Data Structures Heaps and Graphs

i206 Fall 2010

John Chuang

Some slides adapted from Marti Hearst, Brian Hayes, or Glenn Brookshear

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Outline

What is a data structure Basic building blocks: arrays and linked lists

Data structures (uses, methods, performance):- List, stack, queue- Dictionary- Tree --> Heap- Graph

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Heap

A specialized binary tree that satisfies- Heap-order property- Complete binary tree property

Useful for implementing priority queue, heap-sort algorithm

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Heap

Heap-order property: for every node v other than the root, the key stored at v is greater than or equal to the key stored at v’s parent.

Complete binary tree property: A binary tree with height h is a complete binary tree if levels 0, 1, 2, …, h-1 of the tree have the maximum number of nodes, and in level h-1, all the internal nodes are to the left of the external nodes, and there is at most one node with one child, which must be a left child.

12,H14,E25,J16,X

9,F

5,A

15,K

8,W11,S

7,Q 20,B

6,Z

4,C

last node

root node

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Heap Methods

Insert- Insert element as last node of the heap- May need to perform up-heap bubbling to restore heap-order property

Remove- Remove and return element at root node- Move last node to root node- May need to perform down-heap bubbling to restore heap-order property

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Example: Insert(2,T)

12,H14,E25,J16,X

9,F

5,A

15,K

8,W11,S

7,Q 20,B

6,Z

4,C

2,T 12,H14,E25,J16,X

9,F

5,A

15,K

8,W11,S

7,Q 2,T

6,Z

4,C

20,B

12,H14,E25,J16,X

9,F

5,A

15,K

8,W11,S

7,Q 6,Z

2,T

4,C

20,B 12,H14,E25,J16,X

9,F

5,A

15,K

8,W11,S

7,Q 6,Z

4,C

2,T

20,B

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Heap Methods

Insert- Insert element as last node of the heap- May need to perform up-heap bubbling to restore heap-order property

Remove- Remove and return element at root node- Move last node to root node- May need to perform down-heap bubbling to restore heap-order property

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Example: Remove

12,H14,E25,J16,X

9,F

5,A

15,K

13,W11,S

7,Q 20,B

6,Z

4,C

12,H14,E25,J16,X

9,F

5,A

15,K

11,S

7,Q 20,B

6,Z

4,C

13,W

12,H14,E25,J16,X

9,F

5,A

15,K

11,S

7,Q 20,B

6,Z

13,W

12,H14,E25,J16,X

9,F

13,W

15,K

11,S

7,Q 20,B

6,Z

5,A

12,H14,E25,J16,X

13,W

9,F

15,K

11,S

7,Q 20,B

6,Z

5,A

13,W14,E25,J16,X

12,H

9,F

15,K

11,S

7,Q 20,B

6,Z

5,A

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Heap Storage

Heap data easily stored in contiguous array

[0][0] [1] [2] [3] [4]

4,C 5,A 6,Z 15,K 9,F …

12,H14,E25,J16,X

9,F

5,A

15,K

8,W11,S

7,Q 20,B

6,Z

4,C

last node

root node

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Heap Running Times

What is the running time of each operation?

InsertO(logN)

RemoveO(logN)

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Heapsort

Given an unsorted list of n elements:- Insert n elements, then- Remove n elements

What is run-time of heapsort algorithm?

How does it compare to insertion sort?

http://www.cs.pomona.edu/~marshall/courses/2002/spring/cs50/BigO/

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Outline

What is a data structure Basic building blocks: arrays and linked lists

Data structures (uses, methods, performance):- List, stack, queue- Dictionary- Tree- Graph

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Internet Graphs

Source: Cheswick and Burch

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Social Network Graphs

American Journal of Sociology, Vol. 100, No. 1. "Chains of affection: The structure of adolescent romantic and sexual networks," Bearman PS, Moody J, Stovel K.

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Graph

A graph consists of a set of nodes (vertices) and a set of links (edges) that establish relationships (connections) between the nodes

Represented/stored using adjacency list or adjacency matrix data structures- Adjacency list for Graph 1: {a,b}, {a,c}, {b,c}- Adjacency matrix for Graph 2:

Edges can be directed/undirected Edges can have weights Tree is a special case of graph

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Graph Algorithms

Search/traversal: breadth-first or depth-first -- O(|V|+|E|)

Routing: shortest path between two points (Dijkstra’s algorithm) -- O(|V|2+|E|)

Minimum spanning tree -- O(|E|) Maximum Flow -- O(|V|3), O(|V|2|E|), O(|V||E|2)

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QuickTime™ and a decompressor

are needed to see this picture.

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Routing

Problem: network routers have to forward data packets toward destination; must determine next hop

Algorithm: Dijkstra’s algorithm- Shortest Path First (SPF) algorithm- Greedy algorithm- Input: graph with nodes and weighted edges

- Output: shortest paths from source node i to every other node; cost of each path

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Dijkstra’s Algorithm

Source: Doug Comer

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Algorithm Intuition

Start at source node Move outward At each step:

- Find node u that- has not been considered before; and- is closest to source

- Compute:- Distance from u to each neighbor v- If distance shorter, make path to v go through u

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Dijkstra’s Algorithm Example

DistancePredecessor

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Node A’s Routing Table

Destination Address

Next Hop (Cost)

B B (2)

C D (3)

D D (1)

E D (2)

F D (4)