Module 4: Priority Queues

Transcription

1 Module 4: Priority Queues CS 240 Data Structures and Data Management T. Biedl K. Lanctot M. Sepehri S. Wild Based on lecture notes by many previous cs240 instructors David R. Cheriton School of Computer Science, University of Waterloo Winter 2018 References: Goodrich & Tamassia 3.2 version :03 Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

2 Outline 1 Dictionaries and Balanced Search Trees ADT Dictionary Review: Binary Search Trees AVL Trees Insertion in AVL Trees Restoring the AVL Property: Rotations Deletion in AVL Trees AVL Example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter 2018

3 Outline 1 Dictionaries and Balanced Search Trees ADT Dictionary Review: Binary Search Trees AVL Trees Insertion in AVL Trees Restoring the AVL Property: Rotations Deletion in AVL Trees AVL Example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter 2018

4 Dictionary ADT A dictionary is a collection of items, each of which contains a key some data, and is called a key-value pair (KVP). Keys can be compared and are (typically) unique. Operations: search(k) insert(k, v) delete(k) optional: join, isempty, size, etc. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

5 Elementary Implementations Common assumptions: Dictionary has n KVPs Each KVP uses constant space Keys can be compared in constant time Unordered array or linked list search Θ(n) insert Θ(1) delete Θ(n) (need to search) Ordered array search Θ(log n) insert Θ(n) delete Θ(n) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

6 Outline 1 Dictionaries and Balanced Search Trees ADT Dictionary Review: Binary Search Trees AVL Trees Insertion in AVL Trees Restoring the AVL Property: Rotations Deletion in AVL Trees AVL Example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter 2018

7 Binary Search Trees (review) Structure A BST is either empty or contains a KVP, left child BST, and right child BST. Ordering Every key k in T.left is less than the root key. Every key k in T.right is greater than the root key Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

8 BST Search and Insert BST-search(k) Compare k to current node, stop if found, else recurse on subtree unless it s empty BST-insert(k, v) Search for k, then insert (k, v) as new node Example: Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

9 BST Delete First search for the node x that contains the key. If x is a leaf, just delete it. If x has one child, move child up Else, swap key at x with key at successor or predecessor node and then delete that node Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

10 Height of a BST BST-search, BST-insert, BST-delete all have cost Θ(h), where h = height of the tree = max. path length from root to leaf If n items are BST-inserted one-at-a-time, how big is h? Worst-case: n 1 = Θ(n) Best-case: log(n) = Θ(log n) Average-case: Θ(log n) (just like recursion depth in quick-sort1) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

11 Outline 1 Dictionaries and Balanced Search Trees ADT Dictionary Review: Binary Search Trees AVL Trees Insertion in AVL Trees Restoring the AVL Property: Rotations Deletion in AVL Trees AVL Example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter 2018

12 AVL Trees Introduced by Adel son-vel skĭı and Landis in 1962, an AVL Tree is a BST with an additional structural property: The heights of the left subtree L and right subtree R differ by at most 1. (The height of an empty tree is defined to be 1.) At each non-empty node, we require height(r) height(l) { 1, 0, 1}: 1 means the tree is left-heavy 0 means the tree is balanced +1 means the tree is right-heavy Need to store at each node the height of the subtree rooted at it Can show: It suffices to store height(r) height(l) at each node. uses fewer bits code gets more complicated, especially for deleting Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

13 Height of an AVL tree Theorem: An AVL tree on n nodes has Θ(log n) height. AVL-search, AVL-insert, AVL-delete all cost Θ(log n) in the worst case! Proof: Define N(h) to be the least number of nodes in a height-h AVL tree. What is a recurrence relation for N(h)? What does this recurrence relation resolve to? Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

14 Outline 1 Dictionaries and Balanced Search Trees ADT Dictionary Review: Binary Search Trees AVL Trees Insertion in AVL Trees Restoring the AVL Property: Rotations Deletion in AVL Trees AVL Example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter 2018

15 AVL insertion To perform AVL-insert(T, k, v): First, insert (k, v) into T using usual BST insertion Then, move up the tree from the new leaf, updating heights. How to get to parents? (We do not want explicit parent links!) during BST-Insert! If the height difference becomes ±2, then call the AVL-fix algorithm to rebalance at that node. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

16 BST Insert with Insertion Path We need to access parents, but do not store parent links explicitly! Let BST-insert keep track of insertion path. BST-insert(r, k, v) // Tree given by reference to root node r 1. P empty stack of references to nodes // path from root 2. P.push(reference to u); // new! 3. u r 4. while u does not point to empty (sub)tree 5. if k < u.key then 6. P.push(reference to u.left); u u.left 7. else if k > u.key then 8. P.push(reference to u.right); u u.right 9. else return (u, P) // k already present 10. Replace empty tree u points to by new leaf with KVP (k, v). // Assigns new leaf to parent s child pointer. 11. return (u, P) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

17 AVL insertion AVL-insert(r, k, v) // BST-insert returns new leaf u and // the path P from the root to u as references. 1. (u, P) BST-insert(r, k, v) 2. u.height 0 3. while ( P.empty()) 4. u P.pop() 5. if ( u.left.height u.right.height = 2) then 6. AVL-fix(u) 7. break // can argue that we are done 8. else 9. local-update-height(u) local-update-height(v) 1. v.height 1 + max{v.left.height, v.right.height} Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

18 Outline 1 Dictionaries and Balanced Search Trees ADT Dictionary Review: Binary Search Trees AVL Trees Insertion in AVL Trees Restoring the AVL Property: Rotations Deletion in AVL Trees AVL Example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter 2018

19 How to fix an unbalanced AVL tree Goal: change the structure without changing the order A B C D Notice that if heights of A, B, C, D differ by at most 1, then the tree is a proper AVL tree. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

20 Right Rotation This is a right rotation on node z: z y y x z x D C A B C D A B Note: Only two edges need to be moved, and two subtree heights updated. Useful to fix left-left imbalance. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

21 Left Rotation This is a left rotation on node z: z y y z x A x B A B C D C D Again, only two edges need to be moved and two heights updated. Useful to fix right-right imbalance. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

22 Pseudocode for rotations A x B y z C D A x B y C z D rotate-right(z) z: subtree of AVL tree, passed by reference to root 1. y z.left 2. z.left y.right 3. y.right z 4. local-update-height(z) 5. local-update-height(y) 6. z y // change root of subtree A z B y C x D A z B y C x D rotate-left(z) z: subtree of AVL tree, passed by reference to root 1. y z.right 2. z.right y.left 3. y.left z 4. local-update-height(z) 5. local-update-height(y) 6. z y // change root of subtree Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

23 Double Right Rotation This is a double right rotation on node z: z x y y z x D A A B C D B C First, a left rotation on the left subtree (y). Second, a right rotation on the whole tree (z). Useful for left-right imbalance. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

24 Double Left Rotation This is a double left rotation on node z: z x y z y A x D A B C D B C Right rotation on right subtree (y), followed by left rotation on the whole tree (z). Useful for right-left imbalance. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

25 Fixing a slightly-unbalanced AVL tree Idea: Identify one of the previous 4 situations, apply rotations AVL-fix(z) z: slightly unbalanced AVL tree, where root z has balance ±2, passed by reference 1. if (z.right.height > z.left.height) then 2. y z.right 3. if (y.left.height > y.right.height) then 4. rotate-right(z.right) // Changes z.right 5. rotate-left(z) 6. else 7. y z.left 8. if (y.right.height > y.left.height) then 9. rotate-left(z.left) 10. rotate-right(z) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

26 Outline 1 Dictionaries and Balanced Search Trees ADT Dictionary Review: Binary Search Trees AVL Trees Insertion in AVL Trees Restoring the AVL Property: Rotations Deletion in AVL Trees AVL Example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter 2018

27 AVL Deletion Remove node with key k; first check by BST-search if present. AVL-delete(r, k) // Requires existence of node in r that has key k. // BST-delete returns (unique and potentially empty) subtree // whose parent was the node removed during deletion and // the path P form the root to that subtree as references. 1. (r, P) BST-delete(r, k) 2. while ( P.empty()) 3. u P.pop() 4. if ( u.left.height u.right.height = 2) then 5. AVL-fix(u) 6. local-update-height(u) 7. // Always continue up the path and fix if needed. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

28 AVL Tree Operations Runtime search: Just like in BSTs, costs Θ(height) insert: Shown already, total cost Θ(height) AVL-fix restores the height of the tree it fixes to what it was, so AVL-fix will be called at most once. delete: First search, then swap with successor (as with BSTs), then move up the tree and apply AVL-fix (as with insert). AVL-fix may be called Θ(height) times. Total cost is Θ(height). Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25

29 Outline 1 Dictionaries and Balanced Search Trees ADT Dictionary Review: Binary Search Trees AVL Trees Insertion in AVL Trees Restoring the AVL Property: Rotations Deletion in AVL Trees AVL Example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter 2018

30 AVL tree operations Example: (The lower numbers indicate T.right.height T.left.height.) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 4 Winter / 25