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Design and Analysis of Algorithms Binary search trees

Design and Analysis of Algorithms Binary search trees. Haidong Xue Summer 2012, at GSU. Operations on a binary search tree. SEARCH(S, k) MINIMUM(S) MAXIMUM(S) SUCCESSOR(S, x) PREDECESSOR(S, x) INSERT(S, x) DELETE(S, x). O(h). O(h). h is the height of the tree. O(h). O(h). O(h) .

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Design and Analysis of Algorithms Binary search trees

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  1. Design and Analysis of Algorithms Binary search trees HaidongXue Summer 2012, at GSU

  2. Operations on a binary search tree • SEARCH(S, k) • MINIMUM(S) • MAXIMUM(S) • SUCCESSOR(S, x) • PREDECESSOR(S, x) • INSERT(S, x) • DELETE(S, x) O(h) O(h) h is the height of the tree O(h) O(h) O(h) O(h) O(h)

  3. What is a binary search tree? • A binary tree • Binary-search-tree property • For each node, all the nodes in its left sub tree is smaller than to or equal to this node; all the nodes in its right sub tree is larger than or equal to this node What the difference between “binary search tree” and a “max-heap”? Not a complete binary tree With a different tree property

  4. What is a binary search tree? 11 6 9 12 7 5 8 10 11 12 2 5 8 2 Yes Yes

  5. What is a binary search tree? 2 1 5 2 3 7 4 6 8 6 5 Yes Yes

  6. What is a binary search tree? 11 11 9 12 9 25 8 12 11 12 8 10 20 78 8 10 13 19 No No

  7. Elements in a binary search tree • Can we use an array to represent a binary search tree? • No • So some tree structure information has to be stored

  8. Elements in a binary search tree • binary search tree node { • Key • Satellite data • Left node (left) • Right node (right) • Parent node (p) }

  9. 11 NIL 9 12 p 11 right left 8 10 11 12 2 p p 9 12 right right left left p p p p 10 8 11 right 12 left right left right right left left NIL NIL NIL NIL NIL NIL NIL p 2 right left Operations are based on this structure NIL NIL

  10. Print all keys in sorted order • Preorder tree walk • Inorder tree walk • Postorder tree walk

  11. Print all keys in sorted order • Preorder-tree-walk (node x){ • If(x==NIL) return; • Access(x); • Preorder-tree-walk(x.left); • Preorder-tree-walk(x.right) }

  12. Print all keys in sorted order • Preorder-tree-walk ( );

  13. Print all keys in sorted order • Inorder-tree-walk (node x){ • If(x==NIL) return; • Inorder-tree-walk( x.left); • Access(x); • Inorder-tree-walk( x.right) }

  14. Print all keys in sorted order • Inorder-tree-walk ( );

  15. Print all keys in sorted order • Postorder-tree-walk (node x){ • If(x==NIL) return; • Postorder-tree-walk( x.left); • Postorder-tree-walk( x.right) • Access(x); }

  16. Print all keys in sorted order • Postorder-tree-walk ( );

  17. Print all keys in sorted order • Preorder tree walk • Inorder tree walk • Postorder tree walk

  18. Print all keys in sorted order 11 • Preorder tree walk • Inorder tree walk • Postorder tree walk 11 12 10 12 8 11 9 9 12 12 12 8 9 10 11 11 8 10 11 12 8 10 9 11 12 12 11 Sorted order from inorder tree walk!

  19. Print all keys in sorted order • Time complexity of inorder tree walk • Access each node once

  20. Searching in a binary search tree TREE-SEARCH • Input: root pointer (x), key (k) • Output: a element whose key is the same as the input key; NIL if no element has the input key • if(x==NIL or x.key==k) return x; • if(k<x.key) return TREE-SEARCH(x.left, k); • Return TREE-SEARCH(x.right, k);

  21. Searching in a binary search tree 11 TREE-SEARCH( , 11) 11 11 9 12 8 10 11 12 2

  22. Searching in a binary search tree 11 TREE-SEARCH( , 11) 11 9 12 8 10 11 11 12 2

  23. Searching in a binary search tree 11 TREE-SEARCH( , 8) 9 TREE-SEARCH( , 8) 11 8 TREE-SEARCH( , 8) 9 12 8 8 10 11 12 2

  24. Searching in a binary search tree 11 TREE-SEARCH( , 20) 11 12 TREE-SEARCH( , 20) 9 12 12 TREE-SEARCH( , 20) 8 10 11 12 TREE-SEARCH( NIL , 20) NIL 2 NIL Means there is no such a node in the tree

  25. Searching in a binary search tree 30 TREE-SEARCH( , 20) 30 11 9 12 Illegal, but never happen if start from the root 8 10 11 12 2

  26. Searching in a binary search tree What’s the worst case? Worst successful search 11 11 TREE-SEARCH( , 2) O(h) 9 12 Worst unsuccessful search 11 TREE-SEARCH( , 1) O(h) 8 10 11 12 2

  27. Searching in a binary search tree Iterative code could be more efficient than recursive code TREE-SEARCH (x, k) if(x==NIL or x.key==k) return x; if(k<x.key) return TREE-SEARCH(x.left, k); Return TREE-SEARCH(x.right, k); • TREE-SEARCH (x, k) • current=x; • While (current!=NIL and current.key!=k){ • if(x.key<k) current=x.left; • else current=x.right; • } • return current;

  28. Searching in a binary search tree 11 TREE-SEARCH( , 8) • TREE-SEARCH (x, k) • current=x; • while (current!=NIL and current.key!=k){ • if(x.key<k) current=x.left; • else current=x.right; • } • return current; 11 9 12 8 8 10 11 12 2

  29. Searching in a binary search tree 11 TREE-SEARCH( , 20) 11 NIL 9 12 8 10 11 12 NIL 2

  30. Minimum and maximum 11 9 12 8 10 11 12 2 As a human, can you tell where is the minima and maxima?

  31. Minimum and maximum The minima of the tree rooted at x is: the minima of x.leftif x.left is not NIL; x if x.left is NIL 11 9 12 TREE-MINIMUM( x ) //the recursive one 1. if (x.left==NIL) return x; 2. return TREE-MINIMUM(x.left); 0. if(x==NIL) return NIL; 8 10 11 12 2 It has some cost, so if x is guaranteed not NIL we can remove it

  32. Minimum and maximum 11 TREE-MINIMUM( ) //recursive 11 9 12 8 10 11 12 2 2

  33. Minimum and maximum The minima of the tree rooted at x is: the leftmost node 11 9 12 • TREE-MINIMUM( x ) //the iterative one • current = x; • while(current.left!=NIL) • current = current.left; • return current; 0. if(x==NIL) return NIL; 8 10 11 12 2

  34. Minimum and maximum 11 TREE-MINIMUM( ) //iterative 11 9 12 8 10 11 12 2 2

  35. Minimum and maximum TREE-MAXIMUM( x ) //the recursive one 0. if(x==NIL) return NIL; 1. if (x.right==NIL) return x; 2. return TREE-MAXIMUM( x.right); TREE-MINIMUM( x ) //the recursive one 0. if(x==NIL) return NIL; 1. if (x.left==NIL) return x; 2. return TREE-MINIMUM(x.left); • TREE-MINIMUM( x ) //the iterative one • 0. if(x==NIL) return NIL; • current = x; • while(current.left!=NIL) • current = current.left; • return current; • TREE-MAXIMUM( x ) //the iterative one • 0. if(x==NIL) return NIL; • current = x; • while(current.right!=NIL) • current = current.right; • return current; O(h) O(h) Time complexity?

  36. Successor and predecessor • What is a successor of x? • What is a successor of x if there is another node has the same key in the binary search tree? 11 9 12 12 12 8 9 10 11 11 8 10 11 12

  37. Successor and predecessor 15 TREE-SUCCESSOR( ) 15 The minimum of the right sub tree 6 18 13 TREE-SUCCESSOR( ) 3 7 17 20 There is no right sub tree 4 2 13 The lowest ancestor whose left child is also an ancestor of or 13 13 9

  38. Successor and predecessor TREE-SUCCESSOR( x ) // When x has a right sub tree if(x.right!=NIl) return TREE-MINIMUM(x); // When x does not have a right sub tree 2. current = x 3. currentParent = x.p 4. while( currentParent!=NIL and currentParent.left!=current ){ current = currentParent; currentParrent = currenParent.p; } 5. return currentParent;

  39. Successor and predecessor 15 15 TREE-SUCCESSOR( ) TREE-MINIMUM 6 18 3 7 17 17 20 4 2 13 9 2 3 4 6 7 15 18 20 9 13 17

  40. Successor and predecessor 9 15 TREE-SUCCESSOR( ) Has no right sub tree 6 18 current == currentParent.left is true 17 20 3 7 4 13 2 13 currentParent 9 current 2 3 4 6 7 15 18 20 9 13 17

  41. Successor and predecessor 13 15 15 TREE-SUCCESSOR( ) No right subtree 6 18 current == currentParent.left is false currentParent == NIL is false 17 20 3 7 current == currentParent.left is false currentParent== NIL is false currentParent 4 13 2 current == currentParent.left is true current 9 2 3 4 6 7 15 18 20 9 13 17

  42. Successor and predecessor 20 15 TREE-SUCCESSOR( ) No right subtree 6 18 current == currentParent.left is false currentParent== NIL is false currentParent 17 20 3 current == currentParent.left is false currentParent== NIL is false 7 current currentParent== NIL is true 4 13 2 • NIL 9 2 3 4 6 7 15 18 20 9 13 17

  43. TREE-SUCCESSOR( x ) // When x has a right sub tree if(x.right!=NIl) return TREE-MINIMUM(x); // When x does not have a right sub tree 2. current = x 3. currentParent = x.p 4. while( !(currentParent==NIL or currentParent.left==current) ){ current = currentParent; currentParrent = currenParent.p; } 5. return currentParent; Time complexity? TREE-PREDECESSOR( x ) // When x has a left sub tree if(x.left!=NIl) return TREE-MAXIMUM(x); // When x does not have a left sub tree 2. current = x 3. currentParent = x.p 4. while( !(currentParent==NIL or currentParent.right==current) ){ current = currentParent; currentParrent = currenParent.p; } 5. return currentParent; O(h)

  44. Insertion and deletion 11 9 19 8 10 18 22 2 As a human, how to insert a element to a binary search tree?

  45. Insertion and deletion TREE-INSERT( T, z ) if(T.root==NIL){ T.root = z; z.p = NIL; } else { INSERT(t.root, z); } • INSERT(x, z) • if(z.key<x.key) • if(z.left==NIL){ • z.p=x; • x.left=z; • } • else • INSERT(x.left, z); • else • if(z.right==NIL){ • z.p=x; • x.right=z; • } • else • INSERT(x.right, z);

  46. Insertion and deletion 3 3 TREE-INSERT( T, ) //recursive 11 Not NIL 9 19 Not NIL 8 10 18 22 Not NIL 2 NIL

  47. Insertion and deletion TREE-INSERT( T, z ) // iterative posParent = NIL; pos = T.root; // try to find a position, start from T.root while(pos!=NIL){ posParent = pos; if(z.key < pos.key) pos = pos.left; else pos = pos.right; } z.p = posParent; if(posParent==NIL); // T is empty T.root = z; else if(z.key<posParent.key) posParent.left = z; else posParent.right = z; Find a position Modify z Modify posParent

  48. Insertion and deletion 3 3 TREE-INSERT( T, ) //iterative 11 NIL posParent 9 19 pos ….. while(pos!=NIL){ posParent = pos; if(z.key < pos.key) pos = pos.left; else pos = pos.right; } … 8 10 18 22 2 NIL

  49. Insertion and deletion The element has less than 2 children 11 The element has two children and its successor is the right child 19 9 18 8 10 22 The element has two children and its successor is not the right child 2 As a human, how to delete a element from a binary search tree?

  50. Insertion and deletion The element has less than one child 11 22 TREE-DELETE( ) // no child 19 9 8 TREE-DELETE( ) // no child 8 10 18 22 2 Replace a tree with another tree TRANSPLANT( T, u, v )

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