1. Binary Heaps and Heapsort

2. MAX-HEAPIFY MAX-HEAPIFY (A, i) l ← LEFT (i) r ← RIGHT(i)
if l ≤ heap-size[A] and A[l] > A[i]
then largest ← l
else largest ← i
if r ≤ heap-size[A] and A[r]>A[largest]
then largest ← r
if largest ≠ i
then exchange A[i] with A[largest]
MAX-HEAPIFY (A, largest)

3. BUILD-HEAP BUILD-HEAP (A) heap-size[A] ← length[A]
for i ← length[A]/2 downto 1
do MAX-HEAPIFY(A, i)

4. BUILD-HEAP Thus, the running time is 2n = O(n) The number of
nodes at height h
Multiplied by their height

5. HEAPSORT HEAPSORT (A) BUILD-HEAP(A) for i ← length[A] downto 2
do exchange A[1] with A[i]
heap-size[A] ← heap-size[A] - 1
MAX-HEAPIFY(A, 1)

6. Priority Queues
HEAP-MAXIMUM (A)
1 return A[1]

7. HEAP-EXTRACT-MAX HEAP-EXTRACT-MAX (A) 1 if heap-size[A] < 1
2 then error “heap underflow”
3 max  A[1]
4 A[1]  A[heap-size[A]]
5 heap-size[A]  heap-size[A] – 1
6 MAX-HEAPIFY (A, 1)
7 return max

8. HEAP-INCREASE-KEY HEAP-INCREASE-KEY(A, i, key) 1 if key < A[i]
then error “new key smaller than current”
3 A[i]  key
4 while i > 1 and A[PARENT(i) < A[i]
5 do exchange A[i]  A[PARENT(i)]
6 i  PARENT(i)

9. MAX-HEAP-INSERT MAX-HEAP-INSERT (A, key)
1 heap-size[A]  heap-size[A] + 1
2 A[heap-size[A]]  -
3 HEAP-INCREASE-KEY(A, heap-size[A], key)