Objectives Data structure: Heaps Data structure: Graphs Submit - - PDF document

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Objectives Data structure: Heaps Data structure: Graphs Submit - - PDF document

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1/25/19 Objectives Data structure: Heaps Data structure: Graphs Submit problem set 2 Jan 25, 2019 CSCI211 - Sprenkle 1 Review What is a priority queue? What is a heap? Properties Implementation What is the process for

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Objectives

  • Data structure: Heaps
  • Data structure: Graphs

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Submit problem set 2

Review

  • What is a priority queue?
  • What is a heap?

Ø Properties Ø Implementation

  • What is the process for finding the smallest

element in a heap?

  • What is the process for adding to a heap?

Ø What is the runtime of adding to a heap?

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Review: Heap Defined

  • Combines benefits of sorted array and list
  • Balanced binary tree

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root

  • Each node has at most 2 children
  • Node value is its key

Heap order: each node’s key is at least as large as its parent’s Note: not a binary search tree

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Review: Implementing a Heap

  • Option 1: Use pointers

Ø Each node keeps

  • Element it stores (key)
  • 3 pointers: 2 children, parent
  • Option 2: No pointers

Ø Requires knowing upper bound on n Ø For node at position i

  • left child is at 2i
  • right child is at 2i+1

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Review: Implementing a Heap

  • Finding the minimal element

Ø First element Ø O(1)

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Review: Heapify-Up

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Heapify-up(H, i): if i > 1 then j=parent(i)=floor(i/2) if key[H[i]] < key[H[j]] then swap array entries H[i] and H[j] Heapify-up(H, j)

Heap Position where node added

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Heapify-Up

  • Claim. Assuming array H is almost a heap with

key of H[i] too small, Heapify-Up fixes the heap property in O(log i) time

Ø Can insert a new element in a heap of n elements in O(log n) time

  • Proof. By induction

Ø If i=1 …

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Heapify-Up

  • Claim. Assuming array H is almost a heap with

key of H[i] too small, Heapify-Up fixes the heap property in O(log i) time

Ø Can insert a new element in a heap of n elements in O(log n) time

  • Proof. By induction

Ø If i=1, is already a heap à O(1) Ø If i>1, …

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Heapify-Up

  • Claim. Assuming array H is almost a heap with

key of H[i] too small, Heapify-Up fixes the heap property in O(log i) time

Ø Can insert a new element in a heap of n elements in O(log n) time

  • Proof. By induction

Ø If i=1, is already a heap à O(1) Ø If i>1,

  • Swaps are O(1)
  • Swaps continue up to root (max) à log i

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Deleting an Element

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Delete at position 3 w

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Deleting an Element

  • Delete at position i
  • Removing an element:

Ø Messes up heap order Ø Leaves a “hole” in the heap

  • Not as straightforward as Heapify-Up
  • Algorithm:
  • 1. Fill in element where hole was
  • Patch hole: move nth element into ith spot
  • 2. Adjust heap to be in order
  • At position i because moved nth item up to i

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Deleting an Element

  • Two “bad” possibilities: element w is

Ø Too small: violation is between it and parent à Heapify-Up Ø Too big: with one or both children à Heapify-Down (example: w becomes 12)

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Delete at position 3 w

Example of OK: 11 deleted, replaced by 16

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Deleting an Element

  • Delete 9
  • Replace with 5 (from other side of heap)
  • But 5 < 6, so need to Heapify-Up

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Example where new key is too small

3 4 7 5 6 2 10

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Heapify-Down

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Heapify-down(H, i): n = length(H) if 2i > n then Terminate with H unchanged else if 2i < n then left=2i and right=2i+1 j be index that minimizes key[H[left]] and key[[H[right]] else if 2i = n then j=2i if key[H[j]] < key[H[i]] then swap array entries H[i] and H[j] Heapify-down(H, j)

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Why can we stop?

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Heapify-Down

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Heapify-down(H, i): n = length(H) if 2i > n then Terminate with H unchanged else if 2i < n then left=2i and right=2i+1 j be index that minimizes key[H[left]] and key[[H[right]] else if 2i = n then j=2i if key[H[j]] < key[H[i]] then swap array entries H[i] and H[j] Heapify-down(H, j)

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i is a leaf – nowhere to go

Practice: Heapify-Down

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Moved 21 to where element was removed

21

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Practice: Heapify-Down

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21 21 7

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Practice: Heapify-Down

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21 7 8 7 21

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Runtime of Heapify-Down?

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Heapify-down(H, i): n = length(H) if 2i > n then Terminate with H unchanged else if 2i < n then left=2i and right=2i+1 j be index that minimizes key[H[left]] and key[[H[right]] else if 2i = n then j=2i if key[H[j]] < key[H[i]] then swap array entries H[i] and H[j] Heapify-down(H, j)

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O(1) O(1) Num swaps: O(log n)

Implementing Priority Queues with Heaps

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Operation Description Run Time StartHeap(N)

Creates an empty heap that can hold N elements

Insert(v)

Inserts item v into heap

FindMin()

Identifies minimum element in heap but does not remove it

Delete(i)

Deletes element in heap at position i

ExtractMin()

Identifies and deletes an element with minimum key from heap

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Implementing Priority Queues with Heaps

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Operation Description Run Time StartHeap(N)

Creates an empty heap that can hold N elements

O(N) Insert(v)

Inserts item v into heap

O(log n) FindMin()

Identifies minimum element in heap but does not remove it

O(1) Delete(i)

Deletes element in heap at position i

O(log n) ExtractMin()

Identifies and deletes an element with minimum key from heap

O(log n)

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Comparing Data Structures

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Operation Heap Unsorted List Sorted List Start(N) O(1) O(1) Insert(v) O(1) O(n) FindMin() O(1) O(1) Delete(i) O(n) O(1) ExtractMin() O(n) O(1)

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Comparing Data Structures

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Operation Heap Unsorted List Sorted List Start(N) O(N) O(1) O(1) Insert(v) O(log n) O(1) O(n) FindMin() O(1) O(1) O(1) Delete(i) O(log n) O(n) O(1) ExtractMin() O(log n) O(n) O(1)

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Putting It All Together…

  • 1. Add elements into PQ with the number’s value

as its priority

  • 2. Then extract the smallest number until done

Ø Come out in sorted order

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What is the running time of sorting numbers using a PQ implemented with a heap?

O(n log n)

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Additional Heap Operations

  • Access elements in PQ by “name”

Ø Maintain additional array Position that stores current position of each element in heap

  • Operations:

Ø Delete(Position[v])

  • Does not increase overall running time

Ø ChangeKey(v, α)

  • Changes key of element v to α
  • Identify position of element v in array (Position array)
  • Change key, heapify

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Key 2 4 5 6 9 20 Value 3542 5143 8712 1264 9123 5954

Process id Priority

GRAPHS

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Undirected Graphs G = (V, E)

  • V = nodes (vertices)
  • E = edges between pairs of nodes
  • Captures pairwise relationship between
  • bjects
  • Graph size parameters: n = |V|, m = |E|

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V = { 1, 2, 3, 4, 5, 6, 7, 8 } E = { 1-2, 1-3, 2-3, 2-4, 2-5, 3-5, 3-7, 3-8, 4-5, 5-6 } n = 8 m = 11

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Social Networks

  • Node: people; Edge: relationship between 2 people
  • Everything Bad Is Good for You: How Today's

Popular Culture Is Actually Making Us Smarter

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Television shows have complex plots, complex social networks

Social network of Game of Thrones

http://www.cs.duke.edu/csed/harambeenet/modules.html

Jan 25, 2019

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Facebook: Visualizing Friends

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http://www.facebook.com/notes/facebook- engineering/visualizing-friendships/469716398919

World Wide Web

  • Web graph

Ø Node: web page Ø Edge: hyperlink from one page to another

30 cnn.com people.com bleacherreport.com netscape.aol.com time.com hbo.com gameofthrones.com

Directed Graph:

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Ecological Food Web

  • Food web graph

Ø Node = species Ø Edge = from prey to predator

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Reference:

https://www.msu.edu/course/isb/202/ebe rtmay/images/foodweb.jpg

Directed Graph:

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Graph Applications

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transportation

Graph

street intersections

Nodes Edges

highways communication computers fiber optic cables World Wide Web web pages hyperlinks social people relationships food web species predator-prey software systems functions function calls scheduling tasks precedence constraints circuits gates wires

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Graph Representation: Adjacency Matrix

  • nn matrix with Auv = 1 if (u, v) is an edge

Ø Two representations of each edge (symmetric matrix) Ø Space? Ø Checking if (u, v) is an edge? Ø Identifying all edges?

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1 2 3 4 5 6 7 8 1 0 1 1 0 0 0 0 0 2 1 0 1 1 1 0 0 0 3 1 1 0 0 1 0 1 1 4 0 1 0 0 1 0 0 0 5 0 1 1 1 0 1 0 0 6 0 0 0 0 1 0 0 0 7 0 0 1 0 0 0 0 1 8 0 0 1 0 0 0 1 0

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Graph Representation: Adjacency Matrix

  • nn matrix with Auv = 1 if (u, v) is an edge

Ø Two representations of each edge (symmetric matrix) Ø Space: Q(n2) Ø Checking if (u, v) is an edge: Q(1) time Ø Identifying all edges: Q(n2) time

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1 2 3 4 5 6 7 8 1 0 1 1 0 0 0 0 0 2 1 0 1 1 1 0 0 0 3 1 1 0 0 1 0 1 1 4 0 1 0 0 1 0 0 0 5 0 1 1 1 0 1 0 0 6 0 0 0 0 1 0 0 0 7 0 0 1 0 0 0 0 1 8 0 0 1 0 0 0 1 0

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Graph Representation: Adjacency List

  • Node indexed array of lists

Ø Two representations of each edge Ø Space? Ø Checking if (u, v) is an edge? Ø Identifying all edges?

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1 2 3 2 3 4 2 5 5 6 7 3 8 8 1 3 4 5 1 2 5 8 7 2 3 4 6 5 3 7

node edges

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What are the extremes?

Graph Representation: Adjacency List

  • Node indexed array of lists

Ø Two representations of each edge Ø Space = 2m + n = O(m + n) Ø Checking if (u, v) is an edge takes O(deg(u)) time Ø Identifying all edges takes Q(m + n) time

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degree = number of neighbors of u node edges

1 2 3 2 3 4 2 5 5 6 7 3 8 8 1 3 4 5 1 2 5 8 7 2 3 4 6 5 3 7

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Paths and Connectivity

  • Def. A path in an undirected graph G = (V, E) is a

sequence P of nodes v1, v2, …, vk-1, vk

Ø Each consecutive pair vi, vi+1 is joined by an edge in E

  • Def. A path is simple if all nodes are distinct
  • Def. An undirected graph is connected if ∀ pair
  • f nodes u and v, there is a path between u and v

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  • Short path
  • Distance

Jan 25, 2019 CSCI211 - Sprenkle