low instead of −1 on each unsuccessful search. All searches will now be unsuccessful, since there is no equality test. The search will proceed to the right half whenever the key is compared to an identical array element, eventually terminating at the right boundary. Repeating the search after reversing the direction of the binary comparison will lead us to the left boundary. Each search takes O(lgn) time, so we can count the occurrences in logarithmic time regardless of the size of the block.

So, we need two rounds of binary_search to find the lower_bound (find the first number no less than the KEY) and the upper_bound (find the first number bigger than the KEY).

int lower_bound(int arr[], int key, int low, int high) {     if (low > high)         //return -1;         return low;      int mid = low + ((high - low) >> 1);     //if (arr[mid] == key) return mid;      //Attention here, we go left for lower_bound when meeting equal values     if (arr[mid] >= key)          return lower_bound(arr, key, low, mid - 1);     else         return lower_bound(arr, key, mid + 1, high); }  int upper_bound(int arr[], int key, int low, int high) {     if (low > high)         //return -1;         return low;      int mid = low + ((high - low) >> 1);     //if (arr[mid] == key) return mid;      //Attention here, we go right for upper_bound when meeting equal values     if (arr[mid] > key)          return upper_bound(arr, key, low, mid - 1);     else         return upper_bound(arr, key, mid + 1, high); } 

Hope it's helpful :)