Dictionary: an abstract data type
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1 2-3 Trees 1
2 Dictionary: an abstract data type A container that maps keys to values Dictionary operations Insert Search Delete Several possible implementations Balanced search trees Hash tables 2
3 2-3 trees A kind of balanced search tree Assume keys are totally ordered (<, >, =) Assume n key/value pairs are stored in the dictionary Time per dictionary operation is O(log n) Support of other useful operations as well 3
4 Basic structure: a tree Keys/value pairs stored only at leaves (no duplicate keys) All leaves at the same level, with keys sorted order Each internal node: has either 2 or 3 children has a guide : the maximum key in its subtree 4
5 Example
6 Let h := height of tree (Recall: height = length of longest path from root to leaf) Claim: n 2 h Proof by induction on h Base case: h = 0, n = 1 Induction step: h > 0, assume claim holds for h 1 Tree has a root node, which has either 2 or 3 children Each of these children is the root of a subtree, which itself is a 2-3 tree of height h 1 By induction hypothesis, if the th subtree has n leaves, then n 2 h 1 [here, = (or 3)] Corollary: h log 2 n n = n 2h h 1 = 2 h 6
7 Serch(): use guides Serch(, p, heght): // Invoke as Serch(, root, h) if (heght > 0) then else if ( p.child0.gide) then return Serch(, p.child0, heght 1) else if ( p.child1.gide or p.child2 = nll) then return Serch(, p.child1, heght 1) else return Serch(, p.child2, heght 1) if = p.gide then return p.vle else return null (or a default value) 7
8 nsert(): Search for, and if it should belong under p: add as a child of p (if not already present) if p now has 4 children: split p into two two nodes, p 1 and p 2, each with two children process p s parent in the same way Special case: no parent create new root, increasing height of tree by 1 Also need to update guides easy Time = O(height) = O(log n) 8
9 Case when p ends up with 4 children p y z p p 1 p 2 y z y z 9
10 Deete(): Search for, and if found under p: remove if p now only has one child: if p is the root: delete p (height decreases by 1) if one of p s siblings has 3 children: borrow one if none of p s siblings has 3 children: one sibling q must have 2 children give p s only child to q delete p process p s parent 10
11 Easy case: borrow from sibling q p y q p y 11
12 Harder case: give away only child q p y q p y 12
13 2-3 trees: summary Assume n items in dictionary Running time for lookup, insert, delete: O(log n) comparisons, plus O(log n) overhead Space: O(n) pointers 13
14 Augmenting 2-3 trees Idea: augment nodes with additional information to support new types of queries Example: store # of items in subtree at each internal node Queries: What is the kth smallest item? How many items are? 14
15 Items may be marked with an attribute, say, active / inactive Store a count of active items in subtree at each internal node Queries: What is the kth smallest active item? How many active items are? 15
16 Attribute flipping Operation FpRnge(, y) flips all attribute bits of items in the range Assume attributes are bits Store an attribute bit at every node: intrenal nodes and leaves effective value of the attribute is the XOR of all bits on path from root to leaf To perform FpRnge(, y): trace paths e, ƒ to, y flip bits at, y, and all roots of internal subtrees 16
17 Example: y 17
18 2-3 Trees: Join and Split Jon(T 1, T 2 ) joins two 2-3 trees in time O(log n) Assume mx(t 1 ) < min(t 2 ) Assume T has height h for = 1, 2 Case 1: h 1 = h 2 new root Time: O(1) T 1 T 2 18
19 Case 2: h 1 < h 2 p T 1 T 2 Attach as the left-most child of p If p now has 4 children, we split p, and proceed up the tree as in nsert Time: O(h 2 h 1 ) = O(log n) Case 3: h 1 > h 2 similar 19
20 Spt(T, ) = (T 1 [ ], T 2 [> ]) T 1 T 2 T 3 T 4 T 5 join from inside out 20
21 Analysis of rejoin step Start with trees T 0, T 1,..., T k with h = heght(t ) h 0 h 1 h k [Monotonicity] There are at most two trees of any given height (except there may be three of height 0) [Diversity] Form new trees T 0, T 1,..., T k with h = heght(t ): T 0 = T 0, T +1 = Jon(T, T +1 ) Total running time: O(S), where k 1 S = ( h +1 h =0 + 1) 21
22 Claim: For = 0.. k 1: h {h, h + 1} From the claim, h +1 h h +1 h + 1, and so k 1 S (h +1 h + 2) = 2k + h k h 0 = O(log n) Example: = = a fresh tree: root has only 2 children 22
23 We prove the claim by induction, but we need to prove more: strengthening the induction hypothesis For = 0.. k 1, we have P() : Q() : h h {h, h + 1} > h +1 = T fresh Base case: = 0 (recall h 0 = h 0) Induction step: assume P(), Q() and prove P( + 1), Q( + 1), where = 0.. k 2 23
24 Case I. Recall: T +1 = Jon(T, T +1 ) 1. Assume h h By logic of Join and (1), either h +1 = h +1 or h +1 = h and T +1 is fresh 3. So P( + 1) holds 4. To prove Q( + 1): h +1 > h +2 = h +1 > h +1 [By monotonicity] = T is fresh [By (2)] +1 24
25 Case II. Recall: T +1 = Jon(T, T +1 ) 1. Assume h > h Must have > 0 3. By Q() and (1), T is fresh, and Join logic implies h +1 = h 4. We have h h + 1 [By P()] h [By monotonicity] and so h h [By (1)] = h + 1 = h By (4), we have h = h +1, and by diversity, monotonicity, and (2): h h Therefore: h +1 = h = h [ = P( + 1)] h +2 [ = Q( + 1)] 25
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