Problem Statement SkewTree (Hard worth 950 points)

In a binary tree, every node has an optional left, and optional right child node. A BST (binary search tree) is a binary tree that satisfies the following properties:

1) Every node has a unique value

2) The value at a given node must be greater than the value at every node in its left subtree

3) The value at a given node must be less than the value at every node in its right subtree



The depth of a node in a BST is defined as such:

1) The top node (root) has depth 0

2) Every node has depth 1 higher than its parent



Usually, when given a list of distinct values, it is best to build a balanced binary search tree - a tree where the size of the left subtree is approximately the same as the size of the right subtree at each node. A balanced tree isn't always the best solution though. Given a int[] values of distinct values and a int[] probs containing the percentage chance of each element being accessed, your method will return the best (lowest) access score achievable by a BST containing the given values. The access score of any BST is computed by adding together the access scores for all of its nodes. The access score for a particular node is calculated by the formula p*(d+1) where p is the probability of that node being accessed, and d is the depth of that node in the tree.

Definition

Class:

SkewTree

Method:

bestScore

Parameters:

int[], int[]

Returns:

int

Method signature:

int bestScore(int[] values, int[] probs)

(be sure your method is public)

Constraints

-

values and probs will both contain between 1 and 50 elements, inclusive.

-

Each element of values and probs will be between 1 and 1000, inclusive.

-

Each element of values will be unique.

-

values and probs will contain the same number of elements.

Examples

0)

{1,2}

{1,2}

Returns: 4

The best tree looks like:

      2

       \

        1

1)

{1,2,4,3,5,6}

{1,2,3,4,5,6}

Returns: 44

Here the best tree is:

          5

         / \

        /   \

       3     6

      / \   

     2   4 

    /

   1

The score is 5*1 + 4*2 + 6*2 + 2*3 + 3*3 + 1*4 = 44

2)

{6,5,3,7,51,36}

{65,13,49,62,34,16}

Returns: 492