CS : Data Structures

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1 CS : Data Structures Michael Schatz Oct 17, 2016 Lecture 19: Trees and Graphs

Assignment 6: Due Sunday Oct 23 @ 10pm Remember: javac Xlint:all &

2 Assignment 6: Due Sunday Oct 10pm Remember: javac Xlint:all & checkstyle *.java & JUnit Solutions should be independently written!

3 Part 1: Trees

4 Balancing Trees Balanced Binary Tree Minimum possible height Unbalanced Tree Non-minimum height What is the height of a balanced binary tree? What is the height of a balanced 3-ary tree? What is the height of a k-ary tree? How much taller is a binary tree than a k-ary tree? What is the maximum height of an unbalanced tree? lg n log 3 n log k n lg n/log k n = lg k n-1 L

5 Tree Heights

6 Tree Heights

7 Tree Heights (~1M leaves) Binary Tree H=20 3-ary Tree H=13

10 Tree Heights (~1M leaves) Binary Tree H=20 3-ary Tree H=13 What is the height with 1M total nodes?

11 Tree Interface public interface Tree<T>{ Position <T> insertroot(t t ) throws TreeNotEmptyException; Why insertroot? Position <T> insertchild(position <T> p, T t) throws InvalidPositionException; boolean empty(); Position <T> root() throws TreeEmptyException; Position <T>[] children(position <T> p) Why children()? throws InvalidPositionException, LeafException; Position <T> parent(position<t> p) throws InvalidPositionException ; boolean leaf (Position <T> p) throws InvalidPositionException ; What should parent(root()) return? T remove(position<t> p) What should remove(root()) do? throws InvalidPositionException, NotALeafException;

12 Tree Traversals + 1 * 2 3 Note here we visit children from left to right, but could go right to left Notice we visit internal nodes 3 times: 1: stepping down from parent 2: after visiting first child 3: after visiting second child Different algs work at different times 1: preorder + 1 * 2 3 2: inorder * 3 3: postorder * +

13 Traversal Implementations How to traverse? B A C D basictraversal(node n): // just entered from top basictraversal(n.left) // just got back from left basictraversal(n.middle) // just got back from middle basictraversal(n.right) // just got back from right return // leave to top E F G H I J K L

print? PreOrderTraversal(Node n): Stack s s.push(n) while (!s.empty()): Node x = s.

14 PreOrder Traversals [A] [D,C,B] [D,C,F,E] [D,C,F] [D,C,J] [D,C] [D] [I, H, G] Stack leads to a Depth-First Search B A C D How to preorder print? PreOrderTraversal(Node n): Stack s s.push(n) while (!s.empty()): Node x = s.pop() print(x) for c in x.children: s.push(c) E F G H I J K L

15 Level Order Traversals [A] [D,C,B] [F,E,D,C] [F,E,D] [I,H,G,F,E] [I,H,G,F] [J,I,H,G] [J,I,H] Queue leads to a Breadth-First Search B A C D How to level order print? (A) (B C D) (E F G H I) (J K L) LevelOrderTraversal(Node n): Queue q s.enqueue(n) while (!q.empty()): Node x = q.dequeue() print(x) for c in x.children: s.enqueue(c) E F G H I J K L

16 Call back interface public interface Tree<T> {... preorder(operation<t> o); inorder(operation<t> o); postorder(operation<t> o);... This works, but we will need 3 separate methods that have almost exactly the same code // The Operation<T> interface would look like this: public interface Operation<T> { void do(position<t> p); public class PrintOperation<T> implements Operation<T> { public void do(position<t> p) { System.out.println(p.get()); PrintOperation op = new PrintOperation(); tree.inorder(op);

17 Multiple Traversals public abstract class Operation<T> { void pre(position<t> p) { void in(position<t> p) { void post(position<t> p) { public interface Tree<T> {... traverse(operation<t> o);... Client extends Operation<T> but overrides just the methods that are needed J // Tree implementation pseudo-code: nicetraversal(node n, Operation o): if n is not null: o.pre(n) nicetraversal(n.left, o) o.in(n) nicetraversal(n.right, o) o.post(n)

18 Implementation (1) public class TreeImplementation<T> implements Tree<T> {... private static class Node<T> implements Position<T> { T data; Node<T> parent; ArrayList<Node<T>> children; public Node(T t) { this.children = new ArrayList<Node<T>>(); this.data = t; public T get() { return this.data; public void put(t t) { this.data = t; Constructor ensures children, data are initialized correctly What other fields might we want to include? (Hint: Position<>) Should set the color field to point to this Tree so Position<> can be checked

19 ArrayList

20 Traversal private void recurse(node<t> n, Operation<T> o) { if (n == null) { return; o.pre(n); for (Node<T> c: n.children) { this.recurse(c, o); // figure out when to call o.in(n) o.post(n); Private helper method working with Node<T> rather than Position<T> public void traverse(operation<t> o) { this.recurse(this.root, o); Just make sure we start at root When should we call o.in()? We don t want to call the in method after we visit the last child. We do want to call the in method even for a node with no children

21 Alternate Implementation parent parent siblings children[0] children[1] children[2] children A B C D A B C D E F G E F G Simple Implementation Overhead managing children[] Less Space Overhead (Except remove may require doubly linked lists)

22 More to come

23 Part 2:Graphs

24 Graphs are Everywhere! Computers in a network, Friends on Facebook, Roads & Cities on GoogleMaps, Webpages on Internet, Cells in your body,

25 Graphs A B Nodes aka vertices People, Proteins, Cities, Genes, Neurons, Sequences, Numbers, Edges aka arcs A is connected to B A is related to B A regulates B A precedes B A interacts with B A activates B

26 Graph Types A A A A B B C B C D C D E D E B E F G H F Cycle C A I J G B C List Tree Directed Acyclic Graph D E Complete

Definitions (1) A B C D Directed Edge E Self Edge (Unusual but usually allowed) Undirected Edge A and B are adjacent An edge connected to a vertex is incident on that vertex The number of edges

27 Definitions (1) A B C D Directed Edge E Self Edge (Unusual but usually allowed) Undirected Edge A and B are adjacent An edge connected to a vertex is incident on that vertex The number of edges incident on a vertex is the degree of that vertex For directed graphs, separately report indegree and outdegree A multigraph allows multiple edges between the same pair of nodes, a simple graph does not allow multiple edges (most common)

28 Definitions (2) A B H I C F G J D E A path is a sequence of edges e 1, e 2, e n in which each edge starts from the vertex the previous edge ended at A path that starts and ends at the same node is a cycle The number of edges in a path is called the length of the path A graph is connected if there is a path between every pair of nodes, otherwise it is disconnected into >1 connected components

29 The Road to the White House Maps Nodes: Cities / Intersections: Name / GPS Location Edges: Roads / Flight Path: Distance, Time, Cost

30 Graph Interface public interface Graph<V,E> {... Position<V> insertvertex(v v); Separate generic types for vertices <V> and edges <E> Position<E> insertedge(position<v> from, Position<V> to, E e) throws InvalidPositionException, InsertionException; V removevertex(position<v> p) throws InvalidPositionException, RemovalException; E removeedge(position<e> p) throws InvalidPositionException; Iterable<Position<V>> vertices(); Iterable<Position<E>> edges(); Iterable<Position<E>> incomingedges(position<v> p) throws InvalidPositionException; Iterable<Position<E>> outgoingedges(position<v> p) throws InvalidPositionException;... Can iterate through all the nodes OR iterate through all the edges as different algorithms may require one or the other

31 Representing Graphs A B Graph nodes C D E Node Node Node Node Node Node F A B C D in: out: E F G Edge Edge Edge Incidence List Good for sparse graphs Compact storage A D B D D F A: C, D, E D: F B: D, E E: F C: F, G G: edges

32 Representing Graphs A B Graph Do we need a separate edges list? nodes C D E Node Node Node Node Node Node F A B C D in: out: E F G Edge Edge Edge Incidence List Good for sparse graphs Compact storage A D B D D F A: C, D, E D: F B: D, E E: F C: F, G G: edges

33 Representing Graphs A B Graph nodes C D E Node Node Node Node Node Node F A B C D in: out: E F G Edge Edge Edge Incidence List Good for sparse graphs Compact storage A D B D D F A: C, D, E D: F B: D, E E: F C: F, G G: Note the labels in the edges are really references to the corresponding node objects!

34 Representing Graphs A B Complexity Analysis If n is the number of vertices, and m is the number of edges, we need O(n + m) space to represent the graph C D E F When we insert a vertex, allocate one object and two empty edge lists: O(1) When we insert an edge we allocate one object and insert the edge into appropriate lists for the incident vertices: O(1) G Incidence List Good for sparse graphs Compact storage A: C, D, E D: F B: D, E E: F C: F, G G: Remove a node? Remove an edge? Find/check edge between nodes? O(1); Only after edges removed O(d) where d is max degree; O(n) worse case O(d); O(n) worst case

Representing Graphs A B C D E F G A B C D E F G A 1 1 1 B 1 1 C 1 1 D 1 E 1 F 1 G Adjacency Matrix Good for dense graphs Fast, Fixed storage: N 2 bits or N 2 weights Incidence List Good for sparse

35 Representing Graphs A B C D E F G A B C D E F G A B 1 1 C 1 1 D 1 E 1 F 1 G Adjacency Matrix Good for dense graphs Fast, Fixed storage: N 2 bits or N 2 weights Incidence List Good for sparse graphs Compact storage: ~8 bytes/edge A: C, D, E D: F B: D, E E: F C: F, G G: Edge List Easy, good if you (mostly) need to iterate through the edges ~16 bytes / edge A,C B,C C,F A,D B,D C,G A,E B,E D,F E,F F,G Tools Graphviz: Gephi: Cytoscape: digraph G { A->B B->C A->C $ dot -Tpdf -o g.pdf g.dot

36 Graph Interface public interface Graph<V,E> {... Position<V> insertvertex(v v); Separate generic types for vertices <V> and edges <E> Position<E> insertedge(position<v> from, Position<V> to, E e) throws InvalidPositionException, InsertionException; V removevertex(position<v> p) throws InvalidPositionException, RemovalException; E removeedge(position<e> p) throws InvalidPositionException; Iterable<Position<V>> vertices(); Iterable<Position<E>> edges(); Iterable<Position<E>> incomingedges(position<v> p) throws InvalidPositionException; Iterable<Position<E>> What will insertedge outgoingedges(position<v> into Graph<String, Integer> p) return? throws InvalidPositionException;... What will insertedge into Graph<Integer, Integer> return?

37 Graph Interface 2 public interface Edge<T> extends Position<T> { public interface Vertex<T> extends Position<T> { public interface Graph<V,E> {... Vertex<V> insertvertex(v v); Edge<E> insertedge(vertex<v> from, Vertex<V> to, E e) throws InvalidVertexException, InsertionException; V removevertex(vertex<v> p) throws InvalidVertexException, RemovalException; E removeedge(edge<e> p) throws InvalidEdgeException; Iterable<Vertex<V>> vertices(); Iterable<Edge<E>> edges();... Now clients can check at compile time if their types are correct What else is missing from the interface?

38 Graph Interface 3 public interface Graph<V,E> {... Vertex<V> fromvertex(edge<e> e)... Vertex<V> tovertex(edge<e> e) Now clients can check their code to see where the edges go! Don t just define an interface, try to use it!

39 Next Steps 1. Reflect on the magic and power of Trees, Graphs, and Sets! 2. Assignment 5 due Sunday October 10pm

40 Welcome to CS Questions?

CS : Data Structures

CS : Data Structures CS 600.226: Data Structures Michael Schatz Oct 14, 2016 Lecture 18: Tree Implementation Assignment 5: Due Sunday Oct 9 @ 10pm Remember: javac Xlint:all & checkstyle *.java & JUnit Solutions should be independently