Data Structures and Algorithms 4 Searching

## 4.1 Sequential Searches

Let's examine how long it will take to find an item matching a key in the collections we have discussed so far. We're interested in:

1. the average time
2. the worst-case time and
3. the best possible time.
However, we will generally be most concerned with the worst-case time as calculations based on worst-case times can lead to guaranteed performance predictions. Conveniently, the worst-case times are generally easier to calculate than average times.

If there are n items in our collection - whether it is stored as an array or as a linked list - then it is obvious that in the worst case, when there is no item in the collection with the desired key, then n comparisons of the key with keys of the items in the collection will have to be made.

## 4.2 Binary Search

However, if we place our items in an array and sort them in either ascending or descending order on the key first, then we can obtain much better performance with an algorithm called binary search.

In binary search, we first compare the key with the item in the middle position of the array. If there's a match, we can return immediately. If the key is less than the middle key, then the item sought must lie in the lower half of the array; if it's greater then the item sought must lie in the upper half of the array. So we repeat the procedure on the lower (or upper) half of the array.

Our FindInCollection function can now be implemented:

```
static void *bin_search( collection c, int low, int high, void *key ) {
int mid;
/* Termination check */
if (low > high) return NULL;
mid = (high+low)/2;
switch (memcmp(ItemKey(c->items[mid]),key,c->size)) {
/* Match, return item found */
case 0: return c->items[mid];
/* key is less than mid, search lower half */
case -1: return bin_search( c, low, mid-1, key);
/* key is greater than mid, search upper half */
case 1: return bin_search( c, mid+1, high, key );
default : return NULL;
}
}

void *FindInCollection( collection c, void *key ) {
/* Find an item in a collection
Pre-condition:
c is a collection created by ConsCollection
c is sorted in ascending order of the key
key != NULL
Post-condition: returns an item identified by key if
one exists, otherwise returns NULL
*/
int low, high;
low = 0; high = c->item_cnt-1;
return bin_search( c, low, high, key );
}
```
Points to note:
1. bin_search is recursive: it determines whether the search key lies in the lower or upper half of the array, then calls itself on the appropriate half.
2. There is a termination condition (two of them in fact!)
1. If low > high then the partition to be searched has no elements in it and
2. If there is a match with the element in the middle of the current partition, then we can return immediately.
3. AddToCollection will need to be modified to ensure that each item added is placed in its correct place in the array. The procedure is simple:
1. Search the array until the correct spot to insert the new item is found,
2. Move all the following items up one position and
3. Insert the new item into the empty position thus created.
4. bin_search is declared static. It is a local function and is not used outside this class: if it were not declared static, it would be exported and be available to all parts of the program. The static declaration also allows other classes to use the same name internally.
 static reduces the visibility of a function an should be used wherever possible to control access to functions!

#### Analysis

 Each step of the algorithm divides the block of items being searched in half. We can divide a set of n items in half at most log2 n times. Thus the running time of a binary search is proportional to log n and we say this is a O(log n) algorithm.
 Binary search requires a more complex program than our original search and thus for small n it may run slower than the simple linear search. However, for large n, Thus at large n, log n is much smaller than n, consequently an O(log n) algorithm is much faster than an O(n) one. Plot of n and log n vs n .

We will examine this behaviour more formally in a later section. First, let's see what we can do about the insertion (AddToCollection) operation.

In the worst case, insertion may require n operations to insert into a sorted list.

1. We can find the place in the list where the new item belongs using binary search in O(log n) operations.
2. However, we have to shuffle all the following items up one place to make way for the new one. In the worst case, the new item is the first in the list, requiring n move operations for the shuffle!

A similar analysis will show that deletion is also an O(n) operation.

If our collection is static, ie it doesn't change very often - if at all - then we may not be concerned with the time required to change its contents: we may be prepared for the initial build of the collection and the occasional insertion and deletion to take some time. In return, we will be able to use a simple data structure (an array) which has little memory overhead.

However, if our collection is large and dynamic, ie items are being added and deleted continually, then we can obtain considerably better performance using a data structure called a tree.

### Key terms

Big Oh
A notation formally describing the set of all functions which are bounded above by a nominated function.
Binary Search
A technique for searching an ordered list in which we first check the middle item and - based on that comparison - "discard" half the data. The same procedure is then applied to the remaining half until a match is found or there are no more items left.