Computer notes - Binary Search

Class No.32 Data Structures http://ecomputernotes.com

Tables and Dictionaries http://ecomputernotes.com

Tables: rows & columns of information A table has several fields (types of information) A telephone book may have fields name, address, phone number A user account table may have fields user id, password, home folder http://ecomputernotes.com Name Address Phone Sohail Aslam 50 Zahoor Elahi Rd, Gulberg-4, Lahore 576-3205 Imran Ahmad 30-T Phase-IV, LCCHS, Lahore 572-4409 Salman Akhtar 131-D Model Town, Lahore 784-3753

Tables: rows & columns of information To find an entry in the table, you only need know the contents of one of the fields (not all of them). This field is the key In a telephone book, the key is usually “name” In a user account table, the key is usually “user id” http://ecomputernotes.com

Tables: rows & columns of information Ideally, a key uniquely identifies an entry If the key is “name” and no two entries in the telephone book have the same name, the key uniquely identifies the entries http://ecomputernotes.com Name Address Phone Sohail Aslam 50 Zahoor Elahi Rd, Gulberg-4, Lahore 576-3205 Imran Ahmad 30-T Phase-IV, LCCHS, Lahore 572-4409 Salman Akhtar 131-D Model Town, Lahore 784-3753

The Table ADT: operations insert : given a key and an entry, inserts the entry into the table find : given a key, finds the entry associated with the key remove : given a key, finds the entry associated with the key, and removes it http://ecomputernotes.com

How should we implement a table? How often are entries inserted and removed? How many of the possible key values are likely to be used? What is the likely pattern of searching for keys? E.g. Will most of the accesses be to just one or two key values? Is the table small enough to fit into memory? How long will the table exist? Our choice of representation for the Table ADT depends on the answers to the following http://ecomputernotes.com

TableNode: a key and its entry For searching purposes, it is best to store the key and the entry separately (even though the key’s value may be inside the entry) “ Saleem” “ Saleem”, “124 Hawkers Lane”, “9675846” “ Yunus” “ Yunus”, “1 Apple Crescent”, “0044 1970 622455” TableNode http://ecomputernotes.com key entry

Implementation 1: unsorted sequential array An array in which TableNodes are stored consecutively in any order insert : add to back of array; (1) find : search through the keys one at a time, potentially all of the keys; ( n ) remove : find + replace removed node with last node; ( n ) 0 … key entry 1 2 3 and so on http://ecomputernotes.com

Implementation 2:sorted sequential array An array in which TableNodes are stored consecutively, sorted by key insert : add in sorted order; ( n ) find : binary search; (log n ) remove : find, remove node and shuffle down; ( n ) 0 … key entry 1 2 3 We can use binary search because the array elements are sorted and so on http://ecomputernotes.com

Searching an Array: Binary Search Binary search is like looking up a phone number or a word in the dictionary Start in middle of book If name you're looking for comes before names on page, look in first half Otherwise, look in second half http://ecomputernotes.com

Binary Search If ( value == middle element ) value is found else if ( value < middle element ) search left-half of list with the same method else search right-half of list with the same method http://ecomputernotes.com

Case 1: val == a[mid] val = 10 low = 0, high = 8 5 7 9 10 13 17 19 1 27 a: low high Binary Search http://ecomputernotes.com 1 2 3 4 5 6 7 0 8 mid mid = (0 + 8) / 2 = 4 10

Case 2: val > a[mid] val = 19 low = 0, high = 8 mid = (0 + 8) / 2 = 4 Binary Search -- Example 2 5 7 9 10 1 a: mid low high http://ecomputernotes.com 13 17 19 27 1 2 3 4 5 6 7 0 8 new low new low = mid+1 = 5 13 17 19 27

Case 3: val < a[mid] val = 7 low = 0, high = 8 mid = (0 + 8) / 2 = 4 Binary Search -- Example 3 10 13 17 19 27 a: mid low high http://ecomputernotes.com 5 7 9 1 1 2 3 4 5 6 7 0 8 new high new high = mid-1 = 3 5 7 9 1

val = 7 Binary Search -- Example 3 (cont) 5 7 9 10 13 17 19 1 27 1 2 3 4 5 6 7 0 8 a: 5 7 9 10 13 17 19 1 27 1 2 3 4 5 6 7 0 8 a: 5 7 9 10 13 17 19 1 27 1 2 3 4 5 6 7 0 8 a:

Binary Search – C++ Code int isPresent(int *arr, int val, int N) { int low = 0; int high = N - 1; int mid; while ( low <= high ){ mid = ( low + high )/2; if (arr[mid]== val) return 1; // found! else if (arr[mid] < val) low = mid + 1; else high = mid - 1; } return 0; // not found } http://ecomputernotes.com

Binary Search: binary tree The search divides a list into two small sub-lists till a sub-list is no more divisible. First half First half An entire sorted list First half Second half Second half http://ecomputernotes.com

Binary Search Efficiency After 1 bisection N/2 items After 2 bisections N/4 = N/2 2 items . . . After i bisections N/2 i = 1 item i = log 2 N http://ecomputernotes.com

Implementation 3: linked list TableNodes are again stored consecutively (unsorted or sorted) insert : add to front; (1 or n for a sorted list ) find : search through potentially all the keys, one at a time; ( n for unsorted or for a sorted list remove : find, remove using pointer alterations; ( n ) key entry and so on http://ecomputernotes.com

Implementation 4: Skip List Overcome basic limitations of previous lists Search and update require linear time Fast Searching of Sorted Chain Provide alternative to BST (binary search trees) and related tree structures. Balancing can be expensive. Relatively recent data structure: Bill Pugh proposed it in 1990. http://ecomputernotes.com

Skip List Representation Can do better than n comparisons to find element in chain of length n http://ecomputernotes.com 20 30 40 50 60 head tail

Skip List Representation Example: n/2 + 1 if we keep pointer to middle element http://ecomputernotes.com 20 30 40 50 60 head tail

Higher Level Chains For general n, level 0 chain includes all elements level 1 every other element, level 2 chain every fourth, etc. level i , every 2 i th element http://ecomputernotes.com 40 50 60 head tail 20 30 26 57 level 1&2 chains

Higher Level Chains Skip list contains a hierarchy of chains In general level i contains a subset of elements in level i-1 40 50 60 head tail 20 30 26 57 level 1&2 chains

Skip List: formally A skip list for a set S of distinct (key, element) items is a series of lists S 0 , S 1 , … , S h such that Each list S i contains the special keys  and  List S 0 contains the keys of S in nondecreasing order Each list is a subsequence of the previous one, i.e., S 0  S 1  …  S h List S h contains only the two special keys

Lecture No.38 Data Structure Dr. Sohail Aslam

Skip List: formally 56 64 78  31 34 44  12 23 26 S 0 64  31 34  23 S 1  31  S 2   S 3

Skip List: Search We search for a key x as follows: We start at the first position of the top list At the current position p , we compare x with y  key ( after ( p )) x  y : we return element ( after ( p )) x  y : we “scan forward” x  y : we “drop down” If we try to drop down past the bottom list, we return NO_SUCH_KEY

Skip List: Search Example: search for 78 S 0 S 1 S 2 S 3  31  64  31 34  23 56 64 78  31 34 44  12 23 26  

To insert an item ( x , o ) into a skip list, we use a randomized algorithm: We repeatedly toss a coin until we get tails, and we denote with i the number of times the coin came up heads If i  h , we add to the skip list new lists S h  1 , … , S i  1 , each containing only the two special keys Skip List: Insertion

To insert an item ( x , o ) into a skip list, we use a randomized algorithm: (cont) We search for x in the skip list and find the positions p 0 , p 1 , …, p i of the items with largest key less than x in each list S 0 , S 1 , … , S i For j  0, …, i , we insert item ( x , o ) into list S j after position p j Skip List: Insertion

Example: insert key 15 , with i  2 Skip List: Insertion   10 36   23 23   S 0 S 1 S 2 p 0 p 1 p 2   S 0 S 1 S 2 S 3   10 36 23 15   15   23 15

Randomized Algorithms A randomized algorithm performs coin tosses (i.e., uses random bits) to control its execution It contains statements of the type b  random () if b <= 0.5 // head do A … else // tail do B … Its running time depends on the outcomes of the coin tosses, i.e, head or tail

Skip List: Deletion To remove an item with key x from a skip list, we proceed as follows: We search for x in the skip list and find the positions p 0 , p 1 , …, p i of the items with key x , where position p j is in list S j We remove positions p 0 , p 1 , …, p i from the lists S 0 , S 1 , … , S i We remove all but one list containing only the two special keys

Skip List: Deletion Example: remove key 34   S 0 S 1 S 2 S 3   45 12 23 34   34   23 34 p 0 p 1 p 2   45 12   23 23   S 0 S 1 S 2

Skip List: Implementation   S 0 S 1 S 2 S 3   45 12 23 34   34   23 34

Implementation: TowerNode TowerNode will have array of next pointers. Actual number of next pointers will be decided by the random procedure. Define MAXLEVEL as an upper limit on number of levels in a node. 40 50 60 head tail 20 30 26 57 Tower Node

Implementation: QuadNode A quad-node stores: item link to the node before link to the node after link to the node below link to the node above This will require copying the key (jitem) at different levels x quad-node

Skip Lists with Quad Nodes 56 64 78  31 34 44  12 23 26    31  64  31 34  23 S 0 S 1 S 2 S 3

Performance of Skip Lists In a skip list with n items The expected space used is proportional to n . The expected search, insertion and deletion time is proportional to log n . Skip lists are fast and simple to implement in practice

Implementation 5: AVL tree An AVL tree, ordered by key insert : a standard insert; (log n ) find : a standard find (without removing, of course); (log n ) remove : a standard remove; (log n ) and so on key entry key entry key entry key entry

Anything better? So far we have find, remove and insert where time varies between constant log n . It would be nice to have all three as constant time operations!

An array in which TableNodes are not stored consecutively Their place of storage is calculated using the key and a hash function Keys and entries are scattered throughout the array. Implementation 6: Hashing key entry Key hash function array index 4 10 123

insert : calculate place of storage, insert TableNode; (1) find : calculate place of storage, retrieve entry; (1) remove : calculate place of storage, set it to null; (1) Hashing key entry 4 10 123 All are constant time (1) !

Hashing We use an array of some fixed size T to hold the data. T is typically prime. Each key is mapped into some number in the range 0 to T-1 using a hash function , which ideally should be efficient to compute.

Example: fruits Suppose our hash function gave us the following values: hashCode("apple") = 5 hashCode("watermelon") = 3 hashCode("grapes") = 8 hashCode("cantaloupe") = 7 hashCode("kiwi") = 0 hashCode("strawberry") = 9 hashCode("mango") = 6 hashCode("banana") = 2 kiwi banana watermelon apple mango cantaloupe grapes strawberry 0 1 2 3 4 5 6 7 8 9

Example Store data in a table array: table[5] = "apple" table[3] = "watermelon" table[8] = "grapes" table[7] = "cantaloupe" table[0] = "kiwi" table[9] = "strawberry" table[6] = "mango" table[2] = "banana" kiwi banana watermelon apple mango cantaloupe grapes strawberry 0 1 2 3 4 5 6 7 8 9

Example Associative array: table["apple"] table["watermelon"] table["grapes"] table["cantaloupe"] table["kiwi"] table["strawberry"] table["mango"] table["banana"] kiwi banana watermelon apple mango cantaloupe grapes strawberry 0 1 2 3 4 5 6 7 8 9

Example Hash Functions If the keys are strings the hash function is some function of the characters in the strings. One possibility is to simply add the ASCII values of the characters: TableSize ABC h Example TableSize i str str h length i )% 67 66 65 ( ) ( : % ] [ ) ( 1 0               

Finding the hash function int hashCode( char* s ) { int i, sum; sum = 0; for(i=0; i < strlen(s); i++ ) sum = sum + s[i]; // ascii value return sum % TABLESIZE; }

Example Hash Functions Another possibility is to convert the string into some number in some arbitrary base b ( b also might be a prime number): T b b b ABC h Example T b i str str h length i i )% 67 66 65 ( ) ( : % ] [ ) ( 2 1 0 1 0                

Example Hash Functions If the keys are integers then key%T is generally a good hash function, unless the data has some undesirable features. For example, if T = 10 and all keys end in zeros, then key%T = 0 for all keys. In general, to avoid situations like this, T should be a prime number.

Collision Suppose our hash function gave us the following values: hash("apple") = 5 hash("watermelon") = 3 hash("grapes") = 8 hash("cantaloupe") = 7 hash("kiwi") = 0 hash("strawberry") = 9 hash("mango") = 6 hash("banana") = 2 • Now what? hash("honeydew") = 6 kiwi banana watermelon apple mango cantaloupe grapes strawberry 0 1 2 3 4 5 6 7 8 9

Collision When two values hash to the same array location, this is called a collision Collisions are normally treated as “first come, first served”—the first value that hashes to the location gets it We have to find something to do with the second and subsequent values that hash to this same location.

Solution for Handling collisions Solution #1: Search from there for an empty location Can stop searching when we find the value or an empty location. Search must be wrap-around at the end.

Solution for Handling collisions Solution #2: Use a second hash function ...and a third, and a fourth, and a fifth, ...

Solution for Handling collisions Solution #3: Use the array location as the header of a linked list of values that hash to this location

Solution 1: Open Addressing This approach of handling collisions is called open addressing ; it is also known as closed hashing . More formally, cells at h 0 (x) , h 1 (x) , h 2 (x) , … are tried in succession where h i (x) = (hash(x) + f(i)) mod TableSize , with f(0) = 0 . The function, f , is the collision resolution strategy.

Linear Probing We use f(i) = i , i.e., f is a linear function of i . Thus location(x) = (hash(x) + i) mod TableSize The collision resolution strategy is called linear probing because it scans the array sequentially (with wrap around) in search of an empty cell.

Linear Probing: insert Suppose we want to add seagull to this hash table Also suppose: hashCode(“seagull”) = 143 table[143] is not empty table[143] != seagull table[144] is not empty table[144] != seagull table[145] is empty Therefore, put seagull at location 145 seagull robin sparrow hawk bluejay owl . . . 141 142 143 144 145 146 147 148 . . .

Linear Probing: insert Suppose you want to add hawk to this hash table Also suppose hashCode(“hawk”) = 143 table[143] is not empty table[143] != hawk table[144] is not empty table[144] == hawk hawk is already in the table, so do nothing. robin sparrow hawk seagull bluejay owl . . . 141 142 143 144 145 146 147 148 . . .

Linear Probing: insert Suppose: You want to add cardinal to this hash table hashCode(“cardinal”) = 147 The last location is 148 147 and 148 are occupied Solution: Treat the table as circular; after 148 comes 0 Hence, cardinal goes in location 0 (or 1, or 2, or ...) robin sparrow hawk seagull bluejay owl . . . 141 142 143 144 145 146 147 148

Linear Probing: find Suppose we want to find hawk in this hash table We proceed as follows: hashCode(“hawk”) = 143 table[143] is not empty table[143] != hawk table[144] is not empty table[144] == hawk ( found! ) We use the same procedure for looking things up in the table as we do for inserting them robin sparrow hawk seagull bluejay owl . . . 141 142 143 144 145 146 147 148 . . .

Linear Probing and Deletion If an item is placed in array[hash(key)+4] , then the item just before it is deleted How will probe determine that the “hole” does not indicate the item is not in the array? Have three states for each location Occupied Empty (never used) Deleted (previously used)

Clustering One problem with linear probing technique is the tendency to form “clusters”. A cluster is a group of items not containing any open slots The bigger a cluster gets, the more likely it is that new values will hash into the cluster, and make it ever bigger. Clusters cause efficiency to degrade.

Quadratic Probing Quadratic probing uses different formula: Use F(i) = i 2 to resolve collisions If hash function resolves to H and a search in cell H is inconclusive, try H + 1 2 , H + 2 2 , H + 3 2 , … Probe array[hash(key)+1 2 ], then array[hash(key)+2 2 ], then array[hash(key)+3 2 ], and so on Virtually eliminates primary clusters

Collision resolution: chaining Each table position is a linked list Add the keys and entries anywhere in the list (front easiest) 4 10 123 No need to change position! key entry key entry key entry key entry key entry

Collision resolution: chaining Advantages over open addressing: Simpler insertion and removal Array size is not a limitation Disadvantage Memory overhead is large if entries are small. 4 10 123 key entry key entry key entry key entry key entry

Applications of Hashing Compilers use hash tables to keep track of declared variables (symbol table). A hash table can be used for on-line spelling checkers — if misspelling detection (rather than correction) is important, an entire dictionary can be hashed and words checked in constant time.

Applications of Hashing Game playing programs use hash tables to store seen positions, thereby saving computation time if the position is encountered again. Hash functions can be used to quickly check for inequality — if two elements hash to different values they must be different.

When is hashing suitable? Hash tables are very good if there is a need for many searches in a reasonably stable table. Hash tables are not so good if there are many insertions and deletions, or if table traversals are needed — in this case, AVL trees are better. Also, hashing is very slow for any operations which require the entries to be sorted e.g. Find the minimum key

Computer notes - Binary Search

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