CS 106X, Lecture 21 Tries; Graphs

reading: Programming Abstractions in C++, Chapter 18

This document is copyright (C) Stanford Computer Science and Nick Troccoli, licensed under Creative Commons Attribution 2.5 License. All rights reserved. Based on slides created by Keith Schwarz, Julie Zelenski, Jerry Cain, Eric Roberts, Mehran Sahami, Stuart Reges, Cynthia Lee, Marty Stepp, Ashley Taylor and others. Plan For Today

• Tries • Announcements • Graphs • Implementing a Graph • Representing Data with Graphs

2 Plan For Today

• Tries • Announcements • Graphs • Implementing a Graph • Representing Data with Graphs

3 The Lexicon

• Lexicons are good for storing words – contains – containsPrefix – add

4 The Lexicon

root

the

art we

all arts you

artsy

5 The Lexicon

root

the contains? art we containsPrefix? add? all arts you

artsy

6 The Lexicon

S T A R T L I N G

S T A R T

7 The Lexicon

• We want to model a set of words as a tree of some kind • The tree should be sorted in some way for efficient lookup • The tree should take advantage of words containing each other to save space and time

8 Tries trie ("try"): A tree structure optimized for "prefix" searches

struct TrieNode { bool isWord; TrieNode* children[26]; // depends on the alphabet };

9 Tries

isWord: false a b c d e … “”

isWord: true a b c d e … “a” isWord: false a b c d e … “ac”

isWord: true a b c d e … “ace”

10 Reading Words

Yellow = word in the trie

A E H S /

A E H S A E H S A E H S / / / / / / / /

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / /

11 Reading Words

A E H S /

A E H S A E H S A E H S / / / / / / / /

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / /

12 Reading Words

A E H S /

A E H S A E H S A E H S / / / / / / / /

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / / h 13 Reading Words

A E H S /

A E H S A E H S A E H S / / / / / / / /

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / / h a 14 Reading Words

A E H S /

A E H S A E H S A E H S / / / / / / / /

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / / h a s 15 Reading Words

A E H S /

A E H S A E H S A E H S / / / / / / / /

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / / h a s h 16 Reading Words

A E H S /

A E H S A E H S A E H S / / / / / / / /

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / / h a s h 17 Reading Words

What are all the A E H S / words represented by this trie? A E H S A E H S A E H S / / / / / / / /

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / /

18 Reading Words

What are all the A E H S / words represented by this trie? A E H S A E H S A E H S / / / / / / / / a as A E H S A E H S A E H S A E H S / / / / / / / / / / / / ha haha A E H S A E H S A E H S he / / / / / / / / / / she

A E H S A E H S / / / / / / / /

19 Reading Words

A E H S A E H S How can we write a / / / / / / / / function to print this?

20 PrintAllWords void printAllWords(TrieNode* root) { printAllWordsHelper(root, ""); } void printAllWordsHelper(TrieNode* root, string prefix) { if (root == nullptr) { return; } if (root->isWord) { cout << prefix << endl; } for (int i = 0; i < 26; i++) { printAllWordsHelper(root->children[i], prefix + char('a' + i)); } }

21 Prefixes

How can we write A E H S / containsPrefix?

A E H S A E H S A E H S containsPrefix(“a”) => true / / / / / / / / containsPrefix(“se”) => false

A E H S A E H S A E H S A E H S / / / / / / / / / / / /

A E H S A E H S A E H S / / / / / / / / / /

A E H S A E H S / / / / / / / /

22 containsPrefix bool containsPrefix(TrieNode* node, string prefix) { if (node == nullptr) { return false; } if (prefix.length() == 0) { return true; } return containsPrefix(node->children[prefix[0] - 'a'], prefix.substr(1)); }

23 Plan For Today

• Tries • Announcements • Graphs • Implementing a Graph • Representing Data with Graphs

24 Announcements

• Brown Institute Data Visualization @ NY Times talk with Kevin Quealy Tues. 11/13 4:30PM in Packard 101

25 Plan For Today

• Tries • Announcements • Graphs • Implementing a Graph • Representing Data with Graphs

26 Linked Structures

2 5 8

27 Linked Structures

2 5 8

28 Linked Structures

1 3 What about What about What about storing bidirectional links? representing cycles? data in the links, like distances or costs? 24

29 Linked Structures: Shortfalls

Representing airline flights and distances:

849 PVD 1843 ORD SFO 142

337 802 LGA 1743 2555 1387 1099 HNL 1233 LAX DFW 1120 MIA

30 Linked Structures: Shortfalls

31 Linked Structures: Shortfalls

32 Linked Structures: Shortfalls

33 Graphs

A graph consists of a set of nodes connected by edges.

Graphs can model: - Sites and links on the web - Disease outbreaks - Social networks - Geographies - Task and dependency graphs

- and more… 34 Graphs

A graph consists of a set of nodes connected by edges.

35 Nodes

Nodes (“Vertices”)

36 Edges

Edges (“Arcs”)

Directed

37 Edges

Edges (“Arcs”)

Undirected

38 Edges

Edges (“Arcs”)

Unweighted

39 Weights

A weight is a cost associated with a given edge. Edges (“Arcs”) 12

Weighted 5 6 1 18 4

12 22

40 Degree

Degree (# edges touching a node) 5

2 6

1 1

41 Degree

In-degree (# edges coming into a node) 2

2 2

1 1

42 Degree

Out-degree (# edges leaving a node) 3

0 4

0 0

43 Degree

Degree = In-degree + Out-degree

2 6

1 1

44 Neighbors

Neighbors (nodes you can directly reach)

45 Neighbors

Neighbors (nodes you can directly reach)

46 Neighbors

Neighbors (nodes you can directly reach)

47 Neighbors

Neighbors (nodes you can directly reach)

48 Neighbors

Neighbors (nodes you can directly reach)

49 Paths

Path (sequence of Paths are edges that go from represented as nodes one node to another) B visited or edges taken

Path length = 3 C (number of D nodes or edges in the path)

F E

50 Paths

Path (sequence of edges that go from one node to another) B

C D Node x is reachable from node y if a path F exists from y to E x.

51 Cycles

Cycle (path that begins and ends at the same node) B

C D

F E

52 Loops

Loop (an edge that connects a node to itself) B

C D

F E

53 Graph Properties

A graph is connected if every node is reachable from every other node.

54 Graph Properties

A graph is complete if every node has a direct edge to every other node.

55 Graph Properties

A graph is acyclic if it does not contain any cycles.

56 Graph Properties

A graph is weighted if its edges have weights, or unweighted if its edges do not have weights.

6 12 2

5 14

weighted unweighted

57 Graph Properties

A graph is directed if its edges have direction, or undirected if its edges do not have direction (aka are bidirectional).

directed undirected

58 Graph Properties

• Connected or unconnected • Acyclic • Directed or undirected • Weighted or unweighted • Complete

Which of these properties do binary trees have?

What about linked lists? 59 One Big, Happy Family

• A binary tree is a graph with some restrictions: – The tree is an unweighted, directed, acyclic graph (DAG). – Each node's in-degree is at most 1, and out-degree is at most 2. – There is exactly one path from the root to every node.

F • A linked list is also a graph: – Unweighted DAG. – In/out degree of at most 1 for all nodes. B K

A B C D A E H

G J

60 Graph examples

• For each, what are the vertices and what are the edges? – Web pages with links – Functions in a program that call each other – Road maps (e.g., Google maps) – Airline routes – Facebook friends – Course pre-requisites – Family trees – Paths through a maze – Chemical bonds

61 Plan For Today

• Tries • Announcements • Graphs • Implementing a Graph • Representing Data with Graphs

62 Adjacency List

• Map> – or Map> Node Set

63 Adjacency Matrix

• Store a boolean grid, rows/columns correspond to nodes – Alternative to Adjacency List

F T F T F

F F F F T

T F F F T

F F T F F

64 Edge List

• Store a Vector (or Set) – Edge struct would have the two nodes Vector

65 Stanford BasicGraph

• The Stanford C++ library includes a BasicGraph class representing a weighted, directed graph. – Based on an older library class named Graph • You can construct a graph and add vertices/edges: #include "basicgraph.h" ... BasicGraph graph; a b graph.addVertex("a"); graph.addVertex("b"); graph.addVertex("c"); graph.addVertex("d"); c d graph.addEdge("a", "c"); graph.addEdge("b", "c"); graph.addEdge("c", "b"); graph.addEdge("b", "d"); graph.addEdge("c", "d"); 66 BasicGraph members

#include "basicgraph.h" // a directed, weighted graph g.addEdge(v1, v2); adds an edge between two vertexes g.addVertex(name); adds a vertex to the graph g.clear(); removes all vertexes/edges from the graph g.getEdgeSet() returns all edges, or all edges that start at v, g.getEdgeSet(v) as a Set of pointers g.getNeighbors(v) returns a set of all vertices that v has an edge to g.getVertex(name) returns pointer to vertex with the given name g.getVertexSet() returns a set of all vertexes g.isNeighbor(v1, v2) returns true if there is an edge from vertex v1 to v2 g.isEmpty() returns true if queue contains no vertexes/edges g.removeEdge(v1, v2); removes an edge from the graph g.removeVertex(name); removes a vertex from the graph g.size() returns the number of vertexes in the graph g.toString() returns a string such as "{a, b, c, a -> b}" 67 Using BasicGraph

• The graph stores a struct of information about each vertex/edge: struct Vertex { struct Edge { string name; Vertex* start; a b Set edges; Vertex* finish; double weight; 3 ...... }; }; c d • These are returned as pointers by various functions: // example usage Vertex* vc = graph.getVertex("c"); for (Vertex* neighbor : graph.getNeighbors(vc)) { cout << neighbor->edges << endl; } Edge* edgeAC = graph.getEdge("a", "c"); cout << edgeAC->start->name << endl; // a cout << edgeAC->end->name << endl; // c 68 Plan For Today

• Tries • Announcements • Graphs • Implementing a Graph • Representing Data with Graphs

69 Twitter Influence

https://about.twitter.com/en_us/company/brand-resources.html 70 Twitter Influence

• Twitter lets a user follow another user to see their posts. • Following is directional (e.g. I can follow you but you don’t have to follow me back L) • Let’s define being influential as having a high number of followers-of-followers. – Reasoning: doesn’t just matter how many people follow you, but whether the people who follow you reach a large audience.

• Write a function mostInfluential that reads a file of Twitter relationships and outputs the most influential user.

https://about.twitter.com/en_us/company/brand-resources.html 71 Recap

• Tries • Announcements • Graphs • Implementing a Graph • Representing Data with Graphs

Next time: Graphs and Graph Algorithms