**Problem**

Please find the problem here.

**Solution:**

This problem is really a hard nut to crack. I know there exist simpler solution, but I am going with my idea and implement that complex idea first. The goal is, really, to prove to myself that my idea works, and that to practice coding and debugging complex algorithm.

It all started with some simple observations. Judging from the numbers, I believe we can’t allow update to take time proportional to the interval length, or otherwise one could easily construct an example to just keep flipping the whole interval and we are screwed with awful performance. This implies we are doing some sort of virtualized storage for the lights. We won’t be actually storing all the lights (or at least not update them during updates)

Second observation is that for a query, we can’t have performance proportional to the number of updates before, for then we will also have quadratic performance by having the first half updates and second half queries. That implies the updates are not stored independently, but rather in some summary form that allows one to query without going through all of them.

With that, my idea is to have an AVL tree that look like this – after updating [1, 5] and [3, 7]

To understand that, first, let’s define elementary interval – think of all update intervals drawn on a straight line, an elementary interval is the interval between two points without any other point in between it. So in our case, it will be all the nodes.

Now, since elementary intervals never overlap, we have a natural ordering for it and we are using that for the binary search tree property. The AVL property can help us to make sure the tree is balanced and therefore the depth is at most O(log n), where n is the number of elementary intervals, which is at most twice the number of updates.

The key idea to this structure is about query. How does splitting all the input interval into elementary interval helps? It does not! The key to this structure is that we need to summarize some information on the edges.

Suppose we know in the interval [1, 3), the lights are on, in the edge from [1, 3) to [6, 8), we think of it as the edge [3, 6), there is only one elementary interval, and it is off, and is [6, 8), it is on. Then following the path from [1, 3) [3, 6), [6, 8), we can calculate the number of on lights is 2 from 2 to 6. The query time in this case will be at most O(log(n)) because that the worst case length for the path from one node to another in a balanced search tree.

The idea sounds good in theory, but in practice, but how would one implement it? Turn out implementing it is an tiring exercise that took me a good deal of time, but it has been a great practice exercise for my own sake. Once I have done with the algorithm I would also like to talk about some techniques I used to implement this, turn out it would be valuable sharing too.

A key observation that enables the implementation is that we don’t need to materialize the edge. A left edge’s information is actually the same information as the summary of the right sub-tree of the children node.

So for each node, we keep the elementary interval of course, the root to the left and right sub-trees as usual, but also the parent pointer (for query), the prev/succ pointers (for merging intervals) and balance (to maintain AVL invariant). The key summaries used is the number of finite segments, the sum of the even numbered segment lengths, and the sum of odd numbered segment lengths, summarized over both sub-trees (and itself included).

Turn out once we have decided what state the keep, the rest is easy (at least to think of). To start with, standard AVL tree operations will take care of left pointer, right pointers and balance. Prev and Succ pointers can be maintained in insertion when a new leaf node is created, its immediate parent must be either a prev or succ (depends on the new node is a leave child or not). In deletion, prev and succ pointers can be maintained easily as we can find out the prev/succ node before deletion and link them back together. All the summaries are updated whenever left or right pointer changes, and combining summary is easy.

Query is easy too, first, find out the interval containing from, the interval containing to, and the node when they start the split. Leveraging the parent pointers and split node it will be easy to find a path. Combine the summary on the way and that’s it.

All is easy to say, hard to code, that’s why it take me so long to get this completed (and that’s why I write so much that I usually don’t)

To aid implementation, the first thing I do is to introduce logging. Writing out the state explicitly make debugging easy. For example, I dump the tree to console after every operation, and that make debugging easier than it is.

Second, I use assertions. In the code, you might see unnecessary checks. For example, I check if the node is its parent’s left child, and then check if it is its parent’s right child. The second check is not necessary, but the good thing is that I add a else branch as a catch all ,and just throw an exception when that happen, now debugging is even easier because the code is now telling me what’s wrong with the data structure.

Third, to go further, I added the invariant checks. It is sometimes hard to make sure the tree is AVL, and by adding those checks every time after a tree operation, again, the code is telling me when it goes wrong.

Fourth, I added randomized testing. I generate random inputs in small sizes and generate expectation by brute force, and that happened me to find the majority of the bugs.

Fifth, you might notice the code is filled with comment, those are on purpose. With multiple days project on non-trivial algorithm, comment is essential to record what I thought. It is also essential during debugging so that I can verify the program is actually in the expected state.

All these are just general implementation techniques, but I found these particularly useful, especially so for this problem.

Unfortunately, with all these algorithmic analysis - the code still cause time limit exceed on the judge server. Guess I either need to optimize this even more (for example, to reduce allocation) or try the simpler approach (which I will).

**Code:**

#include "stdafx.h" #pragma warning(disable : 4996) // http://www.spoj.com/problems/LITE/ #include "SPOJ_LITE_1.h" //#define VERIFY_BALANCE //#define LOG_SUMMARY //#define LOG_AVL_OPERATIONS //#define LOG_QUERY_STEPS #include <iostream> #include <algorithm> #include <map> #include <vector> #include <stack> using namespace std; const int left_too_heavy = 2; const int left_heavy = 1; const int balance = 0; const int right_heavy = -1; const int right_too_heavy = -2; const int bias_left = 1; const int bias_right = -1; const int intMax = 2147483647; const int intMin = ~intMax; class SegmentTree { public: SegmentTree(); ~SegmentTree(); void insert_interval(int from, int to); int query(int from, int to) const; // Debug only void print() const; void print_successor_chain() const; private: class SegmentTreeNode { public: SegmentTreeNode(int from, int to); inline int get_from() const; inline int get_to() const; inline int get_subtree_from() const; inline int get_subtree_to() const; inline int get_finite_segment_count() const; inline int get_odd_segment_length() const; inline int get_even_segment_length() const; void set_interval(int from, int to); SegmentTreeNode* get_left() const; SegmentTreeNode* get_right() const; SegmentTreeNode* get_parent() const; void set_parent_null() { this->parent = NULL; } void set_left(SegmentTreeNode* left); void set_right(SegmentTreeNode* right); // Debug only void print(int indent) const; // Encapsulation is meaningless for these int balance; SegmentTreeNode* prev; SegmentTreeNode* succ; private: void summarize(); private: SegmentTreeNode* left; SegmentTreeNode* right; SegmentTreeNode* parent; int from; int to; // Summaries for sub-tree int subtree_from; int subtree_to; int finite_segment_count; int even_segment_length; int odd_segment_length; }; SegmentTree::SegmentTreeNode* insert_elementary_interval(int from, int to); bool delete_elementary_interval(SegmentTreeNode* interval_to_delete); void split_elementary_interval(SegmentTree::SegmentTreeNode* interval_to_split, int splitting_value); void merge_elementary_intervals(SegmentTree::SegmentTreeNode* before, SegmentTree::SegmentTreeNode* after); void insert_elementary_interval(SegmentTreeNode* current_node, int from, int to, SegmentTreeNode*& inserted_node, bool& chain_updated, SegmentTreeNode*& subtree_root, int& height_change); SegmentTree::SegmentTreeNode* find_containing_interval(int value) const; void delete_easy_node(SegmentTreeNode* current_node, SegmentTreeNode* toDelete, SegmentTreeNode*& subtree_root, int& height_change); pair<SegmentTree::SegmentTreeNode*, int> rotate_left(SegmentTreeNode* current_node); pair<SegmentTree::SegmentTreeNode*, int> rotate_right(SegmentTreeNode* current_node); void delete_subtree(SegmentTreeNode* subtree_root); #ifdef VERIFY_BALANCE int verify_balance(SegmentTreeNode* subtree_root) const; #endif // VERIFY_BALANCE SegmentTreeNode* root; }; SegmentTree::SegmentTreeNode::SegmentTreeNode(int from, int to) : from(from), to(to), left(NULL), right(NULL), prev(NULL), succ(NULL), balance(0) { this->summarize(); } int SegmentTree::SegmentTreeNode::get_from() const { return this->from; } int SegmentTree::SegmentTreeNode::get_to() const { return this->to; } int SegmentTree::SegmentTreeNode::get_subtree_from() const { return this->subtree_from; } int SegmentTree::SegmentTreeNode::get_subtree_to() const { return this->subtree_to; } int SegmentTree::SegmentTreeNode::get_finite_segment_count() const { return this->finite_segment_count; } int SegmentTree::SegmentTreeNode::get_odd_segment_length() const { return this->odd_segment_length; } int SegmentTree::SegmentTreeNode::get_even_segment_length() const { return this->even_segment_length; } void SegmentTree::SegmentTreeNode::set_interval(int from, int to) { this->from = from; this->to = to; this->summarize(); if (this->parent != NULL) { this->parent->summarize(); } } SegmentTree::SegmentTreeNode* SegmentTree::SegmentTreeNode::get_left() const { return this->left; } SegmentTree::SegmentTreeNode* SegmentTree::SegmentTreeNode::get_right() const { return this->right; } SegmentTree::SegmentTreeNode* SegmentTree::SegmentTreeNode::get_parent() const { return this->parent; } void SegmentTree::SegmentTreeNode::set_left(SegmentTree::SegmentTreeNode* left) { this->left = left; if (this->left != NULL) { this->left->parent = this; } this->summarize(); } void SegmentTree::SegmentTreeNode::set_right(SegmentTree::SegmentTreeNode* right) { this->right = right; if (this->right != NULL) { this->right->parent = this; } this->summarize(); } void SegmentTree::SegmentTreeNode::summarize() { if (this->from == intMin || this->to == intMax) { this->finite_segment_count = 0; } else { this->finite_segment_count = 1; } if (this->left == NULL) { this->subtree_from = this->from; } else { this->subtree_from = this->left->subtree_from; this->finite_segment_count += this->left->finite_segment_count; } if (this->right == NULL) { this->subtree_to = this->to; } else { this->finite_segment_count += this->right->finite_segment_count; this->subtree_to = this->right->subtree_to; } bool middle_is_even = true; this->even_segment_length = 0; this->odd_segment_length = 0; if (this->left != NULL) { this->even_segment_length += this->left->even_segment_length; this->odd_segment_length += this->left->odd_segment_length; middle_is_even = this->left->finite_segment_count % 2 == 0; } if (middle_is_even) { if (this->from != intMin && this->to != intMax) { this->even_segment_length += this->to - this->from; } } else { if (this->from != intMin && this->to != intMax) { this->odd_segment_length += this->to - this->from; } } if (this->right != NULL) { if (middle_is_even) { this->odd_segment_length += this->right->even_segment_length; this->even_segment_length += this->right->odd_segment_length; } else { this->even_segment_length += this->right->even_segment_length; this->odd_segment_length += this->right->odd_segment_length; } } } void SegmentTree::SegmentTreeNode::print(int indent) const { for (int i = 0; i < indent; i++) { cout << " "; } if (this != NULL) { #ifdef LOG_SUMMARY cout << "[" << this->from << ", " << this->to << ")" << " has balance " << this->balance << " covering [" << this->subtree_from << ", " << this->subtree_to << ") with " << this->finite_segment_count << " finite segments " << even_segment_length << "/" << odd_segment_length << endl; #else cout << "[" << this->from << ", " << this->to << ")" << " has balance " << this->balance << endl; #endif // LOG SUMMARY this->left->print(indent + 2); this->right->print(indent + 2); } else { cout << "(null)" << endl; } } SegmentTree::SegmentTree() { this->root = new SegmentTreeNode(intMin, intMax); } SegmentTree::~SegmentTree() { this->delete_subtree(this->root); } void SegmentTree::insert_interval(int from, int to) { #ifdef LOG_AVL_OPERATIONS cout << "------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl; cout << "Insert user interval [" << from << ", " << to << ")" << endl; #endif // LOG_AVL_OPERATIONS SegmentTree::SegmentTreeNode* from_containing_interval = this->find_containing_interval(from); if (from == from_containing_interval->get_from()) { this->merge_elementary_intervals(from_containing_interval->prev, from_containing_interval); } else { this->split_elementary_interval(from_containing_interval, from); } SegmentTree::SegmentTreeNode* to_containing_interval = this->find_containing_interval(to + 1); if (to + 1 == to_containing_interval->get_from()) { this->merge_elementary_intervals(to_containing_interval->prev, to_containing_interval); } else { this->split_elementary_interval(to_containing_interval, to + 1); } } void SegmentTree::split_elementary_interval(SegmentTree::SegmentTreeNode* interval_to_split, int splitting_value) { #ifdef LOG_AVL_OPERATIONS cout << "splitting elementary intervals [" << interval_to_split->get_from() << ", " << interval_to_split->get_to() << ") at " << splitting_value << endl; #endif // LOG_AVL_OPERATIONS int from = interval_to_split->get_from(); int to = interval_to_split->get_to(); bool successor_updated = this->delete_elementary_interval(interval_to_split); this->insert_elementary_interval(from, splitting_value); this->insert_elementary_interval(splitting_value, to); } void SegmentTree::merge_elementary_intervals(SegmentTree::SegmentTreeNode* before, SegmentTree::SegmentTreeNode* after) { #ifdef LOG_AVL_OPERATIONS cout << "merging elementary intervals [" << before->get_from() << ", " << before->get_to() << ") and [" << after->get_from() << ", " << after->get_to() << ")" << endl; #endif // LOG_AVL_OPERATIONS int from = before->get_from(); int to = after->get_to(); bool successor_updated = delete_elementary_interval(before); if (successor_updated) { after = before; } delete_elementary_interval(after); insert_elementary_interval(from, to); } SegmentTree::SegmentTreeNode* SegmentTree::insert_elementary_interval(int from, int to) { #ifdef LOG_AVL_OPERATIONS cout << "------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl; cout << "Insert [" << from << ", " << to << ")" << endl; #endif // LOG_AVL_OPERATIONS SegmentTreeNode* inserted_node; bool chain_updated = false; SegmentTreeNode* subtree_root; int height_change; this->insert_elementary_interval(this->root, from, to, inserted_node, chain_updated, subtree_root, height_change); this->root = subtree_root; this->root->set_parent_null(); #ifdef VERIFY_BALANCE this->verify_balance(this->root); #endif // VERIFY_BALANCE #ifdef LOG_AVL_OPERATIONS this->print(); #endif return inserted_node; } void SegmentTree::insert_elementary_interval(SegmentTree::SegmentTreeNode* current_node, int from, int to, SegmentTree::SegmentTreeNode*& inserted_node, bool& chain_updated, SegmentTree::SegmentTreeNode*& subtree_root, int& height_change) { if (current_node == NULL) { inserted_node = new SegmentTree::SegmentTreeNode(from, to); subtree_root = inserted_node; height_change = 1; return; } else { if (to <= current_node->get_from()) { SegmentTree::SegmentTreeNode* left_subtree_root = NULL; int left_height_change = 0; this->insert_elementary_interval(current_node->get_left(), from, to, inserted_node, chain_updated, left_subtree_root, left_height_change); if (!chain_updated && left_subtree_root == inserted_node) { chain_updated = true; SegmentTree::SegmentTreeNode* prev = current_node->prev; if (prev != NULL) { prev->succ = left_subtree_root; left_subtree_root->prev = prev; } left_subtree_root->succ = current_node; current_node->prev = left_subtree_root; } current_node->set_left(left_subtree_root); if (left_height_change == 1) { int result_height_change = current_node->balance == right_heavy ? 0 : 1; current_node->balance += bias_left; if (current_node->balance == left_too_heavy) { if (left_subtree_root->balance == right_heavy) { pair<SegmentTree::SegmentTreeNode*, int> left_rotation_result = this->rotate_left(left_subtree_root); if (left_rotation_result.second != 0) { throw 1; } current_node->set_left(left_rotation_result.first); } pair<SegmentTree::SegmentTreeNode*, int> left_rotation_result = this->rotate_right(current_node); result_height_change += left_rotation_result.second; current_node = left_rotation_result.first; } subtree_root = current_node; height_change = result_height_change; return; } else { if (left_height_change != 0) { throw 2; } subtree_root = current_node; height_change = 0; return; } } else if (from >= current_node->get_to()) { SegmentTree::SegmentTreeNode* right_subtree_root = NULL; int right_height_change = 0; this->insert_elementary_interval(current_node->get_right(), from, to, inserted_node, chain_updated, right_subtree_root, right_height_change); if (!chain_updated && right_subtree_root == inserted_node) { chain_updated = true; SegmentTree::SegmentTreeNode* succ = current_node->succ; if (succ != NULL) { succ->prev = right_subtree_root; right_subtree_root->succ = succ; } right_subtree_root->prev = current_node; current_node->succ = right_subtree_root; } current_node->set_right(right_subtree_root); if (right_height_change == 1) { int result_height_change = current_node->balance == left_heavy ? 0 : 1; current_node->balance += bias_right; if (current_node->balance == right_too_heavy) { if (right_subtree_root->balance == left_heavy) { pair<SegmentTree::SegmentTreeNode*, int> right_rotation_result = this->rotate_right(right_subtree_root); if (right_rotation_result.second != 0) { throw 3; } current_node->set_right(right_rotation_result.first); } pair<SegmentTree::SegmentTreeNode*, int> right_rotation_result = this->rotate_left(current_node); result_height_change += right_rotation_result.second; current_node = right_rotation_result.first; } subtree_root = current_node; height_change = result_height_change; return; } else { if (right_height_change != 0) { throw 4; } subtree_root = current_node; height_change = 0; return; } } else { throw 5; } } } bool SegmentTree::delete_elementary_interval(SegmentTree::SegmentTreeNode* interval_to_delete) { #ifdef LOG_AVL_OPERATIONS cout << "------------------------------------------------------------------------------------------------------------------------------------------------------------------------------" << endl; cout << "Delete [" << interval_to_delete->get_from() << ", " << interval_to_delete->get_to() << ")" << endl; #endif // LOG_AVL_OPERATIONS SegmentTree::SegmentTreeNode* prev = interval_to_delete->prev; SegmentTree::SegmentTreeNode* succ = interval_to_delete->succ; // Step 1: Find the leaf node to delete int replace_from; int replace_to; bool need_replace = false; SegmentTree::SegmentTreeNode* leaf_node_to_delete = interval_to_delete; if (interval_to_delete->get_left() != NULL && interval_to_delete->get_right() != NULL) { leaf_node_to_delete = succ; prev = interval_to_delete; succ = succ->succ; need_replace = true; replace_from = leaf_node_to_delete->get_from(); replace_to = leaf_node_to_delete->get_to(); } // Step 2: Delete the node without two children recursively SegmentTree::SegmentTreeNode* subtree_root = NULL; int height_change = 0; this->delete_easy_node(this->root, leaf_node_to_delete, subtree_root, height_change); this->root = subtree_root; if (this->root != NULL) { this->root->set_parent_null(); } // Step 3: Replace values if (need_replace) { interval_to_delete->set_interval(replace_from, replace_to); } if (prev != NULL) { prev->succ = succ; } if (succ != NULL) { succ->prev = prev; } #ifdef VERIFY_BALANCE this->verify_balance(this->root); #endif // VERIFY_BALANCE #ifdef LOG_AVL_OPERATIONS this->print(); #endif // LOG_AVL_OPERATIONS return need_replace; } void SegmentTree::delete_easy_node(SegmentTree::SegmentTreeNode* current_node, SegmentTree::SegmentTreeNode* to_delete, SegmentTree::SegmentTreeNode*& subtree_root, int& height_change) { if (current_node == to_delete) { SegmentTree::SegmentTreeNode* next = to_delete->get_left(); if (next == NULL) { next = to_delete->get_right(); } delete to_delete; subtree_root = next; height_change = -1; return; } else { if (to_delete->get_to() <= current_node->get_from()) { SegmentTree::SegmentTreeNode* left_subtree_root = NULL; int left_height_change = 0; this->delete_easy_node(current_node->get_left(), to_delete, left_subtree_root, left_height_change); current_node->set_left(left_subtree_root); if (left_height_change == -1) { int result_height_change = current_node->balance == left_heavy ? -1 : 0; current_node->balance -= bias_left; if (current_node->balance == right_too_heavy) { if (current_node->get_right() != NULL && current_node->get_right()->balance == left_heavy) { pair<SegmentTree::SegmentTreeNode*, int> right_rotation_result = this->rotate_right(current_node->get_right()); if (right_rotation_result.second != 0) { throw 6; } current_node->set_right(right_rotation_result.first); } pair<SegmentTree::SegmentTreeNode*, int> left_rotation_result = this->rotate_left(current_node); result_height_change += left_rotation_result.second; current_node = left_rotation_result.first; } subtree_root = current_node; height_change = result_height_change; return; } else { if (left_height_change != 0) { throw 7; } subtree_root = current_node; height_change = 0; return; } } else if (current_node->get_to() <= to_delete->get_from()) { SegmentTree::SegmentTreeNode* right_subtree_root = NULL; int right_height_change = 0; this->delete_easy_node(current_node->get_right(), to_delete, right_subtree_root, right_height_change); current_node->set_right(right_subtree_root); if (right_height_change == -1) { int result_height_change = current_node->balance == right_heavy ? -1 : 0; current_node->balance -= bias_right; if (current_node->balance == left_too_heavy) { if (current_node->get_left() != NULL && current_node->get_left()->balance == right_heavy) { pair<SegmentTree::SegmentTreeNode*, int> left_rotation_result = this->rotate_left(current_node->get_left()); if (left_rotation_result.second != 0) { throw 8; } current_node->set_left(left_rotation_result.first); } pair<SegmentTree::SegmentTreeNode*, int> right_rotation_result = this->rotate_right(current_node); result_height_change += right_rotation_result.second; current_node = right_rotation_result.first; } subtree_root = current_node; height_change = result_height_change; return; } else { if (right_height_change != 0) { throw 9; } subtree_root = current_node; height_change = 0; return; } } else { throw 10; } } } /** * From * * D * E B * C A * * to * * B * D A * E C * */ pair<SegmentTree::SegmentTreeNode*, int> SegmentTree::rotate_left(SegmentTree::SegmentTreeNode* current_node) { // Step 1: Obtain node references SegmentTree::SegmentTreeNode* d = current_node; SegmentTree::SegmentTreeNode* b = current_node->get_right(); if (b == NULL) { throw 11; } SegmentTree::SegmentTreeNode* c = b->get_left(); // Step 2: Setup links d->set_right(c); b->set_left(d); // Step 3: Account for balances and height changes // // Let the height of d be h // if (d->balance == right_too_heavy) { // d was right too heavy // Therefore, the height of b was h - 1 (because d was right too heavy) // the height of e was h - 3 (because d was right too heavy) if (b->balance == right_too_heavy) { // Now we further know // the height of a is h - 2 (b was right too heavy) // the height of c is h - 4 (b was right too heavy) // // Therefore, after change // d is left heavy (c is h - 4 and e is h - 3), and therefore the height of d is h - 2 // b is balance (a and d are both h - 2), and therefore the height of b is h - 1 d->balance = left_heavy; b->balance = balance; return pair<SegmentTree::SegmentTreeNode*, int>(b, -1); } else if (b->balance == right_heavy) { // Now we further know // the height of a is h - 2 (b is right heavy) // the height of c is h - 3 (b is right heavy) // // Therefore, after change // d is balance (c and e are both h - 3), and therefore the height of d is h - 2 // b is balance (a and d are both h - 2), and therefore the height of b is h - 1 d->balance = balance; b->balance = balance; return pair<SegmentTree::SegmentTreeNode*, int>(b, -1); } else if (b->balance == balance) { // Now we further know // the height of a is h - 2 (b was balance) // the height of c is h - 2 (b was balance) // // Therefore, after change // d is right heavy (c is h - 2 and e is h - 3), and therefore the height of d is h - 1 // b is left heavy (a is h - 2 and d is h - 1), and therefore the height of b is h d->balance = right_heavy; b->balance = left_heavy; return pair<SegmentTree::SegmentTreeNode*, int>(b, 0); } else { throw 12; } } else if (d->balance == right_heavy) { // d was right heavy // Therefore, the height of b was h - 1 (because d was right heavy) // the height of e was h - 2 (because d was right heavy) if (b->balance == right_heavy) { // Now we further know // the height of a is h - 2 (b was right heavy) // the height of c is h - 3 (b was right heavy) // // Therefore, after change // d is left heavy (c is h - 3 and e is h - 2), and therefore the height of d is h - 1 // b is left heavy (a is h - 2 and d is h - 1), and therefore the height of b is h d->balance = left_heavy; b->balance = left_heavy; return pair<SegmentTree::SegmentTreeNode*, int>(b, 0); } else if (b->balance == balance) { // Now we further know // the height of a is h - 2 (b was balance) // the height of c is h - 2 (b was balance) // // Therefore, after change // d is balance (both c and e are h - 2), and therefore the height of d is h - 1 // b is left heavy (a is h - 2 and d is h - 1), and therefore the height of b is h d->balance = balance; b->balance = left_heavy; return pair<SegmentTree::SegmentTreeNode*, int>(b, 0); } else if (b->balance == left_heavy) { // Now we further know // the height of a is h - 3 (b was left heavy) // the height of c is h - 2 (b was left heavy) // // Therefore, after change // d is balance (both c and e are h - 2), and therefore the height of d is h - 1 // b is balance (a is h - 3 and d is h - 1), and therefore the height of b is h d->balance = balance; b->balance = left_too_heavy; return pair<SegmentTree::SegmentTreeNode*, int>(b, 0); } } throw 13; } /** * From * * D * B E * A C * * to * * B * A D * C E * */ pair<SegmentTree::SegmentTreeNode*, int> SegmentTree::rotate_right(SegmentTree::SegmentTreeNode* current_node) { // Step 1: Obtain node references SegmentTree::SegmentTreeNode* d = current_node; SegmentTree::SegmentTreeNode* b = current_node->get_left(); if (b == NULL) { throw 14; } SegmentTree::SegmentTreeNode* c = b->get_right(); // Step 2: Setup links d->set_left(c); b->set_right(d); // Step 3: Account for balances and height changes // // Let the height of d be h // if (d->balance == left_too_heavy) { // d was left too heavy // Therefore, the height of b was h - 1 (because d was left too heavy) // the height of e was h - 3 (because d was left too heavy) if (b->balance == left_too_heavy) { // Now we further know // the height of a is h - 2 (b was left too heavy) // the height of c is h - 4 (b was left too heavy) // // Therefore, after change // d is right heavy (c is h - 4 and e is h - 3), and therefore the height of d is h - 2 // b is balance (a and d are both h - 2), and therefore the height of b is h - 1 d->balance = right_heavy; b->balance = balance; return pair<SegmentTree::SegmentTreeNode*, int>(b, -1); } else if (b->balance == left_heavy) { // Now we further know // the height of a is h - 2 (b is left heavy) // the height of c is h - 3 (b is left heavy) // // Therefore, after change // d is balance (c and e are both h - 3), and therefore the height of d is h - 2 // b is balance (a and d are both h - 2), and therefore the height of b is h - 1 d->balance = balance; b->balance = balance; return pair<SegmentTree::SegmentTreeNode*, int>(b, -1); } else if (b->balance == balance) { // Now we further know // the height of a is h - 2 (b was balance) // the height of c is h - 2 (b was balance) // // Therefore, after change // d is left heavy (c is h - 2 and e is h - 3), and therefore the height of d is h - 1 // b is right heavy (a is h - 2 and d is h - 1), and therefore the height of b is h d->balance = left_heavy; b->balance = right_heavy; return pair<SegmentTree::SegmentTreeNode*, int>(b, 0); } else { throw 15; } } else if (d->balance == left_heavy) { // d was left heavy // Therefore, the height of b was h - 1 (because d was left heavy) // the height of e was h - 2 (because d was left heavy) if (b->balance == left_heavy) { // Now we further know // the height of a is h - 2 (b was left heavy) // the height of c is h - 3 (b was left heavy) // // Therefore, after change // d is right heavy (c is h - 3 and e is h - 2), and therefore the height of d is h - 1 // b is right heavy (a is h - 2 and d is h - 1), and therefore the height of b is h d->balance = right_heavy; b->balance = right_heavy; return pair<SegmentTree::SegmentTreeNode*, int>(b, 0); } else if (b->balance == balance) { // Now we further know // the height of a is h - 2 (b was balance) // the height of c is h - 2 (b was balance) // // Therefore, after change // d is balance (both c and e are h - 2), and therefore the height of d is h - 1 // b is right heavy (a is h - 2 and d is h - 1), and therefore the height of b is h d->balance = balance; b->balance = right_heavy; return pair<SegmentTree::SegmentTreeNode*, int>(b, 0); } else if (b->balance == right_heavy) { // Now we further know // the height of a is h - 3 (b was right heavy) // the height of c is h - 2 (b was right heavy) // // Therefore, after change // d is balance (both c and e are h - 2), and therefore the height of d is h - 1 // b is balance (a is h - 3 and d is h - 1), and therefore the height of b is h d->balance = balance; b->balance = right_too_heavy; return pair<SegmentTree::SegmentTreeNode*, int>(b, 0); } } throw 16; } SegmentTree::SegmentTreeNode* SegmentTree::find_containing_interval(int value) const { if (this->root == NULL) { return NULL; } SegmentTree::SegmentTreeNode* cursor = root; while (true) { if (cursor == NULL) { throw 18; } if (value < cursor->get_from()) { cursor = cursor->get_left(); } else if (value >= cursor->get_to()) { cursor = cursor->get_right(); } else if (cursor->get_from() <= value && value < cursor->get_to()) { return cursor; } else { throw 19; } } } void SegmentTree::print() const { this->root->print(0); this->print_successor_chain(); } void SegmentTree::print_successor_chain() const { SegmentTree::SegmentTreeNode* node = root; if (root != NULL) { while (true) { if (node->prev != NULL) { node = node->prev; } else { break; } } while (node != NULL) { cout << "[" << node->get_from() << ", " << node->get_to() << ") - "; node = node->succ; } cout << endl; } } // The idea of query is as follow // // First, find a path from the interval node containing from going to the other interval node containing to. // that a path exist and is unique is because this is a tree with all elementary intervals // // Second, from a path, we can read out a sequence of number a follow: // for a node on the path, we read out the from and to // for an edge on the path which is a left tree edge, we read out the to of the child node and the from of the parent node // similarily, for an edge on the path which is a right tree edge, we read out the from of the child node and the to of the parent node // naturally, this numbers are connected. // We simply concatenate the summaries. // // Third, the key observation is that the edge is already summarized! // The summary of a left edge is the value of the right child of the child node of the edge. // The summary of a right edge is the value of the left child of the child node of the edge. // // Last but not least, we need to figure out whether an interval is on or off, that is determined by its rank in the tree // int SegmentTree::query(int query_from, int query_to) const { #ifdef LOG_QUERY_STEPS cout << "Query [" << query_from << ", " << query_to << "]" << endl; #endif // LOG_QUERY_STEPS // Step 0: Special case - if the tree has no user interval, return 0, the algorithm is fooled to believe the initial infinite segment is rank 0 if (this->root->get_left() == NULL && this->root->get_right() == NULL) { return 0; } // Step 1: Walk the tree to find the node containing from, the node containing to, and the node where the path diverges. SegmentTreeNode* from_cursor = this->root; SegmentTreeNode* to_cursor = this->root; SegmentTreeNode* split_node = NULL; bool from_found = false; bool to_found = false; int from_rank = 1; // offset 1 so that we account for the initial infinite segment if (from_cursor->get_left() != 0) { from_rank += from_cursor->get_left()->get_finite_segment_count(); } while (!from_found || !to_found) { if (from_cursor == to_cursor) { split_node = from_cursor; } if (!from_found) { if (query_from < from_cursor->get_from()) { from_cursor = from_cursor->get_left(); if (from_cursor == NULL) { throw 20; } from_rank--; if (from_cursor->get_right() != NULL) { from_rank -= from_cursor->get_right()->get_finite_segment_count(); } } else if (query_from >= from_cursor->get_to()) { from_cursor = from_cursor->get_right(); if (from_cursor == NULL) { throw 21; } from_rank++; if (from_cursor->get_left() != NULL) { from_rank += from_cursor->get_left()->get_finite_segment_count(); } } else { from_found = true; } } if (!to_found) { if (query_to < to_cursor->get_from()) { to_cursor = to_cursor->get_left(); } else if (query_to >= to_cursor->get_to()) { to_cursor = to_cursor->get_right(); } else { to_found = true; } } } // Step 2: Find a path of summaries when we walk from the left node to the right node // Step 2.1) Record the path from from node to split node (split node not included) // To do this, we start from the left node and walk up until we reach the split node // We always takes the first node and all edges, but if we are changing direction then // we will not take the intermediate node. stack<pair<SegmentTree::SegmentTreeNode*, int> > right_path; vector<pair<SegmentTree::SegmentTreeNode*, int> > path; int last_direction = 0; // 0 => first move, 1 => left, 2 => right while (from_cursor != split_node) { SegmentTree::SegmentTreeNode* parent = from_cursor->get_parent(); if (parent->get_left() == from_cursor) { if (last_direction == 0 || last_direction == 1) { path.push_back(pair<SegmentTree::SegmentTreeNode*, int>(from_cursor, 0)); } last_direction = 1; if (from_cursor->get_right() != NULL) { path.push_back(pair<SegmentTree::SegmentTreeNode*, int>(from_cursor->get_right(), 1)); } } else if (parent->get_right() == from_cursor) { if (last_direction == 0 || last_direction == 2) { path.push_back(pair<SegmentTree::SegmentTreeNode*, int>(from_cursor, 0)); } last_direction = 2; if (from_cursor->get_left() != NULL) { path.push_back(pair<SegmentTree::SegmentTreeNode*, int>(from_cursor->get_left(), 1)); } } else { throw 23; } from_cursor = parent; } // Step 2.2: The split node is taken now, it is unconditional path.push_back(pair<SegmentTree::SegmentTreeNode*, int>(split_node, 0)); // Step 2.3: for the right hand side, we do the same thing, except since we have to walk backwards from right_node to split_node // so we need to use a stack and reverse it on its way back. last_direction = 0; while (to_cursor != split_node) { SegmentTree::SegmentTreeNode* parent = to_cursor->get_parent(); if (parent->get_left() == to_cursor) { if (last_direction == 0 || last_direction == 1) { right_path.push(pair<SegmentTree::SegmentTreeNode*, int>(to_cursor, 0)); } last_direction = 1; if (to_cursor->get_right() != NULL) { right_path.push(pair<SegmentTree::SegmentTreeNode*, int>(to_cursor->get_right(), 1)); } } else if (parent->get_right() == to_cursor) { if (last_direction == 0 || last_direction == 2) { right_path.push(pair<SegmentTree::SegmentTreeNode*, int>(to_cursor, 0)); } last_direction = 2; if (to_cursor->get_left() != NULL) { right_path.push(pair<SegmentTree::SegmentTreeNode*, int>(to_cursor->get_left(), 1)); } } else { throw 24; } to_cursor = parent; } while (right_path.size() > 0) { path.push_back(right_path.top()); right_path.pop(); } // Step 3: Walk the path and concatenate the summary // We need to keep track of whether the next elementary interval in forward order is on bool next_is_on = from_rank % 2 == 1; int on_count = 0; bool first_segment = true; int last_segment_ends = -1; // Step 3.1: Determine whether we should walk the first segment forward or backward SegmentTree::SegmentTreeNode* first_node = path.begin()->first; if (first_node == split_node) { // we should move forward last_segment_ends = first_node->get_from(); } else { SegmentTree::SegmentTreeNode* parent = first_node->get_parent(); if (parent->get_left() == first_node) { // we should move forward last_segment_ends = first_node->get_from(); } else if (parent->get_right() == first_node) { // we should move backward last_segment_ends = first_node->get_to(); // In this case we are walking backward, so we are counting the next is on of the next segment next_is_on = !next_is_on; } else { throw 0; } } for (vector<pair<SegmentTree::SegmentTreeNode*, int> >::iterator pi = path.begin(); pi != path.end(); pi++) { SegmentTree::SegmentTreeNode* current = pi->first; if (pi->second == 0) { bool forward = last_segment_ends == pi->first->get_from(); if (!forward && last_segment_ends != pi->first->get_to()) { throw 25; } // This is a node if (next_is_on) { if (forward) { on_count += current->get_to() - current->get_from(); } } else { if (!forward) { on_count -= current->get_to() - current->get_from(); } } // Adjustment for the first and last segment for input if (first_segment) { if (forward) { if (next_is_on) { // Suppose we are in [1, 3), moving forward // and from == 2 // we should be counting only 1 value // adjustment is to reduce [1, 2) values on_count -= (query_from - current->get_from()); } } else { if (!next_is_on) { // Suppose we are in [1, 2, 3), moving backward // and from == 2 // we should be counting only 1 value // adjustment is to get back [2, 3) values on_count += (current->get_to() - query_from); } } } if (pi + 1 == path.end()) { if (forward) { if (next_is_on) { // Suppose we are in [4, 5, 6, 7), moving forward // and to == 5 // we should be counting only 2 value // adjustment is to reduce [6, 7) values on_count -= (current->get_to() - (query_to + 1)); } } else { if (!next_is_on) { // Suppose we are in [4, 5, 6, 7), moving backward // and to == 5 // we should be counting only 1 value // adjustment is to get back [4, 5] values on_count += ((query_to + 1) - current->get_from()); } } } // Updating state before moving forward first_segment = false; next_is_on = !next_is_on; if (forward) { last_segment_ends = current->get_to(); } else { last_segment_ends = current->get_from(); } #ifdef LOG_QUERY_STEPS cout << "Moving through node [" << current->get_from() << ", " << current->get_to() << ") in " << (forward ? "forward" : "backward") << endl; cout << "Found " << on_count << " so far " << endl; #endif // LOG_QUERY_STEPS } else if (pi->second == 1) { bool forward = last_segment_ends == current->get_subtree_from(); if (!forward && last_segment_ends != current->get_subtree_to()) { throw 26; } // This is an edge if (next_is_on) { if (forward) { on_count += current->get_even_segment_length(); } else { if (current->get_finite_segment_count() % 2 == 1) { // The last segment is even, and we know the last segment is off on_count -= current->get_odd_segment_length(); } else { on_count -= current->get_even_segment_length(); } } } else { if (forward) { on_count += current->get_odd_segment_length(); } else { if (current->get_finite_segment_count() % 2 == 1) { // The last segment is even, and we know the last segment is on on_count -= current->get_even_segment_length(); } else { on_count -= current->get_odd_segment_length(); } } } if (current->get_finite_segment_count() % 2 == 1) { next_is_on = !next_is_on; } if (forward) { last_segment_ends = current->get_subtree_to(); } else { last_segment_ends = current->get_subtree_from(); } #ifdef LOG_QUERY_STEPS cout << "Moving through edge [" << current->get_subtree_from() << ", " << current->get_subtree_to() << ") in " << (forward ? "forward" : "backward") << endl; cout << "Found " << on_count << " so far " << endl; #endif // LOG_QUERY_STEPS } else { throw 27; } } return on_count; } void SegmentTree::delete_subtree(SegmentTree::SegmentTreeNode* subtree_root) { if (subtree_root == NULL) { return; } this->delete_subtree(subtree_root->get_left()); this->delete_subtree(subtree_root->get_right()); delete subtree_root; } #ifdef VERIFY_BALANCE int SegmentTree::verify_balance(SegmentTree::SegmentTreeNode* subtree_root) const { if (subtree_root == NULL) { return 0; } int left_height = this->verify_balance(subtree_root->get_left()); int right_height = this->verify_balance(subtree_root->get_right()); int expected_balance = left_height - right_height; if (subtree_root->balance != expected_balance) { cout << "Bad tree:" << endl; this->print(); throw 0; } return max(left_height, right_height) + 1; } #endif //VERIFY_BALANCE int SPOJ_LITE_1() { int num_lights; int num_operations; scanf("%d %d", &num_lights, &num_operations); SegmentTree tree; for (int q = 0; q < num_operations; q++) { int operation_id; int from; int to; scanf("%d %d %d", &operation_id, &from, &to); if (operation_id == 0) { tree.insert_interval(from, to); } else { int query_result = tree.query(from, to); printf("%d\n", query_result); } } return 0; }

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