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Data Structures Splay Trees

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Data Structures Splay Trees CSE 326: Data Structures Splay Trees James Fogarty Autumn 2007 Lecture 10 AVL Trees Revisited • Balance condition: Left and right subtrees of every node have heights differing by at most 1 – Strong enough : Worst case depth is O(log n) – Easy to maintain : one single or double rotation • Guaranteed O(log n) running time for – Find ? – Insert ? – Delete ? – buildTree ? Θ(n log n) 2 Single and Double Rotations a b Z h h X h Y a b c Z h h W h -1 X h-1 Y 3 AVL Trees Revisited •What extra info did we maintain in each node? •Wherewere rotations performed? • How did we locate this node? 4 Other Possibilities? • Could use different balance conditions, different ways to maintain balance, different guarantees on running time, … • Why aren’t AVL trees perfect? Extra info, complex logic to detect imbalance, recursive bottom-up implementation • Many other balanced BST data structures – Red-Black trees – AA trees – Splay Trees – 2-3 Trees – B-Trees –… 5 Splay Trees • Blind adjusting version of AVL trees – Why worry about balances? Just rotate anyway! • Amortized time per operations is O(log n) • Worst case time per operation is O(n) – But guaranteed to happen rarely Insert/Find always rotate node to the root! SAT/GRE Analogy question: AVL is to Splay trees as ___________ is to __________ Leftish heap : Skew heap 6 Recall: Amortized Complexity If a sequence of M operations takes O(M f(n)) time, we say the amortized runtime is O(f(n)). • Worst case time per operation can still be large, say O(n) • Worst case time for any sequence of M operations is O(M f(n)) Average time per operation for any sequence is O(f(n)) Amortized complexity is worst-case guarantee over sequences of operations. 7 Recall: Amortized Complexity • Is amortized guarantee any weaker than worstcase? Yes, it is only for sequences • Is amortized guarantee any stronger than averagecase? Yes, guarantees no bad sequences • Is average case guarantee good enough in practice? No, adversarial input, bad day, … • Is amortized guarantee good enough in practice? Yes, again, no bad sequences 8 The Splay Tree Idea 10 If you’re forced to make 17 a really deep access: Since you’re down there anyway, fix up a lot of deep nodes! 5 2 9 3 9 Find/Insert in Splay Trees 1. Find or insert a node k 2. Splay k to the root using: zig-zag, zig-zig, or plain old zig rotation Why could this be good?? 1. Helps the new root, k o Great if k is accessed again 2. And helps many others! o Great if many others on the path are accessed 10 Splaying node k to the root: Need to be careful! One option (that we won’t use) is to repeatedly use AVL single rotation until k becomes the root: (see Section 4.5.1 for details) p k q F r s E p s q D A B F k r A E B C C D 11 Splaying node k to the root: Need to be careful! What’s bad about this process? r is pushed almost as low as k was Bad seq: find(k), find(r), find(…), … p k q F s p r E q s A B D F k r A E B C D C 12 Splay: Zig-Zag* g k p g p X k W X Y Z W Helps those in blue Y Z Hurts those in red *Just like an… Which nodes improve depth? AVL double rotation k and its original children13 Splay: Zig-Zig* g k p p W Z k g X Y Y Z W X *Is this just two AVL single rotations in a row? Not quite – we rotate g and p, then p and k Why does this help? 14 Same number of nodes helped as hurt. But later rotations help the whole subtree. Special Case for Root: Zig root p k root k p Z X X Y Y Z Relative depth of p, Y, Z? Relative depth of everyone else? Down 1 level Much better Why not drop zig-zig and just zig all the way? Zig only helps one child! 15 Splaying Example: Find(6) 1 1 2 2 ? 3 3 Find(6) Zig-zig 4 6 Think of as if created by inserting 6,5,4,3,2,1 – each 5 5 took constant time – a LOT of savings so far to amortize 6 4 those bad accesses 16 over Still Splaying 6 1 1 2 6 ? 3 3 Zig-zig 6 2 5 5 4 4 17 Finally… 1 6 6 1 ? 3 3 Zig 2 5 2 5 4 4 18 Another Splay: Find(4) 6 6 1 1 ? 3 4 Find(4) Zig-zag 2 5 3 5 4 2 19 Example Splayed Out 6 4 1 1 6 ? 4 3 5 Zig-zag 3 5 2 2 20 But Wait… What happened here? Didn’t two find operations take linear time instead of logarithmic? What about the amortized O(log n) guarantee? That still holds, though we must take into account the previous steps used to create this tree. In fact, a splay tree, by construction, will never look like the example we started 21with! Why Splaying Helps • If a node n on the access path is at depth d before the splay, it’s at about depth d/2 after the splay • Overall, nodes which are low on the access path tend to move closer to the root • Splaying gets amortized O(log n) performance. (Maybe not now, but soon, and for the rest of the operations.) 22 Practical Benefit of Splaying • No heights to maintain, no imbalance to check for – Less storage per node, easier to code • Data accessed once, is often soon accessed again – Splaying does implicit caching by bringing it to the root 23 Splay Operations: Find • Find the node in normal BST manner • Splay the node to the root – if node not found, splay what would have been its parent What if we didn’t splay? Amortized guarantee fails! Bad sequence: find(leaf k), find(k), find(k), … 24 Splay Operations: Insert • Insert the node in normal BST manner • Splay the node to the root What if we didn’t splay? Amortized guarantee fails! Bad sequence: insert(k), find(k), find(k), … 25 Splay Operations: Remove Everything else splayed, so we’d better do that for remove k find(k) delete k L R L R < k > k Now what? 26 Join Join(L, R): given two trees such that (stuff in L) < (stuff in R), merge them: splay L max L R R Splay on the maximum element in L, then attach R Similar to BST delete – find max = find element with no right child Does this work to join any two trees? No, need L < R 27 Delete Example Delete(4) 6 4 6 1 9 find(4) 1 6 1 9 4 7 2 9 2 7 Find max 2 7 2 2 1 6 1 6 9 9 7 7 28 Splay Tree Summary • All operations are in amortized O(log n) time • Splaying can be done top-down; this may be better because: – only one pass Like what? Skew heaps! (don’t need to wait) – no recursion or parent pointers necessary – we didn’t cover top-down in class • Splay trees are very effective search trees – Relatively simple What happens to node that never get accessed? – No extra fields required (tend to drop to the bottom) – Excellent locality properties: frequently accessed keys are cheap to find 29 E Splay E B H A D F I C G I H A B G F C D E 30 Splay E E A I B H A C G I D F B H C G D F E 31.
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