DSA1 | Week 10

Heaps

Goal

Maintain a set S with support for  the following operations.

insert(S,x): O(1)

Goal

findmin(S): O(n)

deletemin(S): O(n)

Maintain a set S with support for  the following operations.

insert(S,x): O(n)

Goal

findmin(S): O(1)

deletemin(S): O(1)

Maintain a set S with support for  the following operations.

insert(S,x): O(log n)

Goal

findmin(S): O(1)

deletemin(S): O(log n)

Maintain a set S with support for  the following operations.

Intuition

For findmin to happen in O(1) time,

the minimum element needs to be in

an easy-to-access location.

For deletemin to happen in O(log n) time,

the second minimum element needs to be
reasonably easy to find.

how hard is it to find the secondmin?

very easy

tedious

how hard is it to find the secondmin?

too organized

too messy

🔑 Idea

Each element has two pointers

to two other elements.

These pointers are hints

within the data structure to 

help you navigate the data.

In particular,

the global min points to two locations.

How can we ensure

that both are candidates for secondmin?

Invariant

If an element (a) points to another one (b),

directly or indirectly,

then a is smaller than b.

Invariant

If an element (a) points to another one (b),

directly or indirectly,

then a is smaller than b.

Invariant

If an element (a) points to another one (b),

directly or indirectly,

then a is smaller than b.

Invariant

If an element (a) points to another one (b),

directly or indirectly,

then a is smaller than b.

Invariant

If an element (a) points to another one (b),

directly or indirectly,

then a is smaller than b.

Invariant

If an element (a) points to another one (b),

directly or indirectly,

then a is smaller than b.

&

all elements point to two other elements!

5

15

32

42

64

100

999

5

15

32

42

64

100

999

15

32

42

64

100

999

15

32

42

64

100

999

15

32

42

64

100

999

15

32

42

64

100

999

15

32

42

64

100

999