Paine Algorithm, Priority Queues, Heaps, and Data Structures Analysis, Lecture notes of Computer Science

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Drunken Paine, Dictionary, Jumble,

Priority Queues, Heaps

Paine Algorithm

 Input: A list of people, people

 Output: A Tom if there is one, NULL if not

findTom(people):

if length(people) is 1: return people[0] //base case

if people[0] knows people[1]: //eliminate one

erson

not_tom = remove people[0] from people

else:

not_tom = remove people[1] from people

candidate = findTom(people) //recursive call (solve smaller

roblem)

if candidate is NULL: return NULL //verify candidate

if not_tom knows candidate & candidate doesn’t know not_tom:

return candidate

else:

return NULL

Inductive proof of correctness

 Base case: Our algorithm is correct for a group of size 1

 True. We designed our algorithm’s base case according to the definition of a Tom in a group of size 1

 Assume our algorithm is correct for a group of size k-

 Done.

 Show that if our algorithm is correct for a group of size k-1, then it

is correct for a group of size k.

 We have an input of size k and we remove one non Tom from the group, then make a recursive call to find a Tom in a group of size k-  This recursive call returns the correct answer (the Tom of the group of size k-1 or NULL)by our previous assumption

continued…

 If the recursive call returns NULL:

 There is no Tom in the group of size k because we eliminated one non Tom and there’s no Tom in the other k-1 people  Our algorithm returns NULL in this case, so it’s correct.

 If the recursive call returns a Tom (called candidate):

 candidate is the only possible Tom of the group of size k  candidate is not the Tom of the group of size k if the non Tom we eliminated doesn’t know candidate or if candidate knows the non Tom.  In this case, there is no Tom in the group of size k, so we return NULL  Otherwise, candidateis a Tom in the group of size k  In this case, we return candidate

 In every possible case, our algorithm returns the correct Tom if

there is one for the group of size k.

 …given that our algorithm is correct for a group of size k-

The SET Abstract Data Type

 S=createSet (n): creates a new empty set structure,

initially empty but capable of holding up to n elements.

 isEmpty (S): checks whether the set S is empty.

 size (S): returns the number of elements in S.

 elementOf (x,S): checks whether the value x is in the

set S.

 enumerate (S): yields the elements of S in some

arbitrary order.

 add (S,x): adds the element x to S, if it is not there

already.

 delete (S,x): removes the element x from S, if it is

there.

Implementing sets

 Can use hashtable:

 “create”, “isEmpty”, and “size” are trivial

 “enumerate”: take all elements in all buckets

 “add” is just “insert”; “delete” is “delete”

 isElement is just “find”

Implementing a dictionary

 Create(n)

 Build an array of prime size a little more than n,

each entry an empty list

 Pick k numbers, mod n, to handle keys of length

k

 Insert(key, value)

 Let u = (a 1 key 1 + … + a (^) k keyk ) mod n

 Insert (key, value) into array[u]

 Find(key)

 Let u = (a 1 key 1 + … + a (^) k keyk ) mod n

 Search for (key, *) in array[u]

 If you find (key, val), return val

 Else return None

 (Modify as appropriate to return list of vals)

Example Application: JUMBLE!

JUMBLE

 Input: list of all 5-letter words in English

 Each word represented as an array of five

characters

 Output: all words for which no other

permutation is a word

Priority Queue Definition

 A priority queue stores a collection of items

 An item is a pair

(k, element)

 Main methods of the Priority Queue ADT

 insert(k, o)

inserts an item with key k and element o

 o = removeMin()

removes the item with smallest key and returns its element

Implementation

 insert running time? removeMin running time?

insert 7 removeMin^ removeMin