## heaps
* heap is a data structure capable of giving you the smallest (or the largest) element in constant time, while adding or removing the smallest (or the largest) element on logarithmic time.
* heaps guarantees both insertion and deletion to have time complexity `O(log N)` (while maintaining get_max/get_min at `O(1)`).
* a heap is a binary tree with these properties:
* it must have all of **its nodes in a specific order**, and
* its shape must be **complete** (all the levels of the tree must be completely filled except maybe for the last one and the last level must have the left-most nodes filled, always).
* a max heap's **root node must** have all its children either **greater than or equal** to its children. a min heap is the opposite. duplicate values are allowed.
* heaps can be represented with linked lists, queues (arrays), or binary trees.
* in the case of an array heap:
* the parent node index is given by `n / 2`
* the left children index is `2 * n`
* the right children index is `2 * n + 1`
* a node is a leaf when its index `> n / 2`
* common applications of heap are: sort (heap sort), getting the top-k elements, and finding the kth element.
```python
class Heap:
def __init__(self):
self.heap = []
def heapify(self, n, i):
largest = i
left_children = 2 * i + 1
right_children = 2 * i + 2
if left_children < n and self.heap[i] < self.heap[left_children]:
largest = left_children
if right_children < n and self.heap[largest] < self.heap[right_children]:
largest = right_children
if largest != i:
self.heap[i], self.heap[largest] = self.heap[largest], self.heap[i]
self.heapify(n, largest)
def insert(self, num):
size = len(self.heap)
if size == 0:
self.heap.append(num)
else:
self.heap.append(num)
for i in range((size // 2) - 1, -1, -1):
self.heapify(size, i)
def delete_node(self, num):
size = len(self.heap)
i = 0
for i in range(size):
if num == self.heap[i]:
break
self.heap[i], self.heap[size - 1] = self.heap[size - 1], self.heap[i]
self.heap.remove(size - 1)
for i in range((len(self.heap) // 2) - 1, -1, -1):
self.heapify(len(self.heap), i)
```
---
### heapify
* it's cheaper to heapify an array of data (`O(N)`) than creating an empty heap and inserting each element (`O(N log(N))`).
* heapify means create a binary tree and then comparing each nodes with their child (and swapping when necessary).
* parents node can simply exchange with their smallest child (so the max number of exchanges is `N/2`) and leaves are left out.
* python's built-in heap differs from the standard implementation of a heap in two ways:
* firstly, it uses zero-based indexing, so it stores the root node at index zero instead of the size of the heap.
* secondly, the built-in module does not offer a direct way to create a max heap, instead, we must modify the values of each eelement when inserting in the heap, and when removing it from the heap.
```python
import heapq
min_heap = [3,1,2]
heapq.heapify(min_heap)
max_heap = [-x for x in min_heap]
heapq.heapify(max_heap)
heapq.heappush(min_heap, 5)
heapq.heappush(min_heap, -5)
min_elem = min_heap[0]
max_elem = -1 * max_heap[0
heapq.heappop(min_heap)
heapq.heappop(max_heap)
size_min_heap = len(min_heap)
size_max_heap = len(max_heap)
```
----
### priority queues
* a priority queue is an abstract data type with the following properties:
1. every item has a priority (usually an integer).
2. an item with a high priority is dequeued before an item with low priority.
3. two items with an equal priority are dequeued based on their order in the queue.
* priority queues can be implemented with a stack, queue, or linked list data structures.
---
### min heaps
* a **min heap** is a complete binary tree where each node is smaller than its children (the root is the min element).
* `insert`:
- insert the element at the bottom, by finding the most rightmost node and checking its children: if left is empty, insert there, otherwise, insert on right.
- then compare this node to each parent, exchanging them until the tree's properties are correct.
* `extract_min`:
- first, remove/return the top and then replace the tree's top with its latest element (the bottom most rightmost).
- then bubble down, swapping it with one of its children until the min-heap is properly restored
- there is no need for order between right and left, so this operation would only take `O(log N)` runtime.
* the code below is an example of an ad-hoc heap class in python.
```python
class MinHeap:
def __init__(self, size):
self.heapsize = size
self.minheap = [0] * (size + 1)
self.realsize = 0
def add(self, element):
if self.realsize + 1 > self.heapsize:
print("Too many elements!")
return False
self.realsize += 1
self.minheap[self.realsize] = element
index = self.realsize
parent = index // 2
while self.minheap[index] < self.minheap[parent] and index > 1:
self.minheap[parent], self.minheap[index] = self.minheap[index], self.minheap[parent]
index = parent
parent = index // 2
def peek(self):
return self.minheap[1]
def pop(self):
if self.realsize < 1:
print("Heap is empty.")
return False
else:
remove_element = self.minheap[1]
self.minheap[1] = self.minheap[self.realsize]
self.realsize -= 1
index = 1
while index <= self.realsize // 2:
left_children = index * 2
right_children = (index * 2) + 1
if self.minheap[index] > self.minheap[left_children] or \
self.minheap[index] > self.minheap[right_children]:
if self.minheap[left_children] < self.minheap[right_children]:
self.minheap[left_children], self.minheap[index] = self.minheap[index], self.minheap[left_children]
index = left_children
else:
self.minheap[right_children], self.minheap[index] = self.minheap[index], self.minheap[right_children]
index = right_children
else:
break
return remove_element
def size(self):
return self.realsize
def __str__(self):
return str(self.minheap[1 : self.realsize + 1])
```
---
### max heaps
* a **max heap** is a complete binary tree where each node is larger than its children (the root is the max element).
* `insert`:
* insert the element at the bottom, at the leftmost node.
* then compare the node to each parent, exchanging them until the tree's properties are correct.
* `extreact_max`:
* remove/return the top and then replace the tree's top with its bottom rightmost element.
* swap up until the max element is on the top.
* the code below is an example of a max heap class built in python:
```python
class MaxHeap:
def __init__(self, heapsize):
self.heapsize = heapsize
self.maxheap = [0] * (heapsize + 1)
self.realsize = 0
def add(self, element):
self.realsize += 1
if self.realsize > self.heapsize:
print("Too many elements!")
self.realsize -= 1
return False
self.maxheap[self.realsize] = element
index = self.realsize
parent = index // 2
while self.maxheap[index] > self.maxheap[parent] and index > 1:
self.maxheap[parent], self.maxheap[index] = self.maxheap[index], self.maxheap[parent]
index = parent
parent = index // 2
def peek(self):
return self.maxheap[1]
def pop(self):
if self.realsize < 1:
print("Heap is empty.")
return False
else:
remove_element = self.maxheap[1]
self.maxheap[1] = self.maxheap[self.realsize]
self.realsize -= 1
index = 1
while (index <= self.realsize // 2):
left_children = index * 2
right_children = (index * 2) + 1
if (self.maxheap[index] < self.maxheap[left_children] or self.maxheap[index] < self.maxheap[right_children]):
if self.maxheap[left_children] > self.maxheap[right_children]:
self.maxheap[left_children], self.maxheap[index] = self.maxheap[index], self.maxheap[left_children]
index = left_children
else:
self.maxheap[right_children], self.maxheap[index] = self.maxheap[index], self.maxheap[right_children]
index = right_children
else:
break
return remove_element
def size(self):
return self.realsize
def __str__(self):
return str(self.maxheap[1 : self.realsize + 1])
```
---
### heap sort
* the core concept of the heap sort involves constructing a heap from our input and then repeatedly removing the min/max element to sort the array.
* the key idea for in-place heap sort involves a balance of these two ideas:
- building a heap from an unsorted array through a "bottom-up heapification" process, and
- using the heap to sort the input array
* heapsort traditionally uses a max-heap to sort the array, although a min-heap also works.
* this is not a stable sort.
```python
def heap_sort(self, array) -> None:
def max_heapify(heap_size, index):
left, right = 2 * index + 1, 2 * index + 2
largest = index
if left < heap_size and array[left] > array[largest]:
largest = left
elif if right < heap_size and array[right] > array[largest]:
largest = right
elif largest != index:
array[index], array[largest] = array[largest], array[index]
max_heapify(heap_size, largest)
for i in range(len(lst) // 2 - 1, -1, -1):
max_heapify(len(array), i)
for i in range(len(array) - 1, 0, -1):
array[i], array[0] = array[0], array[i]
max_heapify(i, 0)
return array
```
----
### compare two tops
```python
def compare_two_tops(array) -> int:
for i in range(len(array)):
array[i] *= -1
heapq.heapify(array)
while len(array) > 1:
val1 = heapq.heappop(array)
val2 = heapq.heappop(array)
if val1 != val2:
heapq.heappush(array, val1 - val2)
if array:
return -heapq.heappop(array)
return 0
```
----
### find non-overlapping intervals
* given an array of `intervals[i] = [start_i, end_i]`, return the minimum the non-overlapping intervals
```python
def non_overlapping_invervals(intervals):
if not intervals:
return 0
result = []
intervals.sort(key=lambda x: x[0])
heapq.heappush(result, intervals[0][-1])
for interval in intervals[1:]:
if result[0] <= interval[0]:
heapq.heappop(result)
heapq.heappush(result, interval[1])
return len(result)
```
----
### top k frequent values
```python
def top_k_frequent_values(list, k):
if k == len(nums):
return nums
# hashmap element: frequency
counter = Counter(nums)
return heapq.nlargest(k, counter.keys(), key=counter.get)
```
----
### kth largest element in a stream
```python
class KthLargest:
def __init__(self, k, nums):
self.k = k
self.heap = nums
heapq.heapify(self.heap)
while len(self.heap) > k:
heapq.heappop(self.heap)
def add(self, val: int) -> int:
heapq.heappush(self.heap, val)
if len(self.heap) > self.k:
heapq.heappop(self.heap)
return self.heap[0]
```
----
### kth smallest element in a matrix
* given an `n x n` matrix where each of the rows and columns is sorted in ascending order, return the `kth` smallest element in the matrix.
```python
def kth_smallest(matrix, k) -> int:
min_heap = []
for row in range(min(k, len(matrix))):
min_heap.append((matrix[row][0], row, 0))
heapq.heapify(min_heap)
while k:
element, row, col = heapq.heappop(min_heap)
if col < len(matrix) - 1:
heapq.heappush(min_heap, (matrix[row][cow + 1], row, col + 1))
k -= 1
return element
```