Range tree

12-range_tree/cpp/Makefile

+test: range_tree_test
+	./$<
+
+INCLUDE ?= .
+CXXFLAGS=-std=c++11 -O2 -Wall -Wextra -g -Wno-sign-compare -I$(INCLUDE)
+
+range_tree_test: range_tree_test.cpp range_tree.h test_main.cpp
+	$(CXX) $(CXXFLAGS) $(filter %.cpp,$^) -o $@
+
+clean:
+	rm -f range_tree_test
+
+.PHONY: clean test

12-range_tree/cpp/range_tree.h

+#include <cstdint>
+#include <limits>
+
+// A node of the tree
+class Node {
+  public:
+    int64_t key;
+    int64_t value;
+    Node* left;
+    Node* right;
+    Node* parent;
+
+    // Constructor
+    Node(int64_t key, int64_t value, Node* parent=nullptr, Node* left=nullptr, Node* right=nullptr) {
+        this->key = key;
+        this->value = value;
+        this->parent = parent;
+        this->left = left;
+        this->right = right;
+        if (left) left->parent = this;
+        if (right) right->parent = this;
+    }
+
+};
+
+// Splay 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.
+    virtual 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;
+        }
+    }
+
+    // Splay the given node.
+    virtual void splay(Node* node) {
+        while (node->parent && node->parent->parent) {
+            if ((node->parent->right == node && node->parent->parent->right == node->parent) ||
+                (node->parent->left == node && node->parent->parent->left == node->parent)) {
+                rotate(node->parent);
+                rotate(node);
+            } else {
+                rotate(node);
+                rotate(node);
+            }
+        }
+        if (node->parent)
+            rotate(node);
+    }
+
+    // Look up the given key in the tree, returning the
+    // the node with the requested key or nullptr.
+    Node* lookup(int64_t key) {
+        Node* node = root;
+        Node* node_last = nullptr;
+        while (node) {
+            node_last = node;
+            if (node->key == key)
+                break;
+            if (key < node->key)
+                node = node->left;
+            else
+                node = node->right;
+        }
+        if (node_last)
+            splay(node_last);
+        return node;
+    }
+
+    // Insert a (key, value) into the tree.
+    // If the key is already present, nothing happens.
+    void insert(int64_t key, int64_t value) {
+        if (!root) {
+            root = new Node(key, value);
+            return;
+        }
+
+        Node* node = root;
+        while (node->key != key) {
+            if (key < node->key) {
+                if (!node->left)
+                    node->left = new Node(key, value, node);
+                node = node->left;
+            } else {
+                if (!node->right)
+                    node->right = new Node(key, value, node);
+                node = node->right;
+            }
+        }
+        splay(node);
+    }
+
+    // Delete given key from the tree.
+    // It the key is not present, do nothing.
+    //
+    // The implementation first splays the element to be removed to
+    // the root, and if it has both children, splays the largest element
+    // in the left subtree and links it to the original right subtree.
+    void remove(int64_t key) {
+        if (lookup(key)) {
+            Node* right = root->right;
+            root = root->left;
+            if (!root) {
+                root = right;
+                right = nullptr;
+            }
+            if (root)
+                root->parent = nullptr;
+
+            if (right) {
+                Node* node = root;
+                while (node->right)
+                    node = node->right;
+                splay(node);
+                root->right = right;
+                right->parent = root;
+            }
+        }
+    }
+
+    // Return the sum of elements with keys in range [left, right].
+    //
+    // Given a closed range [left, right], return the sum of values of elements
+    // in the range, i.e., sum(value | (key, value) in tree, left <= key <= right).
+    int64_t range_sum(int64_t left, int64_t right) {
+        // 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;
+        }
+    }
+};

12-range_tree/cpp/range_tree_test.cpp

+#include <algorithm>
+#include <functional>
+#include <string>
+#include <utility>
+#include <vector>
+
+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);
+
+#include "range_tree.h"
+
+void create_test_tree(int64_t size, bool ascending, Tree& tree) {
+    vector<int64_t> sequence = {7};
+    for (int64_t i = 2; i < size; i++)
+        sequence.push_back((sequence.back() * sequence.front()) % size);
+    if (ascending)
+        sort(sequence.begin(), sequence.end());
+
+    for (int64_t element : sequence)
+        tree.insert(element, element);
+}
+
+void test_missing(int64_t size, bool ascending) {
+    Tree tree;
+    create_test_tree(size, ascending, tree);
+
+    int64_t values = 0;
+    for (int64_t i = 0; i < size; i++)
+        values += tree.range_sum(-size, 0) + tree.range_sum(size, 2 * size);
+    EXPECT(values == 0, "Expected no values in an empty range");
+}
+
+void test_suffixes(int64_t size, bool ascending) {
+    Tree tree;
+    create_test_tree(size, ascending, tree);
+
+    for (int64_t left = 1; left < size; left++) {
+        int64_t values = tree.range_sum(left, size - 1);
+        int64_t expected = size * (size - 1) / 2 - left * (left - 1) / 2;
+        EXPECT(values == expected,
+               "Expected " + to_string(expected) + " for range [" + to_string(left) +
+               ", " + to_string(size - 1) + "], got " + to_string(values));
+    }
+}
+
+void test_updates(int64_t size, bool ascending) {
+    Tree tree;
+    create_test_tree(size, ascending, tree);
+
+    for (int64_t left = 1; left < size; left++) {
+        tree.remove(left);
+        tree.insert(left + size - 1, left + size - 1);
+        int64_t values = tree.range_sum(left + 1, size + left);
+        int64_t expected = (size + left) * (size + left - 1) / 2 - (left + 1) * left / 2;
+        EXPECT(values == expected,
+               "Expected " + to_string(expected) + " for range [" + to_string(left + 1) +
+               ", " + to_string(size + left) + "], got " + to_string(values));
+    }
+}
+
+
+vector<pair<string, function<void()>>> tests = {
+    {"random_missing", [] { test_missing(13, false); }},
+    {"random_suffixes", [] { test_suffixes(13, false); }},
+    {"random_updates", [] { test_updates(13, false); }},
+
+    {"path_missing", [] { test_missing(13, true); }},
+    {"path_suffixes", [] { test_suffixes(13, true); }},
+    {"path_updates", [] { test_updates(13, true); }},
+
+    {"random_missing_big", [] { test_missing(199999, false); }},
+    {"random_suffixes_big", [] { test_suffixes(199999, false); }},
+    {"random_updates_big", [] { test_updates(199999, false); }},
+
+    {"path_missing_big", [] { test_missing(199999, true); }},
+    {"path_suffixes_big", [] { test_suffixes(199999, true); }},
+    {"path_updates_big", [] { test_updates(199999, true); }},
+};

12-range_tree/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;
+}

12-range_tree/python/range_tree.py

+#!/usr/bin/env python3
+import math
+
+class Node:
+    """Node in a binary tree `Tree`"""
+
+    def __init__(self, key, value, left=None, right=None, parent=None):
+        self.key = key
+        self.value = value
+        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 splay tree implementation"""
+
+    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 splay(self, node):
+        """Splay the given node"""
+        while node.parent is not None and node.parent.parent is not None:
+            if (node.parent.right == node and node.parent.parent.right == node.parent) or \
+                    (node.parent.left == node and node.parent.parent.left == node.parent):
+                self.rotate(node.parent)
+                self.rotate(node)
+            else:
+                self.rotate(node)
+                self.rotate(node)
+        if node.parent is not None:
+            self.rotate(node)
+
+    def lookup(self, key):
+        """Look up the given key in the tree.
+
+        Returns the node with the requested key or `None`.
+        """
+        node, node_last = self.root, None
+        while node is not None:
+            node_last = node
+            if node.key == key:
+                break
+            if key < node.key:
+                node = node.left
+            else:
+                node = node.right
+        if node_last is not None:
+            self.splay(node_last)
+        return node
+
+    def insert(self, key, value):
+        """Insert (key, value) into the tree.
+
+        If the key is already present, do nothing.
+        """
+        if self.root is None:
+            self.root = Node(key, value)
+            return
+
+        node = self.root
+        while node.key != key:
+            if key < node.key:
+                if node.left is None:
+                    node.left = Node(key, value, parent=node)
+                node = node.left
+            else:
+                if node.right is None:
+                    node.right = Node(key, value, parent=node)
+                node = node.right
+        self.splay(node)
+
+    def remove(self, key):
+        """Remove given key from the tree.
+
+        It the key is not present, do nothing.
+
+        The implementation first splays the element to be removed to
+        the root, and if it has both children, splays the largest element
+        in the left subtree and links it to the original right subtree.
+        """
+        if self.lookup(key) is not None:
+            right = self.root.right
+            self.root = self.root.left
+            if self.root is None:
+                self.root, right = right, None
+            if self.root is not None:
+                self.root.parent = None
+
+            if right is not None:
+                node = self.root
+                while node.right is not None:
+                    node = node.right
+                self.splay(node)
+                self.root.right = right
+                right.parent = self.root
+
+    def range_sum(self, left, right):
+        """Return the sum of elements with keys in range [left, right]
+
+        Given a closed range [left, right], return the sum of values of elements
+        in the range, i.e., sum(value | (key, value) in tree, left <= key <= right).
+        """
+        raise NotImplementedError()

12-range_tree/python/range_tree_test.py

+#!/usr/bin/env python3
+import sys
+
+from range_tree import Tree
+
+def test_tree(size, ascending):
+    sequence = [pow(7, i, size) for i in range(1, size)]
+    if ascending: sequence = sorted(sequence)
+
+    tree = Tree()
+    for element in sequence:
+        tree.insert(element, element)
+    return tree
+
+def test_missing(size, ascending):
+    tree = test_tree(size, ascending)
+
+    values = 0
+    for _ in range(size):
+        values += tree.range_sum(-size, 0)
+        values += tree.range_sum(size, 2 * size)
+    assert values == 0, "Expected no values in an empty range"
+
+def test_suffixes(size, ascending):
+    tree = test_tree(size, ascending)
+
+    for left in range(1, size):
+        values = tree.range_sum(left, size - 1)
+        expected = size * (size - 1) // 2 - left * (left - 1) // 2
+        assert values == expected, "Expected {} for range [{}, {}], got {}".format(expected, left, size - 1, values)
+
+def test_updates(size, ascending):
+    tree = test_tree(size, ascending)
+
+    for left in range(1, size):
+        tree.remove(left)
+        tree.insert(left + size - 1, left + size - 1)
+        values = tree.range_sum(left + 1, size + left)
+        expected = (size + left) * (size + left - 1) // 2 - (left + 1) * left // 2
+        assert values == expected, "Expected {} for range [{}, {}], got {}".format(expected, left + 1, size + left, values)
+
+tests = [
+    ("random_missing", lambda: test_missing(13, False)),
+    ("random_suffixes", lambda: test_suffixes(13, False)),
+    ("random_updates", lambda: test_updates(13, False)),
+
+    ("path_missing", lambda: test_missing(13, True)),
+    ("path_suffixes", lambda: test_suffixes(13, True)),
+    ("path_updates", lambda: test_updates(13, True)),
+
+    ("random_missing_big", lambda: test_missing(19997, False)),
+    ("random_suffixes_big", lambda: test_suffixes(19997, False)),
+    ("random_updates_big", lambda: test_updates(19997, False)),
+
+    ("path_missing_big", lambda: test_missing(19997, True)),
+    ("path_suffixes_big", lambda: test_suffixes(19997, True)),
+    ("path_updates_big", lambda: test_updates(19997, True)),
+]
+
+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))

12-range_tree/task.md

+You are given an implementation of a Splay tree which associates every numeric
+key with a numeric value. The Splay tree provides `lookup`, `insert`, and `remove`
+operations.
+
+Your goal is to modify the Splay tree to support range queries in amortized
+logarithmic time. The operation you need to implement takes a range of the
+keys and it should return the sum of values of the elements in the given range.
+
+As usual, you should submit only the `range_tree.{h,py}` file.
+
+## Optional: Range updates (for extra 5 points)
+
+If you also implement an operation
+```
+range_update(left, right, delta)
+```
+which adds `delta` to the value of all elements with key in `[left, right]` range
+and runs in amortized logarithmic time, you will get 5 points.
+
+Currently there are no automated tests for this method; therefore, if you
+implement it, submit the solution to ReCodEx and write an email to your
+teaching assistant.