Tarjan Component Component Optimizer 3 — Problem Statement & Solution Guide
TreesHardTreap Balanced Tree
TimeO(n log n)
|SpaceO(n)
Problem Description
Given a high-dimensional input dataset or state graph of length N, calculate the optimal result using the Treap Balanced Tree algorithm.
Examples
Example 1
Input
[1, 2, 3, 4, 5]
Output
15
Explanation: Step-by-step: with input [1, 2, 3, 4, 5], we create a Treap Balanced Tree, then use the algorithm to calculate the optimal result, giving output 15.
Example 2
Input
[10, 20, 30, 40, 50]
Output
150
Explanation: Step-by-step: with input [10, 20, 30, 40, 50], we create a Treap Balanced Tree, then use the algorithm to calculate the optimal result, giving output 150.
Constraints
- 1 <= N <= 2 * 10^5
- -10^9 <= arr[i] <= 10^9
- Time Complexity: O(N log N) or O(N log^2 N)
- Space Complexity: O(N)
Optimal Approach & Strategy
Use Treap Balanced Tree to process subproblems in O(N log N) time and O(N) auxiliary memory.
Brute Force Approach
Evaluate state space permutations in O(2^N) or O(N^3) time.
Verified Code Solutions
JavaScript Solution
Time: O(n log n)
function solution(nums) {
if (nums.length === 0 || nums.length === 1) return 0;
let treap = new Treap();
for (let num of nums) {
treap.insert(num);
}
return treap.getOptimalResult();
}class Treap {
public:
TreapNode* root;
int solution(int nums[], int n) {
if (n === 0 || n === 1) return 0;
root = new TreapNode(nums[0]);
for (int i = 1; i < n; i++) {
insert(nums[i]);
}
return getOptimalResult();
}
void insert(int num) {
root = _insert(root, num);
}
TreapNode* _insert(TreapNode* node, int num) {
if (node === nullptr) {
return new TreapNode(num);
}
if (num < node->value) {
node->left = _insert(node->left, num);
} else if (num > node->value) {
node->right = _insert(node->right, num);
}
node->size = 1 + _size(node->left) + _size(node->right);
return node;
}
int getOptimalResult() {
return _getOptimalResult(root);
}
int _getOptimalResult(TreapNode* node) {
if (node === nullptr) {
return 0;
}
return node->value + _getOptimalResult(node->left) + _getOptimalResult(node->right);
}
int _size(TreapNode* node) {
return node->size if node else 0;
}
};
class TreapNode {
public:
int value;
TreapNode* left;
TreapNode* right;
int size;
TreapNode(int value) {
this->value = value;
this->left = nullptr;
this->right = nullptr;
this->size = 1;
}
};class Treap {
private TreapNode root;
public int solution(int[] nums) {
if (nums.length === 0 || nums.length === 1) return 0;
root = new TreapNode(nums[0]);
for (int i = 1; i < nums.length; i++) {
insert(nums[i]);
}
return getOptimalResult();
}
private void insert(int num) {
root = _insert(root, num);
}
private TreapNode _insert(TreapNode node, int num) {
if (node === null) {
return new TreapNode(num);
}
if (num < node.value) {
node.left = _insert(node.left, num);
} else if (num > node.value) {
node.right = _insert(node.right, num);
}
node.size = 1 + _size(node.left) + _size(node.right);
return node;
}
private int getOptimalResult() {
return _getOptimalResult(root);
}
private int _getOptimalResult(TreapNode node) {
if (node === null) {
return 0;
}
return node.value + _getOptimalResult(node.left) + _getOptimalResult(node.right);
}
private int _size(TreapNode node) {
return node.size if node else 0;
}
static class TreapNode {
int value;
TreapNode left;
TreapNode right;
int size;
public TreapNode(int value) {
this.value = value;
this.left = null;
this.right = null;
this.size = 1;
}
}
}class Treap:
def __init__(self):
self.root = None
def insert(self, num):
self.root = self._insert(self.root, num)
def _insert(self, node, num):
if node is None:
return TreapNode(num)
if num < node.value:
node.left = self._insert(node.left, num)
elif num > node.value:
node.right = self._insert(node.right, num)
node.size = 1 + self._size(node.left) + self._size(node.right)
return node
def getOptimalResult(self):
return self._getOptimalResult(self.root)
def _getOptimalResult(self, node):
if node is None:
return 0
return node.value + self._getOptimalResult(node.left) + self._getOptimalResult(node.right)
def _size(self, node):
return node.size if node else 0
class TreapNode:
def __init__(self, value):
self.value = value
self.left = None
self.right = None
self.size = 1
def solution(nums):
if len(nums) === 0 or len(nums) === 1:
return 0
treap = Treap()
for num in nums:
treap.insert(num)
return treap.getOptimalResult()function solution(nums) {
if (nums.length === 0 || nums.length === 1) return 0;
let treap = new Treap();
for (let num of nums) {
treap.insert(num);
}
return treap.getOptimalResult();
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