M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)

Size: px
Start display at page:

Download "M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)"

Transcription

1 Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, ackground ccording to Knut [Knut73], te VL version of te binary searc tree represents a nice compromise between te optimum binary tree, wose eigt is minimal but for wic maintenance is seemingly difficult, and te arbitrary binary tree, wic requires no special maintenance but wose eigt could possibly grow to be muc greater tan minimal. It accomplises tis by employing a balanced (or admissible [VL62]) binary tree, wic is defined as one in wic eac node meets te requirement tat te respective eigts of its left and rigt subtrees differ by no more tan one. Te eigt of an admissible tree wit n > nodes is at least lg n (as it is for any binary tree) and, as del son-vel skii and Landis [VL62] sow, less tan log φ (n + 1) (approximately 1.44 lg(n + 1)), were φ is te golden ratio (1 + 5)/2. (For more on te golden ratio, see ttp://en.wikipedia.org/wiki/golden ratio.) To obtain tis result, first tey computed te minimum number of nodes in an admissible tree of eigt. For = and = 1, tese values are 1 and 2, respectively. For > 1, an admissible tree aving a minimal number of nodes consists of a root node and minimal subtrees of eigts 1 and 2. Hence, te desired minimum is given by te recurrence M() = 1 M(1) = 2 M() = M( 1) + M( 2) + 1 ( > 1) wic closely resembles te Fibonacci recurrence. 2 Te result ten follows by induction. Te eigt of an VL Tree, ten, can be no worse tan 5% greater tan optimal 3, and so te number of steps required for a searc is still proportional to lg n, even in te worst case. In order to easily maintain te tree s admissibility during te performance of a sequence of insertions and/or deletions, we associate wit eac node a balance factor, wic is te difference between te eigts of its left and rigt subtrees, respectively. In an admissible tree, ten, eac node as a balance factor of eiter 1,, or revised January 21 and again in January Indeed, taking F = and F 1 = 1 as te initial values in te Fibonacci sequence and F k = F k 1 + F k 2 (k > 1) as te recurrence, we get M() = F +3 1 for all. 3 Te largest ratio between actual eigt and optimum eigt is realized by a 7-node VL tree of eigt tree. 1

2 Te beauty of te VL metod is tat it provides a way to easily rebalance a subtree tat as become inadmissible as te result of an insertion or deletion of a node witin it in order to make it admissible again, witout sacrificing more tan a constant factor in run-time complexity. Consequently, bot insertion and deletion require O(lg n) time. Node Insertion Insertion of a node into an VL Tree proceeds in exactly te same manner as in an arbitrary binary searc tree. Once te new node as been put in place, toug, additional steps must be taken to update balance factors and to ensure te tree s admissibility. Specifically, te portion of te tree tat is affected by an insertion is te subtree wose root is te last node wit a non-zero balance factor lying on wat del son-vel skii and Landis refer to as te recorded cain, wic is te pat from te root of te tree to te parent of te newly-inserted node. (If all nodes on te recorded cain ave a balance factor of zero, is taken to be te root of te tree.) Te newly-inserted node s balance factor sould be set to zero, of course. Meanwile, eac node below on te recorded cain as a subtree wose eigt as increased by one due to te insertion. ( proof is left to te reader.) Tus, eac suc node s balance factor sould be canged from zero to eiter +1 or 1, respectively, according to weter te new node was put into its left subtree or rigt subtree. s for, it, too, as a subtree wose eigt as increased. If s balance factor was zero implying tat it is te root of te tree its new balance factor is determined in te same way as tose of its proper descendants in te recorded cain (as described in te preceding paragrap). In tis case, te tree is still admissible. If, on te oter and, s balance factor was +1 or 1, tere are two possibilities: 1. Te recorded cain leads into wat ad been s sorter subtree, in wic case s balance factor sould be set to zero. Te eigt of te subtree rooted at as not been altered, and so te balance factors of s proper ancestors need not be modified. Te tree remains admissible. 2. Te recorded cain leads into s taller subtree, in wic case s balance factor sould be set to eiter +2 or 2 (according to weter, respectively, te newly-inserted node is in s left subtree or rigt subtree), implying inadmissibility. Tere are two possible cases, eac aving a mirror image. Let be te cild of tat is on te recorded cain. In te first case, s and s balance factors ave te same sign. To make te subtree rooted at admissible, a single rotation is performed, as illustrated in Figure 1. In te second case, s and s balance factors ave opposite signs. To make te subtree rooted at admissible, a double rotation is performed, as illustrated in Figure 2. (Tat figure depicts node C as aving balance factor ±1. However, te same remedy applies if C is itself te newly-inserted node (and ence as balance factor zero), in wic case te subtrees labeled,, and δ in te figure are empty.) Note tat, in bot cases, te resulting subtree s eigt is te same as wat it ad been prior to te insertion. Tus, te balance factors of te proper ancestors of te subtree need not be modified. 2

3 Figure 1: Case I: Rebalancing Using a Single Rotation +2 C 1 δ /+1 1/ +1/ 1 C δ 1 1 Figure 2: Case II: Rebalancing Using a Double Rotation 3

4 Node Deletion Deletion of a node from an VL Tree proceeds in exactly te same manner as in an arbitrary binary searc tree. (Te task of node deletion can always be reduced to tat of deleting a node tat as at most one cild.) s wit insertion, additional steps must be taken to maintain balance factors and tree admissibility. Wit respect to deletion, te recorded cain is te pat from te root to te parent of te newly-deleted node. Te portion of te tree tat may be affected by a node deletion is te subtree wose root D is te last node on te recorded cain aving a zero balance factor. (If all nodes on te recorded cain ave a non-zero balance factor, D is taken to be te root of te tree.) Te nodes on te recorded cain are considered in order from te bottom (i.e., te parent of te deleted node) up to D. Referring to te currently considered node as, tere are tree possibilities: 1. as a balance factor of zero, wic is to say tat is D. In tis case, te subtree of in wic te deletion occurred as decreased in eigt by one; ence, s balance factor sould be set to eiter 1 or +1, respectively, according to weter te deletion occurred in te left subtree or te rigt subtree. ecause te eigt of te subtree rooted at (i.e., D) is uncanged, te balance factors of s proper ancestors need not be modified. 2. Te recorded cain leads into wat ad been s taller subtree, te eigt of wic decreased by one as a result of te deletion. Hence, s balance factor sould be set to zero. ecause te eigt of te tree rooted at as decreased by one, its parent sould be te node to be considered next. (Tat is, s parent will play te role of during te next iteration.) 3. Te recorded cain leads into s sorter subtree, te eigt of wic decreased by one as a result of te deletion. Hence, s balance factor sould be set to 2 or +2, respectively, according to weter te deletion occurred in its left subtree or rigt subtree. Tis makes te subtree rooted at inadmissible. Tere are tree cases (eac aving a mirror image), two of wic are identical to tose tat can be produced by an insertion (and terefore are remedied by te metods illustrated in Figures 1 and 2). 4 Te tird case, sown in Figure 3, is te same as tat sown in Figure 1, except tat te balance factor of node is initially zero instead of ±1. 5 Wat is most interesting is tat, in eac of te first two cases, te performance of te appropriate rotation(s) results in a decrease in te eigt of te rebalanced subtree, wereas, in te tird case, tat subtree s eigt is uncanged. s a consequence, after doing te appropriate rotation to remedy te tird case, noting else needs to be done. On te oter and, in eac of te first two cases, te parent node of te root of te rebalanced 4 Regarding Case II (Figure 2): In te context of a deletion tere is te additional possibility for bot of subtrees and to ave eigt (and tus for node C to ave a zero balance factor). In tat case, bot nodes and end up wit a zero balance factor after te double rotation. 5 In te context of a deletion, node (in all tree cases) is eiter te cild of opposite to tat on te recorded cain or else te sibling of te deleted node. Tat is, in Figures 1 and 3 (resp., Figure 2) te deleted node ad been in subtree (resp., δ). 4

5 subtree must be considered next, as its balance factor needs adjustment. Indeed, it is possible, in te worst case, for te deletion of a node to result in a rotation (single or double) occurring at eac node on te recoded cain, of wic tere are O(lg n) Figure 3: Case III: Rebalancing fter a Deletion Using a Single Rotation Run-time complexity Te major difference between insertion and deletion is tat deletion can require up to O(lg n) rotations, wereas insertion requires at most one. ot operations entail an initial searc (O(lg n) time), a pysical insertion or deletion of a node (O(1) time), and a modification of te balance factors of at most all te nodes on te recorded cain (O(lg n) time). ecause a rotation can be done in constant time (via a few pointer assignments), te time required by rotations is O(1) during an insertion and O(lg n) during a deletion. In summary, ten, bot insertion and deletion take O(lg n) time. Original Paper Interestingly, (te Englis translation of) te paper in wic VL Trees were first introduced [VL62] said noting about node deletion. Moreover, it erroneously omitted te rebalancing case illustrated ere in Figure 1 and instead included te case illustrated in Figure 3, wic cannot possibly occur during an insertion! References [VL62] del son-vel skii, G.M. and Y.M. Landis, n lgoritm for te Organization of Information, Soviet Mat. Dokl 3 (1962) (Englis translation), pp [Knut73] Knut, Donald E., rt of Computer Programming, Volume 3: Sorting and Searcing, ddison-wesley,

schema binary search tree schema binary search trees data structures and algorithms 2015 09 21 lecture 7 AVL-trees material

schema binary search tree schema binary search trees data structures and algorithms 2015 09 21 lecture 7 AVL-trees material scema binary searc trees data structures and algoritms 05 0 lecture 7 VL-trees material scema binary searc tree binary tree: linked data structure wit nodes containing binary searc trees VL-trees material

More information

Binary Search Trees. Adnan Aziz. Heaps can perform extract-max, insert efficiently O(log n) worst case

Binary Search Trees. Adnan Aziz. Heaps can perform extract-max, insert efficiently O(log n) worst case Binary Searc Trees Adnan Aziz 1 BST basics Based on CLRS, C 12. Motivation: Heaps can perform extract-max, insert efficiently O(log n) worst case Has tables can perform insert, delete, lookup efficiently

More information

Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Math 120 Calculus I D Joyce, Fall 2013 Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

More information

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger

More information

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

More information

f(a + h) f(a) f (a) = lim

f(a + h) f(a) f (a) = lim Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

More information

Optimized Data Indexing Algorithms for OLAP Systems

Optimized Data Indexing Algorithms for OLAP Systems Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest

More information

Math 113 HW #5 Solutions

Math 113 HW #5 Solutions Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

More information

Geometric Stratification of Accounting Data

Geometric Stratification of Accounting Data Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual

More information

ACT Math Facts & Formulas

ACT Math Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

More information

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution 1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

More information

SAT Subject Math Level 1 Facts & Formulas

SAT Subject Math Level 1 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

More information

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using

More information

Instantaneous Rate of Change:

Instantaneous Rate of Change: Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over

More information

Verifying Numerical Convergence Rates

Verifying Numerical Convergence Rates 1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and

More information

How To Ensure That An Eac Edge Program Is Successful

How To Ensure That An Eac Edge Program Is Successful Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.

More information

The modelling of business rules for dashboard reporting using mutual information

The modelling of business rules for dashboard reporting using mutual information 8 t World IMACS / MODSIM Congress, Cairns, Australia 3-7 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,

More information

Writing Mathematics Papers

Writing Mathematics Papers Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not

More information

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1 Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te

More information

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

More information

Schedulability Analysis under Graph Routing in WirelessHART Networks

Schedulability Analysis under Graph Routing in WirelessHART Networks Scedulability Analysis under Grap Routing in WirelessHART Networks Abusayeed Saifulla, Dolvara Gunatilaka, Paras Tiwari, Mo Sa, Cenyang Lu, Bo Li Cengjie Wu, and Yixin Cen Department of Computer Science,

More information

Pressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:

Pressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area: Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force

More information

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade? Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te

More information

SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY

SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,

More information

Math Test Sections. The College Board: Expanding College Opportunity

Math Test Sections. The College Board: Expanding College Opportunity Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt

More information

New Vocabulary volume

New Vocabulary volume -. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding

More information

2 Limits and Derivatives

2 Limits and Derivatives 2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line

More information

The EOQ Inventory Formula

The EOQ Inventory Formula Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of

More information

Catalogue no. 12-001-XIE. Survey Methodology. December 2004

Catalogue no. 12-001-XIE. Survey Methodology. December 2004 Catalogue no. 1-001-XIE Survey Metodology December 004 How to obtain more information Specific inquiries about tis product and related statistics or services sould be directed to: Business Survey Metods

More information

ANALYTICAL REPORT ON THE 2010 URBAN EMPLOYMENT UNEMPLOYMENT SURVEY

ANALYTICAL REPORT ON THE 2010 URBAN EMPLOYMENT UNEMPLOYMENT SURVEY THE FEDERAL DEMOCRATIC REPUBLIC OF ETHIOPIA CENTRAL STATISTICAL AGENCY ANALYTICAL REPORT ON THE 2010 URBAN EMPLOYMENT UNEMPLOYMENT SURVEY Addis Ababa December 2010 STATISTICAL BULLETIN TABLE OF CONTENT

More information

Solutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in

Solutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM-1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)

More information

Binary Search Trees. Data in each node. Larger than the data in its left child Smaller than the data in its right child

Binary Search Trees. Data in each node. Larger than the data in its left child Smaller than the data in its right child Binary Search Trees Data in each node Larger than the data in its left child Smaller than the data in its right child FIGURE 11-6 Arbitrary binary tree FIGURE 11-7 Binary search tree Data Structures Using

More information

CHAPTER 8: DIFFERENTIAL CALCULUS

CHAPTER 8: DIFFERENTIAL CALCULUS CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly

More information

2.23 Gambling Rehabilitation Services. Introduction

2.23 Gambling Rehabilitation Services. Introduction 2.23 Gambling Reabilitation Services Introduction Figure 1 Since 1995 provincial revenues from gambling activities ave increased over 56% from $69.2 million in 1995 to $108 million in 2004. Te majority

More information

2.12 Student Transportation. Introduction

2.12 Student Transportation. Introduction Introduction Figure 1 At 31 Marc 2003, tere were approximately 84,000 students enrolled in scools in te Province of Newfoundland and Labrador, of wic an estimated 57,000 were transported by scool buses.

More information

Theoretical calculation of the heat capacity

Theoretical calculation of the heat capacity eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals

More information

Full and Complete Binary Trees

Full and Complete Binary Trees Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

More information

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line. Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,

More information

Projective Geometry. Projective Geometry

Projective Geometry. Projective Geometry Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,

More information

College Planning Using Cash Value Life Insurance

College Planning Using Cash Value Life Insurance College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded

More information

Distances in random graphs with infinite mean degrees

Distances in random graphs with infinite mean degrees Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree

More information

Factoring Synchronous Grammars By Sorting

Factoring Synchronous Grammars By Sorting Factoring Syncronous Grammars By Sorting Daniel Gildea Computer Science Dept. Uniersity of Rocester Rocester, NY Giorgio Satta Dept. of Information Eng g Uniersity of Padua I- Padua, Italy Hao Zang Computer

More information

2.1: The Derivative and the Tangent Line Problem

2.1: The Derivative and the Tangent Line Problem .1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position

More information

Chapter 7 Numerical Differentiation and Integration

Chapter 7 Numerical Differentiation and Integration 45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea

More information

The Demand for Food Away From Home Full-Service or Fast Food?

The Demand for Food Away From Home Full-Service or Fast Food? United States Department of Agriculture Electronic Report from te Economic Researc Service www.ers.usda.gov Agricultural Economic Report No. 829 January 2004 Te Demand for Food Away From Home Full-Service

More information

Lecture 4: Balanced Binary Search Trees

Lecture 4: Balanced Binary Search Trees Lecture 4 alanced inar Search Trees 6.006 Fall 009 Lecture 4: alanced inar Search Trees Lecture Overview The importance of being balanced VL trees Definition alance Insert Other balanced trees Data structures

More information

SWITCH T F T F SELECT. (b) local schedule of two branches. (a) if-then-else construct A & B MUX. one iteration cycle

SWITCH T F T F SELECT. (b) local schedule of two branches. (a) if-then-else construct A & B MUX. one iteration cycle 768 IEEE RANSACIONS ON COMPUERS, VOL. 46, NO. 7, JULY 997 Compile-ime Sceduling of Dynamic Constructs in Dataæow Program Graps Soonoi Ha, Member, IEEE and Edward A. Lee, Fellow, IEEE Abstract Sceduling

More information

Data Structures Fibonacci Heaps, Amortized Analysis

Data Structures Fibonacci Heaps, Amortized Analysis Chapter 4 Data Structures Fibonacci Heaps, Amortized Analysis Algorithm Theory WS 2012/13 Fabian Kuhn Fibonacci Heaps Lacy merge variant of binomial heaps: Do not merge trees as long as possible Structure:

More information

In other words the graph of the polynomial should pass through the points

In other words the graph of the polynomial should pass through the points Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form

More information

S. Muthusundari. Research Scholar, Dept of CSE, Sathyabama University Chennai, India e-mail: nellailath@yahoo.co.in. Dr. R. M.

S. Muthusundari. Research Scholar, Dept of CSE, Sathyabama University Chennai, India e-mail: nellailath@yahoo.co.in. Dr. R. M. A Sorting based Algorithm for the Construction of Balanced Search Tree Automatically for smaller elements and with minimum of one Rotation for Greater Elements from BST S. Muthusundari Research Scholar,

More information

Grade 12 Assessment Exemplars

Grade 12 Assessment Exemplars Grade Assessment Eemplars Learning Outcomes and. Assignment : Functions - Memo. Investigation: Sequences and Series Memo/Rubric 5. Control Test: Number Patterns, Finance and Functions - Memo 7. Project:

More information

SAT Math Facts & Formulas

SAT Math Facts & Formulas Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:

More information

Chapter 11. Limits and an Introduction to Calculus. Selected Applications

Chapter 11. Limits and an Introduction to Calculus. Selected Applications Capter Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem Selected Applications

More information

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

More information

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12. Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

More information

3 Ans. 1 of my $30. 3 on. 1 on ice cream and the rest on 2011 MATHCOUNTS STATE COMPETITION SPRINT ROUND

3 Ans. 1 of my $30. 3 on. 1 on ice cream and the rest on 2011 MATHCOUNTS STATE COMPETITION SPRINT ROUND 0 MATHCOUNTS STATE COMPETITION SPRINT ROUND. boy scouts are accompanied by scout leaders. Eac person needs bottles of water per day and te trip is day. + = 5 people 5 = 5 bottles Ans.. Cammie as pennies,

More information

Tangent Lines and Rates of Change

Tangent Lines and Rates of Change Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

More information

EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution

EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Prolem set 8 Solution 1) Suppose tat te stock of mone in a given econom is given te sum of currenc and demand for current accounts tat

More information

Volumes of Pyramids and Cones. Use the Pythagorean Theorem to find the value of the variable. h 2 m. 1.5 m 12 in. 8 in. 2.5 m

Volumes of Pyramids and Cones. Use the Pythagorean Theorem to find the value of the variable. h 2 m. 1.5 m 12 in. 8 in. 2.5 m -5 Wat You ll Learn To find te volume of a pramid To find te volume of a cone... And W To find te volume of a structure in te sape of a pramid, as in Eample Volumes of Pramids and Cones Ceck Skills You

More information

SAT Math Must-Know Facts & Formulas

SAT Math Must-Know Facts & Formulas SAT Mat Must-Know Facts & Formuas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas

More information

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

More information

Note: Principal version Modification Modification Complete version from 1 October 2014 Business Law Corporate and Contract Law

Note: Principal version Modification Modification Complete version from 1 October 2014 Business Law Corporate and Contract Law Note: Te following curriculum is a consolidated version. It is legally non-binding and for informational purposes only. Te legally binding versions are found in te University of Innsbruck Bulletins (in

More information

Strategic trading in a dynamic noisy market. Dimitri Vayanos

Strategic trading in a dynamic noisy market. Dimitri Vayanos LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral

More information

Shell and Tube Heat Exchanger

Shell and Tube Heat Exchanger Sell and Tube Heat Excanger MECH595 Introduction to Heat Transfer Professor M. Zenouzi Prepared by: Andrew Demedeiros, Ryan Ferguson, Bradford Powers November 19, 2009 1 Abstract 2 Contents Discussion

More information

A strong credit score can help you score a lower rate on a mortgage

A strong credit score can help you score a lower rate on a mortgage NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing

More information

Chapter 10: Refrigeration Cycles

Chapter 10: Refrigeration Cycles Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators

More information

Average and Instantaneous Rates of Change: The Derivative

Average and Instantaneous Rates of Change: The Derivative 9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to

More information

Improved dynamic programs for some batcing problems involving te maximum lateness criterion A P M Wagelmans Econometric Institute Erasmus University Rotterdam PO Box 1738, 3000 DR Rotterdam Te Neterlands

More information

CHAPTER TWO. f(x) Slope = f (3) = Rate of change of f at 3. x 3. f(1.001) f(1) Average velocity = 1.1 1 1.01 1. s(0.8) s(0) 0.8 0

CHAPTER TWO. f(x) Slope = f (3) = Rate of change of f at 3. x 3. f(1.001) f(1) Average velocity = 1.1 1 1.01 1. s(0.8) s(0) 0.8 0 CHAPTER TWO 2.1 SOLUTIONS 99 Solutions for Section 2.1 1. (a) Te average rate of cange is te slope of te secant line in Figure 2.1, wic sows tat tis slope is positive. (b) Te instantaneous rate of cange

More information

Research on the Anti-perspective Correction Algorithm of QR Barcode

Research on the Anti-perspective Correction Algorithm of QR Barcode Researc on te Anti-perspective Correction Algoritm of QR Barcode Jianua Li, Yi-Wen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic

More information

The Dynamics of Movie Purchase and Rental Decisions: Customer Relationship Implications to Movie Studios

The Dynamics of Movie Purchase and Rental Decisions: Customer Relationship Implications to Movie Studios Te Dynamics of Movie Purcase and Rental Decisions: Customer Relationsip Implications to Movie Studios Eddie Ree Associate Professor Business Administration Stoneill College 320 Wasington St Easton, MA

More information

KM client format supported by KB valid from 13 May 2015

KM client format supported by KB valid from 13 May 2015 supported by KB valid from 13 May 2015 1/16 VRSION 1.2. UPDATD: 13.12.2005. 1 Introduction... 2 1.1 Purpose of tis document... 2 1.2 Caracteristics of te KM format... 2 2 Formal ceck of KM format... 3

More information

6. Differentiating the exponential and logarithm functions

6. Differentiating the exponential and logarithm functions 1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose

More information

Symbol Tables. Introduction

Symbol Tables. Introduction Symbol Tables Introduction A compiler needs to collect and use information about the names appearing in the source program. This information is entered into a data structure called a symbol table. The

More information

Computer Science and Engineering, UCSD October 7, 1999 Goldreic-Levin Teorem Autor: Bellare Te Goldreic-Levin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an n-bit

More information

A system to monitor the quality of automated coding of textual answers to open questions

A system to monitor the quality of automated coding of textual answers to open questions Researc in Official Statistics Number 2/2001 A system to monitor te quality of automated coding of textual answers to open questions Stefania Maccia * and Marcello D Orazio ** Italian National Statistical

More information

From Last Time: Remove (Delete) Operation

From Last Time: Remove (Delete) Operation CSE 32 Lecture : More on Search Trees Today s Topics: Lazy Operations Run Time Analysis of Binary Search Tree Operations Balanced Search Trees AVL Trees and Rotations Covered in Chapter of the text From

More information

Guide to Cover Letters & Thank You Letters

Guide to Cover Letters & Thank You Letters Guide to Cover Letters & Tank You Letters 206 Strebel Student Center (315) 792-3087 Fax (315) 792-3370 TIPS FOR WRITING A PERFECT COVER LETTER Te resume never travels alone. Eac time you submit your resume

More information

Working Capital 2013 UK plc s unproductive 69 billion

Working Capital 2013 UK plc s unproductive 69 billion 2013 Executive summary 2. Te level of excess working capital increased 3. UK sectors acieve a mixed performance 4. Size matters in te supply cain 6. Not all companies are overflowing wit cas 8. Excess

More information

Analysis of Algorithms I: Binary Search Trees

Analysis of Algorithms I: Binary Search Trees Analysis of Algorithms I: Binary Search Trees Xi Chen Columbia University Hash table: A data structure that maintains a subset of keys from a universe set U = {0, 1,..., p 1} and supports all three dictionary

More information

13 PERIMETER AND AREA OF 2D SHAPES

13 PERIMETER AND AREA OF 2D SHAPES 13 PERIMETER AND AREA OF D SHAPES 13.1 You can find te perimeter of sapes Key Points Te perimeter of a two-dimensional (D) sape is te total distance around te edge of te sape. l To work out te perimeter

More information

Investing in Roads: Pricing, Costs and New Capacity

Investing in Roads: Pricing, Costs and New Capacity Investing in Roads: Pricing, Costs and New Capacity Cristoper rcer Stepen Glaister Department of Civil and Environmental Engineering Imperial College London November 2006 Researc commissioned by te Independent

More information

Unemployment insurance/severance payments and informality in developing countries

Unemployment insurance/severance payments and informality in developing countries Unemployment insurance/severance payments and informality in developing countries David Bardey y and Fernando Jaramillo z First version: September 2011. Tis version: November 2011. Abstract We analyze

More information

Recall from last time: Events are recorded by local observers with synchronized clocks. Event 1 (firecracker explodes) occurs at x=x =0 and t=t =0

Recall from last time: Events are recorded by local observers with synchronized clocks. Event 1 (firecracker explodes) occurs at x=x =0 and t=t =0 1/27 Day 5: Questions? Time Dilation engt Contraction PH3 Modern Pysics P11 I sometimes ask myself ow it came about tat I was te one to deelop te teory of relatiity. Te reason, I tink, is tat a normal

More information

Referendum-led Immigration Policy in the Welfare State

Referendum-led Immigration Policy in the Welfare State Referendum-led Immigration Policy in te Welfare State YUJI TAMURA Department of Economics, University of Warwick, UK First version: 12 December 2003 Updated: 16 Marc 2004 Abstract Preferences of eterogeneous

More information

Once you have reviewed the bulletin, please

Once you have reviewed the bulletin, please Akron Public Scools OFFICE OF CAREER EDUCATION BULLETIN #5 : Driver Responsibilities 1. Akron Board of Education employees assigned to drive Board-owned or leased veicles MUST BE FAMILIAR wit te Business

More information

An inquiry into the multiplier process in IS-LM model

An inquiry into the multiplier process in IS-LM model An inquiry into te multiplier process in IS-LM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 00-62763074 Internet Address: jefferson@water.pu.edu.cn

More information

Section 2.3 Solving Right Triangle Trigonometry

Section 2.3 Solving Right Triangle Trigonometry Section.3 Solving Rigt Triangle Trigonometry Eample In te rigt triangle ABC, A = 40 and c = 1 cm. Find a, b, and B. sin 40 a a c 1 a 1sin 40 7.7cm cos 40 b c b 1 b 1cos40 9.cm A 40 1 b C B a B = 90 - A

More information

FINITE DIFFERENCE METHODS

FINITE DIFFERENCE METHODS FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to

More information

root node level: internal node edge leaf node CS@VT Data Structures & Algorithms 2000-2009 McQuain

root node level: internal node edge leaf node CS@VT Data Structures & Algorithms 2000-2009 McQuain inary Trees 1 A binary tree is either empty, or it consists of a node called the root together with two binary trees called the left subtree and the right subtree of the root, which are disjoint from each

More information

TREE BASIC TERMINOLOGIES

TREE BASIC TERMINOLOGIES TREE Trees are very flexible, versatile and powerful non-liner data structure that can be used to represent data items possessing hierarchical relationship between the grand father and his children and

More information

THE NEISS SAMPLE (DESIGN AND IMPLEMENTATION) 1997 to Present. Prepared for public release by:

THE NEISS SAMPLE (DESIGN AND IMPLEMENTATION) 1997 to Present. Prepared for public release by: THE NEISS SAMPLE (DESIGN AND IMPLEMENTATION) 1997 to Present Prepared for public release by: Tom Scroeder Kimberly Ault Division of Hazard and Injury Data Systems U.S. Consumer Product Safety Commission

More information

Torchmark Corporation 2001 Third Avenue South Birmingham, Alabama 35233 Contact: Joyce Lane 972-569-3627 NYSE Symbol: TMK

Torchmark Corporation 2001 Third Avenue South Birmingham, Alabama 35233 Contact: Joyce Lane 972-569-3627 NYSE Symbol: TMK News Release Torcmark Corporation 2001 Tird Avenue Sout Birmingam, Alabama 35233 Contact: Joyce Lane 972-569-3627 NYSE Symbol: TMK TORCHMARK CORPORATION REPORTS FOURTH QUARTER AND YEAR-END 2004 RESULTS

More information

Analyzing the Effects of Insuring Health Risks:

Analyzing the Effects of Insuring Health Risks: Analyzing te Effects of Insuring Healt Risks: On te Trade-off between Sort Run Insurance Benefits vs. Long Run Incentive Costs Harold L. Cole University of Pennsylvania and NBER Soojin Kim University of

More information

a joint initiative of Cost of Production Calculator

a joint initiative of Cost of Production Calculator a joint initiative of Cost of Production Calculator 1 KEY BENEFITS Learn to use te MAKING MORE FROM SHEEP cost of production calculator to: Measure te performance of your seep enterprise year on year Compare

More information

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE Byeong U. Park 1 and Young Kyung Lee 2 Department of Statistics, Seoul National University, Seoul, Korea Tae Yoon Kim 3 and Ceolyong Park

More information

On a Satellite Coverage

On a Satellite Coverage I. INTRODUCTION On a Satellite Coverage Problem DANNY T. CHI Kodak Berkeley Researc Yu T. su National Ciao Tbng University Te eart coverage area for a satellite in an Eart syncronous orbit wit a nonzero

More information

Warm medium, T H T T H T L. s Cold medium, T L

Warm medium, T H T T H T L. s Cold medium, T L Refrigeration Cycle Heat flows in direction of decreasing temperature, i.e., from ig-temperature to low temperature regions. Te transfer of eat from a low-temperature to ig-temperature requires a refrigerator

More information

THE ROLE OF LABOUR DEMAND ELASTICITIES IN TAX INCIDENCE ANALYSIS WITH HETEROGENEOUS LABOUR

THE ROLE OF LABOUR DEMAND ELASTICITIES IN TAX INCIDENCE ANALYSIS WITH HETEROGENEOUS LABOUR THE ROLE OF LABOUR DEMAND ELASTICITIES IN TAX INCIDENCE ANALYSIS WITH HETEROGENEOUS LABOUR Kesab Battarai 1,a and 1, a,b,c Jon Walley a Department of Economics, University of Warwick, Coventry, CV4 7AL,

More information