Sorting, Merge Sort and the Divide-and-Conquer Technique
|
|
- Bridget Flowers
- 7 years ago
- Views:
Transcription
1 Inf2B gorithms and Data Structures Note 7 Sorting, Merge Sort and the Divide-and-Conquer Technique This and a subsequent next ecture wi mainy be concerned with sorting agorithms. Sorting is an extremey important agorithmic probem that often appears as part of a more compicated tas. Quite frequenty, huge amounts of data have to be sorted. It is therefore worthwhie to put some effort into designing efficient sorting agorithms. In this ecture, we wi mainy consider the mergesort agorithm. mergesort is based on the principe of divide-and-conquer. Therefore in this note we wi aso discuss divide-and-conquer agorithms and their anaysis in genera. 7.1 The Sorting Probem The probem we are considering here is that of sorting a coection of items by their eys, which are assumed to be comparabe. We assume that the items are stored in an array that we wi aways denote by. The number of items to be sorted is aways denoted by n. Of course it is aso conceivabe that the items are stored in a ined ist or some other data structure, but we focus on arrays. Most of our agorithms can be adapted for sorting ined ists, athough this may go aong with a oss in efficiency. By assuming we wor with arrays, we assume we may eep a the items in memory at the same time. This is not the case for arge-scae sorting, as we wi see in a ater ecture. We aready saw the insertionsort sorting agorithm in Lecture 2 of this thread. It is dispayed again here as gorithm 1. Reca that the asymptotic worst-case gorithm insertionsort) 1. for j 1 to.ength 1 do 2. a [j] 3. i j 1 4. whie i 0 and [i].ey > a.ey do 5. [i +1] [i] 6. i i 1 7. [i +1] a gorithm 1 running time of insertionsort is n 2 ). For a sorting agorithm, this is quite poor. We wi see a coupe of agorithms that do much better Important characteristics of Sorting gorithms The main characteristic we wi be interested in for the various sorting agorithms we study is worst-case running-time we wi aso refer to best-case and averagecase from time to time). However, we wi aso be interested in whether our sorting agorithm is in-pace and whether it is stabe. Definition 2 n sorting agorithm is said to be an in-pace agorithm if it can be simpy) impemented on the input array, with ony O1) extra space extra variabes). Ceary it is usefu to have an in-pace agorithm if we want to minimize the amount of space used by our agorithm. The issue of minimizing space ony reay becomes important when we have a arge amount of data, for appications such as web-indexing. Definition 3 sorting agorithm is said to be stabe if for every i, j such that i<j and [i].ey = [j].ey, we are guaranteed that [i] comes before [j] in the output array. Stabiity is a desirabe property for a sorting agorithm. We won t reay see why it is usefu in Inf2B if you tae the 3rd-year gorithms and Data Structures course, you see that it can ead to a powerfu reduction in running-time for some cases of sorting 1 ). If we refer bac to insertionsort, we can see that it is ceary an in-pace sorting agorithm, as it operates directy on the input array. so, notice that the test [i].ey > a.ey in ine 4. is a strict inequaity and so that when [i] is inserted into the aready sorted) array [1...i 1], it wi be inserted after any eement with the same ey within [1...i 1]. Hence insertionsort is stabe. 7.3 Merge Sort The idea of merge sort is very simpe: spit the array to be sorted into haves, sort both haves recursivey, and then merge the two sorted subarrays together to one sorted array. gorithm 4 impements this idea in a straightforward manner. To mae the recursive impementation possibe, we introduce two additiona parameters i and j maring the boundaries of the subarray to be sorted; merge- Sort, i, j) sorts the subarray [i... j] =h[i], [i +1],..., [j]i. To sort the whoe array, we obviousy have to ca mergesort, 0, n 1). The ey to MergeSort is the merge) method, caed as merge, i, mid, j) in gorithm 4. ssuming that i appe mid < j and that the subarrays [i... mid] and [mid+1...j] are both sorted, merge, i, mid, j) sorts the whoe subarray [i...j]. The merging is achieved by first defining a new array B to hod the sorted data. Then we initiaise an index for the [i... mid] subarray to i, and index ` for the [mid +1...j] subarray to mid +1. We wa these indices up the array, at each step storing the minimum of [] and [`] in B, and then incrementing that index and the current index of B. 1 This is the Radix-sort agorithm, if you want to oo this up. 2
2 Inf2B gorithms and Data Structures Note 7 gorithm mergesort, i, j) 1. if i<jthen 2. mid b i+j 2 c 3. mergesort, i, mid) 4. mergesort, mid +1, j) 5. merge, i, mid, j) gorithm 4 gorithm 5 shows an impementation of merge). Note that at the end of the initia merge oop on ine 3) there might be some entries eft in the ower haf of or the upper haf but not both; so at most one of the oops on ines 11 and 15 woud be executed these just append any eft over entries to the end of B). Figure 6 shows the first few steps of merge) on an exampe array. Let us anayse the running time of merge. Let n = j i+1, i.e., n is the number of entries in the subarray [i...j]. There are no nested oops, and ceary each of the oops of merge is iterated at most n times. Thus the running time of merge is On). Since the ast oop is certainy executed n times, the running time is aso n) and thus n). ctuay, the running time of merge is n) no matter what the input array is i.e., even in the best case). Now et us oo at mergesort. gain we et n = j i +1. We get the foowing expression for the running time T mergesort n): T mergesort 1) = 1), T mergesort n) = 1) + T mergesort dn/2e)+t mergesort bn/2c)+t merge n). gorithm merge, i, mid, j) 1. initiaise new array B of ength j i i; ` mid +1; m 0 3. whie appe mid and ` appe j do merge haves of into B) 4. if [].ey appe [`].ey then 5. B[m] [] ese 8. B[m] [`] 9. ` ` m m whie appe mid do copy anything eft in ower haf of to B) 12. B[m] [] m m whie ` appe j do copy anything eft in upper haf of to B) 16. B[m] [`] 17. ` ` m m for m =0to j i do copy B bac to ) 20. [m + i] B[m] Since we aready now that the running time of merge is n), we can simpify this to T mergesort n) =T mergesort dn/2e)+t mergesort bn/2c)+ n). gorithm 5 We wi see in 7.5 that soving this recurrence yieds B T mergesort n) = n gn)). Our anaysis wi actuay show that the running time of mergesort is n gn)) no matter what the input is, i.e., even in the best case. The runtime of n gn)) is not surprising. The recurrence shows that for an input of size n the cost is that of the two recursive cas pus a cost proportiona to n. Now each recursive ca is to inputs of size essentiay n/2. Each of these maes two recursive cas for size n/4 and we aso charge a cost proportiona to n/2. So the cost of the next eve is a one off cost proportiona to n taing account of the two n/2 one off costs) and 4 cas to size n/4. So we see that at each recursive ca we pay a one off cost that is proportiona to n and then mae 2 r cas to inputs of size n/2 r. The base case is reached when n/2 r =1, i.e., when r =gn). So we have gn) eves each of which has a cost proportiona to n. 3 =i =mid B B Figure 6. The first few steps of merge) on two component subarrays. 4 m
3 Inf2B gorithms and Data Structures Note 7 Thus the tota cost is proportiona to n gn). We wi prove the correctness of this setch argument beow. Since n gn) grows much sower than n 2, mergesort is much more efficient than insertionsort. s simpe experiments show, this can aso be observed in practice. So usuay mergesort is preferabe to insertionsort. However, there are situations where this is not so cear. Suppose we want to sort an array that is amost sorted, with ony a few items that are sighty out of pace. For such amost sorted arrays, insertionsort wors in amost inear time, because very few items have to be moved. mergesort, on the other hand, uses time n gn)) no matter what the input array oos ie. insertionsort aso tends to be faster on very sma input arrays, because there is a certain overhead caused by the recursive cas in mergesort. Maybe most importanty, mergesort wastes a ot of space, because merge copies the whoe array to an intermediate array. Ceary mergesort is not an in-pace agorithm in the sense of Definition 2. s to the question of stabiity for mergesort, ine 4 of merge achieves stabiity for mergesort ony because it has a non-strict inequaity for testing when we shoud put the item [] from the eft subarray) down before the item [`] from the right subarray). In some paces you may see non-stabe versions of mergesort, where a strict inequaity is used in ine Divide-and-Conquer gorithms The divide-and-conquer technique invoves soving a probem in the foowing way: 1) Divide the input instance into severa instances of the same probem of smaer size. 2) Recursivey sove the probem on these smaer instances. 3) Combine the soutions for the smaer instances to a soution for the origina instance so after the recursive cas, we do some extra wor to find the soution for our origina instance). Obviousy, mergesort is an exampe of a divide-and-conquer agorithm. Suppose now that we want to anayse the running time of a divide-andconquer agorithm. Denote the size of the input by n. Furthermore, suppose that the input instance is spit into a instances of sizes n 1,...,n a, for some constant a 1. Then we get the foowing recurrence for a running time T n): T n) =T n 1 )+...+ T n a )+fn), where fn) is the time required by steps 1) for setting up the recursions) and 3) for the extra wor ). Typicay, n 1,...,n are a of the form bn/bc or dn/be for some b>1. Disregarding foors and ceiings for a moment, we get a recurrence of the form: T n) =atn/b)+fn). 5 If we want to tae foors and ceiings into account, we have to repace this by T n) =a 1 T bn/bc)+a 2 T dn/be)+fn). It is usuay the case that the asymptotic growth rate of the soution to the recurrence does not depend on foors and ceiings. Throughout this ecture note we wi usuay disregard the foors and ceiings this can be justified formay). To specify the recurrence fuy, we need aso to say what happens for sma instances that can no onger be divided, and we must specify the function f. We can aways assume that for input instances of size beow some constant n 0 typicay, n 0 =2), the agorithm requires 1) steps. typica function f occurring in the anaysis of divide-and-conquer agorithms has an asymptotic growth rate of n ), for some constant. To concude: The anaysis of divide-and-conquer agorithms usuay yieds recurrences of the form 1) if n<n 0, T n) = 7.1) atn/b)+ n ) if n n 0, where n 0,a2 N, 2 N 0, and b 2 R with b>1 are constants. We normay assume that n is a power of b so that n/b is an integer as the recurrence unwinds. Without this assumption we must use foor or ceiing functions as appropriate. The fact is that the asymptotic growth rate for the type of recurrence shown; see CLRS for a discussion.) 7.5 Soving Recurrences Exampe 7. Reca that for the running time of merge sort we got the foowing recurrence: 1) if n<2, T mergesort n) = T mergesort dn/2e)+t mergesort bn/2c)+ n) if n 2. We first assume that n is a power of 2, say, n =2`; in this case we can omit foors and ceiings. Then T n) = 2T n/2) + n) = 2 2T n/4) + n/2) + n) = 4T n/4) + 2 n/2) + n) = 4 2T n/8) + n/4) +2 n/2) + n) = 8T n/8) + 4 n/4) + 2 n/2) + n). X 1 = 2 T n/2 )+ 2 i n/2 i ). X` 1 = 2`T n/2`)+ 2 i n/2 i ) 6
4 Inf2B gorithms and Data Structures Note 7 X` 1 = nt 1) + 2 i n/2 i ) gn) 1 X = n 1) + 2 i n/2 i ) = n)+ n gn)) = n gn)). Stricty speaing the argument just given, usuay referred to as unwinding the recurrence, is incompete those dots in the midde). We have taen it on trust that the ast ines are the correct outcome. We can easiy fi in this gap using induction e.g. by proving that the caimed soution is indeed correct). However as ong as the unwinding process is carried out carefuy the induction step is a matter of routine and is usuay omitted. You shoud however be abe to suppy it if ased. though we have ony treated the specia case when n is a power of 2, we can now use a tric to prove that we have T n) = n gn) for a n. We mae the reasonabe assumption that T n) appe T n +1) this can be proved from the recurrence for T n) by induction). Now notice that for every n, there is some exponent such that n appe 2 < 2n. This is the fact we need. We observe that T n) appe T 2 )=O2 g2 )) = O2n g2n)) = On g n). Liewise for every n 1 there is some 0 such that n/2 < 2 0 appe n. Hence T n) = 2 0 )g2 0 )) = n/2)gn/2)) = ngn). Putting these two together we have T n) = ng n). n aternative approach is to use the fu recurrence directy though this is rather tedious. Note though that once we have a good guess at what the soution is we can prove its correctness by using induction and the fu recurrence, we do need to unwind the compicated version. Before moving on it is worth expaining one possibe error that is sometimes made when unwinding recurrences. We have T n) = 2T n/2) + n) = 2 2T n/4) + n/2) + n) = 4T n/4) + 2 n/2) + n) It is very tempting to simpify the ast ine to 4T n/4) + n) on the basis that 2 n/2) + n) = n) and do iewise at each stage. It is certainy true that 2 n/2) + n) = n). However such repacements are ony justified if we have a constant number of ) terms. In our case we unwind the recurrence to obtain gn) terms and this is not a constant. You are strongy encouraged to unwind the recurrence with the n) repaced by cn where c>0 is some constant. Observe that at each stage we obtain a mutipe of cn as an additive term cn, 2cn, 3cn etc.). Of course after any fixed number of stages we just have a constant mutipe of n. But after ogn) stages this is not the case. By further expoiting the idea used in this exampe, one can prove the foowing genera theorem that gives soutions to recurrences of the form 7.1). Reca that og b a) denotes the ogarithm of a with base b, i.e., we have c =og b a) ) b c = a. 7 For exampe, og 2 8) = 3, og 3 9) = 2, og 4 8) = 1.5. The foowing theorem is caed the Master Theorem for soving recurrences). It maes ife easy by aowing us to read-off the ) expression of a recurrence without going through a proof as we did for mergesort: Theorem 8. Let n 0 2 N, 2 N 0 and a, b 2 R with a>0 and b>1, and et T : N! N satisfy the foowing recurrence: 1), if n<n 0 ; T n) = at n/b)+ n ), if n n 0. Let e =og b a); we ca e the critica exponent. Then 8 < n e ), if <e I); T n) = n e gn)), if = e II); : n ), if >e III). The theorem remains true if we repace at n/b) in the recurrence by a 1 T bn/bc) + a 2 T dn/be) for a 1,a 2 0 with a 1 + a 2 = a. In some situations we do not have a n ) expression for the extra cost part of the recurrence but do have a On ) expression. This is fine, the Mater Theorem sti appies but gives us just an upper bound for the runtime, i.e., in the three cases for the vaue of T n) repace by O. Exampes 9. 1) Now reconsider the recurrence for the running time of mergesort: 1), T mergesort n) = T mergesort dn/2e)+t mergesort bn/2c)+ n), if n 2. In the setting of the Master Theorem, we have n 0 =2, =1, a =2, b =2. Thus e = og b a) = og 2 2) = g2) = 1. Hence T mergesort n) = n gn)) by case II). 2) Reca the recurrence for the running time of binary search: 1), T binarysearch n) = T binarysearch bn/2c)+ 1), if n 2. Here we tae n 0 =2, =0, a =1, b =2. Thus e =og b a) =og 2 1) = 0 and therefore T binarysearch n) 2 gn)) by case II) of the Master Theorem. 3) Let T be a function satisfying 1), T n) = 3T n/2) + n), if n n 0. Then e =og b a) =og 2 3) = g3. Hence T n) 2 n g3) ) by case I). 4) Let T be a function satisfying 1), T n) = 7T n/2) + n 4 ), if n n 0. Then e =og b a) =og 2 7) < 3. Hence T n) 2 n 4 ) by case III) of the Master Theorem. 8
5 Exercises 1. Sove the foowing recurrences using the Master Theorem: a) T n) =3T n/2) + n 5 b) T n) =4T n/3) + n c) T n) =8T n/3) + n 2 d) T n) =8T n/3) + n e) T n) =8T n/3) + ngn) Remar: The Master Theorem is not directy appicabe in this case. Find a way to use it anyway. In each case T n) is assumed to be 1) for some appropriate base case. 2. Let T n) be the function satisfying the foowing recurrence: 3, if n =1; T n) = T n 1) + 2n, if n 2. Sove the recurrence and give an expicit formua for T. Try to derive a genera formua for the asymptotic growth rate of functions satisfying recurrences of the form 1), if n appe n 0 ; T n) = T n )+fn), if n>n 0, where, n 0 2 N such that 1 appe appe n 0 and f : N! N is a function. 3. Prove that P gn) 1 2 i n/2 i )= ngn)), this was used in soving the recurrence for T mergesort n). This is much easier than it oos, remember that to say f = g) means there is an n 0 2 N and constants c 1,c 2 2 R both stricty bigger than 0 such that c 1 gn) appe fn) appe c 2 gn) for a n n 0. Reca aso the formua for summing geometric series: P s ri =1 r i+1 )/1 r). 9
Fast Robust Hashing. ) [7] will be re-mapped (and therefore discarded), due to the load-balancing property of hashing.
Fast Robust Hashing Manue Urueña, David Larrabeiti and Pabo Serrano Universidad Caros III de Madrid E-89 Leganés (Madrid), Spain Emai: {muruenya,darra,pabo}@it.uc3m.es Abstract As statefu fow-aware services
More informationBudgeting Loans from the Social Fund
Budgeting Loans from the Socia Fund tes sheet Pease read these notes carefuy. They expain the circumstances when a budgeting oan can be paid. Budgeting Loans You may be abe to get a Budgeting Loan if:
More informationDiscounted Cash Flow Analysis (aka Engineering Economy)
Discounted Cash Fow Anaysis (aka Engineering Economy) Objective: To provide economic comparison of benefits and costs that occur over time Assumptions: Future benefits and costs can be predicted A Benefits,
More informationTeaching fractions in elementary school: A manual for teachers
Teaching fractions in eementary schoo: A manua for teachers H. Wu Apri 0, 998 [Added December, 200] I have decided to resurrect this fie of 998 because, as a reativey short summary of the basic eements
More informationSecure Network Coding with a Cost Criterion
Secure Network Coding with a Cost Criterion Jianong Tan, Murie Médard Laboratory for Information and Decision Systems Massachusetts Institute of Technoogy Cambridge, MA 0239, USA E-mai: {jianong, medard}@mit.edu
More informationNCH Software Warp Speed PC Tune-up Software
NCH Software Warp Speed PC Tune-up Software This user guide has been created for use with Warp Speed PC Tune-up Software Version 1.xx NCH Software Technica Support If you have difficuties using Warp Speed
More informationMarket Design & Analysis for a P2P Backup System
Market Design & Anaysis for a P2P Backup System Sven Seuken Schoo of Engineering & Appied Sciences Harvard University, Cambridge, MA seuken@eecs.harvard.edu Denis Chares, Max Chickering, Sidd Puri Microsoft
More informationTeamwork. Abstract. 2.1 Overview
2 Teamwork Abstract This chapter presents one of the basic eements of software projects teamwork. It addresses how to buid teams in a way that promotes team members accountabiity and responsibiity, and
More informationAn Idiot s guide to Support vector machines (SVMs)
An Idiot s guide to Support vector machines (SVMs) R. Berwick, Viage Idiot SVMs: A New Generation of Learning Agorithms Pre 1980: Amost a earning methods earned inear decision surfaces. Linear earning
More information3.5 Pendulum period. 2009-02-10 19:40:05 UTC / rev 4d4a39156f1e. g = 4π2 l T 2. g = 4π2 x1 m 4 s 2 = π 2 m s 2. 3.5 Pendulum period 68
68 68 3.5 Penduum period 68 3.5 Penduum period Is it coincidence that g, in units of meters per second squared, is 9.8, very cose to 2 9.87? Their proximity suggests a connection. Indeed, they are connected
More informationMany algorithms, particularly divide and conquer algorithms, have time complexities which are naturally
Recurrence Relations Many algorithms, particularly divide and conquer algorithms, have time complexities which are naturally modeled by recurrence relations. A recurrence relation is an equation which
More informationMulti-Robot Task Scheduling
Proc of IEEE Internationa Conference on Robotics and Automation, Karsruhe, Germany, 013 Muti-Robot Tas Scheduing Yu Zhang and Lynne E Parer Abstract The scheduing probem has been studied extensivey in
More informationPricing Internet Services With Multiple Providers
Pricing Internet Services With Mutipe Providers Linhai He and Jean Warand Dept. of Eectrica Engineering and Computer Science University of Caifornia at Berkeey Berkeey, CA 94709 inhai, wr@eecs.berkeey.edu
More informationEarly access to FAS payments for members in poor health
Financia Assistance Scheme Eary access to FAS payments for members in poor heath Pension Protection Fund Protecting Peope s Futures The Financia Assistance Scheme is administered by the Pension Protection
More information1B11 Operating Systems. Input/Output and Devices
University Coege London 1B11 Operating Systems Input/Output and s Prof. Steve R Wibur s.wibur@cs.uc.ac.uk Lecture Objectives How do the bits of the I/O story fit together? What is a device driver? 1B11-5
More informationPay-on-delivery investing
Pay-on-deivery investing EVOLVE INVESTment range 1 EVOLVE INVESTMENT RANGE EVOLVE INVESTMENT RANGE 2 Picture a word where you ony pay a company once they have deivered Imagine striking oi first, before
More informationBetting on the Real Line
Betting on the Rea Line Xi Gao 1, Yiing Chen 1,, and David M. Pennock 2 1 Harvard University, {xagao,yiing}@eecs.harvard.edu 2 Yahoo! Research, pennockd@yahoo-inc.com Abstract. We study the probem of designing
More informationBreakeven analysis and short-term decision making
Chapter 20 Breakeven anaysis and short-term decision making REAL WORLD CASE This case study shows a typica situation in which management accounting can be hepfu. Read the case study now but ony attempt
More informationAustralian Bureau of Statistics Management of Business Providers
Purpose Austraian Bureau of Statistics Management of Business Providers 1 The principa objective of the Austraian Bureau of Statistics (ABS) in respect of business providers is to impose the owest oad
More informationSAT 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 informationLADDER SAFETY Table of Contents
Tabe of Contents SECTION 1. TRAINING PROGRAM INTRODUCTION..................3 Training Objectives...........................................3 Rationae for Training.........................................3
More informationELECTRONIC FUND TRANSFERS YOUR RIGHTS AND RESPONSIBILITIES
ELECTRONIC FUND TRANSFERS YOUR RIGHTS AND RESPONSIBILITIES The Eectronic Fund Transfers we are capabe of handing for consumers are indicated beow, some of which may not appy your account Some of these
More informationSAT 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 informationChapter 1 Structural Mechanics
Chapter Structura echanics Introduction There are many different types of structures a around us. Each structure has a specific purpose or function. Some structures are simpe, whie others are compex; however
More informationAvaya Remote Feature Activation (RFA) User Guide
Avaya Remote Feature Activation (RFA) User Guide 03-300149 Issue 5.0 September 2007 2007 Avaya Inc. A Rights Reserved. Notice Whie reasonabe efforts were made to ensure that the information in this document
More informationSimultaneous Routing and Power Allocation in CDMA Wireless Data Networks
Simutaneous Routing and Power Aocation in CDMA Wireess Data Networks Mikae Johansson *,LinXiao and Stephen Boyd * Department of Signas, Sensors and Systems Roya Institute of Technoogy, SE 00 Stockhom,
More informationNCH Software Bolt PDF Printer
NCH Software Bot PDF Printer This user guide has been created for use with Bot PDF Printer Version 1.xx NCH Software Technica Support If you have difficuties using Bot PDF Printer pease read the appicabe
More informationCapacity of Multi-service Cellular Networks with Transmission-Rate Control: A Queueing Analysis
Capacity of Muti-service Ceuar Networs with Transmission-Rate Contro: A Queueing Anaysis Eitan Atman INRIA, BP93, 2004 Route des Lucioes, 06902 Sophia-Antipois, France aso CESIMO, Facutad de Ingeniería,
More informationDEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We show that if H is an effectivey competey decomposabe computabe torsion-free abeian group, then
More informationPricing and Revenue Sharing Strategies for Internet Service Providers
Pricing and Revenue Sharing Strategies for Internet Service Providers Linhai He and Jean Warand Department of Eectrica Engineering and Computer Sciences University of Caifornia at Berkeey {inhai,wr}@eecs.berkeey.edu
More informationBetting Strategies, Market Selection, and the Wisdom of Crowds
Betting Strategies, Market Seection, and the Wisdom of Crowds Wiemien Kets Northwestern University w-kets@keogg.northwestern.edu David M. Pennock Microsoft Research New York City dpennock@microsoft.com
More information7. Dry Lab III: Molecular Symmetry
0 7. Dry Lab III: Moecuar Symmetry Topics: 1. Motivation. Symmetry Eements and Operations. Symmetry Groups 4. Physica Impications of Symmetry 1. Motivation Finite symmetries are usefu in the study of moecues.
More informationIncome Protection Solutions. Policy Wording
Income Protection Soutions Poicy Wording Wecome to Aviva This booket tes you a you need to know about your poicy, incuding: what to do if you need to caim what s covered, and expanations of some of the
More informationEnergy Density / Energy Flux / Total Energy in 3D
Lecture 5 Phys 75 Energy Density / Energy Fux / Tota Energy in D Overview and Motivation: In this ecture we extend the discussion of the energy associated with wave otion to waves described by the D wave
More informationLife Contingencies Study Note for CAS Exam S. Tom Struppeck
Life Contingencies Study Note for CAS Eam S Tom Struppeck (Revised 9/19/2015) Introduction Life contingencies is a term used to describe surviva modes for human ives and resuting cash fows that start or
More information25 Indirect taxes, subsidies, and price controls
5 Indirect taxes, subsidies, and price contros By the end of this chapter, you shoud be abe to: define and give exampes of an indirect tax expain the difference between a specific tax and a percentage
More informationFRAME BASED TEXTURE CLASSIFICATION BY CONSIDERING VARIOUS SPATIAL NEIGHBORHOODS. Karl Skretting and John Håkon Husøy
FRAME BASED TEXTURE CLASSIFICATION BY CONSIDERING VARIOUS SPATIAL NEIGHBORHOODS Kar Skretting and John Håkon Husøy University of Stavanger, Department of Eectrica and Computer Engineering N-4036 Stavanger,
More informationSNMP Reference Guide for Avaya Communication Manager
SNMP Reference Guide for Avaya Communication Manager 03-602013 Issue 1.0 Feburary 2007 2006 Avaya Inc. A Rights Reserved. Notice Whie reasonabe efforts were made to ensure that the information in this
More information5. Introduction to Robot Geometry and Kinematics
V. Kumar 5. Introduction to Robot Geometry and Kinematics The goa of this chapter is to introduce the basic terminoogy and notation used in robot geometry and kinematics, and to discuss the methods used
More informationThe guaranteed selection. For certainty in uncertain times
The guaranteed seection For certainty in uncertain times Making the right investment choice If you can t afford to take a ot of risk with your money it can be hard to find the right investment, especiay
More informationDistribution of Income Sources of Recent Retirees: Findings From the New Beneficiary Survey
Distribution of Income Sources of Recent Retirees: Findings From the New Beneficiary Survey by Linda Drazga Maxfied and Virginia P. Rena* Using data from the New Beneficiary Survey, this artice examines
More informationThe Use of Cooling-Factor Curves for Coordinating Fuses and Reclosers
he Use of ooing-factor urves for oordinating Fuses and Recosers arey J. ook Senior Member, IEEE S& Eectric ompany hicago, Iinois bstract his paper describes how to precisey coordinate distribution feeder
More informationAdvanced ColdFusion 4.0 Application Development - 3 - Server Clustering Using Bright Tiger
Advanced CodFusion 4.0 Appication Deveopment - CH 3 - Server Custering Using Bri.. Page 1 of 7 [Figures are not incuded in this sampe chapter] Advanced CodFusion 4.0 Appication Deveopment - 3 - Server
More informationIncome Protection Options
Income Protection Options Poicy Conditions Introduction These poicy conditions are written confirmation of your contract with Aviva Life & Pensions UK Limited. It is important that you read them carefuy
More informationTERM INSURANCE CALCULATION ILLUSTRATED. This is the U.S. Social Security Life Table, based on year 2007.
This is the U.S. Socia Security Life Tabe, based on year 2007. This is avaiabe at http://www.ssa.gov/oact/stats/tabe4c6.htm. The ife eperiences of maes and femaes are different, and we usuay do separate
More informationChapter 3: JavaScript in Action Page 1 of 10. How to practice reading and writing JavaScript on a Web page
Chapter 3: JavaScript in Action Page 1 of 10 Chapter 3: JavaScript in Action In this chapter, you get your first opportunity to write JavaScript! This chapter introduces you to JavaScript propery. In addition,
More informationNormalization of Database Tables. Functional Dependency. Examples of Functional Dependencies: So Now what is Normalization? Transitive Dependencies
ISM 602 Dr. Hamid Nemati Objectives The idea Dependencies Attributes and Design Understand concepts normaization (Higher-Leve Norma Forms) Learn how to normaize tabes Understand normaization and database
More informationELECTRONIC FUND TRANSFERS YOUR RIGHTS AND RESPONSIBILITIES. l l l
ELECTRONIC FUND TRANSFERS YOUR RIGHTS AND RESPONSIBILITIES The Eectronic Fund Transfers we are capabe of handing for consumers are indicated beow, some of which may not appy your account Some of these
More informationOligopoly in Insurance Markets
Oigopoy in Insurance Markets June 3, 2008 Abstract We consider an oigopoistic insurance market with individuas who differ in their degrees of accident probabiities. Insurers compete in coverage and premium.
More informationPENALTY TAXES ON CORPORATE ACCUMULATIONS
H Chapter Six H PENALTY TAXES ON CORPORATE ACCUMULATIONS INTRODUCTION AND STUDY OBJECTIVES The accumuated earnings tax and the persona hoding company tax are penaty taxes designed to prevent taxpayers
More informationA Supplier Evaluation System for Automotive Industry According To Iso/Ts 16949 Requirements
A Suppier Evauation System for Automotive Industry According To Iso/Ts 16949 Requirements DILEK PINAR ÖZTOP 1, ASLI AKSOY 2,*, NURSEL ÖZTÜRK 2 1 HONDA TR Purchasing Department, 41480, Çayırova - Gebze,
More informationIn some states, however, the couple must live apart for a period of months or years before they can obtain a no fault divorce.
What is a "no faut" divorce? "No faut" divorce describes any divorce where the spouse asking for a divorce does not have to prove that the other spouse did something wrong. A states aow no faut divorces.
More informationA Description of the California Partnership for Long-Term Care Prepared by the California Department of Health Care Services
2012 Before You Buy A Description of the Caifornia Partnership for Long-Term Care Prepared by the Caifornia Department of Heath Care Services Page 1 of 13 Ony ong-term care insurance poicies bearing any
More informationIEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 12, DECEMBER 2013 1
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 12, DECEMBER 2013 1 Scaabe Muti-Cass Traffic Management in Data Center Backbone Networks Amitabha Ghosh, Sangtae Ha, Edward Crabbe, and Jennifer
More informationA New Statistical Approach to Network Anomaly Detection
A New Statistica Approach to Network Anomay Detection Christian Caegari, Sandrine Vaton 2, and Michee Pagano Dept of Information Engineering, University of Pisa, ITALY E-mai: {christiancaegari,mpagano}@ietunipiit
More information1 Basic concepts in geometry
1 asic concepts in geometry 1.1 Introduction We start geometry with the simpest idea a point. It is shown using a dot, which is abeed with a capita etter. The exampe above is the point. straight ine is
More informationLecture 7 Datalink Ethernet, Home. Datalink Layer Architectures
Lecture 7 Dataink Ethernet, Home Peter Steenkiste Schoo of Computer Science Department of Eectrica and Computer Engineering Carnegie Meon University 15-441 Networking, Spring 2004 http://www.cs.cmu.edu/~prs/15-441
More informationCI/SfB Ro8. (Aq) September 2012. The new advanced toughened glass. Pilkington Pyroclear Fire-resistant Glass
CI/SfB Ro8 (Aq) September 2012 The new advanced toughened gass Pikington Pyrocear Fire-resistant Gass Pikington Pyrocear, fire-resistant screens in the façade: a typica containment appication for integrity
More informationIntroduction to Algorithms March 10, 2004 Massachusetts Institute of Technology Professors Erik Demaine and Shafi Goldwasser Quiz 1.
Introduction to Algorithms March 10, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Quiz 1 Quiz 1 Do not open this quiz booklet until you are directed
More informationEnergy Performance Certificate
Energy Performance Certificate Sampe EPC TheDomesticEnergyAssessorcom Dweing type: Date of assessment: Date of certificate: Reference number: Tota foor area: Mid-terrace house 09 June 2007 31 August 2007
More informationOperation Guide 3440 3441
MO1504-EB 2015 CASIO COMPUTER CO., LTD. Operation Guide 3440 3441 Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua
More informationNCH Software MoneyLine
NCH Software MoneyLine This user guide has been created for use with MoneyLine Version 2.xx NCH Software Technica Support If you have difficuties using MoneyLine pease read the appicabe topic before requesting
More informationArt of Java Web Development By Neal Ford 624 pages US$44.95 Manning Publications, 2004 ISBN: 1-932394-06-0
IEEE DISTRIBUTED SYSTEMS ONLINE 1541-4922 2005 Pubished by the IEEE Computer Society Vo. 6, No. 5; May 2005 Editor: Marcin Paprzycki, http://www.cs.okstate.edu/%7emarcin/ Book Reviews: Java Toos and Frameworks
More informationPROPAGATION OF SURGE WAVES ON NON-HOMOGENEOUS TRANSMISSION LINES INDUCED BY LIGHTNING STROKE
Advances in Eectrica and Eectronic Engineering 98 PROPAGATION OF SRGE WAVES ON NON-HOMOGENEOS TRANSMISSION LINES INDCED BY LIGHTNING STROKE Z. Benešová, V. Kotan WB Pisen, Facuty of Eectrica Engineering,
More informationChapter 3: Investing: Your Options, Your Risks, Your Rewards
Chapter 3: Investing: Your Options, Your Risks, Your Rewards Page 1 of 10 Chapter 3: Investing: Your Options, Your Risks, Your Rewards In This Chapter What is inside a mutua fund? What is a stock? What
More informationAnnual Notice of Changes for 2016
Easy Choice Best Pan (HMO) offered by Easy Choice Heath Pan, Inc. Annua Notice of Changes for 2016 You are currenty enroed as a member of Easy Choice Best Pan (HMO). Next year, there wi be some changes
More informationUncertain Bequest Needs and Long-Term Insurance Contracts 1
Uncertain Bequest Needs and Long-Term Insurance Contracts 1 Wenan Fei (Hartford Life Insurance) Caude Fuet (Université du Québec à Montréa and CIRPEE) Harris Schesinger (University of Aabama) Apri 22,
More informationKey Features of Life Insurance
Key Features of Life Insurance Life Insurance Key Features The Financia Conduct Authority is a financia services reguator. It requires us, Aviva, to give you this important information to hep you to decide
More informationDistributed Strategic Interleaving with Load Balancing
Distributed Strategic Intereaving with Load Baancing J.A. Bergstra 1,2 and C.A. Middeburg 1,3 1 Programming Research Group, University of Amsterdam, P.O. Box 41882, 1009 DB Amsterdam, the Netherands 2
More informationAPPENDIX 10.1: SUBSTANTIVE AUDIT PROGRAMME FOR PRODUCTION WAGES: TROSTON PLC
Appendix 10.1: substantive audit programme for production wages: Troston pc 389 APPENDIX 10.1: SUBSTANTIVE AUDIT PROGRAMME FOR PRODUCTION WAGES: TROSTON PLC The detaied audit programme production wages
More informationNCH Software PlayPad Media Player
NCH Software PayPad Media Payer This user guide has been created for use with PayPad Media Payer Version 2.xx NCH Software Technica Support If you have difficuties using PayPad Media Payer pease read the
More informationASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007
ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007 FRANÇOIS SIMENHAUS Université Paris 7, Mathématiques, case 7012, 2, pace Jussieu, 75251 Paris, France
More informationDEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS
1 DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS 2 ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We show that if H is an effectivey competey decomposabe computabe torsion-free abeian group,
More information500 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013
500 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 3, MARCH 203 Cognitive Radio Network Duaity Agorithms for Utiity Maximization Liang Zheng Chee Wei Tan, Senior Member, IEEE Abstract We
More informationAA Fixed Rate ISA Savings
AA Fixed Rate ISA Savings For the road ahead The Financia Services Authority is the independent financia services reguator. It requires us to give you this important information to hep you to decide whether
More informationFinance 360 Problem Set #6 Solutions
Finance 360 Probem Set #6 Soutions 1) Suppose that you are the manager of an opera house. You have a constant margina cost of production equa to $50 (i.e. each additiona person in the theatre raises your
More informationThe Computation of American Option Price Sensitivities using a Monotone Multigrid Method for Higher Order B Spline Discretizations
The Computation of American Option Price Sensitivities using a Monotone Mutigrid Method for Higher Order B Spine Discretizations Markus Hotz October 26, 2004 Abstract In this paper a fast sover for discrete
More informationTechnical Support Guide for online instrumental lessons
Technica Support Guide for onine instrumenta essons This is a technica guide for Music Education Hubs, Schoos and other organisations participating in onine music essons. The guidance is based on the technica
More informationHybrid Process Algebra
Hybrid Process Agebra P.J.L. Cuijpers M.A. Reniers Eindhoven University of Technoogy (TU/e) Den Doech 2 5600 MB Eindhoven, The Netherands Abstract We deveop an agebraic theory, caed hybrid process agebra
More informationMean-field Dynamics of Load-Balancing Networks with General Service Distributions
Mean-fied Dynamics of Load-Baancing Networks with Genera Service Distributions Reza Aghajani 1, Xingjie Li 2, and Kavita Ramanan 1 1 Division of Appied Mathematics, Brown University, Providence, RI, USA.
More informationRisk Margin for a Non-Life Insurance Run-Off
Risk Margin for a Non-Life Insurance Run-Off Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas August 15, 2011 Abstract For sovency purposes insurance companies need to cacuate so-caed best-estimate reserves
More informationDesign and Analysis of a Hidden Peer-to-peer Backup Market
Design and Anaysis of a Hidden Peer-to-peer Backup Market Sven Seuken, Denis Chares, Max Chickering, Mary Czerwinski Kama Jain, David C. Parkes, Sidd Puri, and Desney Tan December, 2015 Abstract We present
More informationGREEN: An Active Queue Management Algorithm for a Self Managed Internet
: An Active Queue Management Agorithm for a Sef Managed Internet Bartek Wydrowski and Moshe Zukerman ARC Specia Research Centre for Utra-Broadband Information Networks, EEE Department, The University of
More informationRisk Margin for a Non-Life Insurance Run-Off
Risk Margin for a Non-Life Insurance Run-Off Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas February 2, 2011 Abstract For sovency purposes insurance companies need to cacuate so-caed best-estimate
More informationAngles formed by 2 Lines being cut by a Transversal
Chapter 4 Anges fored by 2 Lines being cut by a Transversa Now we are going to nae anges that are fored by two ines being intersected by another ine caed a transversa. 1 2 3 4 t 5 6 7 8 If I asked you
More informationScheduling in Multi-Channel Wireless Networks
Scheduing in Muti-Channe Wireess Networks Vartika Bhandari and Nitin H. Vaidya University of Iinois at Urbana-Champaign, USA vartikab@acm.org, nhv@iinois.edu Abstract. The avaiabiity of mutipe orthogona
More informationWINMAG Graphics Management System
SECTION 10: page 1 Section 10: by Honeywe WINMAG Graphics Management System Contents What is WINMAG? WINMAG Text and Graphics WINMAG Text Ony Scenarios Fire/Emergency Management of Fauts & Disabement Historic
More informationSPOTLIGHT. A year of transformation
WINTER ISSUE 2014 2015 SPOTLIGHT Wecome to the winter issue of Oasis Spotight. These newsetters are designed to keep you upto-date with news about the Oasis community. This quartery issue features an artice
More informationCaibration of a Lib rate and the SABR-LMM
Caibration of a Libor Market Mode with Stochastic Voatiity Master s Thesis by Hendrik Hüsbusch Submitted in Partia Fufiment for the Degree of Master of Science in Mathematics Supervisor: PD. Dr. Vokert
More informationTraffic classification-based spam filter
Traffic cassification-based spam fiter Ni Zhang 1,2, Yu Jiang 3, Binxing Fang 1, Xueqi Cheng 1, Li Guo 1 1 Software Division, Institute of Computing Technoogy, Chinese Academy of Sciences, 100080, Beijing,
More informationTake me to your leader! Online Optimization of Distributed Storage Configurations
Take me to your eader! Onine Optimization of Distributed Storage Configurations Artyom Sharov Aexander Shraer Arif Merchant Murray Stokey sharov@cs.technion.ac.i, {shraex, aamerchant, mstokey}@googe.com
More informationFLAC Legal Divorce v2 band_layout 1 26/06/2014 20:01 Page 1 July 2014 divorce
Juy 2014 divorce Divorce has been avaiabe in the Repubic of Ireand since 1997. Once a divorce is obtained, the parties (the ex-spouses) are free to remarry or enter into a civi partnership. Famiy Law (Divorce)
More information802.11 Power-Saving Mode for Mobile Computing in Wi-Fi hotspots: Limitations, Enhancements and Open Issues
802.11 Power-Saving Mode for Mobie Computing in Wi-Fi hotspots: Limitations, Enhancements and Open Issues G. Anastasi a, M. Conti b, E. Gregori b, A. Passarea b, Pervasive Computing & Networking Laboratory
More informationNCH Software FlexiServer
NCH Software FexiServer This user guide has been created for use with FexiServer Version 1.xx NCH Software Technica Support If you have difficuties using FexiServer pease read the appicabe topic before
More informationWEBSITE ACCOUNT USER GUIDE SECURITY, PASSWORD & CONTACTS
WEBSITE ACCOUNT USER GUIDE SECURITY, PASSWORD & CONTACTS Password Reset Process Navigate to the og in screen Seect the Forgot Password ink You wi be asked to enter the emai address you registered with
More informationNCH Software WavePad Sound Editor
NCH Software WavePad Sound Editor This user guide has been created for use with WavePad Sound Editor Version 6.xx NCH Software Technica Support If you have difficuties using WavePad Sound Editor pease
More informationCLOUD service providers manage an enterprise-class
IEEE TRANSACTIONS ON XXXXXX, VOL X, NO X, XXXX 201X 1 Oruta: Privacy-Preserving Pubic Auditing for Shared Data in the Coud Boyang Wang, Baochun Li, Member, IEEE, and Hui Li, Member, IEEE Abstract With
More informationFigure 1. A Simple Centrifugal Speed Governor.
ENGINE SPEED CONTROL Peter Westead and Mark Readman, contro systems principes.co.uk ABSTRACT: This is one of a series of white papers on systems modeing, anaysis and contro, prepared by Contro Systems
More informationChapter 2 Traditional Software Development
Chapter 2 Traditiona Software Deveopment 2.1 History of Project Management Large projects from the past must aready have had some sort of project management, such the Pyramid of Giza or Pyramid of Cheops,
More informationFlow Pattern Map for In Tube Evaporation and Condensation
Fow Pattern Map for In Tube Evaporation and Condensation Lászó Garbai Dr. *, Róbert Sánta ** * Budapest University of Technoogy and Economics, Hungary ** Coege of Appied Sciences Subotica Tech, Serbia
More information