4. Parallel Structure, Sn str problems 句子平行结构 句子结构问题

常见结构类型常见的结构类型包括：1. 并列结构（Parallel Structure）：由两个或多个并列的独立主题或句子组成，它们在结构上相似或平行。

例如：“I like swimming, reading, and playing soccer.”2. 顺序结构（Sequential Structure）：按照时间或顺序的先后关系，将事件或观点依次排列。

例如：“First, we need to gather the materials. Then, we can start the project. Finally, we can evaluate our results.”3. 对比结构（Comparison Structure）：通过对比两个或多个对象、观点或概念之间的相似点和差异点，进行论述。

例如：“Apples are sweet, while lemons are sour.”4. 因果结构（Causal Structure）：描述一个事件或情况的原因和结果之间的关系。

例如：“Eating too much junk food can cause health problems.”5. 问题解决结构（Problem-Solution Structure）：首先提出一个问题，并提供解决该问题的一个或多个解决方案。

例如：“The problem is that students are not motivated to study.One solution could be introducing interactive teaching methods.”6. 比较与对照结构（Compare and Contrast Structure）：通过将两个或多个事物进行比较和对照，来展示它们的相似点和差异点。

例如：“Dogs and cats are both popular pets, but dogs requiremore exercise than cats.”7. 描述性结构（Descriptive Structure）：通过对一个人、地点、物品或事物的特征、外貌、性质等进行详细描述。

Open Problems ListArising from MathsCSP Workshop,Oxford,March2006Version0.3,April25,20061Complexity and Tractability of CSPQuestion1.0(The Dichotomy Conjecture)Let B be a relational structure.The problem of deciding whether a given relational structure has a homomorphism to B is denoted CSP(B).For which(finite)structures is CSP(B)decidable in polynomial time?Is it true that for anyfinite structure B the problem CSP(B)is either decidable in polynomial time or NP-complete?Communicated by:Tomas Feder&Moshe Vardi(1993) Question1.1A relational structure B is called hereditarily tractable if CSP(B )is tractable for all substructures B of B.Which structures B are hereditarily tractable?Communicated by:Pavol Hell Question1.2A weak near-unanimity term is defined to be one that satisfies the following identities:f(x,...,x)=x and f(x,y,....y)=f(y,x,y,....y)=...=f(y,...,y,x).Is CSP(B)tractable for any(finite)structure B which is preserved by a weak near-unanimity term?Communicated by:Benoit Larose,Matt Valeriote Question1.3A constraint language1S is called globally tractable for a problem P,if P(S)is tractable,and it is called(locally)tractable if for everyfinite L⊆S,P(L)is tractable.These two notions of tractability do not coincide in the Abduction problem(see talk by Nadia Creignou).•For which computational problems related to the CSP do these two notions of tractability coincide?•In particular,do they coincide for the standard CSP decision problem?Communicated by:Nadia Creignou 1That is,a(possibly infinite)set of relations over somefixed set.1Question1.4(see also Question3.5)It has been shown that when a structure B has bounded pathwidth duality the corresponding problem CSP(B)is in the complexity class NL (see talk by Victor Dalmau).Is the converse also true(modulo some natural complexity-theoretic assumptions)?Communicated by:Victor Dalmau Question1.5Is there a good(numerical)parameterization for constraint satisfaction problems that makes themfixed-parameter tractable?Question1.6Further develop techniques based on delta-matroids to complete the com-plexity classification of the Boolean CSP(with constants)with at most two occurrences per variable(see talk by Tomas Feder).Communicated by:Tomas Feder Question1.7Classify the complexity of uniform Boolean CSPs(where both structure and constraint relations are specified in the input).Communicated by:Heribert Vollmer Question1.8The microstructure graph of a binary CSP has vertices for each variable/value pair,and edges that join all pairs of vertices that are compatible with the constraints.What properties of this graph are sufficient to ensure tractability?Are there properties that do not rely on the constraint language or the constraint graph individually?2Approximability and Soft ConstraintsQuestion2.1Is it true that Max CSP(L)is APX-complete whenever Max CSP(L)is NP-hard?Communicated by:Peter Jonsson Question2.2Prove or disprove that Max CSP(L)is in PO if the core of L is super-modular on some lattice,and otherwise this problem is APX-complete.The above has been proved for languages with domain size3,and for languages contain-ing all constants by a computer-assisted case analysis(see talk by Peter Jonsson).Develop techniques that allow one to prove such results without computer-assisted analysis.Communicated by:Peter Jonsson Question2.3For some constraint languages L,the problem Max CSP(L)is hard to approximate better than the random mindless algorithm on satisfiable or almost satisfiable instances.Such problems are called approximation resistant(see talk by Johan Hastad).Is a single random predicate over Boolean variables with large arity approximation resistant?What properties of predicates make a CSP approximation resistant?What transformations of predicates preserve approximation resistance?Communicated by:Johan Hastad2Question2.4Many optimisation problems involving constraints(such as Max-Sat,Max CSP,Min-Ones SAT)can be represented using soft constraints where each constraint is specified by a cost function assigning some measure of cost to each tuple of values in its scope.Are all tractable classes of soft constraints characterized by their multimorphisms?(see talk by Peter Jeavons)Communicated by:Peter Jeavons 3AlgebraQuestion3.1The Galois connection between sets of relations and sets of operations that preserve them has been used to analyse several different computational problems such as the satisfiability of the CSP,and counting the number of solutions.How can we characterise the computational goals for which we can use this Galois connection?Communicated by:Nadia Creignou Question3.2For any relational structure B=(B,R1,...,R k),let co-CSP(B)denote the class of structures which do not have a homomorphism to B.It has been shown that the question of whether co-CSP(B)is definable in Datalog is determined by P ol(B),the polymorphisms of the relations of B(see talk by Andrei Bulatov).Let B be a core,F the set of all idempotent polymorphisms of B and V the variety generated by the algebra(B,F).Is it true that co-CSP(B)is definable in Datalog if and only if V omits types1and2(that is,the local structure of anyfinite algebra in V does not contain a G-set or an affine algebra)?Communicated by:Andrei Bulatov Question3.3Does every tractable clone of polynomials over a group contain a Mal’tsev operation?Communicated by:Pascal Tesson Question3.4Classify(w.r.t.tractability of corresponding CSPs)clones of polynomials of semigroups.Communicated by:Pascal Tesson Question3.5Is it true that for any structure B which is invariant under a near-unanimity operation the problem CSP(B)is in the complexity class NL?Does every such structure have bounded pathwidth duality?(see also Question1.4)Both results are known to hold for a2-element domain(Dalmau)and for majority operations(Dalmau,Krokhin).Communicated by:Victor Dalmau,Benoit Larose3Question3.6Is it decidable whether a given structure is invariant under a near-unanimity function(of some arity)?Communicated by:Benoit Larose Question3.7Let L be afixedfinite lattice.Given an integer-valued supermodular func-tion f on L n,is there an algorithm that maximizes f in polynomial time in n if the function f is given by an oracle?The answer is yes if L is a distributive lattice(see“Supermodular Functions and the Complexity of Max-CSP”,Cohen,Cooper,Jeavons,Krokhin,Discrete Applied Mathemat-ics,2005).More generally,the answer is yes if L is obtained fromfinite distributive lattices via Mal’tsev products(Krokhin,Larose–see talk by Peter Jonsson).The smallest lattice for which the answer is not known is the3-diamond.Communicated by:Andrei Krokhin Question3.8Find the exact relationship between width and relational width.(It is known that one is bounded if and and only if the other is bounded.)Also,what types of width are preserved under natural algebraic constructions?Communicated by:Victor Dalmau 4LogicQuestion4.1The(basic)Propositional Circumscription problem is defined as fol-lows:Input:a propositional formulaφwith atomic relations from a set S,and a clause c.Question:is c satisfied in every minimal model ofφ?It is conjectured(Kirousis,Kolaitis)that there is a trichotomy for this problem,that it iseither in P,coNP-complete or inΠP2,depending on the choice of S.Does this conjecturehold?Communicated by:Nadia Creignou Question4.2The Inverse Satisfiability problem is defined as follows: Input:afinite set of relations S and a relation R.Question:is R expressible by a CNF(S)-formula without existential variables?A dichotomy theorem was obtained by Kavvadias and Sideri for the complexity of this problem with constants.Does a dichotomy hold without the constants?Are the Schaefer cases still tractable?Communicated by:Nadia Creignou4Question4.3Let LFP denote classes of structures definable infirst-order logic with a least-fixed-point operator,let HOM denote classes of structures which are closed under homomorphisms,and let co-CSP denote classes of structures defined by not having a homomorphism to somefixed target structure.•Is LFP∩HOM⊆Datalog?•Is LFP∩co-CSP⊆Datalog?(forfinite target structures)•Is LFP∩co-CSP⊆Datalog?(forω-categorical target structures)Communicated by:Albert Atserias,Manuel BodirskyQuestion4.4(see also Question3.2)Definability of co-CSP(B)in k-Datalog is a sufficient condition for tractability of CSP(B),which is sometimes referred to as having width k. There is a game-theoretic characterisation of definability in k-Datalog in terms of(∃,k)-pebble games(see talk by Phokion Kolaitis).•Is there an algorithm to decide for a given structure B whether co-CSP(B)is definable in k-Datalog(for afixed k)?•Is the width hierarchy strict?The same question when B isω-categorical,but not necessarilyfinite?Communicated by:Phokion Kolaitis,Manuel BodirskyQuestion4.5Find a good logic to capture CSP with“nice”(e.g.,ω-categorical)infinite templates.Communicated by:Iain Stewart 5Graph TheoryQuestion5.1The list homomorphism problem for a(directed)graph H is equivalent to the problem CSP(H∗)where H∗equals H together with all unary relations.•It is conjectured that the list homomorphism problem for a reflexive digraph is tractable if H has the X-underbar property(which is the same as having the bi-nary polymorphism min w.r.t.some total ordering on the set of vertices),and NP-complete otherwise.•It is conjectured that the list homomorphism problem for an irreflexive digraph is tractable if H is preserved by a majority operation,and NP-complete otherwise. Do these conjectures hold?Communicated by:Tomas Feder&Pavol Hell5Question5.2“An island of tractability?”Let A m be the class of all relational structures of the form(A,E1,...,E m)where each E i is an irreflexive symmetric binary relation and the relations E i together satisfy the following‘fullness’condition:any two distinct elements x,y are related in exactly one of the relations E i.Let B m be the single relational structure({1,...,m},E1,...,E m)where each E i is the symmetric binary relation containing all pairs xy except the pair ii.(Note that the relations E i are not irreflexive.)The problem CSP(A m,B m)is defined as:Given A∈A m,is there a homomorphism from A to B m?When m=2,this problem is solvable in polynomial time-it is the recognition problem for split graphs(see“Algorithmic Graph Theory and Perfect Graphs”,M.C.Golumbic, Academic Press,New York,1980)When m>3,this problem is NP-complete(see“Full constraint satisfaction problems”,T.Feder and P.Hell,to appear in SIAM Journal on Computing).What happens when m=3?Is this an“island of tractability”?Quasi-polynomial algorithms are known for this problem(see“Full constraint satisfaction problems”,T. Feder and P.Hell,,to appear in SIAM Journal on Computing,and“Two algorithms for list matrix partitions”,T.Feder,P.Hell,D.Kral,and J.Sgall,SODA2005).Note that a similar problem for m=3was investigated in“The list partition problem for graphs”, K.Cameron,E.E.Eschen,C.T.Hoang and R.Sritharan,SODA2004.Communicated by:Tomas Feder&Pavol Hell Question5.3Finding the generalized hypertree-width,w(H)of a hypergraph H is known to be NP-complete.However it is possible to compute a hypertree-decomposition of H in polynomial time,and the hypertree-width of H is at most3w(H)+1(see talk by Georg Gottlob).Are there other decompositions giving better approximations of the generalized hypertree-width that can be found in polynomial time?Communicated by:Georg Gottlob Question5.4It is known that a CSP whose constraint hypergraph has bounded fractional hypertree width is tractable(see talk by Daniel Marx).Is there a hypergraph property more general than bounded fractional hypertree width that makes the associated CSP polynomial-time solvable?Are there classes of CSP that are tractable due to structural restrictions and have unbounded fractional hypertree width?Communicated by:Georg Gottlob,Daniel Marx Question5.5Prove that there exist two functions f1(w),f2(w)such that,for every w, there is an algorithm that constructs in time n f1(w)a fractional hypertree decomposition of width at most f2(w)for any hypergraph of fractional hypertree width at most w(See talk by Daniel Marx).Communicated by:Daniel Marx6Question5.6Turn the connection between the Robber and Army game and fractional hypertree width into an algorithm for approximating fractional hypertree width.Communicated by:Daniel Marx Question5.7Close the complexity gap between(H,C,K)-colouring and (H,C,K)-colouring (see talk by Dimitrios Thilikos)Find a tight characterization for thefixed-parameter tractable(H,C,K)-colouring problems.•For the(H,C,K)-colouring problems,find nice properties for the non-parameterisedpart(H−C)that guaranteefixed-parameter tractability.•Clarify the role of loops in the parameterised part C forfixed-parameter hardnessresults.Communicated by:Dimitrios Thilikos6Constraint Programming and ModellingQuestion6.1In a constraint programming system there is usually a search procedure that assigns values to particular variables in some order,interspersed with a constraint propagation process which modifies the constraints in the light of these assignments.Is it possible to choose an ordering for the variables and values assigned which changes each problem instance as soon as possible into a new instance which is in a tractable class? Can this be done efficiently?Are there useful heuristics?Question6.2The time taken by a constraint programming system tofind a solution toa given instance can be dramatically altered by modelling the problem differently.Can the efficiency of different constraint models be objectively compared,or does it depend entirely on the solution algorithm?Question6.3For practical constraint solving it is important to eliminate symmetry,in order to avoid wasted search effort.Under what conditions is it tractable to detect the symmetry in a given problem in-stance?7Notes•Representations of constraints-implicit representation-effect on complexity•Unique games conjecture-structural restrictions that make it false-connectionsbetween definability and approximation•MMSNP-characterise tractable problems apart from CSP7•Migrate theoretical results to tools•What restrictions do practical problems actually satisfy?•Practical parallel algorithms-does this align with tractable classes?•Practically relevant constraint languages(”global constraints”)•For what kinds of problems do constraint algorithms/heuristics give good results?8。

Codingbat Python Questions and Answers Section 2This document is prepared and can be used only for educational purposes. All questions are taken from http:// which contains great questions about Python and Java. Please go to original website and solve questions there. Codingbat also presents a report tool which educators can see students’ results.The answers are done by me in my spare times, Codingbat says all answers are true. I used some of questionsin internship application reviews. You can use questions, see for usage details. You can use answers of me or this document it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.Document is prepared with Google Docs and syntax highlighter is GeSHi, which is a great open source tool.If you have any questions, please email me*******************Have fun with Python!Samet AtdagThis document contains 27 questions in these sections:●Warmup-2●Logic-2●String-2●List-2Warmup-21.string_timesGiven a string and a non-negative int n, return a larger string that is n copies of the original string.string_times('Hi', 2) → 'HiHi'string_times('Hi', 3) → 'HiHiHi'string_times('Hi', 1) → 'Hi'My solution:def string_times(str, n):result = ''i=0while i<n:result += stri += 1return result2.front_timesGiven a string and a non-negative int n, we'll say that the front of the string is the first 3 chars, or whatever is there if the string is less than length 3. Return n copies of the front;front_times('Chocolate', 2) → 'ChoCho'front_times('Chocolate', 3) → 'ChoChoCho'front_times('Abc', 3) → 'AbcAbcAbc'My solution:def front_times(str, n):if len(str)<3:front = strelse:front = str[:3]result = ''for i in range(n):result += frontreturn result3.string_bitsGiven a string, return a new string made of every other char starting with the first, so "Hello" yields "Hlo".string_bits('Hello') → 'Hlo'string_bits('Hi') → 'H'string_bits('Heeololeo') → 'Hello'My solution:def string_bits(str):result = ""for i in range(0, len(str)):if i %2 == 0:result = result + str[i]return result4.string_splosionGiven a non-empty string like "Code" return a string like "CCoCodCode".string_splosion('Code') → 'CCoCodCode'string_splosion('abc') → 'aababc'string_splosion('ab') → 'aab'My solution:def string_splosion(str):result = ''for i in range(len(str)+1):result += str[:i]return resultst2Given a string, return the count of the number of times that a substring length 2 appears in the string and also as the last 2 chars of the string, so "hixxxhi" yields 1 (we won't count the end substring).last2('hixxhi') → 1last2('xaxxaxaxx') → 1last2('axxxaaxx') → 2My solution:def last2(str):if len(str)<2:return0last2 = str[len(str)-2:]count = 0for i in range(len(str)-2):sub = str[i:i+2]if sub == last2:count = count + 1return count6.array_count9Given an array of ints, return the number of 9's in the array.array_count9([1, 2, 9]) → 1array_count9([1, 9, 9]) → 2array_count9([1, 9, 9, 3, 9]) → 3My solution:def array_count9(nums):result = 0for i in nums:if i == 9:result += 1return result7.array_front9Given an array of ints, return True if one of the first 4 elements in the array is a 9. The array length may be less than 4.array_front9([1, 2, 9, 3, 4]) → Truearray_front9([1, 2, 3, 4, 9]) → Falsearray_front9([1, 2, 3, 4, 5]) → FalseMy solution:def array_front9(nums):result = Falsefor i in range(4):if len(nums)> i:if nums[i] == 9:result = Truereturn result8.array123Given an array of ints, return True if .. 1, 2, 3, .. appears in the array somewhere.array123([1, 1, 2, 3, 1]) → Truearray123([1, 1, 2, 4, 1]) → Falsearray123([1, 1, 2, 1, 2, 3]) → TrueMy solution:def array123(nums):if len(nums)<3:return Falsefor i in range(len(nums)-2):if nums[i] == 1and nums[i+1] == 2and nums[i+2] == 3:result = Truebreakelse:result = Falsereturn result9.string_matchGiven 2 strings, a and b, return the number of the positions where they contain the same length 2 substring. So "xxcaazz" and "xxbaaz" yields 3, since the "xx", "aa", and "az" substrings appear in the same place in both strings.string_match('xxcaazz', 'xxbaaz') → 3string_match('abc', 'abc') → 2string_match('abc', 'axc') → 0My solution:def string_match(a, b):r = len(a)if len(a)<len(b)else len(b)result = 0for i in range(r):if a[i:i+2] == b[i:i+2]and len(a[i:i+2]) == 2and len(b[i:i+2]) == 2:result += 1return resultLogic-210.make_bricksWe want to make a row of bricks that is goal inches long. We have a numberof small bricks (1 inch each) and big bricks (5 inches each). Return True if it is possible to make the goal by choosing from the given bricks. This is a little harder than it looks and can be done without any loops.make_bricks(3, 1, 8) → Truemake_bricks(3, 1, 9) → Falsemake_bricks(3, 2, 10) → TrueMy solution:def make_bricks(small, big, goal):number_of_fives = goal/5number_of_ones = goal - (5*number_of_fives)if number_of_fives <= big and number_of_ones <= small:return Trueelif(number_of_ones+5<=small)and(big*5 + number_of_ones + 5 == goal): return Trueelse:return False11.lone_sumGiven 3 int values, a b c, return their sum. However, if one of the values is the same as another of the values, it does not count towards the sum.lone_sum(1, 2, 3) → 6lone_sum(3, 2, 3) → 2lone_sum(3, 3, 3) → 0My solution:def lone_sum(a, b, c):if a == b == c:return0elif a == b:return celif a == c:return belif b == c:return aelse:return a + b + c12.lucky_sumGiven 3 int values, a b c, return their sum. However, if one of the values is 13 then it does not count towards the sum and values to its right do not count. So for example, if b is 13, then both b and c do not count.lucky_sum(1, 2, 3) → 6lucky_sum(1, 2, 13) → 3lucky_sum(1, 13, 3) → 1My solution:def lucky_sum(a, b, c):if a == 13:return0elif b == 13:return aelif c == 13:return a + belse:return a + b + c13.no_teen_sumGiven 3 int values, a b c, return their sum. However, if any of the values is a teen -- in the range 13..19 inclusive -- then that value counts as 0, except 15 and 16 donot count as a teens. Suggestion: decompose a separate "def fix_teem(n):" below no_teen_sum() that takes in an int value and returns that value fixed for the teen rule. In this way, you avoid repeating the teen code 3 times.no_teen_sum(1, 2, 3) → 6no_teen_sum(2, 13, 1) → 3no_teen_sum(2, 1, 14) → 3My solution:def no_teen_sum(a, b, c):return fix_teem(a) + fix_teem(b) + fix_teem(c)def fix_teem(n):if n>=13and n<=19:if n == 15or n == 16:return nelse:return0else:return n14.round_sumFor this problem, we'll round an int value up to the next multiple of 10 if its rightmost digit is 5 or more, so 15 rounds up to 20. Alternately, round down to the previous multiple of 10 if its rightmost digit is less than 5, so 12 rounds down to 10. Given 3 ints, a b c, return the sum of their rounded values. To avoid code repetition, decompose a separate "def round10(num):" below your round_sum() method and call it 3 times.round_sum(16, 17, 18) → 60round_sum(12, 13, 14) → 30round_sum(6, 4, 4) → 10My solution:def round_sum(a, b, c):return round10(a) + round10(b) + round10(c)def round10(num):if num%10>=5:return num/10*10 + 10else:return num/10*1015.close_farGiven three ints, a b c, return True if one of b or c is "close" (differing from a byat most 1), while the other is "far", differing from both other values by 2 or more. Note: abs(num) computes the absolute value of a number.close_far(1, 2, 10) → Trueclose_far(1, 2, 3) → Falseclose_far(4, 1, 3) → TrueMy solution:def close_far(a, b, c):if abs(b-a)<= 1:close = belif abs(c-a)<= 1:close = celse:return Falseif(close == b)and(abs(c-a)>= 2)and(abs(c-b)>=2):return Trueelif(close == c)and(abs(b-a)>= 2)and(abs(b-c)>=2):return Trueelse:return FalseString-216.double_charGiven a string, return a string where for every char in the original, there are two chars.double_char('The') → 'TThhee'double_char('AAbb') → 'AAAAbbbb'double_char('Hi-There') → 'HHii--TThheerree'My solution:def double_char(str):result = ''for i in str:result += iresult += ireturn result17.count_hiReturn the number of times that the string "hi" appears anywhere in the givenstring.count_hi('abc hi ho') → 1count_hi('ABChi hi') → 2count_hi('hihi') → 2My solution:def count_hi(str):a = str.split('hi')return len(a) - 118.cat_dogReturn True if the string "cat" and "dog" appear the same number of times in the given string.cat_dog('catdog') → Truecat_dog('catcat') → Falsecat_dog('1cat1cadodog') → TrueMy solution:def cat_dog(str):a = str.split('cat')b = str.split('dog')if len(a) == len(b):return Trueelse:return False19.count_codeReturn the number of times that the string "code" appears anywhere in the given string, except we'll accept any letter for the 'd', so "cope" and "cooe" count.count_code('aaacodebbb') → 1count_code('codexxcode') → 2count_code('cozexxcope') → 2My solution:def count_code(str):result = 0for i in range(len(str)-3):if str[i:i+2] == 'co'and str[i+3] == 'e':result += 1return result20.end_otherGiven two strings, return True if either of the strings appears at the very endof the other string, ignoring upper/lower case differences (in other words, the computation should not be "case sensitive"). Note: s.lower() returns the lowercase version of a string.end_other('Hiabc', 'abc') → Trueend_other('AbC', 'HiaBc') → Trueend_other('abc', 'abXabc') → TrueMy solution:def end_other(a, b):a = a.lower()b = b.lower()return(b.endswith(a)or a.endswith(b))21.xyz_thereReturn True if the given string contains an appearance of "xyz" where the xyz is not directly preceeded by a period (.). So "xxyz" counts but "x.xyz" does not.xyz_there('abcxyz') → Truexyz_there('abc.xyz') → Falsexyz_there('xyz.abc') → TrueMy solution:def xyz_there(str):str = str.replace('.xyz', '')if'xyz'in str:return Trueelse:return FalseList-222.count_evensReturn the number of even ints in the given array. Note: the % "mod" operator computes the remainder, e.g. 5 % 2 is 1.count_evens([2, 1, 2, 3, 4]) → 3count_evens([2, 2, 0]) → 3count_evens([1, 3, 5]) → 0My solution:def count_evens(nums):result = 0for i in nums:if i%2 == 0:result += 1return result23.big_diffGiven an array length 1 or more of ints, return the difference between the largest and smallest values in the array. Note: the built-in min(v1, v2) and max(v1, v2) functions return the smaller or larger of two values.big_diff([10, 3, 5, 6]) → 7big_diff([7, 2, 10, 9]) → 8big_diff([2, 10, 7, 2]) → 8My solution:def big_diff(nums):maxx = nums[0]minn = nums[0]for i in range(len(nums)):if nums[i]> maxx:maxx = nums[i]if nums[i]< minn:minn = nums[i]return maxx - minn24.centered_averageReturn the "centered" average of an array of ints, which we'll say is the mean average of the values, except not counting the largest and smallest values in the array. Use int division to produce the final average. You may assume that the array is length 3 or more.centered_average([1, 2, 3, 4, 100]) → 3centered_average([1, 1, 5, 5, 10, 8, 7]) → 5centered_average([-10, -4, -2, -4, -2, 0]) → -3def centered_average(nums):total = 0number_of_excepts = 2centered = numscentered.remove(max(nums))centered.remove(min(nums))for i in centered:total += ireturn total/len(centered)25.sum13Return the sum of the numbers in the array, returning 0 for an empty array. Except the number 13 is very unlucky, so it does not count and numbers that come immediately after a 13 also do not count.sum13([1, 2, 2, 1]) → 6sum13([1, 1]) → 2sum13([1, 2, 2, 1, 13]) → 6My solution:def sum13(nums):sum = 0for i in range(0, nums.count(13)):if nums.count(13):after = nums.index(13)nums.remove(13)if after <len(nums):nums.pop(after)for i in nums:sum += ireturn sum26.sum67Return the sum of the numbers in the array, except ignore sections of numbers starting with a 6 and extending to the next 7 (every 6 will be followed by at least one 7). Return 0 for no numbers.sum67([1, 2, 2]) → 5sum67([1, 2, 2, 6, 99, 99, 7]) → 5sum67([1, 1, 6, 7, 2]) → 4def sum67(nums):dontadd = 0sum = 0for i in range(0, len(nums)):if dontadd == 0:if nums[i] == 6:dontadd = 1else:sum += nums[i]else:if nums[i] == 7:dontadd = 0else:passreturn sum27.has22Given an array of ints, return True if the array contains a 2 next to a 2 somewhere.has22([1, 2, 2]) → Truehas22([1, 2, 1, 2]) → Falsehas22([2, 1, 2]) → FalseMy solution:def has22(nums):indices = []for i in range(0, len(nums)):if nums[i] == 2:indices.append(i)for i in range(0, len(indices)-1):if indices[i+1] - indices[i] == 1:return Truereturn False。

Dynamic ProgrammingDynamic Programming 是一種解問題的方法及技巧。

大致是：將一個大問題切割成若干小問題，recursive 地解決各個小問題( 並將其解以一種table 儲存起來“tabular method”) ，然後再利用這些“小問題”之解答，回來解決原本的大問題。

它與divide and conquer 有點不太一樣，等一下看了幾個例子再體會其差異( page 385 Figure 15.7 之例可看出差異)。

例: shortest path problem ( multistage graph )找s 到t 之shortest path若用dynamic programming方式解之如下: ( Forward Approach )於是就找到由s到t 之最短路徑( 如何“讀”出來? )以下用數學式子寫出上面的recursive 想法:P i(j) = min cost path from vertex j ( in stage i ) to vertex t.Cost i(j) = 上述path之cost ( 總distance )這叫define the optimal value functionrecursive relation 或( recurrence relation ) :Cost i(j) = min l∈stage i+1{d(j , l) + Cost i+1(l)}initial condition ( 或稱boundary condition)假設共有k個stage (s在第1 stage t 在第k stage)Cost k-1(j) =而Answer : Cost1(s)time complexity : O( |V|+|E| )space complexity : 作table 儲存Cost i(j) → O( |V| )但input 需O( |V| + |E| )有了以上的數學式子，顯示出dynamic Programming 想法再下去才可著手程式的大鋼(註: 等一下會說明“程式”絕不能寫成recursive 程式)dynamic Programming 有時也可用一種 backward approach 圖示如下 :也可類似define 及寫出 optimal value function ，recursive relation ，boundary condition ，answer ，time 及 space complexity.Define P i (j) = a shortest path from vertex s to vertex j (in stage i)BCost i (j) = 上述path 之cost (總distance)Recursive solution :BCost i (j) = 1min −∈i stagel { BCost i-1(l) + d(l, j) } k..., 3,4,5, i i stage j =∈∀initial condition :BCost i (j) = edge j) (s, ),( {這一個若有若無j s d ∞Anser BCost k (t)同樣在計算過程中，要在table 中記錄下在每一個state vertex 應作的 Optional )由s到此點之最短路徑長Time complexity ?Space complexity ?同樣在實際implement程式時，除了要適當的data structure外也是不可以recursive程式來作，必須以iterative方式用一種“table”儲存資料，以供後面重覆讀取資料(等一下再看幾個例子再說明)一個簡化的Knapsack problem: (就是課本p. 360 的 rod cutting problem) 有n items 各具重量 分別是 1, 2, 3, 4, 5, … n 各有profit p 1, p 2 p 3 p 4 p 5 p n 有一個 knapsack 能載重 n （n 是integer ）目標： 在載重範圍內，儘量拿取item ，同一樣東西可以重複拿取，希望所獲 得的profit 越大越好。

Proceedings of the IASTED International ConferenceParallel and Distributed Computing and SystemsNovember3-6,1999,MIT,Boston,USAParallel Refinement of Unstructured MeshesJos´e G.Casta˜n os and John E.SavageDepartment of Computer ScienceBrown UniversityE-mail:jgc,jes@AbstractIn this paper we describe a parallel-refinement al-gorithm for unstructuredfinite element meshes based on the longest-edge bisection of triangles and tetrahedrons. This algorithm is implemented in P ARED,a system that supports the parallel adaptive solution of PDEs.We dis-cuss the design of such an algorithm for distributed mem-ory machines including the problem of propagating refine-ment across processor boundaries to obtain meshes that are conforming and non-degenerate.We also demonstrate that the meshes obtained by this algorithm are equivalent to the ones obtained using the serial longest-edge refine-ment method.Wefinally report on the performance of this refinement algorithm on a network of workstations.Keywords:mesh refinement,unstructured meshes,finite element methods,adaptation.1.IntroductionThefinite element method(FEM)is a powerful and successful technique for the numerical solution of partial differential equations.When applied to problems that ex-hibit highly localized or moving physical phenomena,such as occurs on the study of turbulence influidflows,it is de-sirable to compute their solutions adaptively.In such cases, adaptive computation has the potential to significantly im-prove the quality of the numerical simulations by focusing the available computational resources on regions of high relative error.Unfortunately,the complexity of algorithms and soft-ware for mesh adaptation in a parallel or distributed en-vironment is significantly greater than that it is for non-adaptive computations.Because a portion of the given mesh and its corresponding equations and unknowns is as-signed to each processor,the refinement(coarsening)of a mesh element might cause the refinement(coarsening)of adjacent elements some of which might be in neighboring processors.To maintain approximately the same number of elements and vertices on every processor a mesh must be dynamically repartitioned after it is refined and portions of the mesh migrated between processors to balance the work.In this paper we discuss a method for the paral-lel refinement of two-and three-dimensional unstructured meshes.Our refinement method is based on Rivara’s serial bisection algorithm[1,2,3]in which a triangle or tetrahe-dron is bisected by its longest edge.Alternative efforts to parallelize this algorithm for two-dimensional meshes by Jones and Plassman[4]use randomized heuristics to refine adjacent elements located in different processors.The parallel mesh refinement algorithm discussed in this paper has been implemented as part of P ARED[5,6,7], an object oriented system for the parallel adaptive solu-tion of partial differential equations that we have devel-oped.P ARED provides a variety of solvers,handles selec-tive mesh refinement and coarsening,mesh repartitioning for load balancing,and interprocessor mesh migration.2.Adaptive Mesh RefinementIn thefinite element method a given domain is di-vided into a set of non-overlapping elements such as tri-angles or quadrilaterals in2D and tetrahedrons or hexahe-drons in3D.The set of elements and its as-sociated vertices form a mesh.With theaddition of boundary conditions,a set of linear equations is then constructed and solved.In this paper we concentrate on the refinement of conforming unstructured meshes com-posed of triangles or tetrahedrons.On unstructured meshes, a vertex can have a varying number of elements adjacent to it.Unstructured meshes are well suited to modeling do-mains that have complex geometry.A mesh is said to be conforming if the triangles and tetrahedrons intersect only at their shared vertices,edges or faces.The FEM can also be applied to non-conforming meshes,but conformality is a property that greatly simplifies the method.It is also as-sumed to be a requirement in this paper.The rate of convergence and quality of the solutions provided by the FEM depends heavily on the number,size and shape of the mesh elements.The condition number(a)(b)(c)Figure1:The refinement of the mesh in using a nested refinement algorithm creates a forest of trees as shown in and.The dotted lines identify the leaf triangles.of the matrices used in the FEM and the approximation error are related to the minimum and maximum angle of all the elements in the mesh[8].In three dimensions,the solid angle of all tetrahedrons and their ratio of the radius of the circumsphere to the inscribed sphere(which implies a bounded minimum angle)are usually used as measures of the quality of the mesh[9,10].A mesh is non-degenerate if its interior angles are never too small or too large.For a given shape,the approximation error increases with ele-ment size(),which is usually measured by the length of the longest edge of an element.The goal of adaptive computation is to optimize the computational resources used in the simulation.This goal can be achieved by refining a mesh to increase its resolution on regions of high relative error in static problems or by re-fining and coarsening the mesh to follow physical anoma-lies in transient problems[11].The adaptation of the mesh can be performed by changing the order of the polynomi-als used in the approximation(-refinement),by modifying the structure of the mesh(-refinement),or a combination of both(-refinement).Although it is possible to replace an old mesh with a new one with smaller elements,most -refinement algorithms divide each element in a selected set of elements from the current mesh into two or more nested subelements.In P ARED,when an element is refined,it does not get destroyed.Instead,the refined element inserts itself into a tree,where the root of each tree is an element in the initial mesh and the leaves of the trees are the unrefined elements as illustrated in Figure1.Therefore,the refined mesh forms a forest of refinement trees.These trees are used in many of our algorithms.Error estimates are used to determine regions where adaptation is necessary.These estimates are obtained from previously computed solutions of the system of equations. After adaptation imbalances may result in the work as-signed to processors in a parallel or distributed environ-ment.Efficient use of resources may require that elements and vertices be reassigned to processors at runtime.There-fore,any such system for the parallel adaptive solution of PDEs must integrate subsystems for solving equations,adapting a mesh,finding a good assignment of work to processors,migrating portions of a mesh according to anew assignment,and handling interprocessor communica-tion efficiently.3.P ARED:An OverviewP ARED is a system of the kind described in the lastparagraph.It provides a number of standard iterativesolvers such as Conjugate Gradient and GMRES and pre-conditioned versions thereof.It also provides both-and -refinement of meshes,algorithms for adaptation,graph repartitioning using standard techniques[12]and our ownParallel Nested Repartitioning(PNR)[7,13],and work mi-gration.P ARED runs on distributed memory parallel comput-ers such as the IBM SP-2and networks of workstations.These machines consist of coarse-grained nodes connectedthrough a high to moderate latency network.Each nodecannot directly address a memory location in another node. In P ARED nodes exchange messages using MPI(Message Passing Interface)[14,15,16].Because each message has a high startup cost,efficient message passing algorithms must minimize the number of messages delivered.Thus, it is better to send a few large messages rather than many small ones.This is a very important constraint and has a significant impact on the design of message passing algo-rithms.P ARED can be run interactively(so that the user canvisualize the changes in the mesh that results from meshadaptation,partitioning and migration)or without directintervention from the user.The user controls the systemthrough a GUI in a distinguished node called the coordina-tor,.This node collects information from all the other processors(such as its elements and vertices).This tool uses OpenGL[17]to permit the user to view3D meshes from different angles.Through the coordinator,the user can also give instructions to all processors such as specify-ing when and how to adapt the mesh or which strategy to use when repartitioning the mesh.In our computation,we assume that an initial coarse mesh is given and that it is loaded into the coordinator.The initial mesh can then be partitioned using one of a num-ber of serial graph partitioning algorithms and distributed between the processors.P ARED then starts the simulation. Based on some adaptation criterion[18],P ARED adapts the mesh using the algorithms explained in Section5.Af-ter the adaptation phase,P ARED determines if a workload imbalance exists due to increases and decreases in the num-ber of mesh elements on individual processors.If so,it invokes a procedure to decide how to repartition mesh el-ements between processors;and then moves the elements and vertices.We have found that PNR gives partitions with a quality comparable to those provided by standard meth-ods such as Recursive Spectral Bisection[19]but which(b)(a)Figure2:Mesh representation in a distributed memory ma-chine using remote references.handles much larger problems than can be handled by stan-dard methods.3.1.Object-Oriented Mesh RepresentationsIn P ARED every element of the mesh is assigned to a unique processor.V ertices are shared between two or more processors if they lie on a boundary between parti-tions.Each of these processors has a copy of the shared vertices and vertices refer to each other using remote ref-erences,a concept used in object-oriented programming. This is illustrated in Figure2on which the remote refer-ences(marked with dashed arrows)are used to maintain the consistency of multiple copies of the same vertex in differ-ent processors.Remote references are functionally similar to standard C pointers but they address objects in a different address space.A processor can use remote references to invoke meth-ods on objects located in a different processor.In this case, the method invocations and arguments destined to remote processors are marshalled into messages that contain the memory addresses of the remote objects.In the destina-tion processors these addresses are converted to pointers to objects of the corresponding type through which the meth-ods are invoked.Because the different nodes are inher-ently trusted and MPI guarantees reliable communication, P ARED does not incur the overhead traditionally associated with distributed object systems.Another idea commonly found in object oriented pro-gramming and which is used in P ARED is that of smart pointers.An object can be destroyed when there are no more references to it.In P ARED vertices are shared be-tween several elements and each vertex counts the number of elements referring to it.When an element is created, the reference count of its vertices is incremented.Simi-larly,when the element is destroyed,the reference count of its vertices is decremented.When the reference count of a vertex reaches zero,the vertex is no longer attached to any element located in the processor and can be destroyed.If a vertex is shared,then some other processor might have a re-mote reference to it.In that case,before a copy of a shared vertex is destroyed,it informs the copies in other processors to delete their references to itself.This procedure insures that the shared vertex can then be safely destroyed without leaving dangerous dangling pointers referring to it in other processors.Smart pointers and remote references provide a simple replication mechanism that is tightly integrated with our mesh data structures.In adaptive computation,the struc-ture of the mesh evolves during the computation.During the adaptation phase,elements and vertices are created and destroyed.They may also be assigned to a different pro-cessor to rebalance the work.As explained above,remote references and smart pointers greatly simplify the task of creating dynamic meshes.4.Adaptation Using the Longest Edge Bisec-tion AlgorithmMany-refinement techniques[20,21,22]have been proposed to serially refine triangular and tetrahedral meshes.One widely used method is the longest-edge bisec-tion algorithm proposed by Rivara[1,2].This is a recursive procedure(see Figure3)that in two dimensions splits each triangle from a selected set of triangles by adding an edge between the midpoint of its longest side to the opposite vertex.In the case that makes a neighboring triangle,,non-conforming,then is refined using the same algorithm.This may cause the refinement to prop-agate throughout the mesh.Nevertheless,this procedure is guaranteed to terminate because the edges it bisects in-crease in length.Building on the work of Rosenberg and Stenger[23]on bisection of triangles,Rivara[1,2]shows that this refinement procedure provably produces two di-mensional meshes in which the smallest angle of the re-fined mesh is no less than half of the smallest angle of the original mesh.The longest-edge bisection algorithm can be general-ized to three dimensions[3]where a tetrahedron is bisected into two tetrahedrons by inserting a triangle between the midpoint of its longest edge and the two vertices not in-cluded in this edge.The refinement propagates to neigh-boring tetrahedrons in a similar way.This procedure is also guaranteed to terminate,but unlike the two dimensional case,there is no known bound on the size of the small-est angle.Nevertheless,experiments conducted by Rivara [3]suggest that this method does not produce degenerate meshes.In two dimensions there are several variations on the algorithm.For example a triangle can initially be bisected by the longest edge,but then its children are bisected by the non-conforming edge,even if it is that is not their longest edge[1].In three dimensions,the bisection is always per-formed by the longest edge so that matching faces in neigh-boring tetrahedrons are always bisected by the same com-mon edge.Bisect()let,and be vertices of the trianglelet be the longest side of and let be the midpoint ofbisect by the edge,generating two new triangles andwhile is a non-conforming vertex dofind the non-conforming triangle adjacent to the edgeBisect()end whileFigure3:Longest edge(Rivara)bisection algorithm for triangular meshes.Because in P ARED refined elements are not destroyed in the refinement tree,the mesh can be coarsened by replac-ing all the children of an element by their parent.If a parent element is selected for coarsening,it is important that all the elements that are adjacent to the longest edge of are also selected for coarsening.If neighbors are located in different processors then only a simple message exchange is necessary.This algorithm generates conforming meshes: a vertex is removed only if all the elements that contain that vertex are all coarsened.It does not propagate like the re-finement algorithm and it is much simpler to implement in parallel.For this reason,in the rest of the paper we will focus on the refinement of meshes.5.Parallel Longest-Edge RefinementThe longest-edge bisection algorithm and many other mesh refinement algorithms that propagate the refinement to guarantee conformality of the mesh are not local.The refinement of one particular triangle or tetrahedron can propagate through the mesh and potentially cause changes in regions far removed from.If neighboring elements are located in different processors,it is necessary to prop-agate this refinement across processor boundaries to main-tain the conformality of the mesh.In our parallel longest edge bisection algorithm each processor iterates between a serial phase,in which there is no communication,and a parallel phase,in which each processor sends and receives messages from other proces-sors.In the serial phase,processor selects a setof its elements for refinement and refines them using the serial longest edge bisection algorithms outlined earlier. The refinement often creates shared vertices in the bound-ary between adjacent processors.To minimize the number of messages exchanged between and,delays the propagation of refinement to until has refined all the elements in.The serial phase terminates when has no more elements to refine.A processor informs an adjacent processor that some of its elements need to be refined by sending a mes-sage from to containing the non-conforming edges and the vertices to be inserted at their midpoint.Each edge is identified by its endpoints and and its remote ref-erences(see Figure4).If and are sharedvertices,(a)(c)(b)Figure4:In the parallel longest edge bisection algo-rithm some elements(shaded)are initially selected for re-finement.If the refinement creates a new(black)ver-tex on a processor boundary,the refinement propagates to neighbors.Finally the references are updated accord-ingly.then has a remote reference to copies of and lo-cated in processor.These references are included in the message,so that can identify the non-conforming edge and insert the new vertex.A similar strategy can be used when the edge is refined several times during the re-finement phase,but in this case,the vertex is not located at the midpoint of.Different processors can be in different phases during the refinement.For example,at any given time a processor can be refining some of its elements(serial phase)while neighboring processors have refined all their elements and are waiting for propagation messages(parallel phase)from adjacent processors.waits until it has no elements to refine before receiving a message from.For every non-conforming edge included in a message to,creates its shared copy of the midpoint(unless it already exists) and inserts the new non-conforming elements adjacent to into a new set of elements to be refined.The copy of in must also have a remote reference to the copy of in.For this reason,when propagates the refine-ment to it also includes in the message a reference to its copies of shared vertices.These steps are illustrated in Figure4.then enters the serial phase again,where the elements in are refined.(c)(b)(a)Figure5:Both processors select(shaded)mesh el-ements for refinement.The refinement propagates to a neighboring processor resulting in more elements be-ing refined.5.1.The Challenge of Refining in ParallelThe description of the parallel refinement algorithm is not complete because refinement propagation across pro-cessor boundaries can create two synchronization prob-lems.Thefirst problem,adaptation collision,occurs when two(or more)processors decide to refine adjacent elements (one in each processor)during the serial phase,creating two(or more)vertex copies over a shared edge,one in each processor.It is important that all copies refer to the same logical vertex because in a numerical simulation each ver-tex must include the contribution of all the elements around it(see Figure5).The second problem that arises,termination detection, is the determination that a refinement phase is complete. The serial refinement algorithm terminates when the pro-cessor has no more elements to refine.In the parallel ver-sion termination is a global decision that cannot be deter-mined by an individual processor and requires a collabora-tive effort of all the processors involved in the refinement. Although a processor may have adapted all of its mesh elements in,it cannot determine whether this condition holds for all other processors.For example,at any given time,no processor might have any more elements to re-fine.Nevertheless,the refinement cannot terminate because there might be some propagation messages in transit.The algorithm for detecting the termination of parallel refinement is based on Dijkstra’s general distributed termi-nation algorithm[24,25].A global termination condition is reached when no element is selected for refinement.Hence if is the set of all elements in the mesh currently marked for refinement,then the algorithmfinishes when.The termination detection procedure uses message ac-knowledgments.For every propagation message that receives,it maintains the identity of its source()and to which processors it propagated refinements.Each prop-agation message is acknowledged.acknowledges to after it has refined all the non-conforming elements created by’s message and has also received acknowledgments from all the processors to which it propagated refinements.A processor can be in two states:an inactive state is one in which has no elements to refine(it cannot send new propagation messages to other processors)but can re-ceive messages.If receives a propagation message from a neighboring processor,it moves from an inactive state to an active state,selects the elements for refinement as spec-ified in the message and proceeds to refine them.Let be the set of elements in needing refinement.A processor becomes inactive when:has received an acknowledgment for every propa-gation message it has sent.has acknowledged every propagation message it has received..Using this definition,a processor might have no more elements to refine()but it might still be in an active state waiting for acknowledgments from adjacent processors.When a processor becomes inactive,sends an acknowledgment to the processors whose propagation message caused to move from an inactive state to an active state.We assume that the refinement is started by the coordi-nator processor,.At this stage,is in the active state while all the processors are in the inactive state.ini-tiates the refinement by sending the appropriate messages to other processors.This message also specifies the adapta-tion criterion to use to select the elements for refinement in.When a processor receives a message from,it changes to an active state,selects some elements for refine-ment either explicitly or by using the specified adaptation criterion,and then refines them using the serial bisection algorithm,keeping track of the vertices created over shared edges as described earlier.When itfinishes refining its ele-ments,sends a message to each processor on whose shared edges created a shared vertex.then listens for messages.Only when has refined all the elements specified by and is not waiting for any acknowledgment message from other processors does it sends an acknowledgment to .Global termination is detected when the coordinator becomes inactive.When receives an acknowledgment from every processor this implies that no processor is re-fining an element and that no processor is waiting for an acknowledgment.Hence it is safe to terminate the refine-ment.then broadcasts this fact to all the other proces-sors.6.Properties of Meshes Refined in ParallelOur parallel refinement algorithm is guaranteed to ter-minate.In every serial phase the longest edge bisectionLet be a set of elements to be refinedwhile there is an element dobisect by its longest edgeinsert any non-conforming element intoend whileFigure6:General longest-edge bisection(GLB)algorithm.algorithm is used.In this algorithm the refinement prop-agates towards progressively longer edges and will even-tually reach the longest edge in each processor.Between processors the refinement also propagates towards longer edges.Global termination is detected by using the global termination detection procedure described in the previous section.The resulting mesh is conforming.Every time a new vertex is created over a shared edge,the refinement propagates to adjacent processors.Because every element is always bisected by its longest edge,for triangular meshes the results by Rosenberg and Stenger on the size of the min-imum angle of two-dimensional meshes also hold.It is not immediately obvious if the resulting meshes obtained by the serial and parallel longest edge bisection al-gorithms are the same or if different partitions of the mesh generate the same refined mesh.As we mentioned earlier, messages can arrive from different sources in different or-ders and elements may be selected for refinement in differ-ent sequences.We now show that the meshes that result from refining a set of elements from a given mesh using the serial and parallel algorithms described in Sections4and5,re-spectively,are the same.In this proof we use the general longest-edge bisection(GLB)algorithm outlined in Figure 6where the order in which elements are refined is not spec-ified.In a parallel environment,this order depends on the partition of the mesh between processors.After showing that the resulting refined mesh is independent of the order in which the elements are refined using the serial GLB al-gorithm,we show that every possible distribution of ele-ments between processors and every order of parallel re-finement yields the same mesh as would be produced by the serial algorithm.Theorem6.1The mesh that results from the refinement of a selected set of elements of a given mesh using the GLB algorithm is independent of the order in which the elements are refined.Proof:An element is refined using the GLBalgorithm if it is in the initial set or refinementpropagates to it.An element is refinedif one of its neighbors creates a non-conformingvertex at the midpoint of one of its edges.Therefinement of by its longest edge divides theelement into two nested subelements andcalled the children of.These children are inturn refined by their longest edge if one of their edges is non-conforming.The refinement proce-dure creates a forest of trees of nested elements where the root of each tree is an element in theinitial mesh and the leaves are unrefined ele-ments.For every element,let be the refinement tree of nested elements rooted atwhen the refinement procedure terminates. Using the GLB procedure elements can be se-lected for refinement in different orders,creating possible different refinement histories.To show that this cannot happen we assume the converse, namely,that two refinement histories and generate different refined meshes,and establish a contradiction.Thus,assume that there is an ele-ment such that the refinement trees and,associated with the refinement histories and of respectively,are different.Be-cause the root of and is the same in both refinement histories,there is a place where both treesfirst differ.That is,starting at the root,there is an element that is common to both trees but for some reason,its children are different.Be-cause is always bisected by the longest edge, the children of are different only when is refined in one refinement history and it is not re-fined in the other.In other words,in only one of the histories does have children.Because is refined in only one refinement his-tory,then,the initial set of elements to refine.This implies that must have been refined because one of its edges became non-conforming during one of the refinement histo-ries.Let be the set of elements that are present in both refinement histories,but are re-fined in and not in.We define in a similar way.For each refinement history,every time an ele-ment is refined,it is assigned an increasing num-ber.Select an element from either or that has the lowest number.Assume that we choose from so that is refined in but not in.In,is refined because a neigh-boring element created a non-conforming ver-tex at the midpoint of their shared edge.There-fore is refined in but not in because otherwise it would cause to be refined in both sequences.This implies that is also in and has a lower refinement number than con-。

Parallel structure模板I think it is a complex question and it should depend on…If…have…,they could/should…,because…On the other hand, if they do not have…,they could/should (not)…,because….So, it is really hard to say which is correct/better and it truly depends on…常用原因：Time---you are busy in working/you enjoy the vacationEconomic situation---have enough money to support/give limited supportPersonality---easygoing/shySelf-discipline---have good SD/have bad SDSituation---public place/private areaSubject---program/contentsExamplesQ1你认为学生是否应该带手机上学？I think it is a complex question and it should depend on student’s self-disciplineIf students have good self-discipline, they could bring their mobile phones to school, because it’s very convenient for students to contact with their families and friends.On the other hand, if they do not have good self-discipline, they could not take their mobile phones to school, because it may distract their attention and play games on the phone during the class.So, it is really hard to say which is correct and it truly depends on student’s self-discipline.Q2你认为大学教育要面向所有学生还是部分学生？I think it is a complex question and it should depend on economic situation of one country.If the country have enough money to support education, they should let every student go to university, because knowledge is powerful and it gives student the chance to lead a better life. On the other hand, if they give limited support, they could not require every student to go to university, because not only the country but also the student s’ families cannot afford the tuition fee in the university.So, it is really hard to say which is better and it truly depends on the country’s economic situation.Q3在旅行中，有些人喜欢直接到达目的地，有些人却喜欢沿途一路观光。

Parallel structure means using the same pattern of words to show that two or more ideas have the same level of importance. This can happen at the word, phrase, or clause level. The usual way to join parallel structures is with the use of coordinating conjunctions such as "and" or "or."With the -ing form (gerund) of words: Parallel: Mary likes hik ing, swimm ing, and bicycl ing.With infinitive phrases:Parallel: Mary likes to hike, to swim, and to ride a bicycle.Example 1Mary likes hik ing, swimm ing, and rid ing a bicycle.Example 2The production manager was asked to write his report quick ly, accurate ly, and thorough ly. Example 3The teacher said that he was a poor student because he wait ed until the last minute tostudy for the exam, complet ed his lab problems in a careless manner, and lack ed motivation.Example 4The coach told the players that they should get a lot of sleep, not eat too much, and do some warm-up exercises before the game. Example 5The dictionary can be used for these purposes: to find word meanings, pronunciations, correct spellings, and irregular verbs.。

平行结构（Parallel Structure）平行结构是指在论证某个观点或论题时，将其分解成具有一定联系的几个方面来加以论述，而这几个方面都是从属于中心论题或观点的，它们之间地位平等，呈平行关系，故称平行结构。

议论文结构中较为常见的是平行结构。

议论文中的平行结构依据平行结构与论点（或论题）的关系，可以将其大致分成如下几类。

一、分解类。

论证中平行结构的几个方面是对中心论点的展开与说明，是中心论点的具体化，恰似文章结构中的总分关系。

如初语教材中《理想的阶梯》一文，文章的中心论点是“奋斗是理想的阶样”，并列的几个部分可以说是对“奋斗”一语的分解与具体化：1、刻苦勤奋，是理想的阶梯；2、珍惜时间，是理想的阶梯；3、迎难而上，是理想的阶梯。

一些抽象的文题，如《知识就是力量》、《时间就是一切》等常用此类结构布局谋篇来进行论证。

二、阐述类。

论证中平行的几个方面是对中心论点的属性和思想意义的揭示与阐发，是作者对某一论题思索认识的结果。

比如某年华东六省中学生作文大赛中的一篇夺魁乍文一一《这山望着那山高新解》，作者在辩证地引出学习、工作应有这种精神后，便用了一组平行结构来阐述它的合义：1、这山望着那山高，是一种永不满足的精神；2、这山望着那山高，是一种精益求精的精神；3、这山望着那山高，是一种进取开拓的精神；4、这山望着那山高，是一种力争上游的精神。

此类形式最能显示出作者思维的深刻程度。

阐释类的议论文，如《谈理想》、《说自信》等，很适宜采用此类结构。

三、利弊类。

议论文常要论述某种主张、做法，而要证明应不应这样做，最好的方法真过于谈作用说危害了。

比如宋代罗大经的《鹤林玉露》中有一篇精短议论文《勤有三益》，作者为了说明“为人当勤劳”这一观点，就采用了如下的平行结构：1、勤劳，可以有所获免饥寒也；（创财）2、勤劳，可以有所劳延寿考也；（健体）3、勤劳，可以有所事远淫邪也。

（修身）又如，舫年高考优秀作文有一篇作文叫《严于律己，宽于待人》，作者在论证时，也用了此种结构：正确态度：宽于待人，严于律己。

Chapter1 Configuration of the control system1.empirical [em'pirikəl]adj.以经验为根据的, 经验主义的2.potentiometer [pə͵ten∫i'ɔmitə] n.【电】电位计, 电势计;【信】分压器3.mimic ['mimik] adj.模仿的, 摹拟的, 假的; 伪装的4.illustrate ['iləstreit] vt.举例说明; 例证5.destination [͵desti'nei∫ən] n.目的地，预定的目的, 目标6.actuate ['æktjueit] vt.开动(机械等); 驱使, 激励(人等)7.circuitry ['sə:kitri] n.电路学; 电路图; 电路系统; 电路8.cabinet ['kæbinit] n.机壳, 壳体，框箱、橱、盒、室、间9.coil [kɔil] n.线组，感应器10.fluidic [flu:`ɪdɪk] adj。

流体的,液体的11.universal [͵ju:ni'və:səl] adj.【机】万能的; 通用的12.exclusively [ɪk'sklu:sɪvlɪ] adv. 排他地，独有地13.annunciator [ə'nʌn∫ieitə] n.信号机[器]; 警报器; (电梯)位置指示器14.amplify ['æmplifai] vt.扩大; 放大; 展开15.drilling ['driliŋ]n.操练, 训练，钻孔16.chamber ['t∫eimbə] n.(小)室, 箱, 盒, 容器，房间, 寝17.duct [dʌkt] n.管；输送管18.pipe [paip] n.管, 导管, 筒, 输送管19.subsequent ['sʌbsikwənt] adj.后来的, 随[其]后的, 其次的, 接着发生的, 连续的20.deliver [di'livə] vt.递送, 陈述, 释放, 发表, 引渡, 投递, 交付21.energize ['enədӡaiz] vt.激励，加强; 给与...以活力[精力]; 使...复活22.deenergize 去激励23.solenoid valve 【化】电磁阀; 电磁调节阀24.hydraulic [hai'drɔ:lik] adj.液压的, 水压的25.pneumatic [nju:'mætik] adj.空气的, 气体的, 气体学的, 可充气的26.cylinder ['silində] n.【机】汽缸; (装氧气等的)钢筒, 钢瓶, (气)罐Chapter2 Industrial Control System Consisted by Magnetic Relays1.pick up v.掘地, 捡起, 获得, 使恢复精神, 加快, 加速2.drop out v.不参与, 离去, 放弃3.occasionally [ə'keɪӡənəlɪ]ad. 有时候, 偶而4.in series adv. 串联连续地5.in parallel adv. 并联，并行的[地], 平行的[地]6.branch out 扩展范围7.cam [kæm] n. 凸轮8.fan-out [计][电] 输出端,扇出9.appreciate [ə'pri:∫ieit]vt. 赏识，感激，提高(价值)10.concreteness n. 具体性，确实的11.adequate ['ædikwit] adj. 足够的，恰当的，让人满意的12.contamination [kən͵tæmi'nei∫ən]n. 污染；污物Chapter3 Solid-state Logic Control1.foregoing [for'goɪŋ] a. 前述的,上述的2.consequent ['kɑnsə,kwɛnt]n.【逻】后件，【数】后项，结果;随后的事3.equivalency [ɪ'kwɪvələnsɪ] n. 相等;等值;等效;等义4.diode ['daɪod] n.二极管；整流子5.forward biased 正向偏压6.cathode ['kæθod]n.【电】阴板、负极7.anode ['ænod] n.【电】正极;阳极8.potential [pə'tɛnʃəl] a.【物】势的,位的;电压的9.transistor [træn'zistə] n.【电】晶体管10.hold-off 保持距离，回绝；推迟，拖延11.duplicate ['djupləkɪt] a.一对的;二倍的，复制的;副本的12.cumbersome ['kʌmbɚsəm] a.累赘的;麻烦的，笨重的13.at a glance [ət ə 'glæns] 一瞥14.current-sinking 电流陷落，电流吸收15.jeopardize ['dʒepədaiz] vt.使濒于危险境地;冒...的危险;危及16.totem-pole 推拉输出电路，图腾柱17.schematic symbol [skɪ'mætɪk] ['simbəl] 简要标志18.inverter [in'və:tə]n.【电】反用换流器，倒相器，交换器，换流器19.trouble-free 无故障的20.demonstrably ['demənstrəbli] ad.可论证地;明确地21.superior to [sju:'piəriə tə] 优于,级别高于22.reliability [rɪ,laɪə'bɪlətɪ] 可靠;可信赖性;可靠程度23.maintenance-free 无需维护，免维修24.mechanical linkage [mɪˈkænɪkəl] ['lɪŋkɪdʒ] 机械联动装置25.inrush current 冲击电流，启动电流26.insulation [,ɪnsə'leʃən] n.【电】绝缘(体)27.life expectancy 平均（预期）寿命28.remarkable [rɪ'mɑrkəb!] a.出众，显著，不平常的29.life span 寿命30.figure out to 计算出；估计；理解31.appreciable [ə'priʃɪəb!] a.可估计的，可以看到(或察觉到)的,相当可观的32.barring ['bɑrɪŋ]prep.除...以外33.thermal shock 热冲击，热震34.overcurrent 过量电流，过载电流35.atmosphere ['ætməsfiə] n.大气，大气层，大气压36.pit [pɪt] n.洼坑,凹处，使留下疤痕37.impervious [im'pə:viəs] a.非渗透性，不能透过的38.airborne ['ɛəbɔ:n] a.空中传播的,风媒的，空运的39.spark [spɑrk]n.【电】火花;火星40.clash [klæʃ] vi砰地相碰撞vt使碰撞作声;撞击作(声)41.load kickback (电机等的)负载,负荷['kikbæk]反冲，逆转42.premium ['primɪəm]n.奖品，额外补贴,津贴43.extensive [ɪk'stɛnsɪv] a.大量的;庞大的44.conserve [kən'sə:v] vt.【物】使守恒，保存;保护45.contest ['kɑntɛst]n.竞赛,比赛lisecond ['mɪlə,sɛkənd]n.毫秒47.microsecond ['maɪkrə,sɛkənd] n.微秒48.nanosecond ['nænə,sɛkənd] n.纳秒49.roughly speaking [,rʌflɪ'spikɪŋ]粗略地说50.override [,ovɚ'raɪd] 优先于，推翻;使无效51.dc power supply 直流电源52.signal converter 信号转换器53.output amplifier 输出放大器54.printed circuit board(PCB) 印刷电路板55.considerable [kən'sɪdərəb!] a.相当大的，值得考虑的56.asset ['æsɛt] n.有利条件，资产，有价值的东西57.extraneous [ɛk'strenɪəs] a.外部的，附加的58.ambient temperature 周围的温度；外围的温度；环境温度59.fan-cooled 风扇致冷60.air conditioned 有空调设备的，用空气调节器调节空气的61.down-time 停工期，故障停机时间62.explore [iksˈplɔ:] vt.勘查, 探测, 勘探63.uncomplicated [ʌn'kɔmplikeitid] a.不复杂的;单一的,简单的plex ['kɔmpleks] a.复杂的,复合的65.illustrate ['iləstreit]vt.举例说明，图解66.route [ru:t]n. 路;路线67.channel['tʃænl]n.电路;通讯管道，频道,波段，路线68.accomplish [ə'kɑmplɪʃ]vt.完成,实现,达到69.stationary ['steiʃənəri] a.静止的，固定的；不动的70.snug [snʌg] a.装备完整的，舒适的71.spring [spriŋ]n. 弹簧，弹力vi.弹开,反弹72.extend [iks'tend]vi.伸展;扩大;延续73.router ['ru:tə]n.安排路线的人，安排路线的人74.frame [freim] n.架构,框架75.retract [ri'trækt]vt.缩回，缩回76.anchore 使（某物）固定不动77.physical layout 实际布置图，物理布局78.shut off ['ʃʌt ɔf]关掉；切断电源79.rod [rɔd]n.杆，棒，拉杆80.shove [ʃʌv]v.推;撞，推挤81.annunciator [ə'nʌnʃieitə]n.信号器，报警器82.shut down ['ʃʌt daun] 停机，停工83.eliminate [i'limineit] v.排除，消除；淘汰bustion [kəm'bʌstʃən] n.燃烧，氧化85.surge 浪涌86.clue [klu:]n.线索,迹象,提示87.bounce eliminator [bauns][i'limineitə] n. 抖动削除器88.guarantee [,gærən'ti]vt.保证;担保89.foregoing [fɔ:'gəuiŋ] a.前面的;前述的,上述的90.drilling apparatus ['driliŋ] [,æpə'retəs]n.钻孔器械91.clamp [klæmp]n.螺丝钳,铁箍,夹钳vt.钳紧;夹住92.vertical ['və:tikəl] a.垂直的;竖的,立式的n.垂直位置93.horizontal [,hɔri'zɔntl] a.水平的;横的;卧式的n.水平位置94.drill [drɪl] n.钻,钻头，训练vt.钻(孔);在...上钻孔95.descend [di'send]vi.下来,下降96.withdraw [wið'drɔ:] vt.抽回;拉开;移开97.deliver [di'livə]vt.投递;传送;运送Chapter4 Flip-Flops Applied in the Industrial Control System1.rudimentary [,ru:də'mentəri:] a.基本的,初步的2.flip-flop ['flipflɔp]n.【电子】触发器，(方向等的)突变3.in turn 依次;轮到4.shift register 移位寄存器5.opposite ['ɔpəzit] a.相反的,对立的plement ['kɑmpləmənt]n.补充物, 补足物; 配对物7.welder ['weldə] n.焊工,电焊机8.initation 模拟9.postpone [post'pon] vt.使延期,延迟,延缓10.appropriate [ə'prəupriət] a.适当的,恰当的,相称的11.output amplifier 输出放大器12.oscillate ['ɔsi,leit]vi.【物】振荡13.positive edge-triggered 上升沿触发14.negative edge-triggered 下降沿触发15.triangle ['traiæŋgl]n.三角形；三角形的16.distinguish [dɪ'stɪŋgwɪʃ]vt.区别;识别17.spin [spɪn] vt.旋转;自旋18.assembly [ə'sɛmblɪ] n.(机械的)装配19.rack [ræk] n.【机】齿条[C],装配齿轮之齿棒20.pinion ['pɪnjən]n.小齿轮[C]21.two-position selector switch 选择开关22.conversely [kən'və:sli] ad.逆;相反地23.terminate ['tə:mineit] v.端接，终止24.throw [θrəu]vt.转动(机器开关)，投,掷,抛,扔25.focus on ['fəukəs ɔn](使)集中于26.infinitely ['infinitli] ad.无限地,无穷地;极其27.absolutely ['æbsəlu:tli] ad.绝对地,完全地28.occur [ə'kə:] vi.出现;存在;被发现29.real-life 实在的，现实的30.notion ['nəuʃən] n.概念;想法;见解31.confusion [kən'fjuʒən] 干扰，混乱32.toggling mode 切换模式33.override [,əuvə'raid] 撤销,推翻;使无效34.refresh [ri'freʃ] v刷新，更新，再生35.four-bit shift register 四位移位寄存器36.simultaneously [saɪməl'tenɪəslɪ]ad. 同时地37.at this instant 在这一刹那38.specific [spɪ'sɪfɪk] a.特殊的,特定的39.pat [pæt]轻拍,轻打，非常恰当的;适时的40.subsequent ['sʌbsɪ,kwɛnt] a.后来的,其后的41.merely ['mɪrlɪ] ad. 只是,仅仅,不过42.widespread ['waɪd,sprɛd] a.普遍的;广泛的43.stand for 代替，代表44.characteristic [,kærəktə'rɪstɪk] a.特有的,独特的n.特性,特征45.inspection [ɪn'spɛkʃən] n.检查,检验;审视46.reject [rɪ'dʒɛkt]vt.拒绝,抵制，去除47.mentally ['mentəli] a.心理上;精神上48.evenly ['ivənlɪ] a.平坦地;均匀地49.constraint [kən'strent] v. n.约束；约束条件50.inspector [in'spektə]测定器，检验员51.keep space 保持车距，保持速度52.momentary ['məumənteri] a.短暂的;瞬间的53.diverter （电阻）分流器，换向器，分压器54.reject bin 废料仓55.dump [dʌmp] 倾倒;抛弃56.swing [swɪŋ]摇摆,摆动,摇荡Chapter5 Feedback Systems and Sevomechanisms1.so far 迄今为止；到某个程度2.self-correcting [,sɛlfkə'rɛktɪŋ] a.自动调整的;自动修正的3.variable ['vɛəriəbl] a.可变的4.intervention [,intə'venʃən] 插入;介入，干预5.feedback system 反馈系统6.closed-loop system 闭环系统7.servo system 伺服系统，随动系统ponent [kəm'pəunənt] n. (机器、设备等的)构成要素;零件;成分9.essential [i'senʃəl] a.必要的，基本的10.open-loop system 开环系统11.via ['vaɪə] prep.经由;取道，通过,凭借12.manual valve 手控阀13.eventually [ɪ'vɛntʃuəlɪ]ad.最终地，最后14.precisely [prɪ'saɪslɪ] ad.精确地,准确地15.numerous ['njumərəs] a.许多的，大量的16.subtle ['sʌtl] a.敏锐的、敏感的，微妙的，精巧的17.upset the balance 打破平衡18.upstream [,ʌp'stri:m] 上游领域，向上游(的);逆流(的)19.flow rate 流量；流量率20.overflow [,əuvə'fləu] n.溢出物;溢出量;超出额21.float [fləut] vi.漂浮,浮动;飘动22.viscosity [vis'kɔsiti]n.粘度, 粘性23.etc. 等等24.tapered valve 锥形阀25.inflow ['infləu]n. 流入26.linkage ['lɪŋkɪdʒ] n. 连接装置；联动装置，连锁27.deviation [,divɪ'eʃən]n. 偏差，偏向28.stray [strei] 偏离29.cause-effective relation 因果关系30.generalize ['dʒenərəlaiz] v. 推断，归纳31.disturbance [dis'tə:bəns]n. 干扰；滋扰32.external [eks'tə:nl] a. 外面的,外部的33.outcome ['autkʌm] n. 出口，输出，结果parer 比较器35.alter ['ɔ:ltə]转换，交流电，变换器36.magnitude ['mægnitju:d]n. 幅度，量;大小;强度;37.polarity [po'lærətɪ]n. 极性，两极38.counterclockwise [,kauntə'klɔkwaiz] a. 逆转(的)，逆时针(方向)的39.displacement [dɪs'plesmənt]n. 移位;置换;取代40.clockwise ['klɔkwaiz] a. 顺时针(方向的)，右旋(的)41.correspondence [,kɔri'spɔndəns] n. 一致;符合，通信,通信联系42.clear-cut ['kliə'kʌt] a. 轮廓清晰的，明确的,清楚的43.portion ['pɔ:ʃən]n.一部分44.actual value 实际值45.desired value 预期值，期待值46.adequately ['ædəkwɪtlɪ] ad.充分地，适当地47.embody [ɪm'bɑdɪ]vt.体现，包含,收录48.concentration [,kɑnsɛn'treʃən]n.浓缩，浓缩物，集中49.humidity [hju'mɪdətɪ]n.湿气,湿度50.viscosity [vɪs'kɑsətɪ] n. 粘质;粘性51.measured value 测量值52.error signal 误差信号53.be classified into 分(类)为...54.drastically ['dræstikəli] ad.彻底地，激烈地55.precisely [prɪ'saɪslɪ] ad. 精确地,准确地56.take a close look at 仔细观察57.amplification [,æmplifi'keiʃən] n. 放大；增强58.damper ['dæmpə] n.气闸；调节风门；减震器59.SCR (Silicon Controlled Rectifier) 晶闸管，可控硅整流器60.triac 三端双向可控硅开关61.Wheatstone bridge 惠斯登电桥,单臂电桥62.thermocouple ['θə:mə'kʌpl]n.热(电)偶,温差电偶63.strain gage 应变仪，电阻丝应变仪64.tachometer [tæ'kɔmitə]n.转速计;流速计65.transducer [træns'dju:sə] n. (电能)变换器;换能器，传感器66.nomenclature ['nəumən,kleitʃə]n.系统命名法，(总称)学术用语;术语表67.unfortunately [ʌn'fɔrtʃənɪtlɪ] ad.不幸地68.universally [,ju:ni'və:səli] ad.通用地,万能地69.deviation [,divɪ'eʃən] n. 误差;偏航，越轨;偏向70.characteristics n.特性、特性曲线71.offset ['ɔf,set] n.偏置，补偿;抵消72.unstable ['ʌn'steibl] a.不稳定的，易变的73.violent ['vaɪələnt] a.猛烈的，极端的74.frantically ['fræntɪklɪ] ad.疯狂地;狂热地75.overreact ['əuvəri'ækt]vi.反应过激76.die out ['daɪ aut] 逐渐消失，灭绝77.settle down ['setl daun]定居；沉淀78.consequence ['kɔnsikwəns] n.结果，后果，影响，重要性79.permanently ['pə:mənəntli]ad.永久地;长期不变地80.destroy [dɪ'strɔɪ] vt.毁坏,破坏81.rack-and-pinion 齿条和齿轮82.fixed pulley 固定滑轮83.movable pulley 动滑轮84.scale [skel]n.刻度;尺度;刻度尺85.potentiometer 电位器,电势计86.instantaneous [,ɪnstən'tenɪəs] a.瞬间的，即刻的，即时的87.profile cutting 成形[定形]切削88.elaborate [ɪ'læbərɪt] a.精心制作的；复杂的89.identical [ai'dentikəl] a.同样的，恒等的90.rigid feeler 硬的触头、探针91.pivot ['pɪvət]n.枢,枢轴;支枢92.transmit [træns'mɪt]vt.传送,传达93.duplicate ['djupləkɪt] a.完全一样的;复制的;副本的94.hard-and-fast 明确的,不可变更的,坚定的95.on-off 开关，离合96.proportional [prə'pɔ:ʃənəl] a.比例的,成比例的97.proportional plus integral(PI) 比例积分98.proportional plus derivative(PD) 比例微分99.proportional plus integral plus derivative(PID) 比例微积分plexity [kəm'plɛksətɪ] 复杂性，错综度101.precise [prɪ'saɪs] a.精确的;准确的;确切的102.adequate ['ædikwit] a.足够的；适当的103.inexpensive [,inik'spensiv] a.花费不多的;价钱低廉的104.reliable [ri'laiəbl] a.可信赖的;可靠的;确实的105.principle ['prinsəpl] n.原理，原则106.involve [in'vɔlv]vt.包含，涉及107.implementation [,implimen'teiʃən] n. 履行;完成108.visualize ['viʒuəlaiz]vt.使看得见;使显现109.bang-bang control开关式控制110.conveniently [kən'vinjəntlɪ]ad.方便地,便利地;合宜地。

普林斯顿算法-回复普林斯顿算法：解密传世之难题【引言】在计算机科学和数学领域，算法是解决问题的步骤和指令的有序集合。

而在这个世界上，有一种特殊的算法备受研究人员的关注，它被称为普林斯顿算法。

普林斯顿算法作为解密传世之难题的独特工具，引起了广泛的讨论和探索。

本文将深入研究普林斯顿算法的原理和应用，并逐步回答关于它的问题。

【正文】【第一部分：普林斯顿算法的基本原理】普林斯顿算法是一种基于分治思想的求解问题的算法。

它的核心思想是将原问题划分为若干个子问题，并通过递归的方式解决这些子问题，最终将它们的解合并得到原问题的解。

普林斯顿算法的基本框架如下：PrincetonAlgorithm(问题P)1. 如果问题P可以直接求解，返回解2. 将问题P划分为若干个子问题P1, P2, ..., Pn3. 递归地解决子问题P1, P2, ..., Pn，得到它们的解S1, S2, ..., Sn4. 将子问题P1, P2, ..., Pn的解S1, S2, ..., Sn合并，得到问题P的解5. 返回问题P的解【第二部分：普林斯顿算法的应用】普林斯顿算法具有广泛的应用领域，特别是在解密传世之难题方面。

传世之难题是指那些被加密保护的信息或者密码，其破解困难度极高。

普林斯顿算法在解密传世之难题时，可以通过将整个加密信息或密码划分为若干个子问题，并通过分别破解子问题来逐步得到整个加密信息或密码。

这种逐步解密的过程，利用了普林斯顿算法的分治思想，可以大大提高破解的效率。

【第三部分：普林斯顿算法的案例分析】为了更好地理解普林斯顿算法的应用，我们以实际案例进行分析。

假设有一个传世之难题，我们得到了一个加密密码，该密码是由两个相同长度的字符串拼接而成。

我们知道密码的总长度是10位，并且已经猜到了其中的一部分字符串。

现在的问题是如何通过普林斯顿算法来破解剩下的密码。

首先，我们将问题划分为两个子问题：猜测字符串的前5位和猜测字符串的后5位。

parallel structure例句Parallel Structure例句Parallel structure是英语写作中非常重要的一个概念，通常被用来表达相似的意思或者一系列并列的想法。

实际上，这个概念在我们的日常生活和工作中也非常常见，比如我们写邮件、制定工作计划、撰写报告等等。

在这篇文章中，我们将探讨parallel structure的用途和相关的例句。

1. 并列事件Parallel structure通常被用来表达一系列并列的事件或行动。

这时，我们需要使用相同的词形来描述不同的行动。

例如：- I like to swim, to run, and to bicycle in my free time.- The boss wants us to work hard, to be punctual, and to be honest in our dealings with clients.在这些句子中，我们使用了to + 动词的形式，以表达并列的行动。

2. 并列的比较和对照Parallel structure也可以用于比较和对照。

在这种情况下，我们也需要使用相同的词形来描述不同的对象。

例如：- The coffee was hot, strong, and bitter, just the way I like it.- The car was fast, sleek, and powerful, but it was also expensive.在这些句子中，我们使用了修饰词的形式，以表述对象的特点和特征。

3. 并列的动词短语Parallel structure还可以用来表达一系列并列的动词短语。

这时，我们需要使用相同的词形来描述不同的动作。

例如：- I need to make an appointment, to finish my paper, and todo some grocery shopping before the end of the day.- He decided to quit his job, to travel the world, and tolive life on his own terms.在这些句子中，我们使用了动词短语的形式，以表达并列的动作。