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# Minimum loss challenge

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Problem statement
Lauren has a chart of distinct projected prices for a house over the next $n$ years, where the price of the house in the $i^{th}$ year is $p_i$. She wants to purchase and resell the house at a minimal loss according to the following rules:

The house cannot be sold at a price greater than or equal to the price it was purchased at (i.e., it must be resold at a loss). The house cannot be resold within the same year it was purchased. Find and print the minimum amount of money Lauren must lose if she buys the house and resells it within the next years.

Note: It’s guaranteed that a valid answer exists.

Input Format
The first line contains an integer, $n$, denoting the number of years of house data. The second line contains space-separated long integers describing the respective values of $p_1, p_2,\dots, p_n$.

Constraints $2\le n \le 2 \times 10^5$ $1 \le p_i \le 10^{16}$ All the prices are distinct. It’s guaranteed that a valid answer exists.

Output Format
Print a single integer denoting the minimum amount of money Lauren must lose if she buys and resells the house within the next $n$ years.

Sample Input 0

 3
5 10 3


Sample Output 0

2


Explanation 0
Lauren buys the house in year $1$ at price $p_1 = 5$ and sells it in year $3$ at $p_3 = 3$ for a minimal loss of $5 - 3 = 2$.

Sample Input 1

 5
20 7 8 2 5


Sample Output 1

2


Explanation 1
Lauren buys the house in year $2$ at price $p_2 = 7$ and sells it in year $5$ at $p_5 = 5$ for a minimal loss of $7 - 5 = 2$.

Sample Input & Output 3
minimum-loss-input
minimum-loss-output

Solution
Naive approach
The naive approach is to iterate over each price $p_i$ and compare it with all the subsequent prices, storing any new found minimum loss based on the two rules stated in the problem statement. This obviously has a running time of $\mathcal{O}(n^2)$.

Binary Search Tree (BST) approach
We can construct a BST from the prices array. The construction has a running time of $\mathcal{O}(nlogn)$. We then go through all the nodes. Starting at node x we traverse up (go to its parent) the BST as long as the current node is not a left child of its parent (or no parent is available). If the current node is a left child then we return the value of its parent. This value is the smallest value that is greater than the value of the original/starting node x. Searching the BST for every node has a total worst case running time of $\mathcal{O}(nlogn)$.

Full code

public class MinimumLoss {

private static class MyBinaryTreeNode {
long value;
MyBinaryTreeNode left = null;
MyBinaryTreeNode right = null;
MyBinaryTreeNode parent = null;

public MyBinaryTreeNode(long value) {
this.value = value;
}

if (newValue <= this.value) {
if (this.left == null) {
this.left = new MyBinaryTreeNode(newValue);
this.left.parent = this;
return this.left;
} else {
// Go left
}
} else {
if (this.right == null) {
this.right = new MyBinaryTreeNode(newValue);
this.right.parent = this;
return this.right;
} else {
// Go right
}
}
}

@Override
public String toString() {
StringBuilder sb = new StringBuilder();
sb.append(this.value);
sb.append("->[");
if (this.left != null) {
sb.append(this.left.toString());
} else {
sb.append(".");
}

sb.append(",");

if (this.right != null) {
sb.append(this.right.toString());
} else {
sb.append(".");
}
sb.append("]");
return sb.toString();
}

public long traverseUp() {
if (this.parent != null) {
if (this.parent.left != null) {
if (!this.parent.left.equals(this)) {
// Parent has a left child, but its not the current node --> go up
return this.parent.traverseUp();
} else {
// Parent has left child and its the current node --> return parent's value
return this.parent.value;
}
} else {
// Parent has no left child --> current node must be right child, so we go up
return this.parent.traverseUp();
}
} else {
// No parent
return -1;
}
}

}

private static long solve(long[] p, int n) {
// Build Binary Search Tree and store the nodes into a LinkedList = (O(nlogn)
MyBinaryTreeNode root = new MyBinaryTreeNode(p);
for (int i = 1; i < n; i++) {
}

// Go through all the nodes x in the BST and for each node go up until the current node i is the left child of its parent. if
// the current node i is a left child then return the value of its parent. This value is the smallest value that is greater than the value
// of the original/starting node x.
long minLoss = Long.MAX_VALUE;
for (MyBinaryTreeNode node : nodes) {
long minLarger = node.traverseUp();
if (minLarger >= 0) {
minLoss = Math.min(minLoss, minLarger - node.value);
}
}

return minLoss;
}

public static void main(String[] args) throws FileNotFoundException {
System.setIn(new FileInputStream(System.getProperty("user.home") + "/" + "in.txt"));
Scanner in = new Scanner(System.in);

int n = in.nextInt();
long[] p = new long[n];
for (int i = 0; i < n; i++) {
p[i] = in.nextLong();
}

System.out.println(solve(p, n));
}
}