1. Josephus problem http://www.cut-the-knot.org/recurrence/flavius.shtml 0 1 2 3 4 0 1 2 3 (J(N-1, K) + K) mod N = J(N, K) J(1, K) = 0 N=5, K=3 Ak K=2 a N = b0b1b2b3...bn J(N, 2) = b1b2b3...bnb0

2. Stack - interface public interface Stack { public int size(); public boolean isEmpty(); public E top() throws EmptyStackException; public void push (E element) throws FullStackException; public E pop() throws EmptyStackException; } ArrayStack A = new ArrayStack (100); A.push(7); o = A.pop();

3. public class ArrayStack implements Stack { protected int capacity; public static final int CAPACITY = 1000; protected E S[]; protected int top = -1; public ArrayStack(int cap) { capacity = cap; S = (E[]) new Object[capacity]; } public void push(E element) throws FullStackException { if (size() == capacity) throw new FullStackException(“full."); S[++top] = element; } public E pop() throws EmptyStackException { E element; if (isEmpty()) throw new EmptyStackException("empty."); element = S[top]; S[top--] = null; return element; } Stack – implementation

4. Queue - interface public interface Queue { public int size(); public boolean isEmpty(); public E front() throws EmptyQueueException; public void enqueue (E element); public E dequeue() throws EmptyQueueException; }

5. Node public class Node { private E element; private Node next; public Node() {this(null, null); } public Node(E e, Node n) { element = e; next = n; } public E getElement() { return element; } public Node getNext() { return next; } public void setElement(E newElem) { element = newElem; } public void setNext(Node newNext) { next = newNext; }

6. Queue public void enqueue(E elem) { Node node = new Node (); node.setElement(elem); node.setNext(null); if (size == 0) front = node; else rear.setNext(node); rear = node; size++; } public E dequeue() throws EmptyQueueException { if (size == 0) throw new EmptyQueueException("Queue is empty."); E tmp = front.getElement(); front = front.getNext(); size--; if (size == 0) rear = null; return tmp; }

7. Queue public void enqueue(E elem) { Node node = new Node (); node.setElement(elem); if (size == 0) node.setNext(node); else { node.setNext(rear.getNext()); rear.setNext(node); } rear = node; size++; } public E dequeue() throws EmptyQueueException { if (size == 0) throw new EmptyQueueException("Queue is empty."); size--; E tmp = rear.getNext().getElement(); if (size == 0) rear = null; else read.setNext(rear.getNext().getNext()); return tmp; }

8. Node - Stack public class NodeStack implements Stack { protected Node top; protected int size; public NodeStack() { top = null; size = 0; } public void push(E elem) { Node v = new Node (elem, top); top = v; size++; } public E top() throws EmptyStackException { if (isEmpty()) throw new EmptyStackException("empty."); return top.getElement(); } public E pop() throws EmptyStackException { if (isEmpty()) throw new EmptyStackException("empty."); E temp = top.getElement(); top = top.getNext(); size--; return temp; }

9. Balík - interface public interface Deque { public int size(); public boolean isEmpty(); public E getFirst() throws EmptyDequeException; public E getLast() throws EmptyDequeException; public void addFirst (E element); public void addLast (E element); public E removeFirst() throws EmptyDequeException; public E removeLast() throws EmptyDequeException; }

10. DLNode public class DLNode { private E element; private DLNode prev, next; public DLNode() { this(null, null, null); } public DLNode(E e, Node p, Node n) { element = e; next = n; prev = p; } public E getElement() { return element; } public Node getNext() { return next; } public void setElement(E newElem) { element = newElem; }..

11. Balík – implementácia public class NodeDeque implements Deque { protected DLNode header, trailer; protected int size; public NodeDeque() { header = new DLNode (); trailer = new DLNode (); header.setNext(trailer); trailer.setPrev(header); size = 0; } public E getFirst() throws EmptyDequeException { if (isEmpty()) throw new EmptyDequeException("Deque is empty."); return header.getNext().getElement(); }

12. Balík - implementácia public void addFirst(E o) { DLNode second = header.getNext(); DLNode first = new DLNode (o, header, second); second.setPrev(first); header.setNext(first); size++; } public E removeLast() throws EmptyDequeException { if (isEmpty()) throw new EmptyDequeException("Deque is empty."); DLNode last = trailer.getPrev(); E o = last.getElement(); DLNode secondtolast = last.getPrev(); trailer.setPrev(secondtolast); secondtolast.setNext(trailer); size--; return o; }

13. IndexList - interface public interface IndexList { public int size(); public boolean isEmpty(); public void add(int i, E e) throws IndexOutOfBoundsException; public E get(int i) throws IndexOutOfBoundsException; public E remove(int i)throws IndexOutOfBoundsException; public E set(int i, E e)throws IndexOutOfBoundsException; }

14. IndexList - implementation public class ArrayIndexList implements IndexList { private E[] A; private int capacity = 16; private int size = 0; public ArrayIndexList() { A = (E[]) new Object[capacity]; } public void add(int r, E e) throws IndexOutOfBoundsException { checkIndex(r, size() + 1); if (size == capacity) {// an overflow capacity *= 2; E[] B =(E[]) new Object[capacity]; for (int i=0; i<size; i++) B[i] = A[i]; A = B; } for (int i=size-1; i>=r; i--) A[i+1] = A[i]; A[r] = e; size++; } public E remove(int r) throws IndexOutOfBoundsException { checkIndex(r, size()); E temp = A[r]; for (int i=r; i<size-1; i++)// shift elements down A[i] = A[i+1]; size--; return temp; }

15. Skip list lepší ako red-black tree alebo hash tabluľky find, insert aj delete sú priemerne O(log n) insert je stabilný, t.j. zachováva pôvodné usporiadanie rôvnakých kľúčov in-place algoritmus – žiadna extra pamäť

16. SkipList node public class SkipNode { public int nodeHeight; public int key; public SkipNode[] Nodes; public SkipNode(int value,int h) { nodeHeight = h; key = value; Nodes = new SkipNode[h+1]; for (int i = 1; i <= h; i++) Nodes[i] = null; } public static final int MaxNodeValue = 65536; public static final int MinNodeValue = 0; }

17. SkipList find public int find(int key) { SkipNode tmp = head; for (int h = curHeight; h >= 1; h-- ) { while ( tmp.Nodes[h].key < key ) tmp = tmp.Nodes[h]; update[h] = tmp; if (tmp.Nodes[h].key == key) return key; } return SkipNode.MinNodeValue - 1; }

18. SkipList Insert

20. public boolean insert(int k) { int level = 0; SkipNode[] update = new SkipNode[maxHeight+1]; ……. // find k… if (tmp.Nodes[1].key == k ) return false;// duplicate key else { for(level = 1; (Math.random() < 0.5) && (level < maxHeight); level++) ; level = newLevel(); if ( level > curHeight ) { for ( int i = curHeight + 1; i <= level; i++ ) update[i] = head; curHeight = level; } tmp = new SkipNode(k,level); for ( int i = 1; i <= level; i++ ) { tmp.Nodes[i] = update[i].Nodes[i]; update[i].Nodes[i] = tmp; } return true; }

21. SkipList delete

22. Stromy AVL Tree (1962) B-Tree (Bayer,1971) 2-3 Red-Black (Sedgewick,1978) Splay Tree (Tarjan,1986) Tree-Forest

23. AVL Node class AVLNode { int x; AVLNode left, right; public AVLNode (int init) { x = init; left = null; right = null; } public int height () { int l = (left != null)?(left.height () + 1):0; int r = (right != null)?(right.height () + 1):0; return (l > r)?l:r; }

24. AVL Tree public AVLNode insert (int k) { if (k < x) if (left == null) left = new AVLNode (k); else left = left.insert (k); else if (right == null) right = new AVLNode (k); else right = right.insert (k); … pokračuje public int balance () { int l = (left != null)?(left.height () + 1):0; int r = (right != null)?(right.height () + 1):0; return (r - l); }

25. Rotate public AVLNode rotateLeft() { AVLNode tmp = right; right = tmp.left; tmp.left = this; return tmp; } public AVLNode rotateRight() { AVLNode tmp = left; left = tmp.right; tmp.right = this; return tmp; }

26. Balancing public AVLNode insert (int k) { if (k < x)... pokračovanie... if (balance () < -1) { if (k < left.x) return rotateRight (); else { left = left.rotateLeft (); return rotateRight (); } else if (balance () > 1) { if (k > right.x) return rotateLeft (); else { right = right.rotateRight (); return rotateLeft(); } return this;

27. Splay Trees Ak je spracovávaný uzol key, aplikuje sa metóda splay(key), Ak strom obsahuje key, splay(key) spôsobí, že vybuble do koreňa, Inak, koreň je najväčší kľúč key, Bublaním sa key približuje do koreňa transformáciami: –Zig-Zag, –Zig-Zig –Zig

28. Splay find/insert public int find(int key) { if (root == null) return -1; splay(key); if(root.key != key) return -1; return root.key; } public void insert(int key) { if (root == null) { root = new BinaryNode(key); return; } splay(key); if (key == root.key) return; // not found BinaryNode n = new BinaryNode(key); if (key < root.key) { n.left = root.left; n.right = root; root.left = null; } else { n.right = root.right; n.left = root; root.right = null; } root = n; }

29. private void moveToRoot(int key) { BinaryNode l, r, t; l = r = header; t = root; header.left = header.right = null; for (;;) { if (key < t.key) { if (t.left == null) break; r.left = t; /* link right */ r = t; t = t.left; } else if (key > t.key) { if (t.right == null) break; l.right = t; /* link left */ l = t; t = t.right; } else break; } l.right = t.left; /* assemble */ r.left = t.right; t.left = header.right; t.right = header.left; root = t; } Top-down T TLTR L R T L R TLTR

30. X Y YLYr XR LR LR X Y YLYr Zig

31. Zig-Zig X Y ZL Zr XR L R Z YR LR X XRYR Y ZL Zr Z

32. Zig-Zag X Y ZL Zr XR L R Z YL L R X XR Y ZL Zr Z YL

33. Assembling X XLXR L R X L R XLXR

34. private void splay(int key) { BinaryNode l, r, t, y; l = r = header; t = root; header.left = header.right = null; for (;;) { if (key < t.key) { if (t.left == null) break; if (key < t.left.key) { y = t.left; /* rotate right */ t.left = y.right; y.right = t; t = y; if (t.left == null) break; } r.left = t; /* link right */ r = t; t = t.left; } else if (key > t.key) {..... } else { break; } l.right = t.left; /* assemble */ r.left = t.right; t.left = header.right; t.right = header.left; root = t; } http://www.ibr.cs.tu-bs.de/courses/ss98/audii/applets/BST/SplayTree-Example.html http://webpages.ull.es/users/jriera/Docencia/AVL/AVL%20tree%20applet.htm http://www.cs.princeton.edu/courses/archive/spr06/cos423/Handouts/self%20adjusting.pdf splay

35. Tree/Forest

36. import java.util.*; public class Forest { private List > broths; public Forest(List > broths) { this.broths = broths; } public String toString(){ String s = ""; char ch = '['; for(Tree e:broths) { s += ch + e.toString(); ch = ','; } s += "]"; return s; } public boolean find(E key) { for(Tree e:broths) { if (e.find(key)) return true; } return false; }

37. Tree/Forest public class Tree { private E elem; private Forest sons; public Tree(E elem) { this(elem,null); } public Tree(E elem, Forest sons) { this.elem = elem; this.sons = sons; } public String toString() { if (sons == null) return elem.toString(); else return "("+elem.toString()+":"+sons.toString()+")"; } public boolean find(E key) { if (elem.equals(key)) return true; else if (sons != null) return sons.find(key); else return false; }

38. Tree/Forest Tree t1 = new Tree (“system"); Tree t2 = new Tree ("system32"); Tree t3 = new Tree ("config"); Tree t123 = new Tree ("WINNT", new Forest (Arrays.asList(t1,t2,t3))); Tree u1 = new Tree ("Universal crack"); Tree u2 = new Tree ("java"); Tree u12 = new Tree ("Program Files", new Forest (Arrays.asList(u1,u2))); Forest disk = new Forest (Arrays.asList(t123,u12)); System.out.println(disk); System.out.println(disk.find("java")); System.out.println(disk.find("kava")); [(WINNT:[system,system32,config]),(Program Files:[Universal crack,java])] true false

39. Position List - interface public interface Position { E element(); } public interface PositionList { public int size(); public boolean isEmpty(); public Position first(); public Position last(); public Position next(Position p) throwsInvalidPositionException, BoundaryViolationException; public Position prev(Position p) throwsInvalidPositionException, BoundaryViolationException; public void addFirst(E e); public void addLast(E e); public void addAfter(Position p, E e) throwsInvalidPositionException, BoundaryViolationException; public void addBefore(Position p, E e) throwsInvalidPositionException, BoundaryViolationException; public E remove(Position p) throws InvalidPositionException; public E set(Position p, E e) throws InvalidPositionException; }

40. Position List - DNode public class DNode implements Position { private DNode prev, next; private E element; public DNode(DNode newPrev, DNode newNext, E elem) { prev = newPrev; next = newNext; element = elem; } public E element() throws InvalidPositionException { if ((prev == null) && (next == null)) throw new InvalidPositionException("Position is not in a list!"); return element; } public DNode getNext() { return next; } public DNode getPrev() { return prev; } public void setNext(DNode newNext) { next = newNext; } public void setPrev(DNode newPrev) { prev = newPrev; } public void setElement(E newElement) { element = newElement; } }

41. Position List – implementácia 1 public class NodePositionList implements PositionList { protected int numElts; protected DNode header, trailer; public NodePositionList() { numElts = 0; header = new DNode (null, null, null); trailer = new DNode (header, null, null); header.setNext(trailer); } protected DNode checkPosition(Position p) throws InvalidPositionException { if (p == null) throw new InvalidPositionException("Null position passed to NodeList"); if (p == header) throw new InvalidPositionException("The header node is not a valid position"); if (p == trailer) throw new InvalidPositionException("The trailer node is not a valid position"); try { DNode temp = (DNode ) p; if ((temp.getPrev() == null) || (temp.getNext() == null)) throw new InvalidPositionException("Position not belong to a valid NodeList"); return temp; } catch (ClassCastException e) { throw new InvalidPositionException("Position is of wrong type for this list"); }

42. public Position first() throws EmptyListException { if (isEmpty()) throw new EmptyListException("List is empty"); return header.getNext(); } public Position prev(Position p) throws InvalidPositionException, BoundaryViolationException { DNode v = checkPosition(p); DNode prev = v.getPrev(); if (prev == header) throw new BoundaryViolationException("Cannot advance past the beginning of list"); return prev; } public void addBefore(Position p, E element) throws InvalidPositionException { DNode v = checkPosition(p); numElts++; DNode newNode = new DNode (v.getPrev(), v, element); v.getPrev().setNext(newNode); v.setPrev(newNode); } public void addFirst(E element) { numElts++; DNode newNode = new DNode (header, header.getNext(), element); header.getNext().setPrev(newNode); header.setNext(newNode); } Position List – implementácia 2

43. public E remove(Position p) throws InvalidPositionException { DNode v = checkPosition(p); numElts--; DNode vPrev = v.getPrev(); DNode vNext = v.getNext(); vPrev.setNext(vNext); vNext.setPrev(vPrev); E vElem = v.element(); // unlink the position from the list and make it invalid v.setNext(null); v.setPrev(null); return vElem; } public E set(Position p, E element) throws InvalidPositionException { DNode v = checkPosition(p); E oldElt = v.element(); v.setElement(element); return oldElt; } Position List – implementácia 3