Search Structures - YUynucc.yu.ac.kr/~hrcho/Courses/FS/Chapter07.pdf · Chapter 7 - 1 ÷ ö O F f j ¶ j æ b Ú Chapter 7 Search Structures ÷ ö O F f j ¶ j æ b Ú Chapter 7 -

Chapter 7 - 1

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Chapter 7

Search Structures

Chapter 7 - 1÷�öOF fj¶�j� æbÚ

TABLE OF CONTENTS

● Binary Search Tree as a Solution

● Optimal Binary Search Trees

● AVL Trees

● Two-Three Trees (2-3 Trees)

● Two-Three-Four Trees (2-3-4 Trees)

● Red-Black Trees

Chapter 7 - 2


1. Binary Search Trees as a Solution

z ��✔ Every element has a unique key✔ Key in a nonempty left subtree < Key of root✔ Key in a nonempty right subtree > Key of root✔ Left & right subtrees are also binary search trees✔ þ�bú�Ó : î² 8.2 ~ î² 8.4

z ��✔ Index r sorted order R�¶V:�¼Ú.✔ ¾R³{��nÒ�c·¢únÒÆúN

z ��✔ Balanced tree r¶ÆN¶±^ ( î² 8.5 ~ î² 8.6 )✔ Too many seeks

– 24�N key Rb'� tree: 5~N seek


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FB

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FIGURE 8.2 Binary search tree representation of the list of keys.

Chapter 7 - 3


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FIGURE 8.3 Linked representation of part of a binary search tree.


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KeyLeftchild

Rightchild Key

Leftchild

Rightchild

FIGURE 8.4 Recordcontents for a linked representation of the binary tree in Fig. 8.2.

Chapter 7 - 4


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LV

FIGURE 8.5 Binary search tree with LV added.


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FIGURE 8.6 Binary search tree showingthe effect of added keys.

Chapter 7 - 5


2. Optimal Binary Search Trees

z ��

✔ BST�� average search length��!✔ �� access probability��.

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Extended Binary Tree

z Internal Path Length I = 0 + 1 + 1+ 2 + 3 = 7

z External Path Length E = 2 + 2 + 4 + 4 + 3 + 2 = 17

z E = I + 2n (n: # of internal nodes)➙ I��, �� E��

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Chapter 7 - 6


Cost of Extended Binary Search Tree

z Key� access probability��

✔ �� = Complete binary tree✔ Smallest value for I: Σi=1 log2i = O(nlog2n)

z Key� access probability��

✔ Key �: a1, a2, …, an (a1 < a2 < … < an)✔ pi: probability of searching ai

✔ Ei: all identifiers x such that ai < x < ai +1✔ qi: �� key�� Ei��

✔ Cost: Σi=1 pi•level(ai) + Σi=0 qi•(level(failure node i) – 1)✔ Σi=1 pi + Σi=0 qi = 1

∑=

n

i 1


Example

z pi = qj = 1/7 for all i and j✔ cost(tree a) = 13/7, cost(tree b) = 15/7

z p1 = .5, p2 = .1, p3 = .05, q0 = .15, q1 = .1, q2 = .05, q3 = .05✔ cost(tree a) = 1.9, cost(tree b) = 1.5

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Chapter 7 - 7


Finding Optimal Binary Search Tree

z �� 1: �� BST�� cost ��

✔ Complexity = O(n N(n))✔ N(n): �� n BST�� = O(4n/n3/2)

z �� 2: Greedy Algorithm✔ Let n identifiers: a1 < a2 < …< an

✔ Tij: optimal BST for ai+1, …, aj, i < j (ak < ak+1)– rij: root of Tij

– wij: weight of Tij = qi + Σ(qk + pk), i+1≤ k ≤ j– By definition, rii = 0 and wii = qi, 0 ≤ i ≤ n

✔ Goal: Find T0n!


T0n� ��

z Tij: optimal BST for ai+1, …, aj and rij = k (i < k ≤ j)

✔ Cost cij = pk + cost(L) + cost(R) + weight(L) + weight(R)– weight(L) = weight(Ti, k-1) = wi, k-1

– weight(R) = weight(Tkj) = wkj

✔ cij = pk + ci, k-1 + ckj + wi, k-1 + wkj = wij + ci, k-1 + ckj

✔ wij + ci, k-1 + ckj = min{wij + ci, k-1 + ckj}, i < l ≤ j✔ ci, k-1 + ckj = min{ci, k-1 + ckj}, i < l ≤ j

ak

/ 5

Chapter 7 - 8


void obst(int p[], int q[], int cost[][M], int root[][M], int weight[][M], int n){

int i, j, k, m, min, minpos;for (i = 0; i < n, i++) {

weight[i][i] = q[i]; root[i][i] = cost[i][i] = 0;cost[i][i+1] = weight[i][i+1] = q[i] + q[i+1] + p[i+1];root[i][i+1] = i + 1;

}weight[n][n] = q[n]; root[n][n] = cost[n][n] = 0;for (m = 2; m <= n; m++)

for (i = 0; i <= n-m; i++) {j = i + m;weight[i][j] = weight[i][j-1] + p[j] + q[j];k = knuth_min(cost, root, i, j);cost[i][j] = weight[i][j] + cost[i][k-1] + cost[k][j];root[i][j] = k;

} }

Program: Function to find an optimal binary search tree


z Example✔ n = 4 and (a1, a2, a3, a4) = (do, for, void, while)✔ (p1, p2, p3, p4) = (3, 3, 1, 1)✔ (q0, q1, q2, q3, q4) = (2, 3, 1, 1, 1)

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Chapter 7 - 9


3. AVL Trees

z Introduction✔ G. M. Adel’son-Vel’skii & E. M. Landis ��

✔ Height-balanced tree – HB(1) tree

z ��

✔ 2 �� Subtree �� depth �� ≤ 1✔ 4 �� ( LL, LR, RL, RR ) �� tree ��

✔ �� 5 ��

✔ � : B C G E F D A �� ( �� 8.11 )✔ Search cost : 1.44log2(N+2) ↔ log2(N+1) of B.S.T


D

B F

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FIGURE 8.10 A completely balanced search tree.

FIGURE 8.11 A search tree constructed using AVL procedures.

Chapter 7 - 10


� Insertion into an AVL tree

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Chapter 7 - 11



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Chapter 7 - 12



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Chapter 7 - 13



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(j) Insert June

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(k) Insert October

Chapter 7 - 14



Jan

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June May

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(k) Insert October



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Sept

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Chapter 7 - 15


4. Two-Three Trees (2-3 Trees)

z Definition✔ Internal node: 2-node or 3-node

– 2-node: one element with two children– 3-node: two element with three children

✔ All external nodes are at the same level.

z Properties✔ n: number of elements, h: height of 2-3 trees✔ 2h – 1 ≤ n ≤ 3h – 1✔ log3(n + 1) ≤ h ≤ log2(n + 1)


Searching a 2-3 Tree

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Chapter 7 - 16


Insertion Into a 2-3 Tree

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Chapter 7 - 17


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Program: Insertion into a 2-3 tree


Deletion From a 2-3 Tree

z ��

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– Inorder predecessor��, or– Inorder successor��

✔ Leaf node��

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• Rotation• Combine

Chapter 7 - 18


Example – ��

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Chapter 7 - 19


Steps in deleting from a leaf node, p

Step 1: Modify p as necessary to reflect its status after the desired element has been deleted.

Step 2: while (p has zero elements && p is not the root) {let r be the parent of p;let q be the left or right sibling of p (as appropriate);if (q is a 3-node) rotate;else combine;p = r;

}

Step 3: If p has zero elements, then p must be the root. The left child of p becomes the new root and p is deleted.


5. Two-Three-Four Trees (2-3-4 Trees)

z Definition: Definition of 2-3 Tree +✔ 4-node: three elements and four children

✔ n: number of elements, h: height of 2-3-4 trees✔ 2h – 1 ≤ n ≤ 4h – 1✔ log4(n + 1) ≤ h ≤ log2(n + 1)

z Node Structure

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Chapter 7 - 20


An Example 2-3-4 Tree

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Insertion Into a 2-3-4 Tree

z 2-3 Tree� Insertion Algorithm��

✔ Leaf node� split�� backward splitting ��

✔ Root�� tree� traverse��, 4-node��

✔ �, ��

– ��: Backward leaf to root pass ��

– ��: Node utilization ��

z 4-node��

✔ It is the root of the 2-3-4 tree.✔ Its parent is a 2-node.✔ Its parent is a 3-node.

Chapter 7 - 21


Transformation: Root 4-node

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Chapter 7 - 22


Transformation: Child of a 3-node

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Program: Insertion into a 2-3-4 tree

Chapter 7 - 23


Deletion From a 2-3-4 Tree

z Backward leaf to root restructuring path��

✔ Tree��, �� 3-/4-node��

✔ Insertion��, ��!

z Algorithm (p: current node, q: child node, r: sibling of q)✔ p = leaf: root�� 2-node. Otherwise, 3-/4-node✔ q ≠ 2-node: No restructuring. Just forward to q.✔ q = 2-node & r = 2-node:

– p = 2-node: p, q, r��

– p ≠ 2-node: p��, q, r�� (insertion��)✔ q = 2-node & r ≠ 2-node: rotate (��)


Deletion Transformation: Rotate

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Chapter 7 - 24


6. Red-Black Trees

z Red-Black Tree: 2-3-4 tree� binary tree ��

✔ Black pointer: child pointer of the original 2-3-4 tree✔ Red pointer: sibling relationship of the 2-3-4 tree

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Properties of Red-Black Tree

z Basic Properties✔ Binary search tree✔ Every root to external path: same number of black links✔ No root to external path: 2 or more consecutive red links

z Rank�� Properties✔ Binary search tree✔ Rank(external node) = 0✔ Rank(parent internal node of external node) = 1✔ rank(x) ≤ rank(parent(x)) ≤ rank(x) + 1✔ rank(x) < rank(grand_parent(x))

Chapter 7 - 25


Properties of Red-Black Tree (Cont’d)

z Intuitive Definition of Rank✔ Suppose a node x of a 2-3-4 tree T.✔ Rank of the corresponding nodes in red-black tree =

height(T) – level(x) + 1

z Every red-black tree RB with n (internal) nodes✔ height(RB) ≤ 2 log2(n + 1)✔ height(RB) ≤ 2rank(RB)✔ rank(RB) ≤ log2(n + 1)

�Definition: Rank of a tree = Rank of the roor


Operations On a Red-Black Tree

z Searching a Red-Black Tree✔ �� BST��

✔ Pointer color��

z Insertion✔ Top Down Insertion: 4-node splitting transformation✔ Bottom Up Insertion: BST��

z Deletion✔ 2-3-4 tree�� red-black tree� ��

z Red-Black Tree��

✔ � , �� link� color�� (��)✔ ��

Chapter 7 - 26


Top Down Insertion

z Transforming a node q with two red links (4-node)✔ q�� link� black link��

✔ q� parent� left child��, parent� left color = red– Right child��

✔ ��, 2�� red link��

– gp → p → q (gp: grand parent of q, p: parent of q)– gp → p� X, p → q� Y��, XY ��

4�� (LL, LR, RL, RR)– AVL �� rotation

z Leaf �� red link��


Transformation: Root 4-node

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Chapter 7 - 27


Transformation: Child of a 2-/3- node

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Chapter 7 - 28


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Bottom Up Insertion

z Algorithm✔ Leaf node��, red link�� leaf node��

✔ ��, red link��.– �� red link: p → q (X), q → r (Y)– s� q� sibling��, XYZ violation ��

• X = L, if p → q is a left link. X = R otherwise.• Y = L, if q → r is a left link. Y = R otherwise.• Z = r (for red) if s ≠ NULL and p → s is a left link.

Z = b (for black) otherwise.✔ LLr� LRr, LLb� LRb��

✔ Bottom up�� color change �� propagation

Chapter 7 - 29


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Chapter 7 - 30


Deletion From a Red-Black Tree

z �� leaf node� parent��?✔ Red link: leaf node� 3-/4- node�� (��)✔ Black link: leaf node� 2-node

– Leaf node�� backward restructuring ��

z Backward restructuring��

✔ 2-3-4 deletion transformation��.✔ Insertion�� transformation.

Documents

Search Structures - YUynucc.yu.ac.kr/~hrcho/Courses/FS/Chapter07.pdf · Chapter 7 - 1 ÷ ö O F f j ¶ j æ b Ú Chapter 7 Search Structures ÷ ö O F f j ¶ j æ b Ú Chapter 7 -