Heap & Graph Search Algorithms

Heap/Priority Queue

Properties

1. 任意节点小于它的所有后裔，最小元素在root上
2. heap总是complete tree. 可以将所有元素放在一个array里。 complete tree: 每一层都fully filled， 除了最后一层，如果最后一层有缺，尽量往左靠。
3. 将根结点最大的heap叫max heap，根结点最小的heap叫min heap。
4. index of lChild = index of parent*2+1
5. index of rChild = index of parent*2 +2
6. unsorted but follow rules above

Operations

1. insert: 先放在最后，持续往上swap。时间复杂度O(log(n))
2. update: 将新元素提升使其符合heap的性质，如果新元素比原来的元素大，和更小的child互换。时间复杂度O(log(n))
3. get/top
4. pop: pop之后，把最后一个元素放在上面
5. heapify: 使得一个unsorted array变成一个heap。时间复杂度O(n)。

经典考题

Find smallest k elements from an unsorted array of size n.

Assumption: What is the relationship between k and n?

Solution 1:

sort it and return the first k elements. O(log(n).

Solution 2:

1. Build a min-heap: heapify it -> O(n).
2. Keep popping out k element O(klogn). 因为pop完还要重新组合。

Solution 3:

1. use a max-heap of size k -> O(logk)，先heapify前k个elements
2. from the k+1_th element to the n_th element, and for the current element x
   Case1: if x < max-heap.top(), max-heap.pop(), and max-heap.insert(x);
   Case2: else, do nothing
3. O(k) + O((n-k)log(k))

Compare Solution 2 and Solution 3

1. Case1: k<<<<<<<n, it depends, how small is k
2. Case2: k = n, it depends, hard to say, depending on the value of k versus n