算法导论参考答案
算法导论课程作业答案
算法导论课程作业答案Introduction to AlgorithmsMassachusetts Institute of Technology 6.046J/18.410J Singapore-MIT Alliance SMA5503 Professors Erik Demaine,Lee Wee Sun,and Charles E.Leiserson Handout10Diagnostic Test SolutionsProblem1Consider the following pseudocode:R OUTINE(n)1if n=12then return13else return n+R OUTINE(n?1)(a)Give a one-sentence description of what R OUTINE(n)does.(Remember,don’t guess.) Solution:The routine gives the sum from1to n.(b)Give a precondition for the routine to work correctly.Solution:The value n must be greater than0;otherwise,the routine loops forever.(c)Give a one-sentence description of a faster implementation of the same routine. Solution:Return the value n(n+1)/2.Problem2Give a short(1–2-sentence)description of each of the following data structures:(a)FIFO queueSolution:A dynamic set where the element removed is always the one that has been in the set for the longest time.(b)Priority queueSolution:A dynamic set where each element has anassociated priority value.The element removed is the element with the highest(or lowest)priority.(c)Hash tableSolution:A dynamic set where the location of an element is computed using a function of the ele ment’s key.Problem3UsingΘ-notation,describe the worst-case running time of the best algorithm that you know for each of the following:(a)Finding an element in a sorted array.Solution:Θ(log n)(b)Finding an element in a sorted linked-list.Solution:Θ(n)(c)Inserting an element in a sorted array,once the position is found.Solution:Θ(n)(d)Inserting an element in a sorted linked-list,once the position is found.Solution:Θ(1)Problem4Describe an algorithm that locates the?rst occurrence of the largest element in a?nite list of integers,where the integers are not necessarily distinct.What is the worst-case running time of your algorithm?Solution:Idea is as follows:go through list,keeping track of the largest element found so far and its index.Update whenever necessary.Running time isΘ(n).Problem5How does the height h of a balanced binary search tree relate to the number of nodes n in the tree? Solution:h=O(lg n) Problem 6Does an undirected graph with 5vertices,each of degree 3,exist?If so,draw such a graph.If not,explain why no such graph exists.Solution:No such graph exists by the Handshaking Lemma.Every edge adds 2to the sum of the degrees.Consequently,the sum of the degrees must be even.Problem 7It is known that if a solution to Problem A exists,then a solution to Problem B exists also.(a)Professor Goldbach has just produced a 1,000-page proof that Problem A is unsolvable.If his proof turns out to be valid,can we conclude that Problem B is also unsolvable?Answer yes or no (or don’t know).Solution:No(b)Professor Wiles has just produced a 10,000-page proof that Problem B is unsolvable.If the proof turns out to be valid,can we conclude that problem A is unsolvable as well?Answer yes or no (or don’t know).Solution:YesProblem 8Consider the following statement:If 5points are placed anywhere on or inside a unit square,then there must exist two that are no more than √2/2units apart.Here are two attempts to prove this statement.Proof (a):Place 4of the points on the vertices of the square;that way they are maximally sepa-rated from one another.The 5th point must then lie within √2/2units of one of the other points,since the furthest from the corners it can be is the center,which is exactly √2/2units fromeach of the four corners.Proof (b):Partition the square into 4squares,each with a side of 1/2unit.If any two points areon or inside one of these smaller squares,the distance between these two points will be at most √2/2units.Since there are 5points and only 4squares,at least two points must fall on or inside one of the smaller squares,giving a set of points that are no more than √2/2apart.Which of the proofs are correct:(a),(b),both,or neither (or don’t know)?Solution:(b)onlyProblem9Give an inductive proof of the following statement:For every natural number n>3,we have n!>2n.Solution:Base case:True for n=4.Inductive step:Assume n!>2n.Then,multiplying both sides by(n+1),we get(n+1)n!> (n+1)2n>2?2n=2n+1.Problem10We want to line up6out of10children.Which of the following expresses the number of possible line-ups?(Circle the right answer.)(a)10!/6!(b)10!/4!(c) 106(d) 104 ·6!(e)None of the above(f)Don’t knowSolution:(b),(d)are both correctProblem11A deck of52cards is shuf?ed thoroughly.What is the probability that the4aces are all next to each other?(Circle theright answer.)(a)4!49!/52!(b)1/52!(c)4!/52!(d)4!48!/52!(e)None of the above(f)Don’t knowSolution:(a)Problem12The weather forecaster says that the probability of rain on Saturday is25%and that the probability of rain on Sunday is25%.Consider the following statement:The probability of rain during the weekend is50%.Which of the following best describes the validity of this statement?(a)If the two events(rain on Sat/rain on Sun)are independent,then we can add up the twoprobabilities,and the statement is true.Without independence,we can’t tell.(b)True,whether the two events are independent or not.(c)If the events are independent,the statement is false,because the the probability of no rainduring the weekend is9/16.If they are not independent,we can’t tell.(d)False,no matter what.(e)None of the above.(f)Don’t know.Solution:(c)Problem13A player throws darts at a target.On each trial,independentlyof the other trials,he hits the bull’s-eye with probability1/4.How many times should he throw so that his probability is75%of hitting the bull’s-eye at least once?(a)3(b)4(c)5(d)75%can’t be achieved.(e)Don’t know.Solution:(c),assuming that we want the probability to be≥0.75,not necessarily exactly0.75.Problem14Let X be an indicator random variable.Which of the following statements are true?(Circle all that apply.)(a)Pr{X=0}=Pr{X=1}=1/2(b)Pr{X=1}=E[X](c)E[X]=E[X2](d)E[X]=(E[X])2Solution:(b)and(c)only。
算法导论15-16-答案.pdf
j
j
w(i, j) = ∑ pl + ∑ ql
l =i
l =i −1
(5-5)
OPTIMAL-BST(p, q, n)
1 for i ← 1 to n + 1
2
do e[i, i-1] ←qi-1
3
w[i, i-1] ←qi-1
4 for l ← 1 to n
5
do for i ← 1 to n – l + 1
15-4 (5-19) Planning a company party. Professor Stewart is consulting for the president of a corporation that is planning a company party. The company has a hierarchical structure; that is, the supervisor relation forms tree rooted at the president. The personal office has ranked each employee with a conviviality rating, which is a real number. In order to make the party fun for all attendees, the president does not want both an employee and his or her immediate supervisor to attend.
To print out all the stations, call PRINT-STATIONS (l, l*, n)
算法导论第三版第二章第一节习题答案
算法导论第三版第⼆章第⼀节习题答案2.1-1：以图2-2为模型，说明INSERTION-SORT在数组A=<31,41,59,26,41,58>上的执⾏过程。
NewImage2.1-2：重写过程INSERTION-SORT，使之按⾮升序（⽽不是按⾮降序）排序。
注意，跟之前升序的写法只有⼀个地⽅不⼀样：NewImage2.1-3：考虑下⾯的查找问题：输⼊：⼀列数A=<a1,a2,…,an >和⼀个值v输出：下标i，使得v=A[i]，或者当v不在A中出现时为NIL。
写出针对这个问题的现⾏查找的伪代码，它顺序地扫描整个序列以查找v。
利⽤循环不变式证明算法的正确性。
确保所给出的循环不变式满⾜三个必要的性质。
（2.1-3 Consider the searching problem:Input: A sequence of n numbers A D ha1; a2; : : : ;ani and a value _.Output: An index i such that _ D AOEi_ or the special value NIL if _ does not appear in A.Write pseudocode for linear search, which scans through the sequence, looking for _. Using a loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulfills the three necessary properties.）LINEAR-SEARCH(A,v)1 for i=1 to A.length2 if v = A[i]3 return i4 return NIL现⾏查找算法正确性的证明。
算法导论习题答案 (5)
Three-hole punch your paper on submissions. You will often be called upon to “give an algorithm” to solve a certain problem. Your write-up should take the form of a short essay. A topic paragraph should summarize the problem you are solving and what your results are. The body of the essay should provide the following:
(a) Argue that this problem exhibits optimal substructure.
Solution: First, notice that linecost(i, j) is defined to be � if the words i through j do not fit on a line to guarantee that no lines in the optimal solution overflow. (This relies on the assumption that the length of each word is not more than M .) Second, notice that linecost(i, j) is defined to be 0 when j = n, where n is the total number of words; only the actual last line has zero cost, not the recursive last lines of subprob lems, which, since they are not the last line overall, have the same cost formula as any other line.
《算法导论(第二版)》(中文版)课后答案
5
《算法导论（第二版） 》参考答案 do z←y 调用之前保存结果 y←INTERVAL-SEARCH-SUBTREE(y, i) 如果循环是由于y没有左子树，那我们返回y 否则我们返回z，这时意味着没有在z的左子树找到重叠区间 7 if y≠ nil[T] and i overlap int[y] 8 then return y 9 else return z 5 6 15.1-5 由 FASTEST-WAY 算法知：
15
lg n
2 lg n1 1 2cn 2 cn (n 2 ) 2 1
4.3-1 a) n2 b) n2lgn c) n3 4.3-4
2
《算法导论（第二版） 》参考答案 n2lg2n 7.1-2 (1)使用 P146 的 PARTION 函数可以得到 q=r 注意每循环一次 i 加 1，i 的初始值为 p 1 ，循环总共运行 (r 1) p 1次，最 终返回的 i 1 p 1 (r 1) p 1 1 r (2)由题目要求 q=(p+r)/2 可知，PARTITION 函数中的 i，j 变量应该在循环中同 时变化。 Partition(A, p, r) x = A[p]; i = p - 1; j = r + 1; while (TRUE) repeat j--; until A[j] <= x; repeat i++; until A[i] >= x; if (i < j) Swap(A, i, j); else return j; 7.3-2 （1）由 QuickSort 算法最坏情况分析得知：n 个元素每次都划 n-1 和 1 个，因 为是 p<r 的时候才调用，所以为Θ （n） （2）最好情况是每次都在最中间的位置分，所以递推式是： N(n)= 1+ 2*N(n/2) 不难得到：N(n) =Θ （n） 7.4-2 T(n)=2*T(n/2)+ Θ （n） 可以得到 T(n) =Θ （n lgn） 由 P46 Theorem3.1 可得：Ω (n lgn)
中科大算法导论第一,二次和第四次作业答案
2.2-3 再次考虑线性查找问题 （见练习2.1-3）。在平均情况 下，需要检查输入序列中的多 少个元素？假定待查找的元素 是数组中任何一个元素的可能 性是相等的。在最坏情况下有 怎样呢？用Θ形式表示的话，线 性查找的平均情况和最坏情况 运行时间怎样？对你的答案加 以说明。 • 线性查找问题 • 输入：一列数A=<a1,a2,…,an>和一 个值v。 • 输出：下标i，使得v=A[i]，或者当 v不在A中出现时为NIL。 • 平均情况下需要查找 (1+2+…+n)/n=(n+1)/2 • 最坏情况下即最后一个元素为待 查找元素，需要查找n个。 • 故平均情况和最坏情况的运行时 间都为Θ(n)。
• 2.3-2改写MERGE过程，使之不使 用哨兵元素，而是在一旦数组L或R 中的所有元素都被复制回数组A后， 就立即停止，再将另一个数组中 余下的元素复制回数组A中。 • MERGE(A,p,q,r) 1. n1←q-p+1 2. n2 ←r-q 3. create arrays L[1..n1] and R[1..n2] 4. for i ←1 to n1 5. do L*i+ ←A*p+i-1] 6. for j ←1 to n2 7. do R*j+ ←A*q+j+ 8. i ←1 9. j ←1
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
k ←p while((i<=n1) and (j<=n2)) do if L[i]<=R[j] do A[k]=L[i] i++ else do A[k]=R[j] j++ k++ while(i<=n1) do A[k++]=L[i++] while(j<=n2) do A[k++]=R[j++]
算法导论第九章习题答案（第三版）IntroductiontoAlgorithm
算法导论第九章习题答案（第三版）IntroductiontoAlgorithm Exercise
9.1-1
对所有的元素，两个⼀组进⾏⽐较，共需n-1次⽐较，可以构成⼀棵⼆叉树，最⼩的元素在树的根结点上，接下来，画出⼆叉树，可以很容易的看出共需lgn-1次⽐较，所以共需n+lgn-2次⽐较才可以找出第⼆⼩的元素。
9.1-2
略。
9.2-1
在randomized-select中，对于长度为0的数组，此时p=r，直接返回A[p],所以不会进⾏递归调⽤。
9.2-2
略。
9.2-3
RANDOMIZED-SELECT(A,p,r,i){
while(true){
if(p==r)
return A[p];
q=RANDOMIZED-PARTITION(A,p,r);
k=q-p+1;
if(i==k)
return A[q];
else if(i<k)
q--;
else{
q++;
i-=k;
}
}
}
9.2-4
每次都以最⼤的元素进⾏划分即可。
9.3-1
数学计算，根据书中例题仿照分析即可。
9.3-3
随机化
9.3-5
类似主元划分，只要把⿊箱⼦输出的值作为主元划分去选择即可。
9.3-6
多重⼆分即可。
9.3-7
算出中位数，之后算出每⼀个数与中位数的差即可。
9.3-8
分别取两个数组的中位数进⾏⽐较，如果两个中位数相等，那么即为所求，否则，取中位数较⼩的⼀个的右边，取较⼤的⼀个的右边，直到就剩4个元素为⽌，这时候只要求这4个元素的中位数即可。
算法导论 第八章答案
{ if(m >n)
cout<<"区间输入不对"<<endl; else { if(n <0) else if(m <=0&& n <= k) else if(n > k && m >0) else if(n > k && m <=0) else
cout<<"个数为"<<0<<endl; cout<<"个数为"<<c[n]<<endl; cout<<"个数为"<<c[k] - c[m -1]<<endl;
cout<<"个数为"<<c[k]<<endl; cout<<"个数为"<<c[n] - c[m -1]<<endl;
}}
return 0;
}
void counting_sort(int*&a, int length, int k, int*&b, int*&c)
{ for(int i =0; i < k +1; i++)
const int k =2504; int* c =new int[k +1];
counting_sort(a, LEN, k, b, c); //这里需要注释掉
//for(int i = 0; i < LEN; i++)
// cout<<b[i]<<endl;