Heaps. (our first advanced data structure)

Transcription

1 Heaps (our first advanced data structure) Data Structures Used in essentially every single programming task that you can think of What are some examples of data structures? What are some example programs? What do they do? They organize data so that it can be effectively accessed. A data structure is not necessarily a method of laying out data in memory It is a way of logically thinking about your data. 1

2 The Heap Data Structure (not heap memory) A container for objects that have key values Operations: Insertion : O(lg n) Extract-min : O(lg n) Heapify : O(n) for batched insertions Arbitrary Deletion : O(lgn) Good for continually getting a minimum value Heap used to improve algorithm Selection sort Continually look for the smallest element The element currently being considered is in blue The current smallest element is in red Sorted elements are in yellow 2

3 Heap used to improve algorithm Selection sort Continually look for the smallest element The element currently being considered is in blue The current smallest element is in red Sorted elements are in yellow Heap used to improve algorithm Selection sort Continually look for the smallest element The element currently being considered is in blue The current smallest element is in red Sorted elements are in yellow 3

4 Heap used to improve algorithm Selection sort Continually look for the smallest element What is the runtime of this algorithm? Can we make it faster? Yes! With a heap: O(n 2 ) à O(n lg n) Insert all elements into a heap: n Extract each element: n * lg n Example : Event Manager Uses a priority queue (synonym for Heap) Example: simulation Example: task manager

5 Heap Implementation Conceptually you should think of a Heap as a binary tree It is actually usually implemented using an array (why?) Heap Property: for any given node x, 1. key[x] key[x s left child], and 2. key[x] key[x s right child] Where is the minimum key? Heap Implementation Note: Heaps are not unique You can have multiple different configurations that hold the same data

6 Level 0 Level 1 8 How do you calculate the index of a given nodes parent? How do you calculate the indices of a given nodes children? Level Level Level 0 Level 1 8 Parent(i) = i/2 if i is even Parent(i) = floor(i/2) if i is odd Children(i) = 2i and 2i + 1 Level Base 1 indices Level

7 Insert: Insert:

8 Insert: Insert:

9 Insert: Insert:

10 Insert: Insert:

11 Extract Min Extract Min

12 Extract Min Extract Min

13 Extract Min Now what? Extract Min

14 Extract Min Extract Min

15 Dijkstra s Algorithm visited = {s} path_lengths[s] = 0 What is the running time? while not all nodes visited: of all edges (v,w) where v is visited, and w is not visited pick an edge that minimizes path_lengths[v] + lvw call the edge (v*, w*) add w* to visited set path_lengths[w*] = path_lengths[v*] + lv*w* How can we improve the running time with a Heap? Dijkstra s Algorithm visited = {s} path_lengths[s] = 0 while not all nodes visited: Store vertices in heap of all edges (v,w) where v is visited, and à O(m lg n) w is not visited pick an edge that minimizes path_lengths[v] + lvw call the edge (v*, w*) add w* to visited set path_lengths[w*] = path_lengths[v*] + lv*w* 15

16 Dijkstra s Algorithm visited = {s} path_lengths[s] = 0 State of the art of Dijkstra s: O(m + n lg n) while not all nodes visited: of all edges (v,w) where v is visited, and w is not visited pick an edge that minimizes path_lengths[v] + lvw call the edge (v*, w*) add w* to visited set path_lengths[w*] = path_lengths[v*] + lv*w* Elements in heap are the vertices not yet visited. à O(n lg m) 16