Graphs, Part 1. Please take a handout & Sit in row H or forward. Week 12, Class 21. CS 60, Spring 2016 B D

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1 Graphs, Part 1 nope nope yep! upload.wikimedia.org/wikipedia/commons/4/45/wikimedia_ommons_monthly_uploads_graph.png upload.wikimedia.org/wikipedia/commons/9/93/giraffe_head_1a_( ).jpg Please take a handout & Sit in row H or forward S 60, Spring 2016 Week 12, lass 21

2 The Seven ridges of Königsberg an you: start at point, cross every bridge only once, and return to point? Leonard Euler No! commons.wikimedia.org/wiki/file:leonhard_euler.jpg Koenigsberg, Map by Merian-Erben 1652 commons.wikimedia.org/wiki/file:image-koenigsberg,_map_by_merian-erben_1652.jpg

3 We need a model Koenigsberg, Map by Merian-Erben 1652 commons.wikimedia.org/wiki/file:image-koenigsberg,_map_by_merian-erben_1652.jpg

4 Graphs! set of nodes/vertices (places), and a set of edges (links) nodes edge

5 This week s theme: Graphs Our goals are to: practice implementing data structures and algorithms functional (Racket) and imperative (Java) implementations learn / implement / compare multiple representations There is more than one way to represent a graph! know when to use the right data structure If we care about the relationships among pieces of data, we might want a graph.

6 This week s theme: Graphs Today why graphs? because relationships matter! graph vocabulary node edge (un)directed (un)weighted connected cycle graph as data structure: interface Racket graphs plus: use-it-or-lose-it! Thursday graph as data structure: representation (in Java) edge list adjacency list adjacency matrix ssignment graphs in Racket & Java, plus: Harry Potter!

7 Graphs represent relationships Like what? node is n edge is nodes edge

8 Graphs represent relationships Like Facebook node is a Facebook user n edge is a friendship

9 Undirected graph important vocabulary! We can traverse the edge in both directions. The relationship is mutual.

10 Undirected graph important vocabulary! We can traverse the edge in both directions. The relationship is mutual.

11 Graphs represent relationships Like Twitter node is a Twitter user n edge is a follow williamjturkel.files.wordpress.com/2011/08/fig-5-niche-twitter-followers jpg

12 irected graph important vocabulary! We can traverse the edge in one direction. The relationship is one way. edge destination edge source

13 irected graph important vocabulary! We can traverse the edge in one direction. The relationship is one way. edge source edge destination

14 Graphs represent relationships Like highways node is a city n edge is a highway from one city to another upload.wikimedia.org/wikipedia/commons/4/45/l-ie_freeway_system.jpg

15 Weighted graph important vocabulary! Information (usually cost ) associated with each edge

16 Graphs represent relationships Like flights node is a city n edge is a flight from one city to another

17 irected weighted graph important vocabulary! Information (usually cost ) associated with each edge edge source edge weight 0 edge destination

18 Undirected irected Unweighted facebook twitter Weighted highways flights weights = distance, time, cost weights = distance, time, cost

19 onnected graph important vocabulary! There is a path between each pair of nodes. In other words, each node is reachable from every other node. If this is true in a directed graph, the graph is strongly connected. connected not connected

20 We ve seen graphs before! 7& 27& 3& 6& 13& 1& 42& 2& 5& ville& (ity&1)& eesburg& (ity&2)& ee&ity& (ity&3)& eesdale& (ity&4)& Etown& (ity&5)&

21 We ve seen graphs before! a list is a a tree is a graph graph there may be more than one path to a node tree connected, directed graph exactly one path to each node linked list connected, directed graph exactly one path to each node each node has at most one child

22 ycle important vocabulary! no cycles there are a finite number of paths to a node cycle there may be an infinite number of paths to a node

23 esigning and implementing a new data structure Interface escribes: what this data structure can do Implementation: encoding escribes: how the structure is stored, using existing data structures Implementation: operations efines: how the structure provides its interface via algorithms over the encoding It should be possible to replace the implementation without modifying the interface.

24 esigning and implementing a new data structure Interface escribes: what this data structure can do Nodes construct set contents get contents Edges construct set source set destination set weight get source get destination get weight Graphs construct empty graph add / remove edge add / remove node test if empty get all edges / nodes search for edge / node is node b reachable from a?

25 Graphs in Racket

26 esigning and implementing a new data structure Interface escribes: what this data structure can do Implementation: encoding escribes: how the structure is stored, using existing data structures Implementation: operations efines: how the structure provides its interface via algorithms over the encoding It should be possible to replace the implementation without modifying the interface. Our Racket graphs will be connected, weighted, directed, and functional

27 esigning and implementing a new data structure Interface escribes: what this data structure can do Nodes construct set contents get contents Edges construct set source set destination set weight get source get destination get weight Graphs construct empty graph add / remove edge add / remove node test if empty get all edges / nodes search for edge / node is node b reachable from a?

28 Our Racket graphs: Interface this interface is not built in to Racket Edges onstructor (make-edge source destination weight) Graphs onstructor (make-empty-graph) Operations (emptygraph? graph) (edge-list graph) Operations (src edge) (dst edge) (weight edge) Modifications return a new graph (add-edge edge graph) (remove-edge edge graph)

29 Our Racket graphs: examples > (make-edge ' ' ) > (src ) ' > (dst ) ' > (weight ) > (edge-list ) '(

30 esigning and implementing a new data structure Interface escribes: what this data structure can do Nodes construct set contents get contents Edges construct set source set destination set weight get source get destination get weight Graphs construct empty graph add / remove edge add / remove node test if empty get all edges / nodes search for edge / node is node b reachable from a?

31 ;; node-list: ;; inputs: graph ;; outputs: list of nodes in graph (no duplicates) (define (node-list graph) (let* ([edges (edge-list graph)] [srcs (map src edges)] [dsts (map dst edges)]) (remove-duplicates (append srcs dsts)))) Q graph operations (emptygraph? graph) (edge-list graph) (add-edge edge graph) (remove-edge edge graph) (src edge) (dst edge) list operations (empty? list) (cons elem list) (first list) (rest list) (map function list) (append list1 list2) (member elem list) (remove-duplicates list) and more: docs.racket-lang.org/reference/pairs.html

32 ;; node-list: ;; inputs: graph ;; outputs: list of nodes in graph (no duplicates) (define (node-list graph) (let* ([edges (edge-list graph)] [srcs (map src edges)] [dsts (map dst edges)]) (remove-duplicates (append srcs dsts)))) Q graph operations (emptygraph? graph) (edge-list graph) (add-edge edge graph) (remove-edge edge graph) (src edge) (dst edge) list operations (empty? list) (cons elem list) (first list) (rest list) (map function list) (append list1 list2) (member elem list) (remove-duplicates list) and more: docs.racket-lang.org/reference/pairs.html

33 ;; contains-edge: ;; inputs: an edge and a graph ;; outputs: true if the graph contains the edge (define (contains-edge edge graph) (list? (member edge (edge-list graph)))) ;; contains-node: ;; inputs: a node and a graph ;; outputs: true if the node is in the graph (define (contains-node node graph) (list? (member node (node-list graph)))) graph operations (emptygraph? graph) (edge-list graph) (add-edge edge graph) (remove-edge edge graph) (src edge) (dst edge) list operations (empty? list) (cons elem list) (first list) (rest list) (map function list) (append list1 list2) (member elem list) (remove-duplicates list) and more: docs.racket-lang.org/reference/pairs.html

34 esigning and implementing a new data structure Interface escribes: what this data structure can do Nodes construct set contents get contents Edges construct set source set destination set weight get source get destination get weight Graphs construct empty graph add / remove edge add / remove node test if empty get all edges / nodes search for edge / node is node b reachable from a?

35 upload.wikimedia.org/wikipedia/commons/8/8c/key_break.jpg

36 etour: Use it or lose it

37 Review: use it or lose it a recursive strategy fill in the pieces ase case(s) Recursive case(s) 1. It: a piece of the problem 2. Lose it: How could we solve a smaller version without it? 3. Use it: How could we solve a smaller version with it? 4. How would we combine the solutions to solve the full problem?

38 > (uniq '(c a l i f o r n i a)) '(c l f o r n i a) > (uniq '(m u d d)) '(m u d) > (uniq '(m i s s i p p i)) '(m s p i)

39 unique: use it or lose it a recursive strategy fill in the pieces ase case(s) empty list empty list Recursive case(s) 1. It: a piece of the problem the first element 2. Smaller version: What does the input look like without it? the rest of the list 3. Lose it: How could we solve a smaller version without it? unique-ify the rest of the list 4. Use it: How could we solve a smaller version with it? pre-pend it to the lose-it solution > (uniq '(c a l i f o r n i a)) '(c l f o r n i a) > (uniq '(m u d d)) '(m u d) > (uniq '(m i s s i p p i)) '(m s p i) 5. How would we combine the solutions to solve the full problem? if it is in lose-it, then lose-it ; else use-it

40 unique: use it or lose it a recursive strategy (define (uniq L) (if (empty? L) '() (let* ([it (first L)] [lose-it (uniq (rest L))] [use-it (cons it lose-it)]) (if (member it lose-it) lose-it use-it))))

41 unique: use it or lose it a recursive strategy (define (uniq L) (if (empty? L) '() (let* ([it (first L)] [lose-it (uniq (rest L))] [use-it (cons it lose-it)]) (if (member it lose-it) lose-it use-it))))

42 The cost of uniq measured in list-element comparisons (define (uniq L) (if (empty? L) '() (let* ([it (first L)] n-1 comparisons [lose-it (uniq (rest L))] [use-it (cons it lose-it)]) (if (member it lose-it) lose-it use-it)))) T(0) = 0 T(n) = n-1 + T(n-1) T(n) = n-1 + T(n-1) = n-1 + n-2 + T(n-2) O(n 2 )

43 reachable?: use it or lose it a recursive strategy fill in the pieces ase case(s) a node is always reachable from itself no nodes are reachable in the empty graph Recursive case(s) 1. It: a piece of the problem an edge, c d 2. Smaller version: What does the input look like without it? the graph without that edge, called sub-graph 3. Lose it: How could we solve a smaller version without it? is a reachable from b in sub-graph? 4. Use it: How could we solve a smaller version with it? is c reachable from a N is b reachable from d in sub-graph? 5. How would we combine the solutions to solve the full problem? use-it OR lose-it

44 reachable?: use it or lose it (define (reachable? a b G) (cond [(equal? a b) true] [(emptygraph? G) false] [else graph operations (emptygraph? graph) (edge-list graph) (add-edge edge graph) (remove-edge edge graph) (src edge) (dst edge) list operations (empty? list) (cons elem list) (first list) (rest list) (map function list) (append list1 list2) (member elem list) (remove-duplicates list) and more: docs.racket-lang.org/reference/pairs.html

45 reachable?: use it or lose it (define (reachable? a b G) (cond [(equal? a b) true] [(emptygraph? G) false] [else (let* ([edges (edge-list G)] [it (first edges)] [subg (remove-edge it G)] [lose-it (reachable? a b subg)] [use-it (and (reachable? a (src it) subg) (reachable? (dst it) b subg))]) (or use-it lose-it))])) graph operations (emptygraph? graph) (edge-list graph) (add-edge edge graph) (remove-edge edge graph) (src edge) (dst edge) list operations (empty? list) (cons elem list) (first list) (rest list) (map function list) (append list1 list2) (member elem list) (remove-duplicates list) and more: docs.racket-lang.org/reference/pairs.html

46 The cost of reachable? measured in number of edges seen, i.e., calls to remove-edge (define (reachable? a b G) (cond [(equal? a b) true] [(emptygraph? G) false] [else (let* ([edges (edge-list G)] [it (first edges)] [subg (remove-edge it G)] [lose-it (reachable? a b subg)] [use-it (and (reachable? a (src it) subg) (reachable? (dst it) b subg))]) (or use-it lose-it))])) T(0) = 0 T(e) = * T(e-1) T(e) = 1 + 3*T(e-1) = 1 + 3*(1 + 3*T(e-2)) = 4 + 9*T(e-2) = 4 + 9*(1 + 3*T(e-3)) = *T(e-3) O(3 e )