POLYNOMIALS and Tree sort

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Lecture 11

• POLYNOMIALS and Tree sort

Prof. Sin-Min Lee Department of Computer Science
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Topics include

• INTRODUCTION
• EVALUATING POLYNOMIAL FUNCTIONS
• Horners method
• Permutation
• Tree sort

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INTRODUCTION

• The problems examined in this lecture are
  polynomial evaluation (with and without
  preprocessing of the coefficients), polynomial
  multiplication, and multiplication of matrices
  and vectors.
• Several algorithms in this lecture use the
  divide-and-conquer method evaluating a
  polynomial with preprocessing of coefficients,
  Strassens matrix multiplication algorithm.

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EVALUATING POLYNOMIAL FUNCTIONS

• Consider the polynomial
• P(x) anxn an-1xn-1 a1x a0
• with real coefficients and ngt1
• Suppose the coefficients of a0, a1, , an and x
  are given and that the problem is the evaluate
  p(x).
• In this section we look at some algorithms and
  some lower bounds for this problem.

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Synthetic Division is a process whereby the
quotient and remainder can be determined when a
polynomial function f is divided by g(x) x - c.
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Use synthetic division to find the quotient and
remainder when
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f(-3) 278
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Algorithms

• The obvious way to solve the problem is to
  compute each term and add it to the sum of the
  others already computed. The following algorithm
  does this

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Algorithm Polynomial EvaluationTerm by Term

• Input The coefficients of polynomial p(x) in
  the array a ngt0, the degree of p and x, the
  point at which to evaluate p.
• Output The value of p(x).
• float poly(float a, int n, float x)
• float p, xpower
• int i
• p a0 xpower 1
• for (i 1 i lt n i)
• xpower xpower x
• pai xpower
• return p
• Note This algorithm does 2n multiplications and
  n additions

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Horners method

• The key to Horners method for evaluating p(x) is
  simply a particular factorization of p
• p(x) (((anx an-1)x an-2)xa1)x a0.
• The computation is done in a short loop with only
  n multiplications and n additions.

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Algorithm Polynomial Evaluation Horners Method

• Input a, n, and x as in Algorithm 12.1.
• Output The value of p(x).
• float hornerPoly(floata, int n, float x)
• float p
• int i
• p an
• for (i n 1 igt 0 i--)
• p p x ai
• return p
• Thus simply by factoring p we have cut the
  number of multiplications in half without
  increasing the number of additions.

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Binary Tree

• A binary tree is a structure in which
• Each node can have at most two children, and
  in which a unique path exists from the root to
  every other node.
• The two children of a node are called the left
  child and the right child, if they exist.

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A Binary Tree

V
Q
L
T
A
E
K
S
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How many leaf nodes?

V
Q
L
T
A
E
K
S
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How many descendants of Q?

V
Q
L
T
A
E
K
S
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How many ancestors of K?

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Implementing a Binary Tree with Pointers and
Dynamic Data

V
Q
L
T
A
E
K
S
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Each node contains two pointers
templatelt class ItemType gt struct TreeNode
ItemType info // Data member
TreeNodeltItemTypegt left // Pointer to
left child TreeNodeltItemTypegt right //
Pointer to right child
NULL A 6000
. left . info . right
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A Binary Search Tree (BST) is . . .

• A special kind of binary tree in which
• 1. Each node contains a distinct data value,
• 2. The key values in the tree can be compared
  using greater than and less than, and
• 3. The key value of each node in the tree is
• less than every key value in its right subtree,
  and greater than every key value in its left
  subtree.

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Shape of a binary search tree . . .

• Depends on its key values and their order of
  insertion.
• Insert the elements J E F T A
  in that order.
• The first value to be inserted is put into the
  root node.

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Inserting E into the BST

• Thereafter, each value to be inserted begins by
  comparing itself to the value in the root node,
  moving left it is less, or moving right if it is
  greater. This continues at each level until it
  can be inserted as a new leaf.

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Inserting F into the BST

• Begin by comparing F to the value in the root
  node, moving left it is less, or moving right if
  it is greater. This continues until it can be
  inserted as a leaf.

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Inserting T into the BST

• Begin by comparing T to the value in the root
  node, moving left it is less, or moving right if
  it is greater. This continues until it can be
  inserted as a leaf.

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Inserting A into the BST

• Begin by comparing A to the value in the root
  node, moving left it is less, or moving right if
  it is greater. This continues until it can be
  inserted as a leaf.

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What binary search tree . . .

• is obtained by inserting
• the elements A E F J T in
  that order?

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Binary search tree . . .

• obtained by inserting
• the elements A E F J T in
  that order.

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Another binary search tree

T
E
A
H
M
P
K
Add nodes containing these values in this
order D B L Q S
V Z
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Is F in the binary search tree?

J
T
E
A
V
M
H
P
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Inorder Traversal A E H J M T Y
Print second
tree

T
E
A
H
M
Y
Print left subtree first
Print right subtree last
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Preorder Traversal J E A H T M Y
Print first
tree

T
E
A
H
M
Y
Print left subtree second
Print right subtree last
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Postorder Traversal A H E M Y T J
Print last
tree

T
E
A
H
M
Y
Print left subtree first
Print right subtree second