Persistent Seg Tree Lazy Prop
D
/**
* @file
* @brief [Persistent segment tree with range updates (lazy
* propagation)](https://en.wikipedia.org/wiki/Persistent_data_structure)
*
* @details
* A normal segment tree facilitates making point updates and range queries in
* logarithmic time. Lazy propagation preserves the logarithmic time with range
* updates. So, a segment tree with lazy propagation enables doing range updates
* and range queries in logarithmic time, but it doesn't save any information
* about itself before the last update. A persistent data structure always
* preserves the previous version of itself when it is modified. That is, a new
* version of the segment tree is generated after every update. It saves all
* previous versions of itself (before every update) to facilitate doing range
* queries in any version. More memory is used ,but the logarithmic time is
* preserved because the new version points to the same nodes, that the previous
* version points to, that are not affected by the update. That is, only the
* nodes that are affected by the update and their ancestors are copied. The
* rest is copied using lazy propagation in the next queries. Thus preserving
* the logarithmic time because the number of nodes copied after any update is
* logarithmic.
*
* @author [Magdy Sedra](https://github.com/MSedra)
*/
#include <iostream> /// for IO operations
#include <memory> /// to manage dynamic memory
#include <vector> /// for std::vector
/**
* @namespace range_queries
* @brief Range queries algorithms
*/
namespace range_queries {
/**
* @brief Range query here is range sum, but the code can be modified to make
* different queries like range max or min.
*/
class perSegTree {
private:
class Node {
public:
std::shared_ptr<Node> left = nullptr; /// pointer to the left node
std::shared_ptr<Node> right = nullptr; /// pointer to the right node
int64_t val = 0,
prop = 0; /// val is the value of the node (here equals to the
/// sum of the leaf nodes children of that node),
/// prop is the value to be propagated/added to all
/// the leaf nodes children of that node
};
uint32_t n = 0; /// number of elements/leaf nodes in the segment tree
std::vector<std::shared_ptr<Node>>
ptrs{}; /// ptrs[i] holds a root pointer to the segment tree after the
/// ith update. ptrs[0] holds a root pointer to the segment
/// tree before any updates
std::vector<int64_t> vec{}; /// values of the leaf nodes that the segment
/// tree will be constructed with
/**
* @brief Creating a new node with the same values of curr node
* @param curr node that would be copied
* @returns the new node
*/
std::shared_ptr<Node> newKid(std::shared_ptr<Node> const &curr) {
auto newNode = std::make_shared<Node>(Node());
newNode->left = curr->left;
newNode->right = curr->right;
newNode->prop = curr->prop;
newNode->val = curr->val;
return newNode;
}
/**
* @brief If there is some value to be propagated to the passed node, value
* is added to the node and the children of the node, if exist, are copied
* and the propagated value is also added to them
* @param i the left index of the range that the passed node holds its sum
* @param j the right index of the range that the passed node holds its sum
* @param curr pointer to the node to be propagated
* @returns void
*/
void lazy(const uint32_t &i, const uint32_t &j,
std::shared_ptr<Node> const &curr) {
if (!curr->prop) {
return;
}
curr->val += (j - i + 1) * curr->prop;
if (i != j) {
curr->left = newKid(curr->left);
curr->right = newKid(curr->right);
curr->left->prop += curr->prop;
curr->right->prop += curr->prop;
}
curr->prop = 0;
}
/**
* @brief Constructing the segment tree with the early passed vector. Every
* call creates a node to hold the sum of the given range, set its pointers
* to the children, and set its value to the sum of the children's values
* @param i the left index of the range that the created node holds its sum
* @param j the right index of the range that the created node holds its sum
* @returns pointer to the newly created node
*/
std::shared_ptr<Node> construct(const uint32_t &i, const uint32_t &j) {
auto newNode = std::make_shared<Node>(Node());
if (i == j) {
newNode->val = vec[i];
} else {
uint32_t mid = i + (j - i) / 2;
auto leftt = construct(i, mid);
auto right = construct(mid + 1, j);
newNode->val = leftt->val + right->val;
newNode->left = leftt;
newNode->right = right;
}
return newNode;
}
/**
* @brief Doing range update, checking at every node if it has some value to
* be propagated. All nodes affected by the update are copied and
* propagation value is added to the leaf of them
* @param i the left index of the range that the passed node holds its sum
* @param j the right index of the range that the passed node holds its sum
* @param l the left index of the range to be updated
* @param r the right index of the range to be updated
* @param value the value to be added to every element whose index x
* satisfies l<=x<=r
* @param curr pointer to the current node, which has value = the sum of
* elements whose index x satisfies i<=x<=j
* @returns pointer to the current newly created node
*/
std::shared_ptr<Node> update(const uint32_t &i, const uint32_t &j,
const uint32_t &l, const uint32_t &r,
const int64_t &value,
std::shared_ptr<Node> const &curr) {
lazy(i, j, curr);
if (i >= l && j <= r) {
std::shared_ptr<Node> newNode = newKid(curr);
newNode->prop += value;
lazy(i, j, newNode);
return newNode;
}
if (i > r || j < l) {
return curr;
}
auto newNode = std::make_shared<Node>(Node());
uint32_t mid = i + (j - i) / 2;
newNode->left = update(i, mid, l, r, value, curr->left);
newNode->right = update(mid + 1, j, l, r, value, curr->right);
newNode->val = newNode->left->val + newNode->right->val;
return newNode;
}
/**
* @brief Querying the range from index l to index r, checking at every node
* if it has some value to be propagated. Current node's value is returned
* if its range is completely inside the wanted range, else 0 is returned
* @param i the left index of the range that the passed node holds its sum
* @param j the right index of the range that the passed node holds its sum
* @param l the left index of the range whose sum should be returned as a
* result
* @param r the right index of the range whose sum should be returned as a
* result
* @param curr pointer to the current node, which has value = the sum of
* elements whose index x satisfies i<=x<=j
* @returns sum of elements whose index x satisfies l<=x<=r
*/
int64_t query(const uint32_t &i, const uint32_t &j, const uint32_t &l,
const uint32_t &r, std::shared_ptr<Node> const &curr) {
lazy(i, j, curr);
if (j < l || r < i) {
return 0;
}
if (i >= l && j <= r) {
return curr->val;
}
uint32_t mid = i + (j - i) / 2;
return query(i, mid, l, r, curr->left) +
query(mid + 1, j, l, r, curr->right);
}
/**
* public methods that can be used directly from outside the class. They
* call the private functions that do all the work
*/
public:
/**
* @brief Constructing the segment tree with the values in the passed
* vector. Returned root pointer is pushed in the pointers vector to have
* access to the original version if the segment tree is updated
* @param vec vector whose values will be used to build the segment tree
* @returns void
*/
void construct(const std::vector<int64_t>
&vec) // the segment tree will be built from the values
// in "vec", "vec" is 0 indexed
{
if (vec.empty()) {
return;
}
n = vec.size();
this->vec = vec;
auto root = construct(0, n - 1);
ptrs.push_back(root);
}
/**
* @brief Doing range update by passing the left and right indexes of the
* range as well as the value to be added.
* @param l the left index of the range to be updated
* @param r the right index of the range to be updated
* @param value the value to be added to every element whose index x
* satisfies l<=x<=r
* @returns void
*/
void update(const uint32_t &l, const uint32_t &r,
const int64_t
&value) // all elements from index "l" to index "r" would
// by updated by "value", "l" and "r" are 0 indexed
{
ptrs.push_back(update(
0, n - 1, l, r, value,
ptrs[ptrs.size() -
1])); // saving the root pointer to the new segment tree
}
/**
* @brief Querying the range from index l to index r, getting the sum of the
* elements whose index x satisfies l<=x<=r
* @param l the left index of the range whose sum should be returned as a
* result
* @param r the right index of the range whose sum should be returned as a
* result
* @param version the version to query on. If equals to 0, the original
* segment tree will be queried
* @returns sum of elements whose index x satisfies l<=x<=r
*/
int64_t query(
const uint32_t &l, const uint32_t &r,
const uint32_t
&version) // querying the range from "l" to "r" in a segment tree
// after "version" updates, "l" and "r" are 0 indexed
{
return query(0, n - 1, l, r, ptrs[version]);
}
/**
* @brief Getting the number of versions after updates so far which is equal
* to the size of the pointers vector
* @returns the number of versions
*/
uint32_t size() // returns the number of segment trees (versions) , the
// number of updates done so far = returned value - 1
// ,because one of the trees is the original segment tree
{
return ptrs.size();
}
};
} // namespace range_queries
/**
* @brief Test implementations
* @returns void
*/
static void test() {
std::vector<int64_t> arr = {-5, 2, 3, 11, -2, 7, 0, 1};
range_queries::perSegTree tree;
std::cout << "Elements before any updates are {";
for (uint32_t i = 0; i < arr.size(); ++i) {
std::cout << arr[i];
if (i != arr.size() - 1) {
std::cout << ",";
}
}
std::cout << "}\n";
tree.construct(
arr); // constructing the original segment tree (version = 0)
std::cout << "Querying range sum on version 0 from index 2 to 4 = 3+11-2 = "
<< tree.query(2, 4, 0) << '\n';
std::cout
<< "Subtract 7 from all elements from index 1 to index 5 inclusive\n";
tree.update(1, 5, -7); // subtracting 7 from index 1 to index 5
std::cout << "Elements of the segment tree whose version = 1 (after 1 "
"update) are {";
for (uint32_t i = 0; i < arr.size(); ++i) {
std::cout << tree.query(i, i, 1);
if (i != arr.size() - 1) {
std::cout << ",";
}
}
std::cout << "}\n";
std::cout << "Add 10 to all elements from index 0 to index 7 inclusive\n";
tree.update(0, 7, 10); // adding 10 to all elements
std::cout << "Elements of the segment tree whose version = 2 (after 2 "
"updates) are {";
for (uint32_t i = 0; i < arr.size(); ++i) {
std::cout << tree.query(i, i, 2);
if (i != arr.size() - 1) {
std::cout << ",";
}
}
std::cout << "}\n";
std::cout << "Number of segment trees (versions) now = " << tree.size()
<< '\n';
std::cout << "Querying range sum on version 0 from index 3 to 5 = 11-2+7 = "
<< tree.query(3, 5, 0) << '\n';
std::cout << "Querying range sum on version 1 from index 3 to 5 = 4-9+0 = "
<< tree.query(3, 5, 1) << '\n';
}
/**
* @brief Main function
* @returns 0 on exit
*/
int main() {
test(); // run self-test implementations
return 0;
}
