Question

Hello this needs to be a small write up with drawings. important bit is the second part describe how the credits in the proof of Lemma 8.3 are used

Illustrate what happens when the sequence 1, 5, 2, 4, 3 is added to an empty ScapegoatTree, and show where the credits described in the proof of Lemma 8.3 go, and how they are used during this sequence of additions

Thanks

Answer 1

Scapegoat Tree:

Scapegoat Tree is a self balancing binary search tree.

Insertion:

Insertion is implemented with the same basic ideas as an unbalanced binary search tree.

An entire subtree rooted at a scapegoat undergoes a balancing operation.

size(left) ≤ α*size(node)
size(right) ≤ α*size(node)

size[x+1] = size[x] + size(sibling) + 1

Where x = this node, x + 1 = parent and size(sibling) is the only function call actually required.

Algorithm Average Worst Case

space O(n) O(n)

Insert O(log n) amortized O(log n)

Delete O(log n) amortized O(log n)

5

/ \

3 4

/ \

1 2

5 = 5 is the root node.

5 have two child 3 & 4.

3 also contain the two child node 1 & 2.

1,2 & 4 is the leaf node.

In Scapegoat tree a counter 'i' always maintains an upper bound on the number of nodes and scapegoat tree is a binary search tree.

left part = left part<root node

right part = right part>root node

consider our sequence 1,5,2,4,3

first insert node "1" then counter i also equals to n or greater .i.e i=1 beccause n = first node i.e 1.n is the no of nodes in the tree.

1) i/2 <=n=""><=>

2) log3/2 i<= log 3/2 2n =""><>

initially i =0 because numbers of nodes are 0.

after inserting first node '1' i.....