CSC 1700 Analysis of Algorithms: Heaps

Transcription

1 CSC 1700 Analysis of Algorithms: Heaps Professor Henry Carter Fall 2016

2 Recap Transform-and-conquer preprocesses a problem to make it simpler/more familiar Three types: Instance simplification Representation change Problem reduction Balanced search trees take advantage of modified representation to make data access more efficient 2

3 Data Structure: Heaps Binary tree structure Two properties: Shape: the tree must be (essentially) complete Heap: the key in each node must be greater than or equal to the keys stored in its children Min-heaps reverse the heap property inequality 3

4 Example heap 4

5 Heap application: priority queue Data structure operations: Add element with a priority Read the highest priority element Remove the highest priority element In a heap, the highest priority element is always the root 5

6 A heap of fun facts For any size n there is a unique heap structure The root is always the largest element Any node considered with its descendants is also a heap A heap can be implemented as an array Store each level sequentially 6

7 Heap insertion 7

8 Max element deletion 8

9 Heap creation (bottom-up) 9

10 Heap creation 10

11 Heap Creation Analysis 11

12 Sorting with heaps Build the array into a heap Repeatedly remove the root Insert elements in reverse order 12

13 Analysis 13

14 Sort Comparisons Bubble Sort Selection Sort Insertion Sort Mergesort Quicksort Heapsort Strategy Brute force Brute force Decreasen-conq Divide-nconq Divide-nconq Transform -n-conq Average efficiency Worst efficiency Θ(n 2 ) Θ(n 2 ) Θ(n 2 ) Θ(n log n) Θ(n log n) Θ(n log n) Θ(n 2 ) Θ(n 2 ) Θ(n 2 ) Θ(n log n) Θ(n 2 ) Θ(n log n) Stable yes no yes yes no no In-place yes yes yes no no yes 14

15 Further representation change: Horner s rule Polynomial evaluation problem Previous discussion revealed the algorithm must be Ω(n) Restricting operations to addition and multiplication, can we implement it in Θ(n)? 15

16 Nested representation 2x 3 + x 2 3x

17 Horner s algorithm Horner(P [0...n],x) input : an array A[] of coe cients and a number x output: The value of the polynomial with coe p P [n] evaluated at x for i n 1 to 0 do p x p + P [i] end cients in A[] 17

18 Binary Exponentiation Borrow the nested representation from Horner s rule Represent the exponent as a polynomial Perform equivalent exponent math to evaluate the polynomial based on the bits of the exponent 18

19 The Binary Polynomial n = p(x) = Example: n = 13 19

20 Exponent Math Multiplying by the base addition in the exponent Squaring the current product doubles the exponent 20

21 Evaluating the polynomial 21

22 Left-to-right Algorithm LeftRightExponentiation(a, b(n)) input :Anumbera and a list of binary digits b(n) in the positive integer n output: The value a n product a for i I 1 downto 0 do product product product end if b i =1then product end return product product a 22

23 Recap Representation change allows for unintuitive performance gains Heaps allow for fast implementations of priority queues and fast sorting Horner s rule provides a nested representation for fast polynomial evaluation Fast exponentiation can be achieved based on a modification of Horner s rule 23

24 Next Time... Levitin Chapter 6.6 Remember, you need to read it BEFORE you come to class! Homework 6.4: 2, 3, 5, 10, : 3, 4, 10 24