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switch to min heap now

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Jidong Xiao
2025-04-08 21:52:36 -04:00
committed by JamesFlare
parent 53824b1147
commit 6bdc38ef77
2 changed files with 197 additions and 1 deletions
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97
lectures/24_priority_queues_II/README.md
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@@ -112,7 +112,102 @@ organized heap data, but incur a O(n log n) cost. Why?
- Heap Sort is a simple algorithm to sort a vector of values: Build a heap and then run n consecutive pop operations, storing each “popped” value in a new vector.
- It is straightforward to show that this requires O(n log n) time.
- Exercise: Implement an in-place heap sort. An in-place algorithm uses only the memory holding the input data – a separate large temporary vector is not needed.
- Heap sort is an in-place sort. An in-place algorithm uses only the memory holding the input data – a separate large temporary vector is not needed.
- The following is the sort algorithm with a main function to test it; the code makes a min heap.
```cpp
/* The heapify function is designed to ensure that a subtree rooted at a given index i
* in an array representation of a min heap maintains the heap property.
*
* Why Not Just Heapify Once? A single call to heapify on the entire array
* wouldn't suffice because heapify is designed to correct violations of
* the heap property starting from a specific node, assuming its subtrees are already heaps.
* Initially, the array doesn't have this structure, so multiple calls are necessary to build the initial min-heap.
* Similarly, during the sorting phase, each extraction disrupts the heap structure, necessitating a call to heapify to restore order.
*/
void heapify(std::vector<int>& nums, int n, int i){
int smallest = i; // assuming i is the smallest
int left = 2*i+1; // i's left child is at this location
int right = 2*i+2; // i's right child is at this location
if(left<n && nums[left]<nums[smallest]){
smallest = left;
}
if(right<n && nums[right]<nums[smallest]){
smallest = right;
}
// after the above, smallest basically will either stay the same, or will be either left or right, depending on nums[left] is larger or nums[right] is larger. largest stays the same if it is already larger than its two children.
// if largest is changed, then we do need to swap.
if(smallest != i){
std::swap(nums[i], nums[smallest]);
heapify(nums, n, smallest);
}
}
// heap sort: O(nlogn)
std::vector<int> sortArray(std::vector<int>& nums) {
int n = nums.size();
// build the heap, starting from the last non-leaf node.
// why we start from the last non-leaf node? because leaf nodes inherently satisfy the heap property, as they have no children.
// By beginning the heapify process from the last non-leaf node and moving upwards:
// We ensure that when we heapify a node, its children are already heapified.
// This bottom-up approach guarantees that each subtree satisfies the heap property before moving to the next node.
for(int i=n/2-1; i>=0; i--){
// heapify the subtree whose root is at i
// i.e., build a min heap, with i being the root; and this heap contains nodes from i to n-1;
heapify(nums, n, i);
}
// now the first one is the largest, swap it to the back
// do this n-1 times.
for(int i=0; i<(n-1); i++){
// build the min heap again, with 0 being the root.
// but only consider n-1-i elements, as the others are already in the right place.
heapify(nums, n-1-i, 0);
}
return nums;
}
// Assuming the heapify and sortArray functions are defined above or included from another file
int main() {
// Sample data to be sorted
std::vector<int> nums = {42, 12, 13, 65, 98, 45, 97, 85, 76, 90};
// Output the original array
std::cout << "Original array:\n";
for (int num : nums) {
std::cout << num << " ";
}
std::cout << std::endl;
// Sort the array using your sortArray function
std::vector<int> sortedNums = sortArray(nums);
// Output the sorted array
std::cout << "\nSorted array:\n";
for (int num : sortedNums) {
std::cout << num << " ";
}
std::cout << std::endl;
return 0;
}
```
The above program prints the following:
```console
$ g++ heap_sort.cpp
$ ./a.out
Original array:
42 12 13 65 98 45 97 85 76 90
Sorted array:
12 42 13 65 90 45 97 85 76 98
```
## 24.9 Summary Notes about Vector-Based Priority Queues

101
lectures/24_priority_queues_II/heap_sort.cpp Normal file
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@@ -0,0 +1,101 @@
#include <iostream>
#include <vector>
/* The heapify function is designed to ensure that a subtree rooted at a given index i
* in an array representation of a heap maintains the heap property.
* While the function doesn't have an explicit base case like some recursive functions,
* it inherently terminates due to the following conditions:
*
* Leaf Node Condition: If the node at index i is a leaf node (i.e., it has no children),
* the function reaches a point where both left and right indices are greater than or equal to n (the size of the heap).
* In this scenario, the conditions left < n and right < n in the if statements evaluating the children will both be false,
* preventing further recursive calls.
*
* Heap Property Satisfaction: If the node at index i is greater than or equal to its children (or if it has no children),
* the heap property is already satisfied. Consequently, the variable largest remains equal to i,
* and the condition largest != i evaluates to false.
* This prevents the swap operation and the subsequent recursive call, leading to termination.
* In essence, the function will return when:
* The node is a leaf node.
* The node's value is greater than or equal to its children's values, maintaining the heap property.
* These implicit conditions ensure that the recursion does not continue indefinitely.
*
* Why Not Just Heapify Once? A single call to heapify on the entire array
* wouldn't suffice because heapify is designed to correct violations of
* the heap property starting from a specific node, assuming its subtrees are already heaps.
* Initially, the array doesn't have this structure, so multiple calls are necessary to build the initial max-heap.
* Similarly, during the sorting phase, each extraction disrupts the heap structure, necessitating a call to heapify to restore order.
* */
void heapify(std::vector<int>& nums, int n, int i){
int smallest = i; // assuming i is the smallest
int left = 2*i+1; // i's left child is at this location
int right = 2*i+2; // i's right child is at this location
if(left<n && nums[left]<nums[smallest]){
smallest = left;
}
if(right<n && nums[right]<nums[smallest]){
smallest = right;
}
// after the above, smallest basically will either stay the same, or will be either left or right, depending on nums[left] is larger or nums[right] is larger. largest stays the same if it is already larger than its two children.
// if largest is changed, then we do need to swap.
if(smallest != i){
std::swap(nums[i], nums[smallest]);
heapify(nums, n, smallest);
}
}
// heap sort: O(nlogn)
std::vector<int> sortArray(std::vector<int>& nums) {
int n = nums.size();
// build the heap, starting from the last non-leaf node.
// why we start from the last non-leaf node? because leaf nodes inherently satisfy the heap property, as they have no children.
// By beginning the heapify process from the last non-leaf node and moving upwards:
// We ensure that when we heapify a node, its children are already heapified.
// This bottom-up approach guarantees that each subtree satisfies the heap property before moving to the next node.
for(int i=n/2-1; i>=0; i--){
// heapify the subtree whose root is at i
// i.e., build a max heap, with i being the root; and this heap contains nodes from i to n-1;
heapify(nums, n, i);
}
// now the first one is the largest, swap it to the back
// do this n-1 times.
for(int i=0; i<(n-1); i++){
// nums[0] is always the largest one
std::swap(nums[0], nums[n-1-i]);
// build the max heap again, with 0 being the root.
// but only consider n-1-i elements, as the others are already in the right place.
heapify(nums, n-1-i, 0);
}
return nums;
}
// Assuming the heapify and sortArray functions are defined above or included from another file
int main() {
// Sample data to be sorted
std::vector<int> nums = {42, 12, 13, 65, 98, 45, 97, 85, 76, 90};
// Output the original array
std::cout << "Original array:\n";
for (int num : nums) {
std::cout << num << " ";
}
std::cout << std::endl;
// Sort the array using your sortArray function
std::vector<int> sortedNums = sortArray(nums);
// Output the sorted array
std::cout << "\nSorted array:\n";
for (int num : sortedNums) {
std::cout << num << " ";
}
std::cout << std::endl;
return 0;
}