Binary Search

Binary Search

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:

a. 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.

b. There is a termination condition (two of them in fact!) i. If low > high then the partition to be searched has no elements in it and

If there is a match with the element in the middle of the current partition, then we can return immediately.

c. 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:

i. Search the array until the correct spot to insert the new item is found,

ii. Move all the following items up one position and

iii. Insert the new item into the empty position thus created.

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.

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.

1. Characteristics

The worst case performance scenario for a linear search is that it needs to loop through the entire collection; either because the item is the last one, or because the item isn't found. In other words, if you have N items in your collection, the worst case scenario to find an item is N iterations. This is known as O(N) using the Big O Notation. The speed of search grows linearly with the number of items within your collection. Linear searches don't require the collection to be sorted. In some cases, you'll know ahead of time that some items will be disproportionally searched for. In such situations, frequently requested items can be kept at the start of the collection. This can result in exceptional performance, regardless of size, for these frequently requested items.

Linear Search 1 Problem:

Given a list of N values, determine whether a given value X occurs in the list. 1 2 3 4 5 6 7 8 17 31 9 73 55 12 19 7

For example, consider the problem of determining whether the value 55 occurs in: There is an obvious, correct algorithm:

start at one end of the list, if the current element doesn't equal the search target,

move to the next element, stopping when a match is found or the opposite end of the list is reached.

Basic principle: divide the list into the current element and everything before (or after) it;

if current isn't a match, search the other case algorithm Linear Search takes number X, list number L, number Sz

# Determines whether the value X occurs within the list L.

# Pre: L must be initialized to hold exactly Sz values

## Walk from the upper end of the list toward the lower end,

# looking for a match:

while

Sz > 0 AND L[Sz] != X Sz := Sz - 1

endwhile if Sz > 0

# See if we walked off the front of the list display true

# if so, no match else display false

if not, got a match

Halt