CSC 8301– Design and Analysis of Algorithms: Heaps · CSC 8301– Design and Analysis of...

Villanova University Department of Computing Sciences

CSC 8301– Design andAnalysis of Algorithms:

Heaps

Professor Henry CarterFall 2016


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

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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

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Example heap

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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

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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

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Heap insertion

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Max element deletion

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Heap creation (bottom-up)

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Heap creation

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Heap Creation Analysis

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Sorting with heaps

• Build the array into a heap

• Repeatedly remove the root

• Insert elements in reverse order

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Analysis

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Sort Comparisons

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Bubble Sort

Selection Sort

Insertion Sort

Mergesort Quicksort Heapsort

Strategy Brute force

Brute force

Decrease-n-conq

Divide-n-conq

Divide-n-conq

Transform-n-conq

Average efficiency

Θ(n2) Θ(n2) Θ(n2) Θ(n log n) Θ(n log n) Θ(n log n)

Worst efficiency

Θ(n2) Θ(n2) Θ(n2) Θ(n log n) Θ(n2) Θ(n log n)

Stable yes no yes yes no no

In-place yes yes yes no no yes


Puzzle

• Imagine a handful of uncooked spaghetti, individual rods whose lengths represent numbers that need to be sorted. Outline a “spaghetti sort”–a sorting algorithm that takes advantage of this unorthodox representation.

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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)?

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Nested representation

• 2x3 + x2 – 3x + 1

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Horner’s algorithm

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Horner(P [0 . . . n], x)input : an array A[] of coe�cients and a number x

output: The value of the polynomial with coe�cients in A[]

evaluated at x

p P [n]

for i n� 1 to 0 dop x ⇤ p+ P [i]

end


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

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The Binary Polynomial

• n =

• p(x) =

• Example: n = 13

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Exponent Math

• Multiplying by the base → addition in the exponent

• Squaring the current product → doubles the exponent

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Evaluating the polynomial

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Left-to-right Algorithm

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LeftRightExponentiation(a, b(n))

input : A number a 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 ⇤ productif bi = 1 then

product product ⇤ aend

end

return product


Challenge

• How would you extend the left-to-right binary exponentiation algorithm to work for every nonnegative integer exponent (including n = 0)?

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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

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Next Time...

• Levitin Chapter 6.6

‣ Remember, you need to read it BEFORE you come to class!

• Homework

‣ 6.4: 3, 6, 7, 8, 10

‣ 6.5: 4, 7

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CSC 8301– Design and Analysis of Algorithms: Heaps · CSC 8301– Design and Analysis of...

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