Splay trees

02-splay_operation/cpp/Makefile

+test: splay_operation_test
+	./$<
+
+CXXFLAGS=-std=c++11 -O2 -Wall -Wextra -g -Wno-sign-compare
+
+splay_operation_test: splay_operation.h splay_operation_test.cpp test_main.cpp
+	$(CXX) $(CXXFLAGS) $^ -o $@
+
+clean::
+	rm -f splay_operation_test
+
+.PHONY: clean test

02-splay_operation/cpp/splay_operation.h

+// A node of the tree
+class Node {
+  public:
+    int key;
+    Node* left;
+    Node* right;
+    Node* parent;
+
+    // Constructor
+    Node(int key, Node* parent=nullptr, Node* left=nullptr, Node* right=nullptr) {
+        this->key = key;
+        this->parent = parent;
+        this->left = left;
+        this->right = right;
+        if (left) left->parent = this;
+        if (right) right->parent = this;
+    }
+};
+
+// Binary tree
+class Tree {
+  public:
+    // Pointer to root of the tree; nullptr if the tree is empty.
+    Node* root;
+
+    Tree(Node* root=nullptr) {
+        this->root = root;
+    }
+
+    // Rotate the given `node` up. Perform a single rotation of the edge
+    // between the node and its parent, choosing left or right rotation
+    // appropriately.
+    void rotate(Node* node) {
+        if (node->parent) {
+            if (node->parent->left == node) {
+                if (node->right) node->right->parent = node->parent;
+                node->parent->left = node->right;
+                node->right = node->parent;
+            } else {
+                if (node->left) node->left->parent = node->parent;
+                node->parent->right = node->left;
+                node->left = node->parent;
+            }
+            if (node->parent->parent) {
+                if (node->parent->parent->left == node->parent)
+                    node->parent->parent->left = node;
+                else
+                    node->parent->parent->right = node;
+            } else {
+                root = node;
+            }
+
+            Node* original_parent = node->parent;
+            node->parent = node->parent->parent;
+            original_parent->parent = node;
+        }
+    }
+
+    // Look up the given key in the tree, returning the
+    // the node with the requested key or nullptr.
+    Node* lookup(int key) {
+        // TODO: Utilize splay suitably.
+        Node* node = root;
+        while (node) {
+            if (node->key == key) {
+                return node;
+            }
+            if (key < node->key)
+                node = node->left;
+            else
+                node = node->right;
+        }
+        return nullptr;
+    }
+
+    // Insert a key into the tree.
+    // If the key is already present, nothing happens.
+    void insert(int key) {
+        // TODO: Utilize splay suitably.
+        if (!root) {
+            root = new Node(key);
+            return;
+        }
+
+        Node* node = root;
+        while (node->key != key) {
+            if (key < node->key) {
+                if (!node->left)
+                    node->left = new Node(key, node);
+                node = node->left;
+            } else {
+                if (!node->right)
+                    node->right = new Node(key, node);
+                node = node->right;
+            }
+        }
+    }
+
+    // Delete given key from the tree.
+    // It the key is not present, do nothing.
+    void remove(int key) {
+        // TODO: Utilize splay suitably.
+        Node* node = root;
+        while (node && node->key != key) {
+            if (key < node->key)
+                node = node->left;
+            else
+                node = node->right;
+        }
+
+        if (node) {
+            if (node->left && node->right) {
+                Node* replacement = node->right;
+                while (replacement->left)
+                    replacement = replacement->left;
+                node->key = replacement->key;
+                node = replacement;
+            }
+
+            Node* replacement = node->left ? node->left : node->right;
+            if (node->parent) {
+                if (node->parent->left == node) node->parent->left = replacement;
+                else node->parent->right = replacement;
+            } else {
+                root = replacement;
+            }
+            if (replacement)
+                replacement->parent = node->parent;
+        }
+    }
+
+    // Splay the given node.
+    // If a single rotation needs to be performed, perform it as the last rotation
+    // (i.e., to move the splayed node to the root of the tree).
+    void splay(Node* node) {
+        // TODO: Implement
+    }
+
+    // Destructor to free all allocated memory.
+    ~Tree() {
+        Node* node = root;
+        while (node) {
+            Node* next;
+            if (node->left) {
+                next = node->left;
+                node->left = nullptr;
+            } else if (node->right) {
+                next = node->right;
+                node->right = nullptr;
+            } else {
+                next = node->parent;
+                delete node;
+            }
+            node = next;
+        }
+    }
+};

02-splay_operation/cpp/splay_operation_test.cpp

+#include <algorithm>
+#include <cassert>
+#include <fstream>
+#include <functional>
+#include <string>
+#include <utility>
+#include <vector>
+
+#include "splay_operation.h"
+
+using namespace std;
+
+// If the condition is not true, report an error and halt.
+#define EXPECT(condition, message) do { if (!(condition)) expect_failed(message); } while (0)
+void expect_failed(const string& message);
+
+// Flatten the tree: return a sorted list of all keys in the tree.
+vector<int> flatten(const Tree& tree) {
+    constexpr int L = 0, R = 1, F = 2;
+
+    Node* node = tree.root;
+    vector<int> flattened, stack = {L};
+    while (!stack.empty()) {
+        if (stack.back() == L) {
+            stack.back() = R;
+            if (node->left) {
+                node = node->left;
+                stack.push_back(L);
+            }
+        } else if (stack.back() == R) {
+            flattened.push_back(node->key);
+            stack.back() = F;
+            if (node->right) {
+                node = node->right;
+                stack.push_back(L);
+            }
+        } else {
+            node = node->parent;
+            stack.pop_back();
+        }
+    }
+    return flattened;
+}
+
+// Test for splay operation with required helpers
+class TestSplay {
+  public:
+    static Node* deserialize_node(const string& text, int& index) {
+        EXPECT(text[index++] == '(', "Internal error during example deserialization");
+        if (text[index] == ')') {
+            index++;
+            return nullptr;
+        } else {
+            int comma = text.find(',', index);
+            int key = stoi(text.substr(index, comma - index));
+            Node* left = deserialize_node(text, (index=comma + 1));
+            Node* right = deserialize_node(text, ++index);
+            EXPECT(text[index++] == ')', "Internal error during example deserialization");
+            return new Node(key, nullptr, left, right);
+        }
+    }
+
+    static Node* deserialize_root(const string& text) {
+        int index = 0;
+        Node* root = deserialize_node(text, index);
+        assert(index == text.size());
+        return root;
+    }
+
+    static string compare(Node* system, Node* gold) {
+        if (!system && gold) {
+            return "expected node with key " + to_string(gold->key) + ", found None";
+        } else if (system && !gold) {
+            return "expected None, found node with key " + to_string(system->key);
+        } else if (system && gold) {
+            if (system->key != gold->key)
+                return "expected node with key " + to_string(gold->key) + ", found " + to_string(system->key);
+            auto result = compare(system->left, gold->left);
+            if (!result.empty()) return result;
+            return compare(system->right, gold->right);
+        }
+        return string();
+    }
+
+    static void test() {
+        ifstream splay_tests_file("splay_tests.txt");
+        EXPECT(splay_tests_file.is_open(), "Cannot open splay_tests.txt file with the tests");
+
+        string original, splayed;
+        int target;
+        while (splay_tests_file >> original >> target >> splayed) {
+            Tree original_tree(deserialize_root(original));
+            Tree splayed_tree(deserialize_root(splayed));
+
+            Node* target_node = original_tree.root;
+            while (target_node && target_node->key != target)
+                if (target < target_node->key)
+                    target_node = target_node->left;
+                else
+                    target_node = target_node->right;
+            EXPECT(target_node, "Internal error during finding the target node in the tree to splay");
+
+            original_tree.splay(target_node);
+            auto error = compare(original_tree.root, splayed_tree.root);
+            EXPECT(error.empty(), "Error running splay on key " + to_string(target) + " of " + original + ": " + error);
+        }
+    }
+};
+
+void test_lookup() {
+    // Insert even numbers
+    Tree tree;
+    for (int i = 0; i < 5000000; i += 2)
+        tree.insert(i);
+
+    // Find non-existing
+    for (int i = 1; i < 5000000; i += 2)
+        for (int j = 0; j < 10; j++)
+            EXPECT(!tree.lookup(i), "Non-existing element was found");
+
+    // Find existing
+    for (int i = 0; i < 5000000; i += 2)
+        for (int j = 0; j < 10; j++)
+            EXPECT(tree.lookup(i), "Existing element was not found");
+}
+
+void test_insert() {
+    // Test validity first
+    {
+        Tree tree;
+        vector<int> sequence = {997};
+        for (int i = 2; i < 1999; i++)
+            sequence.push_back((sequence.back() * sequence.front()) % 1999);
+        for (const auto& i : sequence)
+            tree.insert(i);
+
+        vector<int> flattened = flatten(tree);
+        sort(sequence.begin(), sequence.end());
+        EXPECT(flattened == sequence, "Incorrect tree after a sequence of inserts");
+    }
+
+    // Test speed
+    {
+        Tree tree;
+        for (int i = 0; i < 5000000; i++)
+            for (int j = 0; j < 10; j++)
+                tree.insert(i);
+    }
+}
+
+void test_remove() {
+    // Test validity first
+    {
+        Tree tree;
+        for (int i = 2; i < 1999 * 2; i++)
+            tree.insert(i);
+
+        vector<int> sequence = {2 * 997};
+        for (int i = 2; i < 1999; i++)
+            sequence.push_back(2 * ((sequence.back() * sequence.front() / 4) % 1999));
+        for (const auto& i : sequence)
+            tree.remove(i + 1);
+
+        vector<int> flattened = flatten(tree);
+        sort(sequence.begin(), sequence.end());
+        EXPECT(flattened == sequence, "Correct tree after a sequence of removes");
+    }
+
+    // Test speed
+    {
+        Tree tree;
+        for (int i = 0; i < 5000000; i++)
+            tree.insert(i);
+
+        // Non-existing elements
+        for (int i = 1; i < 5000000; i += 2)
+            for (int j = 0; j < 10; j++)
+                tree.remove(i);
+
+        // Existing elements
+        for (int i = 2; i < 5000000; i += 2)
+            for (int j = 0; j < 10; j++)
+                tree.remove(i);
+    }
+}
+
+vector<pair<string, function<void()>>> tests = {
+    { "splay", TestSplay::test },
+    { "lookup", test_lookup },
+    { "insert", test_insert },
+    { "remove", test_remove },
+};

02-splay_operation/cpp/splay_tests.txt

+(3,(1,(0,(),()),(2,(),())),(4,(),())) 4 (4,(3,(1,(0,(),()),(2,(),())),()),())
+(6,(5,(),()),(8,(7,(),()),(9,(),()))) 5 (5,(),(6,(),(8,(7,(),()),(9,(),()))))
+(3,(1,(0,(),()),(2,(),())),(7,(5,(4,(),()),(6,(),())),(8,(),()))) 8 (8,(7,(3,(1,(0,(),()),(2,(),())),(5,(4,(),()),(6,(),()))),()),())
+(11,(9,(7,(6,(),()),(8,(),())),(10,(),())),(13,(12,(),()),(14,(),()))) 10 (10,(9,(7,(6,(),()),(8,(),())),()),(11,(),(13,(12,(),()),(14,(),()))))
+(8,(6,(5,(),()),(7,(),())),(10,(9,(),()),(12,(11,(),()),(13,(),())))) 9 (9,(8,(6,(5,(),()),(7,(),())),()),(10,(),(12,(11,(),()),(13,(),()))))
+(16,(12,(11,(),()),(14,(13,(),()),(15,(),()))),(18,(17,(),()),(19,(),()))) 11 (11,(),(12,(),(16,(14,(13,(),()),(15,(),())),(18,(17,(),()),(19,(),())))))
+(3,(1,(0,(),()),(2,(),())),(7,(5,(4,(),()),(6,(),())),(11,(9,(8,(),()),(10,(),())),(12,(),())))) 12 (12,(3,(1,(0,(),()),(2,(),())),(11,(7,(5,(4,(),()),(6,(),())),(9,(8,(),()),(10,(),()))),())),())
+(17,(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(16,(),()))),(19,(18,(),()),(20,(),()))) 16 (16,(15,(11,(9,(8,(),()),(10,(),())),(13,(12,(),()),(14,(),()))),()),(17,(),(19,(18,(),()),(20,(),()))))
+(9,(7,(6,(),()),(8,(),())),(15,(13,(11,(10,(),()),(12,(),())),(14,(),())),(17,(16,(),()),(18,(),())))) 14 (14,(9,(7,(6,(),()),(8,(),())),(13,(11,(10,(),()),(12,(),())),())),(15,(),(17,(16,(),()),(18,(),()))))
+(23,(19,(17,(15,(14,(),()),(16,(),())),(18,(),())),(21,(20,(),()),(22,(),()))),(25,(24,(),()),(26,(),()))) 18 (18,(17,(15,(14,(),()),(16,(),())),()),(23,(19,(),(21,(20,(),()),(22,(),()))),(25,(24,(),()),(26,(),()))))
+(8,(6,(5,(),()),(7,(),())),(12,(10,(9,(),()),(11,(),())),(14,(13,(),()),(16,(15,(),()),(17,(),()))))) 13 (13,(8,(6,(5,(),()),(7,(),())),(12,(10,(9,(),()),(11,(),())),())),(14,(),(16,(15,(),()),(17,(),()))))
+(22,(16,(14,(13,(),()),(15,(),())),(18,(17,(),()),(20,(19,(),()),(21,(),())))),(24,(23,(),()),(25,(),()))) 17 (17,(16,(14,(13,(),()),(15,(),())),()),(22,(18,(),(20,(19,(),()),(21,(),()))),(24,(23,(),()),(25,(),()))))
+(14,(12,(11,(),()),(13,(),())),(20,(16,(15,(),()),(18,(17,(),()),(19,(),()))),(22,(21,(),()),(23,(),())))) 15 (15,(14,(12,(11,(),()),(13,(),())),()),(16,(),(20,(18,(17,(),()),(19,(),())),(22,(21,(),()),(23,(),())))))
+(28,(24,(20,(19,(),()),(22,(21,(),()),(23,(),()))),(26,(25,(),()),(27,(),()))),(30,(29,(),()),(31,(),()))) 19 (19,(),(28,(20,(),(24,(22,(21,(),()),(23,(),())),(26,(25,(),()),(27,(),())))),(30,(29,(),()),(31,(),()))))
+(3,(1,(0,(),()),(2,(),())),(7,(5,(4,(),()),(6,(),())),(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(16,(),()))))) 16 (16,(7,(3,(1,(0,(),()),(2,(),())),(5,(4,(),()),(6,(),()))),(15,(11,(9,(8,(),()),(10,(),())),(13,(12,(),()),(14,(),()))),())),())
+(25,(15,(13,(12,(),()),(14,(),())),(19,(17,(16,(),()),(18,(),())),(23,(21,(20,(),()),(22,(),())),(24,(),())))),(27,(26,(),()),(28,(),()))) 24 (24,(15,(13,(12,(),()),(14,(),())),(23,(19,(17,(16,(),()),(18,(),())),(21,(20,(),()),(22,(),()))),())),(25,(),(27,(26,(),()),(28,(),()))))
+(11,(9,(8,(),()),(10,(),())),(21,(15,(13,(12,(),()),(14,(),())),(19,(17,(16,(),()),(18,(),())),(20,(),()))),(23,(22,(),()),(24,(),())))) 20 (20,(11,(9,(8,(),()),(10,(),())),(19,(15,(13,(12,(),()),(14,(),())),(17,(16,(),()),(18,(),()))),())),(21,(),(23,(22,(),()),(24,(),()))))
+(33,(29,(23,(21,(20,(),()),(22,(),())),(27,(25,(24,(),()),(26,(),())),(28,(),()))),(31,(30,(),()),(32,(),()))),(35,(34,(),()),(36,(),()))) 28 (28,(27,(23,(21,(20,(),()),(22,(),())),(25,(24,(),()),(26,(),()))),()),(29,(),(33,(31,(30,(),()),(32,(),())),(35,(34,(),()),(36,(),())))))
+(9,(7,(6,(),()),(8,(),())),(13,(11,(10,(),()),(12,(),())),(19,(17,(15,(14,(),()),(16,(),())),(18,(),())),(21,(20,(),()),(22,(),()))))) 18 (18,(13,(9,(7,(6,(),()),(8,(),())),(11,(10,(),()),(12,(),()))),(17,(15,(14,(),()),(16,(),())),())),(19,(),(21,(20,(),()),(22,(),()))))
+(31,(21,(19,(18,(),()),(20,(),())),(27,(25,(23,(22,(),()),(24,(),())),(26,(),())),(29,(28,(),()),(30,(),())))),(33,(32,(),()),(34,(),()))) 26 (26,(21,(19,(18,(),()),(20,(),())),(25,(23,(22,(),()),(24,(),())),())),(31,(27,(),(29,(28,(),()),(30,(),()))),(33,(32,(),()),(34,(),()))))
+(17,(15,(14,(),()),(16,(),())),(27,(23,(21,(19,(18,(),()),(20,(),())),(22,(),())),(25,(24,(),()),(26,(),()))),(29,(28,(),()),(30,(),())))) 22 (22,(17,(15,(14,(),()),(16,(),())),(21,(19,(18,(),()),(20,(),())),())),(27,(23,(),(25,(24,(),()),(26,(),()))),(29,(28,(),()),(30,(),()))))
+(39,(35,(31,(29,(27,(26,(),()),(28,(),())),(30,(),())),(33,(32,(),()),(34,(),()))),(37,(36,(),()),(38,(),()))),(41,(40,(),()),(42,(),()))) 30 (30,(29,(27,(26,(),()),(28,(),())),()),(35,(31,(),(33,(32,(),()),(34,(),()))),(39,(37,(36,(),()),(38,(),())),(41,(40,(),()),(42,(),())))))
+(8,(6,(5,(),()),(7,(),())),(12,(10,(9,(),()),(11,(),())),(16,(14,(13,(),()),(15,(),())),(18,(17,(),()),(20,(19,(),()),(21,(),())))))) 17 (17,(12,(8,(6,(5,(),()),(7,(),())),(10,(9,(),()),(11,(),()))),(16,(14,(13,(),()),(15,(),())),())),(18,(),(20,(19,(),()),(21,(),()))))
+(30,(20,(18,(17,(),()),(19,(),())),(24,(22,(21,(),()),(23,(),())),(26,(25,(),()),(28,(27,(),()),(29,(),()))))),(32,(31,(),()),(33,(),()))) 25 (25,(20,(18,(17,(),()),(19,(),())),(24,(22,(21,(),()),(23,(),())),())),(30,(26,(),(28,(27,(),()),(29,(),()))),(32,(31,(),()),(33,(),()))))
+(16,(14,(13,(),()),(15,(),())),(26,(20,(18,(17,(),()),(19,(),())),(22,(21,(),()),(24,(23,(),()),(25,(),())))),(28,(27,(),()),(29,(),())))) 21 (21,(16,(14,(13,(),()),(15,(),())),(20,(18,(17,(),()),(19,(),())),())),(26,(22,(),(24,(23,(),()),(25,(),()))),(28,(27,(),()),(29,(),()))))
+(38,(34,(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(32,(31,(),()),(33,(),())))),(36,(35,(),()),(37,(),()))),(40,(39,(),()),(41,(),()))) 29 (29,(28,(26,(25,(),()),(27,(),())),()),(34,(30,(),(32,(31,(),()),(33,(),()))),(38,(36,(35,(),()),(37,(),())),(40,(39,(),()),(41,(),())))))
+(14,(12,(11,(),()),(13,(),())),(18,(16,(15,(),()),(17,(),())),(24,(20,(19,(),()),(22,(21,(),()),(23,(),()))),(26,(25,(),()),(27,(),()))))) 19 (19,(18,(14,(12,(11,(),()),(13,(),())),(16,(15,(),()),(17,(),()))),()),(20,(),(24,(22,(21,(),()),(23,(),())),(26,(25,(),()),(27,(),())))))
+(36,(26,(24,(23,(),()),(25,(),())),(32,(28,(27,(),()),(30,(29,(),()),(31,(),()))),(34,(33,(),()),(35,(),())))),(38,(37,(),()),(39,(),()))) 27 (27,(26,(24,(23,(),()),(25,(),())),()),(36,(28,(),(32,(30,(29,(),()),(31,(),())),(34,(33,(),()),(35,(),())))),(38,(37,(),()),(39,(),()))))
+(22,(20,(19,(),()),(21,(),())),(32,(28,(24,(23,(),()),(26,(25,(),()),(27,(),()))),(30,(29,(),()),(31,(),()))),(34,(33,(),()),(35,(),())))) 23 (23,(22,(20,(19,(),()),(21,(),())),()),(32,(24,(),(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(31,(),())))),(34,(33,(),()),(35,(),()))))
+(44,(40,(36,(32,(31,(),()),(34,(33,(),()),(35,(),()))),(38,(37,(),()),(39,(),()))),(42,(41,(),()),(43,(),()))),(46,(45,(),()),(47,(),()))) 31 (31,(),(40,(32,(),(36,(34,(33,(),()),(35,(),())),(38,(37,(),()),(39,(),())))),(44,(42,(41,(),()),(43,(),())),(46,(45,(),()),(47,(),())))))
+(3,(1,(0,(),()),(2,(),())),(7,(5,(4,(),()),(6,(),())),(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(19,(17,(16,(),()),(18,(),())),(20,(),())))))) 20 (20,(3,(1,(0,(),()),(2,(),())),(11,(7,(5,(4,(),()),(6,(),())),(9,(8,(),()),(10,(),()))),(19,(15,(13,(12,(),()),(14,(),())),(17,(16,(),()),(18,(),()))),()))),())
+(37,(23,(21,(20,(),()),(22,(),())),(27,(25,(24,(),()),(26,(),())),(31,(29,(28,(),()),(30,(),())),(35,(33,(32,(),()),(34,(),())),(36,(),()))))),(39,(38,(),()),(40,(),()))) 36 (36,(27,(23,(21,(20,(),()),(22,(),())),(25,(24,(),()),(26,(),()))),(35,(31,(29,(28,(),()),(30,(),())),(33,(32,(),()),(34,(),()))),())),(37,(),(39,(38,(),()),(40,(),()))))
+(15,(13,(12,(),()),(14,(),())),(29,(19,(17,(16,(),()),(18,(),())),(23,(21,(20,(),()),(22,(),())),(27,(25,(24,(),()),(26,(),())),(28,(),())))),(31,(30,(),()),(32,(),())))) 28 (28,(15,(13,(12,(),()),(14,(),())),(19,(17,(16,(),()),(18,(),())),(27,(23,(21,(20,(),()),(22,(),())),(25,(24,(),()),(26,(),()))),()))),(29,(),(31,(30,(),()),(32,(),()))))
+(49,(45,(35,(33,(32,(),()),(34,(),())),(39,(37,(36,(),()),(38,(),())),(43,(41,(40,(),()),(42,(),())),(44,(),())))),(47,(46,(),()),(48,(),()))),(51,(50,(),()),(52,(),()))) 44 (44,(35,(33,(32,(),()),(34,(),())),(43,(39,(37,(36,(),()),(38,(),())),(41,(40,(),()),(42,(),()))),())),(49,(45,(),(47,(46,(),()),(48,(),()))),(51,(50,(),()),(52,(),()))))
+(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(25,(19,(17,(16,(),()),(18,(),())),(23,(21,(20,(),()),(22,(),())),(24,(),()))),(27,(26,(),()),(28,(),()))))) 24 (24,(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(23,(19,(17,(16,(),()),(18,(),())),(21,(20,(),()),(22,(),()))),()))),(25,(),(27,(26,(),()),(28,(),()))))
+(45,(31,(29,(28,(),()),(30,(),())),(41,(35,(33,(32,(),()),(34,(),())),(39,(37,(36,(),()),(38,(),())),(40,(),()))),(43,(42,(),()),(44,(),())))),(47,(46,(),()),(48,(),()))) 40 (40,(31,(29,(28,(),()),(30,(),())),(39,(35,(33,(32,(),()),(34,(),())),(37,(36,(),()),(38,(),()))),())),(45,(41,(),(43,(42,(),()),(44,(),()))),(47,(46,(),()),(48,(),()))))
+(23,(21,(20,(),()),(22,(),())),(37,(33,(27,(25,(24,(),()),(26,(),())),(31,(29,(28,(),()),(30,(),())),(32,(),()))),(35,(34,(),()),(36,(),()))),(39,(38,(),()),(40,(),())))) 32 (32,(23,(21,(20,(),()),(22,(),())),(31,(27,(25,(24,(),()),(26,(),())),(29,(28,(),()),(30,(),()))),())),(33,(),(37,(35,(34,(),()),(36,(),())),(39,(38,(),()),(40,(),())))))
+(57,(53,(49,(43,(41,(40,(),()),(42,(),())),(47,(45,(44,(),()),(46,(),())),(48,(),()))),(51,(50,(),()),(52,(),()))),(55,(54,(),()),(56,(),()))),(59,(58,(),()),(60,(),()))) 48 (48,(47,(43,(41,(40,(),()),(42,(),())),(45,(44,(),()),(46,(),()))),()),(57,(49,(),(53,(51,(50,(),()),(52,(),())),(55,(54,(),()),(56,(),())))),(59,(58,(),()),(60,(),()))))
+(9,(7,(6,(),()),(8,(),())),(13,(11,(10,(),()),(12,(),())),(17,(15,(14,(),()),(16,(),())),(23,(21,(19,(18,(),()),(20,(),())),(22,(),())),(25,(24,(),()),(26,(),())))))) 22 (22,(9,(7,(6,(),()),(8,(),())),(17,(13,(11,(10,(),()),(12,(),())),(15,(14,(),()),(16,(),()))),(21,(19,(18,(),()),(20,(),())),()))),(23,(),(25,(24,(),()),(26,(),()))))
+(43,(29,(27,(26,(),()),(28,(),())),(33,(31,(30,(),()),(32,(),())),(39,(37,(35,(34,(),()),(36,(),())),(38,(),())),(41,(40,(),()),(42,(),()))))),(45,(44,(),()),(46,(),()))) 38 (38,(33,(29,(27,(26,(),()),(28,(),())),(31,(30,(),()),(32,(),()))),(37,(35,(34,(),()),(36,(),())),())),(43,(39,(),(41,(40,(),()),(42,(),()))),(45,(44,(),()),(46,(),()))))
+(21,(19,(18,(),()),(20,(),())),(35,(25,(23,(22,(),()),(24,(),())),(31,(29,(27,(26,(),()),(28,(),())),(30,(),())),(33,(32,(),()),(34,(),())))),(37,(36,(),()),(38,(),())))) 30 (30,(21,(19,(18,(),()),(20,(),())),(25,(23,(22,(),()),(24,(),())),(29,(27,(26,(),()),(28,(),())),()))),(35,(31,(),(33,(32,(),()),(34,(),()))),(37,(36,(),()),(38,(),()))))
+(55,(51,(41,(39,(38,(),()),(40,(),())),(47,(45,(43,(42,(),()),(44,(),())),(46,(),())),(49,(48,(),()),(50,(),())))),(53,(52,(),()),(54,(),()))),(57,(56,(),()),(58,(),()))) 46 (46,(41,(39,(38,(),()),(40,(),())),(45,(43,(42,(),()),(44,(),())),())),(55,(51,(47,(),(49,(48,(),()),(50,(),()))),(53,(52,(),()),(54,(),()))),(57,(56,(),()),(58,(),()))))
+(17,(15,(14,(),()),(16,(),())),(21,(19,(18,(),()),(20,(),())),(31,(27,(25,(23,(22,(),()),(24,(),())),(26,(),())),(29,(28,(),()),(30,(),()))),(33,(32,(),()),(34,(),()))))) 26 (26,(17,(15,(14,(),()),(16,(),())),(21,(19,(18,(),()),(20,(),())),(25,(23,(22,(),()),(24,(),())),()))),(31,(27,(),(29,(28,(),()),(30,(),()))),(33,(32,(),()),(34,(),()))))
+(51,(37,(35,(34,(),()),(36,(),())),(47,(43,(41,(39,(38,(),()),(40,(),())),(42,(),())),(45,(44,(),()),(46,(),()))),(49,(48,(),()),(50,(),())))),(53,(52,(),()),(54,(),()))) 42 (42,(37,(35,(34,(),()),(36,(),())),(41,(39,(38,(),()),(40,(),())),())),(51,(47,(43,(),(45,(44,(),()),(46,(),()))),(49,(48,(),()),(50,(),()))),(53,(52,(),()),(54,(),()))))
+(29,(27,(26,(),()),(28,(),())),(43,(39,(35,(33,(31,(30,(),()),(32,(),())),(34,(),())),(37,(36,(),()),(38,(),()))),(41,(40,(),()),(42,(),()))),(45,(44,(),()),(46,(),())))) 34 (34,(29,(27,(26,(),()),(28,(),())),(33,(31,(30,(),()),(32,(),())),())),(39,(35,(),(37,(36,(),()),(38,(),()))),(43,(41,(40,(),()),(42,(),())),(45,(44,(),()),(46,(),())))))
+(63,(59,(55,(51,(49,(47,(46,(),()),(48,(),())),(50,(),())),(53,(52,(),()),(54,(),()))),(57,(56,(),()),(58,(),()))),(61,(60,(),()),(62,(),()))),(65,(64,(),()),(66,(),()))) 50 (50,(49,(47,(46,(),()),(48,(),())),()),(63,(55,(51,(),(53,(52,(),()),(54,(),()))),(59,(57,(56,(),()),(58,(),())),(61,(60,(),()),(62,(),())))),(65,(64,(),()),(66,(),()))))
+(8,(6,(5,(),()),(7,(),())),(12,(10,(9,(),()),(11,(),())),(16,(14,(13,(),()),(15,(),())),(20,(18,(17,(),()),(19,(),())),(22,(21,(),()),(24,(23,(),()),(25,(),()))))))) 21 (21,(8,(6,(5,(),()),(7,(),())),(16,(12,(10,(9,(),()),(11,(),())),(14,(13,(),()),(15,(),()))),(20,(18,(17,(),()),(19,(),())),()))),(22,(),(24,(23,(),()),(25,(),()))))
+(42,(28,(26,(25,(),()),(27,(),())),(32,(30,(29,(),()),(31,(),())),(36,(34,(33,(),()),(35,(),())),(38,(37,(),()),(40,(39,(),()),(41,(),())))))),(44,(43,(),()),(45,(),()))) 37 (37,(32,(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(31,(),()))),(36,(34,(33,(),()),(35,(),())),())),(42,(38,(),(40,(39,(),()),(41,(),()))),(44,(43,(),()),(45,(),()))))
+(20,(18,(17,(),()),(19,(),())),(34,(24,(22,(21,(),()),(23,(),())),(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(32,(31,(),()),(33,(),()))))),(36,(35,(),()),(37,(),())))) 29 (29,(20,(18,(17,(),()),(19,(),())),(24,(22,(21,(),()),(23,(),())),(28,(26,(25,(),()),(27,(),())),()))),(34,(30,(),(32,(31,(),()),(33,(),()))),(36,(35,(),()),(37,(),()))))
+(54,(50,(40,(38,(37,(),()),(39,(),())),(44,(42,(41,(),()),(43,(),())),(46,(45,(),()),(48,(47,(),()),(49,(),()))))),(52,(51,(),()),(53,(),()))),(56,(55,(),()),(57,(),()))) 45 (45,(40,(38,(37,(),()),(39,(),())),(44,(42,(41,(),()),(43,(),())),())),(54,(50,(46,(),(48,(47,(),()),(49,(),()))),(52,(51,(),()),(53,(),()))),(56,(55,(),()),(57,(),()))))
+(16,(14,(13,(),()),(15,(),())),(20,(18,(17,(),()),(19,(),())),(30,(24,(22,(21,(),()),(23,(),())),(26,(25,(),()),(28,(27,(),()),(29,(),())))),(32,(31,(),()),(33,(),()))))) 25 (25,(16,(14,(13,(),()),(15,(),())),(20,(18,(17,(),()),(19,(),())),(24,(22,(21,(),()),(23,(),())),()))),(30,(26,(),(28,(27,(),()),(29,(),()))),(32,(31,(),()),(33,(),()))))
+(50,(36,(34,(33,(),()),(35,(),())),(46,(40,(38,(37,(),()),(39,(),())),(42,(41,(),()),(44,(43,(),()),(45,(),())))),(48,(47,(),()),(49,(),())))),(52,(51,(),()),(53,(),()))) 41 (41,(36,(34,(33,(),()),(35,(),())),(40,(38,(37,(),()),(39,(),())),())),(50,(46,(42,(),(44,(43,(),()),(45,(),()))),(48,(47,(),()),(49,(),()))),(52,(51,(),()),(53,(),()))))
+(28,(26,(25,(),()),(27,(),())),(42,(38,(32,(30,(29,(),()),(31,(),())),(34,(33,(),()),(36,(35,(),()),(37,(),())))),(40,(39,(),()),(41,(),()))),(44,(43,(),()),(45,(),())))) 33 (33,(28,(26,(25,(),()),(27,(),())),(32,(30,(29,(),()),(31,(),())),())),(38,(34,(),(36,(35,(),()),(37,(),()))),(42,(40,(39,(),()),(41,(),())),(44,(43,(),()),(45,(),())))))
+(62,(58,(54,(48,(46,(45,(),()),(47,(),())),(50,(49,(),()),(52,(51,(),()),(53,(),())))),(56,(55,(),()),(57,(),()))),(60,(59,(),()),(61,(),()))),(64,(63,(),()),(65,(),()))) 49 (49,(48,(46,(45,(),()),(47,(),())),()),(62,(54,(50,(),(52,(51,(),()),(53,(),()))),(58,(56,(55,(),()),(57,(),())),(60,(59,(),()),(61,(),())))),(64,(63,(),()),(65,(),()))))
+(14,(12,(11,(),()),(13,(),())),(18,(16,(15,(),()),(17,(),())),(22,(20,(19,(),()),(21,(),())),(28,(24,(23,(),()),(26,(25,(),()),(27,(),()))),(30,(29,(),()),(31,(),())))))) 23 (23,(14,(12,(11,(),()),(13,(),())),(22,(18,(16,(15,(),()),(17,(),())),(20,(19,(),()),(21,(),()))),())),(24,(),(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(31,(),())))))
+(48,(34,(32,(31,(),()),(33,(),())),(38,(36,(35,(),()),(37,(),())),(44,(40,(39,(),()),(42,(41,(),()),(43,(),()))),(46,(45,(),()),(47,(),()))))),(50,(49,(),()),(51,(),()))) 39 (39,(38,(34,(32,(31,(),()),(33,(),())),(36,(35,(),()),(37,(),()))),()),(48,(40,(),(44,(42,(41,(),()),(43,(),())),(46,(45,(),()),(47,(),())))),(50,(49,(),()),(51,(),()))))
+(26,(24,(23,(),()),(25,(),())),(40,(30,(28,(27,(),()),(29,(),())),(36,(32,(31,(),()),(34,(33,(),()),(35,(),()))),(38,(37,(),()),(39,(),())))),(42,(41,(),()),(43,(),())))) 31 (31,(26,(24,(23,(),()),(25,(),())),(30,(28,(27,(),()),(29,(),())),())),(40,(32,(),(36,(34,(33,(),()),(35,(),())),(38,(37,(),()),(39,(),())))),(42,(41,(),()),(43,(),()))))
+(60,(56,(46,(44,(43,(),()),(45,(),())),(52,(48,(47,(),()),(50,(49,(),()),(51,(),()))),(54,(53,(),()),(55,(),())))),(58,(57,(),()),(59,(),()))),(62,(61,(),()),(63,(),()))) 47 (47,(46,(44,(43,(),()),(45,(),())),()),(60,(56,(48,(),(52,(50,(49,(),()),(51,(),())),(54,(53,(),()),(55,(),())))),(58,(57,(),()),(59,(),()))),(62,(61,(),()),(63,(),()))))
+(22,(20,(19,(),()),(21,(),())),(26,(24,(23,(),()),(25,(),())),(36,(32,(28,(27,(),()),(30,(29,(),()),(31,(),()))),(34,(33,(),()),(35,(),()))),(38,(37,(),()),(39,(),()))))) 27 (27,(22,(20,(19,(),()),(21,(),())),(26,(24,(23,(),()),(25,(),())),())),(36,(28,(),(32,(30,(29,(),()),(31,(),())),(34,(33,(),()),(35,(),())))),(38,(37,(),()),(39,(),()))))
+(56,(42,(40,(39,(),()),(41,(),())),(52,(48,(44,(43,(),()),(46,(45,(),()),(47,(),()))),(50,(49,(),()),(51,(),()))),(54,(53,(),()),(55,(),())))),(58,(57,(),()),(59,(),()))) 43 (43,(42,(40,(39,(),()),(41,(),())),()),(56,(52,(44,(),(48,(46,(45,(),()),(47,(),())),(50,(49,(),()),(51,(),())))),(54,(53,(),()),(55,(),()))),(58,(57,(),()),(59,(),()))))
+(34,(32,(31,(),()),(33,(),())),(48,(44,(40,(36,(35,(),()),(38,(37,(),()),(39,(),()))),(42,(41,(),()),(43,(),()))),(46,(45,(),()),(47,(),()))),(50,(49,(),()),(51,(),())))) 35 (35,(34,(32,(31,(),()),(33,(),())),()),(44,(36,(),(40,(38,(37,(),()),(39,(),())),(42,(41,(),()),(43,(),())))),(48,(46,(45,(),()),(47,(),())),(50,(49,(),()),(51,(),())))))
+(68,(64,(60,(56,(52,(51,(),()),(54,(53,(),()),(55,(),()))),(58,(57,(),()),(59,(),()))),(62,(61,(),()),(63,(),()))),(66,(65,(),()),(67,(),()))),(70,(69,(),()),(71,(),()))) 51 (51,(),(68,(60,(52,(),(56,(54,(53,(),()),(55,(),())),(58,(57,(),()),(59,(),())))),(64,(62,(61,(),()),(63,(),())),(66,(65,(),()),(67,(),())))),(70,(69,(),()),(71,(),()))))

02-splay_operation/cpp/test_main.cpp

+#include <cstdlib>
+#include <functional>
+#include <iostream>
+#include <string>
+#include <utility>
+#include <vector>
+
+using namespace std;
+
+extern vector<pair<string, function<void()>>> tests;
+
+void expect_failed(const string& message) {
+    cerr << "Test error: " << message << endl;
+    exit(1);
+}
+
+int main(int argc, char* argv[]) {
+    vector<string> required_tests;
+
+    if (argc > 1) {
+        required_tests.assign(argv + 1, argv + argc);
+    } else {
+        for (const auto& test : tests)
+            required_tests.push_back(test.first);
+    }
+
+    for (const auto& required_test : required_tests) {
+        bool found = false;
+        for (const auto& test : tests)
+            if (required_test == test.first) {
+                cerr << "Running test " << required_test << endl;
+                test.second();
+                found = true;
+                break;
+            }
+        if (!found) {
+            cerr << "Unknown test " << required_test << endl;
+            return 1;
+        }
+    }
+
+    return 0;
+}

02-splay_operation/python/splay_operation.py

+#!/usr/bin/env python3
+
+class Node:
+    """Node in a binary tree `Tree`"""
+
+    def __init__(self, key, left=None, right=None, parent=None):
+        self.key = key
+        self.parent = parent
+        self.left = left
+        self.right = right
+        if left is not None: left.parent = self
+        if right is not None: right.parent = self
+
+class Tree:
+    """A simple binary search tree"""
+
+    def __init__(self, root=None):
+        self.root = root
+
+    def rotate(self, node):
+        """ Rotate the given `node` up.
+
+        Performs a single rotation of the edge between the given node
+        and its parent, choosing left or right rotation appropriately.
+        """
+        if node.parent is not None:
+            if node.parent.left == node:
+                if node.right is not None: node.right.parent = node.parent
+                node.parent.left = node.right
+                node.right = node.parent
+            else:
+                if node.left is not None: node.left.parent = node.parent
+                node.parent.right = node.left
+                node.left = node.parent
+            if node.parent.parent is not None:
+                if node.parent.parent.left == node.parent:
+                    node.parent.parent.left = node
+                else:
+                    node.parent.parent.right = node
+            else:
+                self.root = node
+            node.parent.parent, node.parent = node, node.parent.parent
+
+    def lookup(self, key):
+        """Look up the given key in the tree.
+
+        Returns the node with the requested key or `None`.
+        """
+        # TODO: Utilize splay suitably.
+        node = self.root
+        while node is not None:
+            if node.key == key:
+                return node
+            if key < node.key:
+                node = node.left
+            else:
+                node = node.right
+        return None
+
+    def insert(self, key):
+        """Insert key into the tree.
+
+        If the key is already present, do nothing.
+        """
+        # TODO: Utilize splay suitably.
+        if self.root is None:
+            self.root = Node(key)
+            return
+
+        node = self.root
+        while node.key != key:
+            if key < node.key:
+                if node.left is None:
+                    node.left = Node(key, parent=node)
+                node = node.left
+            else:
+                if node.right is None:
+                    node.right = Node(key, parent=node)
+                node = node.right
+
+    def remove(self, key):
+        """Remove given key from the tree.
+
+        It the key is not present, do nothing.
+        """
+        # TODO: Utilize splay suitably.
+        node = self.root
+        while node is not None and node.key != key:
+            if key < node.key:
+                node = node.left
+            else:
+                node = node.right
+
+        if node is not None:
+            if node.left is not None and node.right is not None:
+                replacement = node.right
+                while replacement.left is not None:
+                    replacement = replacement.left
+                node.key = replacement.key
+                node = replacement
+
+            replacement = node.left if node.left is not None else node.right
+            if node.parent is not None:
+                if node.parent.left == node: node.parent.left = replacement
+                else: node.parent.right = replacement
+            else:
+                self.root = replacement
+            if replacement is not None:
+                replacement.parent = node.parent
+
+    def splay(self, node):
+        """Splay the given node.
+
+        If a single rotation needs to be performed, perform it as the last rotation
+        (i.e., to move the splayed node to the root of the tree).
+        """
+        # TODO: Implement
+        raise NotImplementedError

02-splay_operation/python/splay_operation_test.py

+#!/usr/bin/env python3
+import itertools
+import math
+import sys
+
+from splay_operation import Tree, Node
+
+def flatten(tree):
+    """Flatten given tree in ascending order."""
+    L, R, F = 0, 1, 2
+
+    node, stack, flattened = tree.root, [L], []
+    while node is not None:
+        if stack[-1] == L:
+            stack[-1] = R
+            if node.left is not None:
+                node = node.left
+                stack.append(L)
+        elif stack[-1] == R:
+            flattened.append(node.key)
+            stack[-1] = F
+            if node.right is not None:
+                node = node.right
+                stack.append(L)
+        else:
+            node = node.parent
+            stack.pop()
+
+    return flattened
+
+def test_splay():
+    def deserialize_tree(string):
+        def deserialize_node(i):
+            assert string[i] == "("
+            i += 1
+            if string[i] == ")":
+                return i + 1, None
+            else:
+                comma = string.find(",", i)
+                comma2, left = deserialize_node(comma + 1)
+                rparen, right = deserialize_node(comma2 + 1)
+                assert string[rparen] == ")"
+                return rparen + 1, Node(int(string[i : comma]), left=left, right=right)
+
+        index, root = deserialize_node(0)
+        assert index == len(string)
+        return Tree(root)
+
+    def compare(system, gold):
+        if system is None and gold is not None:
+            return "expected node with key {}, found None".format(gold.key)
+        elif system is not None and gold is None:
+            return "expected None, found node with key {}".format(system.key)
+        elif system is not None and gold is not None:
+            if system.key != gold.key:
+                return "expected node with key {}, found {}".format(gold.key, system.key)
+            return compare(system.left, gold.left) or compare(system.right, gold.right)
+
+    with open("splay_tests.txt", "r") as splay_tests_file:
+        for line in splay_tests_file:
+            original_serialized, target_serialized, splayed_serialized = line.rstrip("\n").split()
+            original = deserialize_tree(original_serialized)
+            splayed = deserialize_tree(splayed_serialized)
+            target = int(target_serialized)
+
+            node = original.root
+            while node is not None and node.key != target:
+                if target < node.key: node = node.left
+                else: node = node.right
+            assert node is not None
+
+            original.splay(node)
+            error = compare(original.root, splayed.root)
+            assert not error, "Error running splay on key {} of {}: {}".format(node.key, original_serialized, error)
+
+def test_lookup():
+    tree = Tree()
+    for elem in range(0, 100000, 2):
+        tree.insert(elem)
+
+    # Find non-existing
+    for elem in range(1, 100000, 2):
+        for _ in range(10):
+            assert tree.lookup(elem) is None, "Non-existing element was found"
+
+    # Find existing
+    for elem in range(0, 100000, 2):
+        for _ in range(10):
+            assert tree.lookup(elem) is not None, "Existing element was not found"
+
+def test_insert():
+    # Test validity first
+    tree = Tree()
+    sequence = [pow(997, i, 1999) for i in range(1, 1999)]
+    for elem in sequence:
+        tree.insert(elem)
+    assert flatten(tree) == sorted(sequence), "Incorrect tree after a sequence of inserts"
+
+    # Test speed
+    tree = Tree()
+    for elem in range(200000):
+        for _ in range(10):
+            tree.insert(elem)
+
+def test_remove():
+    # Test validity first
+    tree = Tree()
+    for elem in range(2, 1999 * 2):
+        tree.insert(elem)
+
+    sequence = [2 * pow(997, i, 1999) for i in range(1, 1999)]
+    for elem in sequence:
+        tree.remove(elem + 1)
+    assert flatten(tree) == sorted(sequence), "Incorrect tree after a sequence of removes"
+
+    # Test speed
+    tree = Tree()
+    for elem in range(0, 100000, 2):
+        tree.insert(elem)
+
+    # Non-existing elements
+    for elem in range(1, 100000, 2):
+        for _ in range(10):
+            tree.remove(elem)
+
+    # Existing elements
+    for elem in range(2, 100000, 2):
+        tree.remove(elem)
+
+tests = [
+    ("splay", test_splay),
+    ("lookup", test_lookup),
+    ("insert", test_insert),
+    ("remove", test_remove),
+]
+
+if __name__ == "__main__":
+    for required_test in sys.argv[1:] or [name for name, _ in tests]:
+        for name, test in tests:
+            if name == required_test:
+                print("Running test {}".format(name), file=sys.stderr)
+                test()
+                break
+        else:
+            raise ValueError("Unknown test {}".format(name))

02-splay_operation/python/splay_tests.txt

+(3,(1,(0,(),()),(2,(),())),(4,(),())) 4 (4,(3,(1,(0,(),()),(2,(),())),()),())
+(6,(5,(),()),(8,(7,(),()),(9,(),()))) 5 (5,(),(6,(),(8,(7,(),()),(9,(),()))))
+(3,(1,(0,(),()),(2,(),())),(7,(5,(4,(),()),(6,(),())),(8,(),()))) 8 (8,(7,(3,(1,(0,(),()),(2,(),())),(5,(4,(),()),(6,(),()))),()),())
+(11,(9,(7,(6,(),()),(8,(),())),(10,(),())),(13,(12,(),()),(14,(),()))) 10 (10,(9,(7,(6,(),()),(8,(),())),()),(11,(),(13,(12,(),()),(14,(),()))))
+(8,(6,(5,(),()),(7,(),())),(10,(9,(),()),(12,(11,(),()),(13,(),())))) 9 (9,(8,(6,(5,(),()),(7,(),())),()),(10,(),(12,(11,(),()),(13,(),()))))
+(16,(12,(11,(),()),(14,(13,(),()),(15,(),()))),(18,(17,(),()),(19,(),()))) 11 (11,(),(12,(),(16,(14,(13,(),()),(15,(),())),(18,(17,(),()),(19,(),())))))
+(3,(1,(0,(),()),(2,(),())),(7,(5,(4,(),()),(6,(),())),(11,(9,(8,(),()),(10,(),())),(12,(),())))) 12 (12,(3,(1,(0,(),()),(2,(),())),(11,(7,(5,(4,(),()),(6,(),())),(9,(8,(),()),(10,(),()))),())),())
+(17,(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(16,(),()))),(19,(18,(),()),(20,(),()))) 16 (16,(15,(11,(9,(8,(),()),(10,(),())),(13,(12,(),()),(14,(),()))),()),(17,(),(19,(18,(),()),(20,(),()))))
+(9,(7,(6,(),()),(8,(),())),(15,(13,(11,(10,(),()),(12,(),())),(14,(),())),(17,(16,(),()),(18,(),())))) 14 (14,(9,(7,(6,(),()),(8,(),())),(13,(11,(10,(),()),(12,(),())),())),(15,(),(17,(16,(),()),(18,(),()))))
+(23,(19,(17,(15,(14,(),()),(16,(),())),(18,(),())),(21,(20,(),()),(22,(),()))),(25,(24,(),()),(26,(),()))) 18 (18,(17,(15,(14,(),()),(16,(),())),()),(23,(19,(),(21,(20,(),()),(22,(),()))),(25,(24,(),()),(26,(),()))))
+(8,(6,(5,(),()),(7,(),())),(12,(10,(9,(),()),(11,(),())),(14,(13,(),()),(16,(15,(),()),(17,(),()))))) 13 (13,(8,(6,(5,(),()),(7,(),())),(12,(10,(9,(),()),(11,(),())),())),(14,(),(16,(15,(),()),(17,(),()))))
+(22,(16,(14,(13,(),()),(15,(),())),(18,(17,(),()),(20,(19,(),()),(21,(),())))),(24,(23,(),()),(25,(),()))) 17 (17,(16,(14,(13,(),()),(15,(),())),()),(22,(18,(),(20,(19,(),()),(21,(),()))),(24,(23,(),()),(25,(),()))))
+(14,(12,(11,(),()),(13,(),())),(20,(16,(15,(),()),(18,(17,(),()),(19,(),()))),(22,(21,(),()),(23,(),())))) 15 (15,(14,(12,(11,(),()),(13,(),())),()),(16,(),(20,(18,(17,(),()),(19,(),())),(22,(21,(),()),(23,(),())))))
+(28,(24,(20,(19,(),()),(22,(21,(),()),(23,(),()))),(26,(25,(),()),(27,(),()))),(30,(29,(),()),(31,(),()))) 19 (19,(),(28,(20,(),(24,(22,(21,(),()),(23,(),())),(26,(25,(),()),(27,(),())))),(30,(29,(),()),(31,(),()))))
+(3,(1,(0,(),()),(2,(),())),(7,(5,(4,(),()),(6,(),())),(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(16,(),()))))) 16 (16,(7,(3,(1,(0,(),()),(2,(),())),(5,(4,(),()),(6,(),()))),(15,(11,(9,(8,(),()),(10,(),())),(13,(12,(),()),(14,(),()))),())),())
+(25,(15,(13,(12,(),()),(14,(),())),(19,(17,(16,(),()),(18,(),())),(23,(21,(20,(),()),(22,(),())),(24,(),())))),(27,(26,(),()),(28,(),()))) 24 (24,(15,(13,(12,(),()),(14,(),())),(23,(19,(17,(16,(),()),(18,(),())),(21,(20,(),()),(22,(),()))),())),(25,(),(27,(26,(),()),(28,(),()))))
+(11,(9,(8,(),()),(10,(),())),(21,(15,(13,(12,(),()),(14,(),())),(19,(17,(16,(),()),(18,(),())),(20,(),()))),(23,(22,(),()),(24,(),())))) 20 (20,(11,(9,(8,(),()),(10,(),())),(19,(15,(13,(12,(),()),(14,(),())),(17,(16,(),()),(18,(),()))),())),(21,(),(23,(22,(),()),(24,(),()))))
+(33,(29,(23,(21,(20,(),()),(22,(),())),(27,(25,(24,(),()),(26,(),())),(28,(),()))),(31,(30,(),()),(32,(),()))),(35,(34,(),()),(36,(),()))) 28 (28,(27,(23,(21,(20,(),()),(22,(),())),(25,(24,(),()),(26,(),()))),()),(29,(),(33,(31,(30,(),()),(32,(),())),(35,(34,(),()),(36,(),())))))
+(9,(7,(6,(),()),(8,(),())),(13,(11,(10,(),()),(12,(),())),(19,(17,(15,(14,(),()),(16,(),())),(18,(),())),(21,(20,(),()),(22,(),()))))) 18 (18,(13,(9,(7,(6,(),()),(8,(),())),(11,(10,(),()),(12,(),()))),(17,(15,(14,(),()),(16,(),())),())),(19,(),(21,(20,(),()),(22,(),()))))
+(31,(21,(19,(18,(),()),(20,(),())),(27,(25,(23,(22,(),()),(24,(),())),(26,(),())),(29,(28,(),()),(30,(),())))),(33,(32,(),()),(34,(),()))) 26 (26,(21,(19,(18,(),()),(20,(),())),(25,(23,(22,(),()),(24,(),())),())),(31,(27,(),(29,(28,(),()),(30,(),()))),(33,(32,(),()),(34,(),()))))
+(17,(15,(14,(),()),(16,(),())),(27,(23,(21,(19,(18,(),()),(20,(),())),(22,(),())),(25,(24,(),()),(26,(),()))),(29,(28,(),()),(30,(),())))) 22 (22,(17,(15,(14,(),()),(16,(),())),(21,(19,(18,(),()),(20,(),())),())),(27,(23,(),(25,(24,(),()),(26,(),()))),(29,(28,(),()),(30,(),()))))
+(39,(35,(31,(29,(27,(26,(),()),(28,(),())),(30,(),())),(33,(32,(),()),(34,(),()))),(37,(36,(),()),(38,(),()))),(41,(40,(),()),(42,(),()))) 30 (30,(29,(27,(26,(),()),(28,(),())),()),(35,(31,(),(33,(32,(),()),(34,(),()))),(39,(37,(36,(),()),(38,(),())),(41,(40,(),()),(42,(),())))))
+(8,(6,(5,(),()),(7,(),())),(12,(10,(9,(),()),(11,(),())),(16,(14,(13,(),()),(15,(),())),(18,(17,(),()),(20,(19,(),()),(21,(),())))))) 17 (17,(12,(8,(6,(5,(),()),(7,(),())),(10,(9,(),()),(11,(),()))),(16,(14,(13,(),()),(15,(),())),())),(18,(),(20,(19,(),()),(21,(),()))))
+(30,(20,(18,(17,(),()),(19,(),())),(24,(22,(21,(),()),(23,(),())),(26,(25,(),()),(28,(27,(),()),(29,(),()))))),(32,(31,(),()),(33,(),()))) 25 (25,(20,(18,(17,(),()),(19,(),())),(24,(22,(21,(),()),(23,(),())),())),(30,(26,(),(28,(27,(),()),(29,(),()))),(32,(31,(),()),(33,(),()))))
+(16,(14,(13,(),()),(15,(),())),(26,(20,(18,(17,(),()),(19,(),())),(22,(21,(),()),(24,(23,(),()),(25,(),())))),(28,(27,(),()),(29,(),())))) 21 (21,(16,(14,(13,(),()),(15,(),())),(20,(18,(17,(),()),(19,(),())),())),(26,(22,(),(24,(23,(),()),(25,(),()))),(28,(27,(),()),(29,(),()))))
+(38,(34,(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(32,(31,(),()),(33,(),())))),(36,(35,(),()),(37,(),()))),(40,(39,(),()),(41,(),()))) 29 (29,(28,(26,(25,(),()),(27,(),())),()),(34,(30,(),(32,(31,(),()),(33,(),()))),(38,(36,(35,(),()),(37,(),())),(40,(39,(),()),(41,(),())))))
+(14,(12,(11,(),()),(13,(),())),(18,(16,(15,(),()),(17,(),())),(24,(20,(19,(),()),(22,(21,(),()),(23,(),()))),(26,(25,(),()),(27,(),()))))) 19 (19,(18,(14,(12,(11,(),()),(13,(),())),(16,(15,(),()),(17,(),()))),()),(20,(),(24,(22,(21,(),()),(23,(),())),(26,(25,(),()),(27,(),())))))
+(36,(26,(24,(23,(),()),(25,(),())),(32,(28,(27,(),()),(30,(29,(),()),(31,(),()))),(34,(33,(),()),(35,(),())))),(38,(37,(),()),(39,(),()))) 27 (27,(26,(24,(23,(),()),(25,(),())),()),(36,(28,(),(32,(30,(29,(),()),(31,(),())),(34,(33,(),()),(35,(),())))),(38,(37,(),()),(39,(),()))))
+(22,(20,(19,(),()),(21,(),())),(32,(28,(24,(23,(),()),(26,(25,(),()),(27,(),()))),(30,(29,(),()),(31,(),()))),(34,(33,(),()),(35,(),())))) 23 (23,(22,(20,(19,(),()),(21,(),())),()),(32,(24,(),(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(31,(),())))),(34,(33,(),()),(35,(),()))))
+(44,(40,(36,(32,(31,(),()),(34,(33,(),()),(35,(),()))),(38,(37,(),()),(39,(),()))),(42,(41,(),()),(43,(),()))),(46,(45,(),()),(47,(),()))) 31 (31,(),(40,(32,(),(36,(34,(33,(),()),(35,(),())),(38,(37,(),()),(39,(),())))),(44,(42,(41,(),()),(43,(),())),(46,(45,(),()),(47,(),())))))
+(3,(1,(0,(),()),(2,(),())),(7,(5,(4,(),()),(6,(),())),(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(19,(17,(16,(),()),(18,(),())),(20,(),())))))) 20 (20,(3,(1,(0,(),()),(2,(),())),(11,(7,(5,(4,(),()),(6,(),())),(9,(8,(),()),(10,(),()))),(19,(15,(13,(12,(),()),(14,(),())),(17,(16,(),()),(18,(),()))),()))),())
+(37,(23,(21,(20,(),()),(22,(),())),(27,(25,(24,(),()),(26,(),())),(31,(29,(28,(),()),(30,(),())),(35,(33,(32,(),()),(34,(),())),(36,(),()))))),(39,(38,(),()),(40,(),()))) 36 (36,(27,(23,(21,(20,(),()),(22,(),())),(25,(24,(),()),(26,(),()))),(35,(31,(29,(28,(),()),(30,(),())),(33,(32,(),()),(34,(),()))),())),(37,(),(39,(38,(),()),(40,(),()))))
+(15,(13,(12,(),()),(14,(),())),(29,(19,(17,(16,(),()),(18,(),())),(23,(21,(20,(),()),(22,(),())),(27,(25,(24,(),()),(26,(),())),(28,(),())))),(31,(30,(),()),(32,(),())))) 28 (28,(15,(13,(12,(),()),(14,(),())),(19,(17,(16,(),()),(18,(),())),(27,(23,(21,(20,(),()),(22,(),())),(25,(24,(),()),(26,(),()))),()))),(29,(),(31,(30,(),()),(32,(),()))))
+(49,(45,(35,(33,(32,(),()),(34,(),())),(39,(37,(36,(),()),(38,(),())),(43,(41,(40,(),()),(42,(),())),(44,(),())))),(47,(46,(),()),(48,(),()))),(51,(50,(),()),(52,(),()))) 44 (44,(35,(33,(32,(),()),(34,(),())),(43,(39,(37,(36,(),()),(38,(),())),(41,(40,(),()),(42,(),()))),())),(49,(45,(),(47,(46,(),()),(48,(),()))),(51,(50,(),()),(52,(),()))))
+(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(25,(19,(17,(16,(),()),(18,(),())),(23,(21,(20,(),()),(22,(),())),(24,(),()))),(27,(26,(),()),(28,(),()))))) 24 (24,(11,(9,(8,(),()),(10,(),())),(15,(13,(12,(),()),(14,(),())),(23,(19,(17,(16,(),()),(18,(),())),(21,(20,(),()),(22,(),()))),()))),(25,(),(27,(26,(),()),(28,(),()))))
+(45,(31,(29,(28,(),()),(30,(),())),(41,(35,(33,(32,(),()),(34,(),())),(39,(37,(36,(),()),(38,(),())),(40,(),()))),(43,(42,(),()),(44,(),())))),(47,(46,(),()),(48,(),()))) 40 (40,(31,(29,(28,(),()),(30,(),())),(39,(35,(33,(32,(),()),(34,(),())),(37,(36,(),()),(38,(),()))),())),(45,(41,(),(43,(42,(),()),(44,(),()))),(47,(46,(),()),(48,(),()))))
+(23,(21,(20,(),()),(22,(),())),(37,(33,(27,(25,(24,(),()),(26,(),())),(31,(29,(28,(),()),(30,(),())),(32,(),()))),(35,(34,(),()),(36,(),()))),(39,(38,(),()),(40,(),())))) 32 (32,(23,(21,(20,(),()),(22,(),())),(31,(27,(25,(24,(),()),(26,(),())),(29,(28,(),()),(30,(),()))),())),(33,(),(37,(35,(34,(),()),(36,(),())),(39,(38,(),()),(40,(),())))))
+(57,(53,(49,(43,(41,(40,(),()),(42,(),())),(47,(45,(44,(),()),(46,(),())),(48,(),()))),(51,(50,(),()),(52,(),()))),(55,(54,(),()),(56,(),()))),(59,(58,(),()),(60,(),()))) 48 (48,(47,(43,(41,(40,(),()),(42,(),())),(45,(44,(),()),(46,(),()))),()),(57,(49,(),(53,(51,(50,(),()),(52,(),())),(55,(54,(),()),(56,(),())))),(59,(58,(),()),(60,(),()))))
+(9,(7,(6,(),()),(8,(),())),(13,(11,(10,(),()),(12,(),())),(17,(15,(14,(),()),(16,(),())),(23,(21,(19,(18,(),()),(20,(),())),(22,(),())),(25,(24,(),()),(26,(),())))))) 22 (22,(9,(7,(6,(),()),(8,(),())),(17,(13,(11,(10,(),()),(12,(),())),(15,(14,(),()),(16,(),()))),(21,(19,(18,(),()),(20,(),())),()))),(23,(),(25,(24,(),()),(26,(),()))))
+(43,(29,(27,(26,(),()),(28,(),())),(33,(31,(30,(),()),(32,(),())),(39,(37,(35,(34,(),()),(36,(),())),(38,(),())),(41,(40,(),()),(42,(),()))))),(45,(44,(),()),(46,(),()))) 38 (38,(33,(29,(27,(26,(),()),(28,(),())),(31,(30,(),()),(32,(),()))),(37,(35,(34,(),()),(36,(),())),())),(43,(39,(),(41,(40,(),()),(42,(),()))),(45,(44,(),()),(46,(),()))))
+(21,(19,(18,(),()),(20,(),())),(35,(25,(23,(22,(),()),(24,(),())),(31,(29,(27,(26,(),()),(28,(),())),(30,(),())),(33,(32,(),()),(34,(),())))),(37,(36,(),()),(38,(),())))) 30 (30,(21,(19,(18,(),()),(20,(),())),(25,(23,(22,(),()),(24,(),())),(29,(27,(26,(),()),(28,(),())),()))),(35,(31,(),(33,(32,(),()),(34,(),()))),(37,(36,(),()),(38,(),()))))
+(55,(51,(41,(39,(38,(),()),(40,(),())),(47,(45,(43,(42,(),()),(44,(),())),(46,(),())),(49,(48,(),()),(50,(),())))),(53,(52,(),()),(54,(),()))),(57,(56,(),()),(58,(),()))) 46 (46,(41,(39,(38,(),()),(40,(),())),(45,(43,(42,(),()),(44,(),())),())),(55,(51,(47,(),(49,(48,(),()),(50,(),()))),(53,(52,(),()),(54,(),()))),(57,(56,(),()),(58,(),()))))
+(17,(15,(14,(),()),(16,(),())),(21,(19,(18,(),()),(20,(),())),(31,(27,(25,(23,(22,(),()),(24,(),())),(26,(),())),(29,(28,(),()),(30,(),()))),(33,(32,(),()),(34,(),()))))) 26 (26,(17,(15,(14,(),()),(16,(),())),(21,(19,(18,(),()),(20,(),())),(25,(23,(22,(),()),(24,(),())),()))),(31,(27,(),(29,(28,(),()),(30,(),()))),(33,(32,(),()),(34,(),()))))
+(51,(37,(35,(34,(),()),(36,(),())),(47,(43,(41,(39,(38,(),()),(40,(),())),(42,(),())),(45,(44,(),()),(46,(),()))),(49,(48,(),()),(50,(),())))),(53,(52,(),()),(54,(),()))) 42 (42,(37,(35,(34,(),()),(36,(),())),(41,(39,(38,(),()),(40,(),())),())),(51,(47,(43,(),(45,(44,(),()),(46,(),()))),(49,(48,(),()),(50,(),()))),(53,(52,(),()),(54,(),()))))
+(29,(27,(26,(),()),(28,(),())),(43,(39,(35,(33,(31,(30,(),()),(32,(),())),(34,(),())),(37,(36,(),()),(38,(),()))),(41,(40,(),()),(42,(),()))),(45,(44,(),()),(46,(),())))) 34 (34,(29,(27,(26,(),()),(28,(),())),(33,(31,(30,(),()),(32,(),())),())),(39,(35,(),(37,(36,(),()),(38,(),()))),(43,(41,(40,(),()),(42,(),())),(45,(44,(),()),(46,(),())))))
+(63,(59,(55,(51,(49,(47,(46,(),()),(48,(),())),(50,(),())),(53,(52,(),()),(54,(),()))),(57,(56,(),()),(58,(),()))),(61,(60,(),()),(62,(),()))),(65,(64,(),()),(66,(),()))) 50 (50,(49,(47,(46,(),()),(48,(),())),()),(63,(55,(51,(),(53,(52,(),()),(54,(),()))),(59,(57,(56,(),()),(58,(),())),(61,(60,(),()),(62,(),())))),(65,(64,(),()),(66,(),()))))
+(8,(6,(5,(),()),(7,(),())),(12,(10,(9,(),()),(11,(),())),(16,(14,(13,(),()),(15,(),())),(20,(18,(17,(),()),(19,(),())),(22,(21,(),()),(24,(23,(),()),(25,(),()))))))) 21 (21,(8,(6,(5,(),()),(7,(),())),(16,(12,(10,(9,(),()),(11,(),())),(14,(13,(),()),(15,(),()))),(20,(18,(17,(),()),(19,(),())),()))),(22,(),(24,(23,(),()),(25,(),()))))
+(42,(28,(26,(25,(),()),(27,(),())),(32,(30,(29,(),()),(31,(),())),(36,(34,(33,(),()),(35,(),())),(38,(37,(),()),(40,(39,(),()),(41,(),())))))),(44,(43,(),()),(45,(),()))) 37 (37,(32,(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(31,(),()))),(36,(34,(33,(),()),(35,(),())),())),(42,(38,(),(40,(39,(),()),(41,(),()))),(44,(43,(),()),(45,(),()))))
+(20,(18,(17,(),()),(19,(),())),(34,(24,(22,(21,(),()),(23,(),())),(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(32,(31,(),()),(33,(),()))))),(36,(35,(),()),(37,(),())))) 29 (29,(20,(18,(17,(),()),(19,(),())),(24,(22,(21,(),()),(23,(),())),(28,(26,(25,(),()),(27,(),())),()))),(34,(30,(),(32,(31,(),()),(33,(),()))),(36,(35,(),()),(37,(),()))))
+(54,(50,(40,(38,(37,(),()),(39,(),())),(44,(42,(41,(),()),(43,(),())),(46,(45,(),()),(48,(47,(),()),(49,(),()))))),(52,(51,(),()),(53,(),()))),(56,(55,(),()),(57,(),()))) 45 (45,(40,(38,(37,(),()),(39,(),())),(44,(42,(41,(),()),(43,(),())),())),(54,(50,(46,(),(48,(47,(),()),(49,(),()))),(52,(51,(),()),(53,(),()))),(56,(55,(),()),(57,(),()))))
+(16,(14,(13,(),()),(15,(),())),(20,(18,(17,(),()),(19,(),())),(30,(24,(22,(21,(),()),(23,(),())),(26,(25,(),()),(28,(27,(),()),(29,(),())))),(32,(31,(),()),(33,(),()))))) 25 (25,(16,(14,(13,(),()),(15,(),())),(20,(18,(17,(),()),(19,(),())),(24,(22,(21,(),()),(23,(),())),()))),(30,(26,(),(28,(27,(),()),(29,(),()))),(32,(31,(),()),(33,(),()))))
+(50,(36,(34,(33,(),()),(35,(),())),(46,(40,(38,(37,(),()),(39,(),())),(42,(41,(),()),(44,(43,(),()),(45,(),())))),(48,(47,(),()),(49,(),())))),(52,(51,(),()),(53,(),()))) 41 (41,(36,(34,(33,(),()),(35,(),())),(40,(38,(37,(),()),(39,(),())),())),(50,(46,(42,(),(44,(43,(),()),(45,(),()))),(48,(47,(),()),(49,(),()))),(52,(51,(),()),(53,(),()))))
+(28,(26,(25,(),()),(27,(),())),(42,(38,(32,(30,(29,(),()),(31,(),())),(34,(33,(),()),(36,(35,(),()),(37,(),())))),(40,(39,(),()),(41,(),()))),(44,(43,(),()),(45,(),())))) 33 (33,(28,(26,(25,(),()),(27,(),())),(32,(30,(29,(),()),(31,(),())),())),(38,(34,(),(36,(35,(),()),(37,(),()))),(42,(40,(39,(),()),(41,(),())),(44,(43,(),()),(45,(),())))))
+(62,(58,(54,(48,(46,(45,(),()),(47,(),())),(50,(49,(),()),(52,(51,(),()),(53,(),())))),(56,(55,(),()),(57,(),()))),(60,(59,(),()),(61,(),()))),(64,(63,(),()),(65,(),()))) 49 (49,(48,(46,(45,(),()),(47,(),())),()),(62,(54,(50,(),(52,(51,(),()),(53,(),()))),(58,(56,(55,(),()),(57,(),())),(60,(59,(),()),(61,(),())))),(64,(63,(),()),(65,(),()))))
+(14,(12,(11,(),()),(13,(),())),(18,(16,(15,(),()),(17,(),())),(22,(20,(19,(),()),(21,(),())),(28,(24,(23,(),()),(26,(25,(),()),(27,(),()))),(30,(29,(),()),(31,(),())))))) 23 (23,(14,(12,(11,(),()),(13,(),())),(22,(18,(16,(15,(),()),(17,(),())),(20,(19,(),()),(21,(),()))),())),(24,(),(28,(26,(25,(),()),(27,(),())),(30,(29,(),()),(31,(),())))))
+(48,(34,(32,(31,(),()),(33,(),())),(38,(36,(35,(),()),(37,(),())),(44,(40,(39,(),()),(42,(41,(),()),(43,(),()))),(46,(45,(),()),(47,(),()))))),(50,(49,(),()),(51,(),()))) 39 (39,(38,(34,(32,(31,(),()),(33,(),())),(36,(35,(),()),(37,(),()))),()),(48,(40,(),(44,(42,(41,(),()),(43,(),())),(46,(45,(),()),(47,(),())))),(50,(49,(),()),(51,(),()))))
+(26,(24,(23,(),()),(25,(),())),(40,(30,(28,(27,(),()),(29,(),())),(36,(32,(31,(),()),(34,(33,(),()),(35,(),()))),(38,(37,(),()),(39,(),())))),(42,(41,(),()),(43,(),())))) 31 (31,(26,(24,(23,(),()),(25,(),())),(30,(28,(27,(),()),(29,(),())),())),(40,(32,(),(36,(34,(33,(),()),(35,(),())),(38,(37,(),()),(39,(),())))),(42,(41,(),()),(43,(),()))))
+(60,(56,(46,(44,(43,(),()),(45,(),())),(52,(48,(47,(),()),(50,(49,(),()),(51,(),()))),(54,(53,(),()),(55,(),())))),(58,(57,(),()),(59,(),()))),(62,(61,(),()),(63,(),()))) 47 (47,(46,(44,(43,(),()),(45,(),())),()),(60,(56,(48,(),(52,(50,(49,(),()),(51,(),())),(54,(53,(),()),(55,(),())))),(58,(57,(),()),(59,(),()))),(62,(61,(),()),(63,(),()))))
+(22,(20,(19,(),()),(21,(),())),(26,(24,(23,(),()),(25,(),())),(36,(32,(28,(27,(),()),(30,(29,(),()),(31,(),()))),(34,(33,(),()),(35,(),()))),(38,(37,(),()),(39,(),()))))) 27 (27,(22,(20,(19,(),()),(21,(),())),(26,(24,(23,(),()),(25,(),())),())),(36,(28,(),(32,(30,(29,(),()),(31,(),())),(34,(33,(),()),(35,(),())))),(38,(37,(),()),(39,(),()))))
+(56,(42,(40,(39,(),()),(41,(),())),(52,(48,(44,(43,(),()),(46,(45,(),()),(47,(),()))),(50,(49,(),()),(51,(),()))),(54,(53,(),()),(55,(),())))),(58,(57,(),()),(59,(),()))) 43 (43,(42,(40,(39,(),()),(41,(),())),()),(56,(52,(44,(),(48,(46,(45,(),()),(47,(),())),(50,(49,(),()),(51,(),())))),(54,(53,(),()),(55,(),()))),(58,(57,(),()),(59,(),()))))
+(34,(32,(31,(),()),(33,(),())),(48,(44,(40,(36,(35,(),()),(38,(37,(),()),(39,(),()))),(42,(41,(),()),(43,(),()))),(46,(45,(),()),(47,(),()))),(50,(49,(),()),(51,(),())))) 35 (35,(34,(32,(31,(),()),(33,(),())),()),(44,(36,(),(40,(38,(37,(),()),(39,(),())),(42,(41,(),()),(43,(),())))),(48,(46,(45,(),()),(47,(),())),(50,(49,(),()),(51,(),())))))
+(68,(64,(60,(56,(52,(51,(),()),(54,(53,(),()),(55,(),()))),(58,(57,(),()),(59,(),()))),(62,(61,(),()),(63,(),()))),(66,(65,(),()),(67,(),()))),(70,(69,(),()),(71,(),()))) 51 (51,(),(68,(60,(52,(),(56,(54,(53,(),()),(55,(),())),(58,(57,(),()),(59,(),())))),(64,(62,(61,(),()),(63,(),())),(66,(65,(),()),(67,(),())))),(70,(69,(),()),(71,(),()))))

02-splay_operation/task.md

+Given an implementation of a binary search tree including parent pointers:
+- implement `splay` method, possibly utilizing the provided `rotate` operation
+  performing a single rotation;
+- update `lookup`, `insert` and `remove` methods to utilize it correctly.