Algorithm 1 Minimum Priority Queue

1:procedure HeapMinimum( $A$ )

2:return $A[1]$

3:end procedure

4:

5:procedure HeapExtractMin( $A$ )

6:if $A.\text{heap-size} < 1$ then

7:error "heap underflow"

8:end if

9: $min = A[1]$

10: $A[1] = A[A.\text{heap-size}]$

11: $A.\text{heap-size} = A.\text{heap-size} - 1$

12:MinHeapify( $A, 1$ )

13:return $min$

14:end procedure

15:

16:procedure HeapDecreaseKey( $A, i, key$ )

17:if $key > A[i]$ then

18:error "new key is larger than the current key"

19:end if

20: $A[i] = key$

21:while $i > 1$ and $A[PARENT(i)] > A[i]$ do

22:exchange $A[i]$ with $A[PARENT(i)]$

23: $i = PARENT(i)$

24:end while

25:end procedure

26:

27:procedure MinHeapInsert( $A, key$ )

28: $A.\text{heap-size} = A.\text{heap-size} + 1$

29: $A[A.\text{heap-size}] = +\infty$

30:HeapDecreaseKey( $A, A.\text{heap-size}, key$ )

31:end procedure