Max-Heapify

Max_Heapify(A,i)

{

l=left(i)

r=right(i)

if(l<=A.heap-size and A[l]>A[i])

largest=l

else largest=i

if(r<=A.heap-size and A[r]>A[largest])

largest=r

if(largest!=i)

{

exchange A[i] with A[largest]

Max_Heapify(A.largest)

}

}

Build Max Heap(A)

void build_maxheap(int *a,int sz){ int i; for(i=sz/2;i>=0;i--) { Max_Heapify(a,i,sz); }}

Heap_Sort(A)

void heapsort(int *a, int sz){ int temp; build_maxheap(a,sz); while(sz>0) { exchange a[sz-1] with a[0] sz--; Max_Heapify(A,0,sz) }}

2)Sort a nearly sorted (or K sorted) arrayGiven an array of n elements, where each element is at most k away from its target position, devise an algorithm that sorts in O(n log k) time. For example, let us consider k is 2, an element at index 7 in the sorted array, can be at indexes 5, 6, 7, 8, 9 in the given array.

1) Given a stream of numbers find k largest elements

Priority Queue

A max priority queue supports the following operations:1) Insert(S,x):- inserts the element x into the priority queue2) Maximum(S):-returns the element with the largest key3) Extract-Max(S):-removes and returns the element with the largest

key4) Increase Key(S,x,k):-increases the value of element x’s key to the

new value k

Heap-Maximumreturn A[0]

Heap-Extract-Max(A){If(A.heapsize<1) error “heap underflow”Max=A[0]A[0]=A[A.heapsize-1]A.heapsize--Max_Heapify(A,0)Return Max}

Heap-Increase-key(A,i,key){A[i]=keywhile(i>0 and A[Parent(i)]<A[i])

exchange A[i]with A[Parent(i)]i=Parent(i)

}

Max-Heap-Insert(A,key){A.heapsize++A[A.heapsize]=-infinityHeap-Increase-key(A,A.heapsize,key)}