算法分析与设计-第10讲

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五年级上册信息技术浙教版第10课问题的分解(教案)

五年级上册第3单元第10课《问题的分解》教学设计【教材分析】本节学习内容选自浙江教育出版社小学信息技术教材五年级上册第三单元《用算法解决问题》中的第 10 课《问题的分解》，帮助学生了解在解决实际问题时要经历分析问题、抽象与建模、算法设计、验证与优化几个步骤，本课以帮助了解分析问题，引出问题分解的学习需求，知道分解问题三个步骤，体验问题界定、分解问题、确定关键问题的过程。

培养学生分析问题的能力发展计算思维。

【学情分析】五年级的学生具有一定的逻辑思维能力和创新能力，在知识储备上，在五年级第一单元当中学习过算法与算法表示，知道了算法的特征，会绘制简单的流程图；在第二单元中学习了算法的控制结构。

因此，本节课的学习从学生学情出发，从生活中的实例和实际问题入手，基于项目化的学习，提升学生的创新创造能力。

通过让学生去分析问题分解的场景，理解问题分解的含义。

结合自己的生活实际经验与自己的知识基础,经历问题界定分解问题、确定关键问题的过程进一步掌握问题分解的方法。

【教学目标】1．理解问题界定的含义，初步体验问题界定的过程。

2．了解问题分解的两种方法，能根据实际生活经验与知识基础尝试对生活实际问题进行规模分解与行动分解。

3．体验将大问题分解成子问题，确定关键问题后进--步经历问题分解的过程，感受问题分解。

4．通过分析问题，在经历问题解决的过程中，提升学生的计算思维能力。

【教学重点】理解分解问题的两种方法。

在问题解决的过程中，能将问题分解为可处理的子问题。

【教学难点】对问题进行方向与边界的界定。

如何确定关键问题，并对关键问题进行进一步分解。

【教学准备】课件、学习单，机器人小车。

算法分析与设计

他把1,2,3,...16 这16个数字填写在4 x 4的方格中。 16 ? ? 13 ? ? 11 ? 9??* ? 15 ? 1
表中有些数字已经显露出来，还有些用?和*代替。请你计算出? 和 * 所代表的数字。并把 * 所代表的数字作为本题答案提交。
素数环问题
素数环是一个计算机程序问题，指的是将从1到n这n个整数围成一个圆环，若其中任意2个相邻的数字相加，结果均为素数，那么这个环就成为素数环。现在要求输入一个n，求n个数围成一圈有多少种素数环，规定第一个数字是1。 143256 165234
例如当n=5，m=4时，面值为1，3，11，15，32的5种邮票可以贴出邮资的最大连续区间是1到70。
➢ 通用的解题法 ➢ 核心在于构造解空间树：
➢ 子集树 ➢ 排列树 ➢ 回溯法是优化的暴力搜索： ➢ 不满足限制条件； ➢ 当前解与最优解进行预计算； ➢ 学习回溯法：心中有树
回溯法
总结
➢ 动态规划适合两个连续步骤之间有联系的问题； ➢ 回溯法几乎适用于所有的问题，但问题之间最好有明确的层次。
总结
➢ 构造心中的解空间树是关键； ➢ 回溯法与函数的局部变量； ➢ 访问解空间树的优化处理；
迷宫问题中的回溯法
➢ 四邻域 ➢ 八邻域
图论问题
无向图： ➢ 连通 ➢ 不连通
有向图： ➢ 弱连通 ➢ 单向连通 ➢ 强连通
最大团问题
连通子图（分支）
最大团问题
给定无向图G=(V,E)，如果UV，且对任意的u,vU，都有(u,v)E，则称U是G的完全子图。G的完全子图U是G 的一个团当且仅当U不包含在G的更大的完全子图中。G中的最大团是指G中所含顶点数最多的团。
yes no yes
➢ 通用的解题法 ➢ 核心在于构造解空间树：

算法分析与设计重点课后习题答案

习题13．设计算法求数组中相差最小的两个元素（称为最接近数）的差。

要求分别给出伪代码和C++描述。

//采用分治法//对数组先进行快速排序//在依次比较相邻的差#include <iostream>using namespace std;int partions(int b[],int low,int high){int prvotkey=b[low];b[0]=b[low];while (low<high){while (low<high&&b[high]>=prvotkey)--high;b[low]=b[high];while (low<high&&b[low]<=prvotkey)++low;b[high]=b[low];}b[low]=b[0];return low;}void qsort(int l[],int low,int high){int prvotloc;if(low<high){prvotloc=partions(l,low,high); //将第一次排序的结果作为枢轴qsort(l,low,prvotloc-1); //递归调用排序由low 到prvotloc-1qsort(l,prvotloc+1,high); //递归调用排序由 prvotloc+1到 high}}void quicksort(int l[],int n){qsort(l,1,n); //第一个作为枢轴，从第一个排到第n个}int main(){int a[11]={0,2,32,43,23,45,36,57,14,27,39};int value=0;//将最小差的值赋值给valuefor (int b=1;b<11;b++)cout<<a[b]<<' ';cout<<endl;quicksort(a,11);for(int i=0;i!=9;++i){if( (a[i+1]-a[i])<=(a[i+2]-a[i+1]) )value=a[i+1]-a[i];elsevalue=a[i+2]-a[i+1];}cout<<value<<endl;return 0;}4．设数组a[n]中的元素均不相等，设计算法找出a[n]中一个既不是最大也不是最小的元素，并说明最坏情况下的比较次数。

算法分析与设计概论

9
How to Study Algorithm？
“Sometimes we have experiences, and sometimes not. Therefore, the better way is to learn more."
10
1.1 算法与程序
算法：是满足下述性质的指令序列。
输入：有零个或多个外部量作为算法的输入。输出：算法产生至少一个量作为输出。确定性：组成算法的每条指令清晰、无歧义。有限性：算法中每条指令的执行次数有限，执行每条指令的时间也有限。
1）第一种解法：
输入：所购买的三种鸡的总数目n 输出：满足问题的解的数目k,公鸡,母鸡,小鸡的只数g[ ],m[ ],s[ ] 1. void chicken_question(int n,int &k,int g[ ],int m[ ],int s[ ]) 2. { int a,b,c; 4. k = 0; 5. for (a=0;a<=n;a++) 6. for (b=0;b<=n;b++) 7. for (c=0;c<=n;c++) { 8. if ((a+b+c==n)&&(5*a+3*b+c/3==n)&&(c%3==0)) { 9. g[k] = a; 10. m[k] = b; 11. s[k] = c; 12. k++; 13. }}}
矩阵。
数组 T：表示售货员的路线，依次存放旅行路线中的城市编号。
售货员的每一条路线，对应于城市编号的一个排列。
n 个城市共有 n! 个排列，采用穷举法逐一计算每一条路线的费用，从中找出费用最小的路线，便可求出问题的解。

《算法设计与分析》课件

常见的贪心算法包括最小生成树算法、Prim算法、Dijkstra算法和拓扑排序等。
贪心算法的时间复杂度和空间复杂度通常都比较优秀，但在某些情况下可能需要额外的空间来保存状态。
动态规划
常见的动态规划算法包括斐波那契数列、背包问题、最长公共子序列和矩阵链乘法等。
动态规划的时间复杂度和空间复杂度通常较高，但通过优化状态转移方程和状态空间可以显著提高效率。
动态规划算法的时间和空间复杂度分析
动态规划算法的时间复杂度通常为O(n^2)，空间复杂度为O(n)。
04 经典问题与算法实现
排序问题
冒泡排序
通过重复地遍历待排序序列，比较相邻元素的大小，交换位置，使得较大的元素逐渐往后移动，最终达到排序的目的。
快速排序
采用分治策略，选取一个基准元素，将比基准元素小的元素移到其左边，比基准元素大的元素移到其右边，然后对左右两边的子序列递归进行此操作。
动态规划是一种通过将原问题分解为若干个子问题，并从子问题的最优解推导出原问题的最优解的算法设计方法。
动态规划的关键在于状态转移方程的建立和状态空间的优化，以减少不必要的重复计算。
回溯算法
01
回溯算法是一种通过穷举所有可能情况来求解问题的算法设计方法。
02
常见的回溯算法包括排列组合、八皇后问题和图的着色问题等。
空间换时间分治策略贪心算法动态规划
通过增加存储空间来减少计算时间，例如使用哈希表解决查找问题。
将问题分解为若干个子问题，递归地解决子问题，最终合并子问题的解以得到原问题的解。
在每一步选择中都采取当前状态下最好或最优（即最有利）的选择，从而希望导致结果是最好或最优的。
通过将问题分解为相互重叠的子问题，并保存子问题的解，避免重复计算，提高算法效率。

智慧树知道网课《算法分析与设计(山东联盟)》课后章节测试满分答案1

第一章测试1【判断题】(10分)一个问题的同一实例可以有不同的表示形式A.错B.对2【判断题】(10分)同一数学模型使用不同的数据结构会有不同的算法，有效性有很大差别。

A.错B.对3【判断题】(10分)问题的两个要素是输入和实例。

A.对B.错4【单选题】(10分)算法与程序的区别是（）A.有穷性B.确定性C.输出D.输入5【单选题】(10分)解决问题的基本步骤是（）。

（1）算法设计（2）算法实现（3）数学建模（4）算法分析（5）正确性证明A.(3)(1)(5)(4)(2)B.(3)(4)(1)(5)(2)C.(1)(2)(3)(4)(5)D.(3)(1)(4)(5)(2)6【单选题】(10分)下面说法关于算法与问题的说法的是（）。

A.算法是一种计算方法，对问题的每个实例计算都能得到正确答案。

B.证明算法不正确，需要证明对任意实例算法都不能正确处理。

C.如果一个算法能应用于问题的任意实例，并保证得到正确解答，称这个算法解答了该问题。

D.同一问题可能有几种不同的算法，解题思路和解题速度也会显著不同。

7【多选题】(10分)下面关于程序和算法的说法正确的是（）。

A.算法的每一步骤必须要有确切的含义，必须是清楚的、无二义的。

B.程序总是在有穷步的运算后终止。

C.程序是算法用某种程序设计语言的具体实现。

D.算法是一个过程，计算机每次求解是针对问题的一个实例求解。

8【多选题】(10分)最大独立集问题和（）问题等价。

A.最大团B.稳定匹配问题C.区间调度问题D.最小顶点覆盖9【多选题】(10分)给定两张喜欢列表，稳定匹配问题的输出是（）。

A.完美匹配B.最大匹配C.稳定匹配D.没有不稳定配对10【单选题】(10分)问题变换的目的有（）。

(1)复杂变简单(2)未知变已知(3)隐式变显式(4)难解变易解（5）以上都是。

A.（5）B.（1）C.（2）D.（3）E.（4）11【单选题】(10分)按照霍纳法则，计算p(x)=a n x n+a n-1x n-1+…+a1x1+a0的数量级为____。

算法分析与设计第二版英文版 (潘彦著) 清华大学出版社课后答案--solu9

This ﬁle contains the exercises,hints,and solutions for Chapter 9of the book ”Introduction to the Design and Analysis of Algorithms,”2nd edition,byA.Levitin.The problems that might be challenging for at least some students are marked by ;those that might be diﬃcult for a majority of students are marked by .Exercises 9.11.Give an instance of the change-making problem for which the greedy al-gorithm does not yield an optimal solution.2.Write a pseudocode of the greedy algorithm for the change-making prob-lem,with an amount n and coin denominations d 1>d 2>...>d m as its input.What is the time eﬃciency class of your algorithm?3.Consider the problem of scheduling n jobs of known durations t 1,...,t n for execution by a single processor.The jobs can be executed in any order,one job at a time.You want to ﬁnd a schedule that minimizes the total time spent by all the jobs in the system.(The time spent by one job in the system is the sum of the time spent by this job in waiting plus the time spent on its execution.)Design a greedy algorithm for this problem. Does the greedy algo-rithm always yield an optimal solution?4.Design a greedy algorithm for the assignment problem (see Section 3.4).Does your greedy algorithm always yield an optimal solution?5.Bridge crossing revisited Consider the generalization of the bridge cross-ing puzzle (Problem 2in Exercises 1.2)in which we have n >1people whose bridge crossing times are t 1,t 2,...,t n .All the other conditions of the problem remain the same:at most two people at the time can cross the bridge (and they move with the speed of the slower of the two)and they must carry with them the only ﬂashlight the group has.Design a greedy algorithm for this problem and ﬁnd how long it willtake to cross the bridge by using this algorithm.Does your algorithm yields a minimum crossing time for every instance of the problem?If it does–prove it,if it does not–ﬁnd an instance with the smallest number of people for which this happens.6.Bachet-Fibonacci weighing problem Find an optimal set of n weights {w 1,w 2,...,w n }so that it would be possible to weigh on a balance scale any integer load in the largest possible range from 1to W ,provided a. weights can be put only on the free cup of the scale.b. weights can be put on both cups of the scale.1课后答案网 w w w .k h d a w .c o m7.a.Apply Prim’s algorithm to the following graph.Include in the priority queue all the vertices not already in the tree.b.Apply Prim’s algorithm to the following graph.Include in the priority queue only the fringe vertices (the vertices not in the current tree which are adjacent to at least one tree vertex).8.The notion of a minimum spanning tree is applicable to a connected weighted graph.Do we have to check a graph’s connectivity before ap-plying Prim’s algorithm or can the algorithm do it by itself?9.a.How can we use Prim’s algorithm to ﬁnd a spanning tree of a connected graph with no weights on its edges?b.Is it a good algorithm for this problem?10. Prove that any weighted connected graph with distinct weights hasexactly one minimum spanning tree.11.Outline an eﬃcient algorithm for changing an element’s value in a min-heap.What is the time eﬃciency of your algorithm?2课后答案网 w h d a w .c o mHints to Exercises 9.11.As coin denominations for your counterexample,you may use,among a multitude of other possibilities,the ones mentioned in the text:d 1=7,d 2=5,d 3=1.2.You may use integer divisions in your algorithm.3.Considering the case of two jobs might help.Of course,after forming a hypothesis,you will have to either prove the algorithm’s optimality for an arbitrary input or ﬁnd a speciﬁc counterexample showing that it is not the case.4.You can apply the greedy approach either to the entire cost matrix or to each of its rows (or columns).5.Simply apply the greedy approach to the situation at hand.You may assume that t 1≤t 2≤...≤t n .6.For both versions of the problem,it is not diﬃcult to get to a hypothesis about the solution’s form after considering the cases of n =1,2,and 3.It is proving the solutions’optimality that is at the heart of this problem.7.a.Trace the algorithm for the graph given.An example can be found in the text of the section.b.After the next fringe vertex is added to the tree,add all the unseen vertices adjacent to it to the priority queue of fringe vertices.8.Applying Prim’s algorithm to a weighted graph that is not connected should help in answering this question.9.a.Since Prim’s algorithm needs weights on a graph’s edges,some weights have to be assigned.b.Do you know other algorithms that can solve this problem?10.Strictly speaking,the wording of the question asks you to prove two things:the fact that at least one minimum spanning tree exists for any weighted connected graph and the fact that a minimum spanning tree is unique if all the weights are distinct numbers.The proof of the former stems from the obvious observation about ﬁniteness of the number of spanning trees for a weighted connected graph.The proof of the latter can be obtained by repeating the correctness proof of Prim’s algorithm with a minor adjustment at the end.11.Consider two cases:the key’s value was decreased (this is the case needed for Prim’s algorithm)and the key’s value was increased.3课后答案网 w w w .k h d a w .c o mSolutions to Exercises 9.11.Here is one of many such instances:For the coin denominations d 1=7,d 2=5,d 3=1and the amount n =10,the greedy algorithm yields one coin of denomination 7and three coins of denomination 1.The actual optimal solution is two coins of denomination 5.2.Algorithm Change (n,D [1..m ])//Implements the greedy algorithm for the change-making problem //Input:A nonnegative integer amount n and//a decreasing array of coin denominations D//Output:Array C [1..m ]of the number of coins of each denomination //in the change or the ”no solution”messagefor i ←1to m doC [i ]← n/D [i ]n ←n mod D [i ]if n =0return Celse return ”no solution”The algorithm’s time eﬃciency is in Θ(m ).(We assume that integer di-visions take a constant time no matter how big dividends are.)Note also that if we stop the algorithm as soon as the remaining amount becomes 0,the time eﬃciency will be in O (m ).3.a.Sort the jobs in nondecreasing order of their execution times and exe-cute them in that order.b.Yes,this greedy algorithm always yields an optimal solution.Indeed,for any ordering (i.e.,permutation)of the jobs i 1,i 2,...,i n ,the total time in the system is given by the formula t i 1+(t i 1+t i 2)+...+(t i 1+t i 2+...+t i n )=nt i 1+(n −1)t i 2+...+t i n .Thus,we have a sum of numbers n,n −1,...,1multiplied by “weights”t 1,t 2,...t n assigned to the numbers in some order.To minimize such a sum,we have to assign smaller t ’s to larger numbers.In other words,the jobs should be executed in nondecreasing order of their execution times.Here is a more formal proof of this fact.We will show that if jobs are ex-ecuted in some order i 1,i 2,...,i n ,in which t i k >t i k +1for some k,then the total time in the system for such an ordering can be decreased.(Hence,no such ordering can be an optimal solution.)Let us consider the other job ordering,which is obtained by swapping the jobs k and k +1.Obvi-ously,the time in the systems will remain the same for all but these two 4课后答案网 w w w .k h d a w .c o mjobs.Therefore,the diﬀerence between the total time in the system for the new ordering and the one before the swap will be[(k −1j =1t i j +t i k +1)+(k −1j =1t i j +t i k +1+t i k )]−[(k −1j =1t i j +t i k )+(k −1j =1t i j +t i k +t i k +1)]=t i k +1−t i k <0.4.a.The all-matrix version:Repeat the following operation n times.Select the smallest element in the unmarked rows and columns of the cost matrix and then mark its row and column.The row-by-row version:Starting with the ﬁrst row and ending with the last row of the cost matrix,select the smallest element in that row which is not in a previously marked column.After such an element is selected,mark its column to prevent selecting another element from the same col-umn.b.Neither of the versions always yields an optimal solution.Here isa simple counterexample:C = 122100 5.Repeat the following step n −2times:Send to the other side the pair of two fastest remaining persons and then return the ﬂashlight with the fastest person.Finally,send the remaining two people together.Assuming that t 1≤t 2≤...≤t n ,the total crossing time will be equal to (t 2+t 1)+(t 3+t 1)+...+(t n −1+t 1)+t n =ni =2t i +(n −2)t 1=n i =1t i +(n −3)t 1.Note:For an algorithm that always yields a minimal crossing time,seeGünter Rote,“Crossing the Bridge at Night,”EATCS Bulletin,vol.78(October 2002),241—246.The solution to the instance of Problem 2in Exercises 1.2shows that the greedy algorithm doesn’t always yield the minimal crossing time for n >3.No smaller counterexample can be given as a simple exhaustive check for n =3demonstrates.(The obvious solution for n =2is the one generated by the greedy algorithm as well.)5课后答案网 w w w .kh d a w .c o m6.a.Let’s apply the greedy approach to the ﬁrst few instances of the problem in question.For n =1,we have to use w 1=1to balance weight 1.For n =2,we simply add w 2=2to balance the ﬁrst previously unattainable weight of 2.The weights {1,2}can balance every integral weights up to their sum 3.For n =3,in the spirit of greedy thinking,we take the next previously unattainable weight:w 3=4.The three weights {1,2,4}allow to weigh any integral load l between 1and their sum 7,with l ’s binary expansion indicating the weights needed for load l :Generalizing these observations,we should hypothesize that for any posi-tive integer n the set of consecutive powers of 2{w i =2i −1,i =1,2,...n }makes it possible to balance every integral load in the largest possible range,which is up to and including n i =12i −1=2n −1.The fact that every integral weight l in the range 1≤l ≤2n −1can be balanced with this set of weights follows immediately from the binary expansion of l,which yields the weights needed for weighing l.(Note that we can obtain the weights needed for a given load l by applying to it the greedy algorithm for the change-making problem with denominations d i =2i −1,i =1,2,...n.)In order to prove that no set of n weights can cover a larger range of consecutive integral loads,it will suﬃce to note that there are just 2n −1nonempty selections of n weights and,hence,no more than 2n −1sums they yield.Therefore,the largest range of consecutive integral loads they can cover cannot exceed 2n −1.[Alternatively,to prove that no set of n weights can cover a larger range of consecutive integral loads,we can prove by induction on i that if any mul-tiset of n weights {w i ,i =1,...,n }–which we can assume without loss of generality to be sorted in nondecreasing order–can balance every integral load starting with 1,then w i ≤2i −1for i =1,2,...,n.The basis checks out immediately:w 1must be 1,which is equal to 21−1.For the general case,assume that w k ≤2k −1for every 1≤k <i.The largest weight the ﬁrst i −1weights can balance is i −1k =1w k ≤ i −1k =12k −1=2i −1−1.If w i were larger than 2i ,then this load could have been balanced neither with the ﬁrst i −1weights (which are too light even taken together)nor with the weights w i ≤...≤w n (which are heavier than 2i even individ-ually).Hence,w i ≤2i −1,which completes the proof by induction.This immediately implies that no n weights can balance every integral load up to the upper limit larger than n i =1w i ≤ n i =12i −1=2n −1,the limit attainable with the consecutive powers of 2weights.]b.If weights can be put on both cups of the scale,then a larger range can 6课后答案网 w w w .k h d a w .be reached with n weights for n >1.(For n =1,the single weight still needs to be 1,of course.)The weights {1,3}enable weighing of every integral load up to 4;the weights {1,3,9}enable weighing of every inte-gral load up to 13,and,in general,the weights {w i =3i −1,i =1,2,...,n }enable weighing of every integral load up to and including their sum of n i =13i −1=(3n −1)/2.A load’s expansion in the ternary system indicates the weights needed.If the ternary expansion contains only 0’s and 1’s,the load requires putting the weights corresponding to the 1’s on the opposite cup of the balance.If the ternary expansion of load l,l ≤(3n −1)/2,contains one or more 2’s,we can replace each 2by (3-1)to represent it in the form l =n i =1βi 3i −1,where βi ∈{0,1,−1},n = log 3(l +1) .In fact,every positive integer can be uniquely represented in this form,obtained from its ternary expansion as described above.For example,5=123=1·31+2·30=1·31+(3−1)·30=2·31−1·30=(3−1)·31−1·30=1·32−1·31−1·30.(Note that if we start with the rightmost 2,after a simpliﬁcation,the new rightmost 2,if any,will be at some position to the left of the starting one.This proves that after a ﬁnite number of such replacements,we will be able to eliminate all the 2’s.)Using the representation l = n i =1βi 3i −1,we can weigh load l by placing all the weights w i =3i −1for negative βi ’s along with the load on one cup of the scale and all the weights w i =3i −1for positive βi ’s on the opposite cup.Now we’ll prove that no set of n weights can cover a larger range of con-secutive integral loads than (3n −1)/2.Each of the n weights can be either put on the left cup of the scale,or put on the right cup,or not to be used at all.Hence,there are 3n −1possible arrangements of the weights on the scale,with each of them having its mirror image (where all the weights are switched to the opposite pan of the scale).Eliminating this symmetry,leaves us withjust (3n −1)/2arrangements,which can weight at most (3n −1)/2diﬀerent integral loads.Therefore,the largest range of consecutive integral loads they can cover cannot exceed (3n −1)/2.7.a.Apply Prim’s algorithm to the following graph:7课后答案网 w w w.k h d a w .c o mthe edges ae,eb,ec,and cd.b.Apply Prim’s algorithm to the following graph:the edges ab,be,ed,dc,ef,ei,ij,cg,gh,il,gk.8.There is no need to check the graph’s connectivity because Prim’s algo-rithm can do it itself.If the algorithm reaches all the graph’s vertices (via edges of ﬁnite lengths),the graph is connected,otherwise,it is not.9.a.The simplest and most logical solution is to assign all the edge weights to 1.8课a w .c o mb.Applying a depth-ﬁrst search (or breadth-ﬁrst search)traversal to get a depth-ﬁrst search tree (or a breadth-ﬁrst search tree),is conceptually simpler and for sparse graphs represented by their adjacency lists faster.10.The number of spanning trees for any weighted connected graph is a pos-itive ﬁnite number.(At least one spanning tree exists,e.g.,the one obtained by a depth-ﬁrst search traversal of the graph.And the number of spanning trees must be ﬁnite because any such tree comprises a subset of edges of the ﬁnite set of edges of the given graph.)Hence,one can always ﬁnd a spanning tree with the smallest total weight among the ﬁnite number of the candidates.Let’s prove now that the minimum spanning tree is unique if all the weights are distinct.We’ll do this by contradiction,i.e.,by assuming that there exists a graph G =(V,E )with all distinct weights but with more than one minimum spanning tree.Let e 1,...,e |V |−1be the list of edges com-posing the minimum spanning tree T P obtained by Prim’s algorithm with some speciﬁc vertex as the algorithm’s starting point and let T be an-other minimum spanning tree.Let e i =(v,u )be the ﬁrst edge in the list e 1,...,e |V |−1of the edges of T P which is not in T (if T P =T ,such edge must exist)and let (v,u )be an edge of T connecting v with a vertex not in the subtree T i −1formed by {e 1,...,e i −1}(if i =1,T i −1consists of vertex v only).Similarly to the proof of Prim’s algorithms correctness,let us replace (v,u )by e i =(v,u )in T .It will create another spanning tree,whose weight is smaller than the weight of T because the weight of e i =(v,u )is smaller than the weight of (v,u ).(Since e i was chosen by Prim’s algorithm,its weight is the smallest among all the weights on the edges connecting the tree vertices of the subtree T i −1and the vertices adjacent to it.And since all the weights are distinct,the weight of (v,u )must be strictly greater than the weight of e i =(v,u ).)This contradicts the assumption that T was a minimum spanning tree.11.If a key’s value in a min-heap was decreased,it may need to be pushedup (via swaps)along the chain of its ancestors until it is smaller than or equal to its parent or reaches the root.If a key’s value in a min-heap was increased,it may need to be pushed down by swaps with the smaller of its current children until it is smaller than or equal to its children or reaches a leaf.Since the height of a min-heap with n nodes is equal to log 2n (by the same reason the height of a max-heap is given by this formula–see Section 6.4),the operation’s eﬃciency is in O (log n ).(Note:The old value of the key in question need not be known,of paring the new value with that of the parent and,if the min-heap condition holds,with the smaller of the two children,will suﬃce.)9课后答案网 w w w.k h d a w .c o mExercises 9.21.Apply Kruskal’s algorithm to ﬁnd a minimum spanning tree of the follow-ing graphs.a.b.2.Indicate whether the following statements are true or false:a.If e is a minimum-weight edge in a connected weighted graph,it must be among edges of at least one minimum spanning tree of the graph.b.If e is a minimum-weight edge in a connected weighted graph,it must be among edges of each minimum spanning tree of the graph.c.If edge weights of a connected weighted graph are all distinct,the graph must have exactly one minimum spanning tree.d.If edge weights of a connected weighted graph are not all distinct,the graph must have more than one minimum spanning tree.3.What changes,if any,need to be made in algorithm Kruskal to make it ﬁnd a minimum spanning forest for an arbitrary graph?(A minimum spanning forest is a forest whose trees are minimum spanning trees of the graph’s connected components.)10课后答案网h d a w .c o m4.Will either Kruskal’s or Prim’s algorithm work correctly on graphs that have negative edge weights?5.Design an algorithm for ﬁnding a maximum spanning tree –a spanning tree with the largest possible edge weight–of a weighted connected graph.6.Rewrite the pseudocode of Kruskal’s algorithm in terms of the operations of the disjoint subsets’ADT.7. Prove the correctness of Kruskal’s algorithm.8.Prove that the time eﬃciency of ﬁnd (x )is in O (log n )for the union-by-size version of quick union.9.Find at least two Web sites with animations of Kruskal’s and Prim’s al-gorithms.Discuss their merits and demerits..10.Design and conduct an experiment to empirically compare the eﬃcienciesof Prim’s and Kruskal’s algorithms on random graphs of diﬀerent sizes and densities.11. Steiner tree Four villages are located at the vertices of a unit squarein the Euclidean plane.You are asked to connect them by the shortest network of roads so that there is a path between every pair of the villages along those roads.Find such a network.11课后答案网ww w.kh d aw .c omHints to Exercises 9.21.Trace the algorithm for the given graphs the same way it is done for another input in the section.2.Two of the four assertions are true,the other two are false.3.Applying Kruskal’s algorithm to a disconnected graph should help to an-swer the question.4.The answer is the same for both algorithms.If you believe that the algorithms work correctly on graphs with negative weights,prove this assertion;it you believe this is not to be the case,give a counterexample for each algorithm.5.Is the general trick of transforming maximization problems to their mini-mization counterparts (see Section6.6)applicable here?6.Substitute the three operations of the disjoint subsets’ADT–makeset (x ),ﬁnd (x ),and union (x,y )–in the appropriate places of the pseudocode given in the section.7.Follow the plan used in Section 9.1to prove the correctness of Prim’s algorithm.8.The argument is very similar to the one made in the section for the union-by-size version of quick ﬁnd.9.You may want to take advantage of the list of desirable characteristics in algorithm visualizations,which is given in Section 2.7.10.n/a11.The question is not trivial because introducing extra points (called Steinerpoints )may make the total length of the network smaller than that of a minimum spanning tree of the square.Solving ﬁrst the problem for three equidistant points might give you an indication how a solution to the problem in question could look like.12课后答案网ww w.kh d aw .c omSolutions to Exercises9.21.a.后课13b.⇒⇒⇒⇒⇒⇒14课c⇒⇒⇒⇒⇒课2.a.True.(Otherwise,Kruskal’s algorithm would be invalid.)b.False.As a simple counterexample,consider a complete graph withthree vertices and the same weight on its three edgesc.True(Problem10in Exercises9.1).15d.False (see,for example,the graph of Problem 1a).3.Since the number of edges in a minimum spanning forest of a graph with |V |vertices and |C |connected components is equal to |V |−|C |(this for-mula is a simple generalization of |E |=|V |−1for connected graphs),Kruskal (G )will never get to |V |−1tree edges unless the graph is con-nected.A simple remedy is to replace the loop while ecounter <|V |−1with while k <|E |to make the algorithm stop after exhausting the sorted list of its edges.4.Both algorithms work correctly for graphs with negative edge weights.One way of showing this is to add to all the weights of a graph with negative weights some large positive number.This makes all the new weights positive,and one can “translate”the algorithms’actions on the new graph to the corresponding actions on the old one.Alternatively,you can check that the proofs justifying the algorithms’correctness do not depend on the edge weights being nonnegative.5.Replace each weight w (u,v )by −w (u,v )and apply any minimum spanning tree algorithm that works on graphs with arbitrary weights (e.g.,Prim’s or Kruskal’s algorithm)to the graph with the new weights.6.Algorithm Kruskal (G )//Kruskal’s algorithm with explicit disjoint-subsets operations //Input:A weighted connected graph G = V,E//Output:E T ,the set of edges composing a minimum spanning tree of G sort E in nondecreasing order of the edge weights w (e i 1)≤...≤w (e i |E |)for each vertex v ∈V make (v )E T ←∅;ecounter ←0//initialize the set of tree edges and its size k ←0//the number of processed edges while ecounter <|V |−1k ←k +1if ﬁnd (u )=ﬁnd (v )//u,v are the endpoints of edge e i kE T ←E T ∪{e i k };ecounter ←ecounter +1union (u,v )return E T 7.Let us prove by induction that each of the forests F i ,i =0,...,|V |−1,of Kruskal’s algorithm is a part (i.e.,a subgraph)of some minimum span-ning tree.(This immediately implies,of course,that the last forest in the sequence,F |V |−1,is a minimum spanning tree itself.Indeed,it contains all vertices of the graph,and it is connected because it is both acyclic and has |V |−1edges.)The basis of the induction is trivial,since F 0is16课后答案网ww w.kh d aw .c ommade up of |V |single-vertex trees and therefore must be a subgraph of any spanning tree of the graph.For the inductive step,let us assume that F i −1is a subgraph of some minimum spanning tree T .We need to prove that F i ,generated from F i −1by Kruskal’s algorithm,is also a part of a minimum spanning tree.We prove this by contradiction by assuming that no minimum spanning tree of the graph can contain F i .Let e i =(v,u )be the minimum weight edge added by Kruskal’s algorithm to forest F i −1to obtain forest F i .(Note that vertices v and u must belong to diﬀerent trees of F i −1–otherwise,edge (v,u )would’ve created a cycle.)By our assumption,e i cannot belong to T .Therefore,if we add e i to T ,a cycle must be formed (see the ﬁgure below).In addition to edge e i =(v,u ),this cycle must contain another edge (v ,u )connecting a vertex v in the same tree of F i −1as v to a vertex u not in that tree.(It is possible that v coincides with v or u coincides with u but not both.)If we now delete the edge (v ,u )from this cycle,we will obtain another spanning tree of the entire graph whose weight is less than or equal to the weight of T since the weight of e i is less than or equal to the weight of (v ,u ).Hence,this spanning tree is a minimum spanning tree,which contradicts the assumption that no minimum spanning tree contains F i .This com-pletes the correctness proof of Kruskal’s algorithm.8.In the union-by-size version of quick-union ,each vertex starts at depth 0of its own tree.The depth of a vertex increases by 1when the tree it is in is attached to a tree with at least as many nodes during a union operation.Since the total number of nodes in the new tree containing the node is at least twice as much as in the old one,the number of such increases cannot exceed log 2n.Therefore the height of any tree (which is the largest depth of the tree’s nodes)generated by a legitimate sequence of unions will not exceed log 2n.Hence,the eﬃciency of ﬁnd (x )is in O (log n )because ﬁnd (x )traverses the pointer chain from the x ’s node to the tree’s root.9.n/a10.n/a17课后答案.kh d aw .c om11.The minimum Steiner tree that solves the problem is shown below.(Theother solution can be obtained by rotating the ﬁgure 90◦.)A popular discussion of Steiner trees can be found in “Last Recreations:Hydras,Eggs,and Other Mathematical Mystiﬁcations”by Martin Gard-ner.In general,no polynomial time algorithm is known for ﬁnding a minimum Steiner tree;moreover,the problem is known to be NP -hard (see Section 11.3).For the state-of-the-art information,see,e.g.,The Steiner Tree Page at /steiner/.18课后答案网ww w.kc omExercises 9.31.Explain what adjustments if any need to be made in Dijkstra’s algorithm and/or in an underlying graph to solve the following problems.a.Solve the single-source shortest-paths problem for directed weighted graphs.b.Find a shortest path between two given vertices of a weighted graph or digraph.(This variation is called the single-pair shortest-path prob-lem .)c.Find the shortest paths to a given vertex from each other vertex of a weighted graph or digraph.(This variation is called the single-destination shortest-paths problem .)d.Solve the single-source shortest-path problem in a graph with nonneg-ative numbers assigned to its vertices (and the length of a path deﬁned as the sum of the vertex numbers on the path).2.Solve the following instances of the single-source shortest-paths problem with vertex a as the source:a.b.3.Give a counterexample that shows that Dijkstra’s algorithm may not work for a weighted connected graph with negative weights.19课案w w.kh d aw .c om4.Let T be a tree constructed by Dijkstra’s algorithm in the process of solving the single-source shortest-path problem for a weighted connected graph G .a.True or false:T is a spanning tree of G ?b.True or false:T is a minimum spanning tree of G ?5.Write a pseudocode of a simpler version of Dijkstra’s algorithm that ﬁnds only the distances (i.e.,the lengths of shortest paths but not shortest paths themselves)from a given vertex to all other vertices of a graph represented by its weight matrix.6. Prove the correctness of Dijkstra’s algorithm for graphs with positive weights.7.Design a linear-time algorithm for solving the single-source shortest-paths problem for dags (directed acyclic graphs)represented by their adjacency lists.8.Design an eﬃcient algorithm for ﬁnding the length of a longest path in a dag.(This problem is important because it determines a lower bound on the total time needed for completing a project composed of precedence-constrained tasks.)9.Shortest-path modeling Assume you have a model of a weighted con-nected graph made of balls (representing the vertices)connected by strings of appropriate lengths (representing the edges).a.Describe how you can solve the single-pair shortest-path problem with this model .b.Describe how you can solve the single-source shortest-paths problem with this model .10.Revisit Problem 6in Exercises 1.3about determining the best route fora subway passenger to take from one designated station to another in a well-developed subway system like those in Washington,DC and London,UK.Write a program for this task.20课后答案网ww w.kh d aw .c om。

算法设计与分析ppt课件

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ACM国际大学生程序设计竞赛
ACM国际大学生程序设计竞赛（英文全称：ACM International Collegiate Programming Contest（ACM-ICPC或 ICPC）是由美国计算机协会（ACM）主办的，一项旨在展示大学生创新能力、团队精神和在压力下编写程序、分析和解决问题能力的年度竞赛。经过30多年的发展， ACM国际大学生程序设计竞赛已经发展成为最具影响力的大学生计算机竞赛。赛事目前由IBM公司赞助。
第3章动态规划 3.1 矩阵连乘问题 3.2 动态规划算法的基本要素 3.3 最长公共子序列 3.4 最大子段和 3.5 凸多边形最优三角剖分 3.6 多边形游戏 3.7 图像压缩 3.8 电路布线 3.9 流水作业调度 3.10 0-1背包问题 3.11 最优二叉搜索树 3.12 动态规划加速原理
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1.1 算法与程序
算法：是满足下述性质的指令序列。
输入：有零个或多个外部量作为算法的输入。输出：算法产生至少一个量作为输出。确定性：组成算法的每条指令清晰、无歧义。有限性：算法中每条指令的执行次数有限，执行
每条指令的时间也有限。
程序：是算法用某种程序设计语言的具体实现。
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教材与参考书
教材：
◦ 算法设计与分析(第三版) 王晓东，2007年 5月，电子工业出版社。
参考书：
◦ 徐士良编，C常用算法程序集，华大学出版社，1998年
◦ 霍红卫编，算法设计与分析西安电子科技大学出版社，2005年
◦ 卢开澄编，计算机算法导引，清华大学出版社，2003年
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部分目录
算法分析是计算机领域的“古老”而“前沿” 的课题。
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算法分析与设计(习题答案)

算法分析与设计教程习题解答第1章算法引论1. 解：算法是一组有穷的规则，它规定了解决某一特定类型问题的一系列计算方法。

频率计数是指计算机执行程序中的某一条语句的执行次数。

多项式时间算法是指可用多项式函数对某算法进行计算时间限界的算法。

指数时间算法是指某算法的计算时间只能使用指数函数限界的算法。

2. 解：算法分析的目的是使算法设计者知道为完成一项任务所设计的算法的优劣，进而促使人们想方设法地设计出一些效率更高效的算法，以便达到少花钱、多办事、办好事的经济效果。

3. 解：事前分析是指求出某个算法的一个时间限界函数（它是一些有关参数的函数）；事后测试指收集计算机对于某个算法的执行时间和占用空间的统计资料。

4. 解：评价一个算法应从事前分析和事后测试这两个阶段进行，事前分析主要应从时间复杂度和空间复杂度这两个维度进行分析；事后测试主要应对所评价的算法作时空性能分布图。

5. 解：①n=11； ②n=12； ③n=982； ④n=39。

第2章递归算法与分治算法1．解：递归算法是将归纳法的思想应用于算法设计之中，递归算法充分地利用了计算机系统内部机能，自动实现调用过程中对于相关且必要的信息的保存与恢复；分治算法是把一个问题划分为一个或多个子问题，每个子问题与原问题具有完全相同的解决思路，进而可以按照递归的思路进行求解。

2．解：通过分治算法的一般设计步骤进行说明。

3．解：int fibonacci(int n) {if(n<=1) return 1;return fibonacci(n-1)+fibonacci(n-2); }4. 解：void hanoi(int n,int a,int b,int c) {if(n>0) {hanoi(n-1,a,c,b); move(a,b);hanoi(n-1,c,b,a); } } 5. 解：①22*2)(−−=n n f n② )log *()(n n n f O =6．解：算法略。

《算法设计与分析》课件

《算法设计与分析》PPT课件
本课程将介绍算法的设计与分析，包括排序算法、查找算法和动态规划算法。通过掌握这些算法，您将能够解决各种复杂的问题。
课程介绍
课程目标和内容概述
掌握算法设计与分析的基本概念和方法，学习不同类型的算法及其应用。
教学方法和要求
通过理论讲解、案例分析和实际编程练习，提高算法设计与分析的能力。
2 背包问题的动态规划解法
学习如何使用动态规划算法解决背包问题，掌握求解最优解的方法。
总结和课程评价
总结
回顾本课程涉及的算法内容，并思考所学知识的实际应用。
课程评价
对本课程的内容、教学方法和教师的表现进行评价和反馈。
算法基础
1 算法概述和分类
了解算法的定义、特性和常见的分类方法，为后续学习打下基础。
2 时间复杂度和空间复杂度
学习如何评估算法的时间和空间效率，并选择最合适的算法。
排序算法
1
插入排序
2
学习插排序算法的思想和实现过程，
掌握其时间复杂度和适用范围。
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冒泡排序
掌握冒泡排序算法的原理和实现方法，了解其时间复杂度和应用场景。
快速排序
了解快速排序算法的原理和分治思想，学会如何选择合适的划分策略。
查找算法
顺序查找
掌握顺序查找算法的基本思想和实现过程，了解其时间复杂度和使用场景。
二分查找
学习二分查找算法的原理和应用，了解其时间复杂度和适用条件。
动态规划算法
1 原理和应用举例
了解动态规划算法的核心原理，并通过实例了解其在解决复杂问题时的应用。

算法设计与分析习题解答(第2版)

第1章算法引论11.1 算法与程序11.2 表达算法的抽象机制11.3 描述算法31.4 算法复杂性分析13小结16习题17第2章递归与分治策略192.1 递归的概念192.2 分治法的基本思想262.3 二分搜索技术272.4 大整数的乘法282.5 Strassen矩阵乘法302.6 棋盘覆盖322.7 合并排序342.8 快速排序372.9 线性时间选择392.10 最接近点对问题432.11 循环赛日程表53小结54习题54第3章动态规划613.1 矩阵连乘问题62目录算法设计与分析(第2版)3.2 动态规划算法的基本要素67 3.3 最长公共子序列713.4 凸多边形最优三角剖分753.5 多边形游戏793.6 图像压缩823.7 电路布线853.8 流水作业调度883.9 0-1背包问题923.10 最优二叉搜索树98小结101习题102第4章贪心算法1074.1 活动安排问题1074.2 贪心算法的基本要素1104.2.1 贪心选择性质1114.2.2 最优子结构性质1114.2.3 贪心算法与动态规划算法的差异1114.3 最优装载1144.4 哈夫曼编码1164.4.1 前缀码1174.4.2 构造哈夫曼编码1174.4.3 哈夫曼算法的正确性1194.5 单源最短路径1214.5.1 算法基本思想1214.5.2 算法的正确性和计算复杂性123 4.6 最小生成树1254.6.1 最小生成树性质1254.6.2 Prim算法1264.6.3 Kruskal算法1284.7 多机调度问题1304.8 贪心算法的理论基础1334.8.1 拟阵1334.8.2 带权拟阵的贪心算法1344.8.3 任务时间表问题137小结141习题141第5章回溯法1465.1 回溯法的算法框架1465.1.1 问题的解空间1465.1.2 回溯法的基本思想1475.1.3 递归回溯1495.1.4 迭代回溯1505.1.5 子集树与排列树1515.2 装载问题1525.3 批处理作业调度1605.4 符号三角形问题1625.5 n后问题1655.6 0\|1背包问题1685.7 最大团问题1715.8 图的m着色问题1745.9 旅行售货员问题1775.10 圆排列问题1795.11 电路板排列问题1815.12 连续邮资问题1855.13 回溯法的效率分析187小结190习题191第6章分支限界法1956.1 分支限界法的基本思想1956.2 单源最短路径问题1986.3 装载问题2026.4 布线问题2116.5 0\|1背包问题2166.6 最大团问题2226.7 旅行售货员问题2256.8 电路板排列问题2296.9 批处理作业调度232小结237习题238第7章概率算法2407.1 随机数2417.2 数值概率算法2447.2.1 用随机投点法计算π值2447.2.2 计算定积分2457.2.3 解非线性方程组2477.3 舍伍德算法2507.3.1 线性时间选择算法2507.3.2 跳跃表2527.4 拉斯维加斯算法2597.4.1 n 后问题2607.4.2 整数因子分解2647.5 蒙特卡罗算法2667.5.1 蒙特卡罗算法的基本思想2667.5.2 主元素问题2687.5.3 素数测试270小结273习题273第8章 NP完全性理论2788.1 计算模型2798.1.1 随机存取机RAM2798.1.2 随机存取存储程序机RASP2878.1.3 RAM模型的变形与简化2918.1.4 图灵机2958.1.5 图灵机模型与RAM模型的关系297 8.1.6 问题变换与计算复杂性归约299 8.2 P类与NP类问题3018.2.1 非确定性图灵机3018.2.2 P类与NP类语言3028.2.3 多项式时间验证3048.3 NP完全问题3058.3.1 多项式时间变换3058.3.2 Cook定理3078.4 一些典型的NP完全问题3108.4.1 合取范式的可满足性问题3118.4.2 3元合取范式的可满足性问题312 8.4.3 团问题3138.4.4 顶点覆盖问题3148.4.5 子集和问题3158.4.6 哈密顿回路问题3178.4.7 旅行售货员问题322小结323习题323第9章近似算法3269.1 近似算法的性能3279.2 顶点覆盖问题的近似算法3289.3 旅行售货员问题近似算法3299.3.1 具有三角不等式性质的旅行售货员问题330 9.3.2 一般的旅行售货员问题3319.4 集合覆盖问题的近似算法3339.5 子集和问题的近似算法3369.5.1 子集和问题的指数时间算法3369.5.2 子集和问题的完全多项式时间近似格式337 小结340习题340第10章算法优化策略34510.1 算法设计策略的比较与选择34510.1.1 最大子段和问题的简单算法34510.1.2 最大子段和问题的分治算法34610.1.3 最大子段和问题的动态规划算法34810.1.4 最大子段和问题与动态规划算法的推广349 10.2 动态规划加速原理35210.2.1 货物储运问题35210.2.2 算法及其优化35310.3 问题的算法特征35710.3.1 贪心策略35710.3.2 对贪心策略的改进35710.3.3 算法三部曲35910.3.4 算法实现36010.3.5 算法复杂性36610.4 优化数据结构36610.4.1 带权区间最短路问题36610.4.2 算法设计思想36710.4.3 算法实现方案36910.4.4 并查集37310.4.5 可并优先队列37610.5 优化搜索策略380小结388习题388第11章在线算法设计39111.1 在线算法设计的基本概念39111.2 页调度问题39311.3 势函数分析39511.4 k 服务问题39711.4.1 竞争比的下界39711.4.2 平衡算法39911.4.3 对称移动算法39911.5 Steiner树问题40311.6 在线任务调度40511.7 负载平衡406小结407习题407词汇索引409参考文献415习题1-1 实参交换1习题1-2 方法头签名1习题1-3 数组排序判定1习题1-4 函数的渐近表达式2习题1-5 O(1) 和 O(2) 的区别2习题1-7 按渐近阶排列表达式2习题1-8 算法效率2习题1-9 硬件效率3习题1-10 函数渐近阶3习题1-11 n !的阶4习题1-12 平均情况下的计算时间复杂性4算法实现题1-1 统计数字问题4算法实现题1-2 字典序问题5算法实现题1-3 最多约数问题6算法实现题1-4 金币阵列问题8算法实现题1-5 最大间隙问题11第2章递归与分治策略14 习题2-1 Hanoi 塔问题的非递归算法14习题2-2 7个二分搜索算法15习题2-3 改写二分搜索算法18习题2-4 大整数乘法的 O(nm log(3/2))算法19习题2-5 5次 n /3位整数的乘法19习题2-6 矩阵乘法21习题2-7 多项式乘积21习题2-8 不动点问题的 O( log n) 时间算法22习题2-9 主元素问题的线性时间算法22习题2-10 无序集主元素问题的线性时间算法22习题2-11 O (1)空间子数组换位算法23习题2-12 O (1)空间合并算法25习题2-13 n 段合并排序算法32习题2-14 自然合并排序算法32习题2-15 最大值和最小值问题的最优算法35习题2-16 最大值和次大值问题的最优算法35习题2-17 整数集合排序35习题2-18 第 k 小元素问题的计算时间下界36习题2-19 非增序快速排序算法37习题2-20 随机化算法37习题2-21 随机化快速排序算法38习题2-22 随机排列算法38习题2-23 算法qSort中的尾递归38习题2-24 用栈模拟递归38习题2-25 算法select中的元素划分39习题2-26 O(n log n) 时间快速排序算法40习题2-27 最接近中位数的 k 个数40习题2-28 X和Y 的中位数40习题2-29 网络开关设计41习题2-32 带权中位数问题42习题2-34 构造Gray码的分治算法43习题2-35 网球循环赛日程表44目录算法设计与分析习题解答（第2版）算法实现题2-1 输油管道问题（习题2-30) 49算法实现题2-2 众数问题（习题2-31) 50算法实现题2-3 邮局选址问题（习题2-32) 51算法实现题2-4 马的Hamilton周游路线问题（习题2-33) 51算法实现题2-5 半数集问题60算法实现题2-6 半数单集问题62算法实现题2-7 士兵站队问题63算法实现题2-8 有重复元素的排列问题63算法实现题2-9 排列的字典序问题65算法实现题2-10 集合划分问题（一）67算法实现题2-11 集合划分问题（二）68算法实现题2-12 双色Hanoi塔问题69算法实现题2-13 标准二维表问题71算法实现题2-14 整数因子分解问题72算法实现题2-15 有向直线2中值问题72第3章动态规划76习题3-1 最长单调递增子序列76习题3-2 最长单调递增子序列的 O(n log n) 算法77习题3-7 漂亮打印78习题3-11 整数线性规划问题79习题3-12 二维背包问题80习题3-14 Ackermann函数81习题3-17 最短行驶路线83习题3-19 最优旅行路线83算法实现题3-1 独立任务最优调度问题（习题3-3) 83算法实现题3-2 最少硬币问题（习题3-4) 85算法实现题3-3 序关系计数问题（习题3-5) 86算法实现题3-4 多重幂计数问题（习题3-6) 87算法实现题3-5 编辑距离问题（习题3-8) 87算法实现题3-6 石子合并问题（习题3-9) 89算法实现题3-7 数字三角形问题（习题3-10) 91算法实现题3-8 乘法表问题（习题3-13) 92算法实现题3-9 租用游艇问题（习题3-15) 93算法实现题3-10 汽车加油行驶问题（习题3-16) 95算法实现题3-11 圈乘运算问题（习题3-18) 96算法实现题3-12 最少费用购物（习题3-20) 102算法实现题3-13 最大长方体问题（习题3-21) 104算法实现题3-14 正则表达式匹配问题（习题3-22) 105算法实现题3-15 双调旅行售货员问题（习题3-23) 110算法实现题3-16 最大 k 乘积问题（习题5-24) 111算法实现题3-17 最小 m 段和问题113算法实现题3-18 红黑树的红色内结点问题115第4章贪心算法123 习题4-2 活动安排问题的贪心选择123习题4-3 背包问题的贪心选择性质123习题4-4 特殊的0-1背包问题124习题4-10 程序最优存储问题124习题4-13 最优装载问题的贪心算法125习题4-18 Fibonacci序列的Huffman编码125习题4-19 最优前缀码的编码序列125习题4-21 任务集独立性问题126习题4-22 矩阵拟阵126习题4-23 最小权最大独立子集拟阵126习题4-27 整数边权Prim算法126习题4-28 最大权最小生成树127习题4-29 最短路径的负边权127习题4-30 整数边权Dijkstra算法127算法实现题4-1 会场安排问题（习题4-1) 128算法实现题4-2 最优合并问题（习题4-5) 129算法实现题4-3 磁带最优存储问题（习题4-6) 130算法实现题4-4 磁盘文件最优存储问题（习题4-7) 131算法实现题4-5 程序存储问题（习题4-8) 132算法实现题4-6 最优服务次序问题（习题4-11) 133算法实现题4-7 多处最优服务次序问题（习题4-12) 134算法实现题4-8 d 森林问题（习题4-14) 135算法实现题4-9 汽车加油问题（习题4-16) 137算法实现题4-10 区间覆盖问题（习题4-17) 138算法实现题4-11 硬币找钱问题（习题4-24) 138算法实现题4-12 删数问题（习题4-25) 139算法实现题4-13 数列极差问题（习题4-26) 140算法实现题4-14 嵌套箱问题（习题4-31) 140算法实现题4-15 套汇问题（习题4-32) 142算法实现题4-16 信号增强装置问题（习题5-17) 143算法实现题4-17 磁带最大利用率问题（习题4-9) 144算法实现题4-18 非单位时间任务安排问题（习题4-15) 145算法实现题4-19 多元Huffman编码问题（习题4-20) 147算法实现题4-20 多元Huffman编码变形149算法实现题4-21 区间相交问题151算法实现题4-22 任务时间表问题151第5章回溯法153习题5\|1 装载问题改进回溯法（一）153习题5\|2 装载问题改进回溯法（二）154习题5\|4 0-1背包问题的最优解155习题5\|5 最大团问题的迭代回溯法156习题5\|7 旅行售货员问题的费用上界157习题5\|8 旅行售货员问题的上界函数158算法实现题5-1 子集和问题（习题5-3) 159算法实现题5-2 最小长度电路板排列问题（习题5-9) 160算法实现题5-3 最小重量机器设计问题（习题5-10) 163算法实现题5-4 运动员最佳匹配问题（习题5-11) 164算法实现题5-5 无分隔符字典问题（习题5-12) 165算法实现题5-6 无和集问题（习题5-13) 167算法实现题5-7 n 色方柱问题（习题5-14) 168算法实现题5-8 整数变换问题（习题5-15) 173算法实现题5-9 拉丁矩阵问题（习题5-16) 175算法实现题5-10 排列宝石问题（习题5-16) 176算法实现题5-11 重复拉丁矩阵问题（习题5-16) 179算法实现题5-12 罗密欧与朱丽叶的迷宫问题181算法实现题5-13 工作分配问题（习题5-18) 183算法实现题5-14 独立钻石跳棋问题（习题5-19) 184算法实现题5-15 智力拼图问题（习题5-20) 191算法实现题5-16 布线问题（习题5-21) 198算法实现题5-17 最佳调度问题（习题5-22) 200算法实现题5-18 无优先级运算问题（习题5-23) 201算法实现题5-19 世界名画陈列馆问题（习题5-25) 203算法实现题5-20 世界名画陈列馆问题（不重复监视）（习题5-26) 207 算法实现题5-21 部落卫队问题（习题5-6) 209算法实现题5-22 虫蚀算式问题211算法实现题5-23 完备环序列问题214算法实现题5-24 离散01串问题217算法实现题5-25 喷漆机器人问题218算法实现题5-26 n 2-1谜问题221第6章分支限界法229习题6-1 0-1背包问题的栈式分支限界法229习题6-2 用最大堆存储活结点的优先队列式分支限界法231习题6-3 团顶点数的上界234习题6-4 团顶点数改进的上界235习题6-5 修改解旅行售货员问题的分支限界法235习题6-6 解旅行售货员问题的分支限界法中保存已产生的排列树237 习题6-7 电路板排列问题的队列式分支限界法239算法实现题6-1 最小长度电路板排列问题一（习题6-8) 241算法实现题6-2 最小长度电路板排列问题二（习题6-9) 244算法实现题6-3 最小权顶点覆盖问题（习题6-10) 247算法实现题6-4 无向图的最大割问题（习题6-11) 250算法实现题6-5 最小重量机器设计问题（习题6-12) 253算法实现题6-6 运动员最佳匹配问题（习题6-13) 256算法实现题6-7 n 后问题（习题6-15) 259算法实现题6-8 圆排列问题（习题6-16) 260算法实现题6-9 布线问题（习题6-17) 263算法实现题6-10 最佳调度问题（习题6-18) 265算法实现题6-11 无优先级运算问题（习题6-19) 268算法实现题6-12 世界名画陈列馆问题（习题6-21) 271算法实现题6-13 骑士征途问题274算法实现题6-14 推箱子问题275算法实现题6-15 图形变换问题281算法实现题6-16 行列变换问题284算法实现题6-17 重排 n 2宫问题285算法实现题6-18 最长距离问题290第7章概率算法296习题7-1 模拟正态分布随机变量296习题7-2 随机抽样算法297习题7-3 随机产生 m 个整数297习题7-4 集合大小的概率算法298习题7-5 生日问题299习题7-6 易验证问题的拉斯维加斯算法300习题7-7 用数组模拟有序链表300习题7-8 O(n 3/2)舍伍德型排序算法300习题7-9 n 后问题解的存在性301习题7-11 整数因子分解算法302习题7-12 非蒙特卡罗算法的例子302习题7-13 重复3次的蒙特卡罗算法303习题7-14 集合随机元素算法304习题7-15 由蒙特卡罗算法构造拉斯维加斯算法305习题7-16 产生素数算法306习题7-18 矩阵方程问题306算法实现题7-1 模平方根问题（习题7-10) 307算法实现题7-2 集合相等问题（习题7-17) 309算法实现题7-3 逆矩阵问题（习题7-19) 309算法实现题7-4 多项式乘积问题（习题7-20) 310算法实现题7-5 皇后控制问题311算法实现题7-6 3-SAT问题314算法实现题7-7 战车问题315算法实现题7-8 圆排列问题317算法实现题7-9 骑士控制问题319算法实现题7-10 骑士对攻问题320第8章NP完全性理论322 习题8-1 RAM和RASP程序322习题8-2 RAM和RASP程序的复杂性322习题8-3 计算 n n 的RAM程序322习题8-4 没有MULT和DIV指令的RAM程序324习题8-5 MULT和DIV指令的计算能力324习题8-6 RAM和RASP的空间复杂性325习题8-7 行列式的直线式程序325习题8-8 求和的3带图灵机325习题8-9 模拟RAM指令325习题8-10 计算2 2 n 的RAM程序325习题8-11 计算 g(m,n)的程序 326习题8-12 图灵机模拟RAM的时间上界326习题8-13 图的同构问题326习题8-14 哈密顿回路327习题8-15 P类语言的封闭性327习题8-16 NP类语言的封闭性328习题8-17 语言的2 O (n k) 时间判定算法328习题8-18 P CO -NP329习题8-19 NP≠CO -NP329习题8-20 重言布尔表达式329习题8-21 关系∝ p的传递性329习题8-22 L ∝ p 330习题8-23 语言的完全性330习题8-24 的CO-NP完全性330习题8-25 判定重言式的CO-NP完全性331习题8-26 析取范式的可满足性331习题8-27 2-SAT问题的线性时间算法331习题8-28 整数规划问题332习题8-29 划分问题333习题8-30 最长简单回路问题334第9章近似算法336习题9-1 平面图着色问题的绝对近似算法336习题9-2 最优程序存储问题336习题9-4 树的最优顶点覆盖337习题9-5 顶点覆盖算法的性能比339习题9-6 团的常数性能比近似算法339习题9-9 售货员问题的常数性能比近似算法340习题9-10 瓶颈旅行售货员问题340习题9-11 最优旅行售货员回路不自相交342习题9-14 集合覆盖问题的实例342习题9-16 多机调度问题的近似算法343习题9-17 LPT算法的最坏情况实例345习题9-18 多机调度问题的多项式时间近似算法345算法实现题9-1 旅行售货员问题的近似算法（习题9-9) 346 算法实现题9-2 可满足问题的近似算法（习题9-20) 348算法实现题9-3 最大可满足问题的近似算法（习题9-21) 349 算法实现题9-4 子集和问题的近似算法（习题9-15) 351算法实现题9-5 子集和问题的完全多项式时间近似算法352算法实现题9-6 实现算法greedySetCover（习题9-13) 352算法实现题9-7 装箱问题的近似算法First Fit（习题9-19) 356算法实现题9-8 装箱问题的近似算法Best Fit（习题9-19) 358算法实现题9-9 装箱问题的近似算法First Fit Decreasing（习题9-19) 360算法实现题9-10 装箱问题的近似算法Best Fit Decreasing（习题9-19) 361算法实现题9-11 装箱问题的近似算法Next Fit361第10章算法优化策略365 习题10-1 算法obst的正确性365习题10-2 矩阵连乘问题的 O(n 2) 时间算法365习题10-6 货物储运问题的费用371习题10-7 Garsia算法371算法实现题10-1 货物储运问题（习题10-3) 374算法实现题10-2 石子合并问题（习题10-4) 374算法实现题10-3 最大运输费用货物储运问题（习题10-5) 375算法实现题10-4 五边形问题377算法实现题10-5 区间图最短路问题（习题10-8) 381算法实现题10-6 圆弧区间最短路问题（习题10-9) 381算法实现题10-7 双机调度问题（习题10-10) 382算法实现题10-8 离线最小值问题（习题10-11) 390算法实现题10-9 最近公共祖先问题（习题10-12) 393算法实现题10-10 达尔文芯片问题395算法实现题10-11 多柱Hanoi塔问题397算法实现题10-12 线性时间Huffman算法400算法实现题10-13 单机调度问题402算法实现题10-14 最大费用单机调度问题405算法实现题10-15 飞机加油问题408第11章在线算法设计410习题11-1 在线算法LFU的竞争性410习题11-4 多读写头磁盘问题的在线算法410习题11-6 带权页调度问题410算法实现题11-1 最优页调度问题（习题11-2) 411算法实现题11-2 在线LRU页调度（习题11-3) 414算法实现题11-3 k 服务问题（习题11-5) 416参考文献422。

算法设计与分析-课后习题集答案

第一章3. 最大公约数为1。

快1414倍。

程序1-2的while 循环体做了10次，程序1-3的while 循环体做了14141次（14142-2循环）8.（1）画线语句的执行次数为log n ⎡⎤⎢⎥。

(log )n O 。

（2）画线语句的执行次数为111(1)(21)16jnii j k n n n ===++=∑∑∑。

3()n O 。

（3）画线语句的执行次数为。

O 。

（4）当n 为奇数时画线语句的执行次数为(1)(1)4n n +-，当n 为偶数时画线语句的执行次数为 (2)4n n +。

2()n O 。

10.（1）当 1n ≥ 时，225825n n n -+≤，所以，可选 5c =，01n =。

对于0n n ≥，22()5825f n n n n =-+≤，所以，22582()-+=O n n n 。

（2）当 8n ≥ 时，2222582524n n n n n -+≥-+≥，所以，可选 4c =，08n =。

对于0n n ≥，22()5824f n n n n =-+≥，所以，22582()-+=Ωn n n 。

（3）由（1）、（2）可知，取14c =，25c =，08n =，当0n n ≥时，有22212582c n n n c n ≤-+≤，所以22582()-+=Θn n n 。

11. (1) 当3n ≥时，3log log n n n <<，所以()20log 21f n n n n =+<，3()log 2g n n n n =+>。

可选212c =，03n =。

对于0n n ≥，()()f n cg n ≤，即()(())f n g n =O 。

（2）当 4n ≥ 时，2log log n n n <<，所以 22()/log f n n n n =<，22()log g n n n n =≥。

可选 1c =，04n =。

算法设计与分析课程设计(完整版)

HUNAN CITY UNIVERSITY 算法设计与分析课程设计题目：求最大值与最小值问题专业：学号：姓名：指导教师：成绩：二0年月日一、问题描述输入一列整数，求出该列整数中的最大值与最小值。

二、课程设计目的通过课程设计，提高用计算机解决实际问题的能力，提高独立实践的能力，将课本上的理论知识和实际有机的结合起来，锻炼分析解决实际问题的能力。

提高适应实际，实践编程的能力。

在实际的编程和调试综合试题的基础上，把高级语言程序设计的思想、编程巧和解题思路进行总结与概括，通过比较系统地练习达到真正比较熟练地掌握计算机编程的基本功，为后续的学习打下基础。

了解一般程序设计的基本思路与方法。

三、问题分析看到这个题目我们最容易想到的算法是直接比较算法：将数组的第 1 个元素分别赋给两个临时变量：fmax:=A[1]; fmin:=A[1]; 然后从数组的第 2 个元素 A[2]开始直到第 n个元素逐个与 fmax 和 fmin 比较，在每次比较中，如果A[i] > fmax，则用 A[i]的值替换 fmax 的值；如果 A[i] < fmin，则用 A[i]的值替换 fmin 的值；否则保持 fmax（fmin）的值不变。

这样在程序结束时的fmax、fmin 的值就分别是数组的最大值和最小值。

这个算法在最好、最坏情况下，元素的比较次数都是 2(n-1)，而平均比较次数也为 2(n-1)。

如果将上面的比较过程修改为：从数组的第 2 个元素 A[2]开始直到第 n 个元素，每个 A[i]都是首先与 fmax 比较，如果 A[i]>fmax,则用 A[i]的值替换 fmax 的值；否则才将 A[i]与 fmin 比较，如果 A[i] < fmin，则用 A[i]的值替换 fmin 的值。

这样的算法在最好、最坏情况下使用的比较次数分别是 n-1 和 2(n-1),而平均比较次数是 3(n-1)/2,因为在比较过程中，将有一半的几率出现 A[i]>fmax 情况。

第二单元第10课《生活和算法》教学设计2023—2024学年浙教版(2020)初中信息技术七年级下册

-加入算法学习小组，与同学互相交流、共同进步。
-利用网络资源，关注算法领域的最新发展动态。
-积极参与算法相关的实践活动，将所学知识应用于实际问题解决。
-结合生活实际，思考算法在生活中的应用，培养算法思维。
-定期回顾和总结所学算法知识，形成完整的知识体系。
-积极参与算法学习相关的讨论和交流，提高算法理解能力。
情感升华：
-结合算法内容，引导学生思考学科与生活的联系，培养学生的社会责任感。
-鼓励学生分享学习算法的心得和体会，增进师生之间的情感交流。
（六）课堂小结（预计用时：2分钟）
-简要回顾本节课学习的算法内容，强调算法重点和难点。
-肯定学生的表现，鼓励他们继续努力。
布置作业：
-根据本节课学习的算法内容，布置适量的课后作业，巩固学习效果。
-提醒学生注意作业要求和时间安排，确保作业质量。
教学资源拓展
1.拓展资源：
-算法相关图书和杂志：提供算法理论、应用案例等深入学习的资料。
-算法动画和教学视频：通过动画和视频形式直观展示算法的原理和应用。
-算法在线课程和讲座：提供系统学习算法的机会，拓宽知识面。
-算法游戏和挑战项目：通过游戏和项目形式，提高算法实践能力。
（二）课堂导入（预计用时：3分钟）
激发兴趣：
-通过展示与算法内容相关的图片、视频或故事，吸引学生的注意力。
-提出问题或设置悬念，引发学生的好奇心和求知欲，引导学生进入算法学习状态。
回顾旧知：
-简要回顾上节课学习的算法内容，帮助学生建立知识之间的联系。
-提出问题，检查学生对旧知的掌握情况，为算法新课学习打下基础。
为了更好地激发学生的学习兴趣，本课程设计注重课程内容的趣味性和实用性。课程以生活实例为载体，让学生在学习过程中感受到算法与生活的紧密联系。同时，课程注重实践环节，让学生在实际操作中掌握算法的应用。此外，课程还设计了丰富的课堂活动，如小组讨论、案例分析等，以增强课程的互动性和趣味性。

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第二单元第10课《生活和算法》教学设计2023—2024学年浙教版(2020)初中信息技术七年级下册

算法分析与设计第二版英文版 (潘彦著) 清华大学出版社课后答案--solu9