第四周--多项式求和

<pre class="cpp" name="code">/*
*Copyright (c) 2015 烟台大学计算机与控制工程学院
*All right reserved.
*标题:数据结构实践——多项式求和
*作者：杨珺
*date:2015年10月13日
*版本:V1.0.1
*操作系统:XP
*运行环境:VC6.0
*问题描述: 用单链表存储一元多项式，并实现两个多项式的加法。<blockquote><p>提示：
<strong>1、存储多项式的数据结构</strong>
　　多项式的通式是<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-3-Frame" role="textbox" aria-readonly="true" style="display: inline-block;"><nobr><span class="math" id="MathJax-Span-1" style="width: 21.44em; display: inline-block;"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 17.28em; height: 0px; font-size: 124%; display: inline-block; position: relative;"><span style="left: 0em; top: -3.97em; position: absolute; clip: rect(2.96em, 1000em, 4.39em, -0.55em);"><span class="mrow" id="MathJax-Span-2"><span class="msubsup" id="MathJax-Span-3"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 0.99em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.84em, -0.55em);"><span class="mi" id="MathJax-Span-4" style="font-family: MathJax_Math-italic-Web;">p</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.51em; top: -1.78em; position: absolute;"><span class="mi" id="MathJax-Span-5" style="font-family: MathJax_Math-italic-Web; font-size: 70.7%;"><span style="font-size:14px;">n</span></span> <span style="width: 0px; height: 2.01em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-6" style="font-family: MathJax_Main-Web;">(</span><span class="mi" id="MathJax-Span-7" style="font-family: MathJax_Math-italic-Web;">x</span><span class="mo" id="MathJax-Span-8" style="font-family: MathJax_Main-Web;">)</span><span class="mo" id="MathJax-Span-9" style="padding-left: 0.27em; font-family: MathJax_Main-Web;">=</span><span class="msubsup" id="MathJax-Span-10" style="padding-left: 0.27em;"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 0.99em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-11" style="font-family: MathJax_Math-italic-Web;">a</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.51em; top: -1.86em; position: absolute;"><span class="mi" id="MathJax-Span-12" style="font-family: MathJax_Math-italic-Web; font-size: 70.7%;"><span style="font-size:14px;">n</span></span> <span style="width: 0px; height: 2.01em; display: inline-block;"></span></span></span></span><span class="msubsup" id="MathJax-Span-13"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 1.05em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-14" style="font-family: MathJax_Math-italic-Web;">x</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.57em; top: -2.37em; position: absolute;"><span class="mi" id="MathJax-Span-15" style="font-family: MathJax_Math-italic-Web; font-size: 70.7%;"><span style="font-size:14px;">n</span></span> <span style="width: 0px; height: 2.01em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-16" style="padding-left: 0.22em; font-family: MathJax_Main-Web;">+</span><span class="msubsup" id="MathJax-Span-17" style="padding-left: 0.22em;"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 1.91em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-18" style="font-family: MathJax_Math-italic-Web;">a</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.51em; top: -2.44em; position: absolute;"><span class="texatom" id="MathJax-Span-19"><span class="mrow" id="MathJax-Span-20"><span style="font-size:14px;"><span class="mi" id="MathJax-Span-21" style="font-family: MathJax_Math-italic-Web; font-size: 70.7%;">n</span><span class="mo" id="MathJax-Span-22" style="font-family: MathJax_Main-Web;">−</span><span class="mn" id="MathJax-Span-23" style="font-family: MathJax_Main-Web;">1</span></span></span></span> <span style="width: 0px; height: 2.59em; display: inline-block;"></span></span></span></span><span class="msubsup" id="MathJax-Span-24"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 1.97em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-25" style="font-family: MathJax_Math-italic-Web;">x</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.57em; top: -2.95em; position: absolute;"><span class="texatom" id="MathJax-Span-26"><span class="mrow" id="MathJax-Span-27"><span style="font-size:14px;"><span class="mi" id="MathJax-Span-28" style="font-family: MathJax_Math-italic-Web; font-size: 70.7%;">n</span><span class="mo" id="MathJax-Span-29" style="font-family: MathJax_Main-Web;">−</span><span class="mn" id="MathJax-Span-30" style="font-family: MathJax_Main-Web;">1</span></span></span></span> <span style="width: 0px; height: 2.59em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-31" style="font-family: MathJax_Main-Web;">+</span><span class="mo" id="MathJax-Span-32" style="font-family: MathJax_Main-Web;">.</span><span class="mo" id="MathJax-Span-33" style="padding-left: 0.16em; font-family: MathJax_Main-Web;">.</span><span class="mo" id="MathJax-Span-34" style="padding-left: 0.16em; font-family: MathJax_Main-Web;">.</span><span class="mo" id="MathJax-Span-35" style="padding-left: 0.16em; font-family: MathJax_Main-Web;">+</span><span class="msubsup" id="MathJax-Span-36"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 0.93em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-37" style="font-family: MathJax_Math-italic-Web;">a</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.51em; top: -2.03em; position: absolute;"><span class="mn" id="MathJax-Span-38" style="font-family: MathJax_Main-Web; font-size: 70.7%;"><span style="font-size:14px;">1</span></span> <span style="width: 0px; height: 2.18em; display: inline-block;"></span></span></span></span><span class="mi" id="MathJax-Span-39" style="font-family: MathJax_Math-italic-Web;">x</span><span class="mo" id="MathJax-Span-40" style="padding-left: 0.22em; font-family: MathJax_Main-Web;">+</span><span class="msubsup" id="MathJax-Span-41" style="padding-left: 0.22em;"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 0.93em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-42" style="font-family: MathJax_Math-italic-Web;">a</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.51em; top: -2.03em; position: absolute;"><span class="mn" id="MathJax-Span-43" style="font-family: MathJax_Main-Web; font-size: 70.7%;"><span style="font-size:14px;">0</span></span> <span style="width: 0px; height: 2.18em; display: inline-block;"></span></span></span></span></span> <span style="width: 0px; height: 3.97em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.48em; overflow: hidden; vertical-align: -0.38em; border-left-color: currentColor; border-left-width: 0em; border-left-style: solid; display: inline-block;"></span></span></nobr></span>

。n次多项式共有n+1项。直观地，可以定义一个数组来存储这n+1个系数。以多项式<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-4-Frame" role="textbox" aria-readonly="true" style="display: inline-block;"><nobr><span class="math" id="MathJax-Span-44" style="width: 19.01em; display: inline-block;"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 15.32em; height: 0px; font-size: 124%; display: inline-block; position: relative;"><span style="left: 0em; top: -3.97em; position: absolute; clip: rect(2.96em, 1000em, 4.39em, -0.55em);"><span class="mrow" id="MathJax-Span-45"><span class="mi" id="MathJax-Span-46" style="font-family: MathJax_Math-italic-Web;">p</span><span class="mo" id="MathJax-Span-47" style="font-family: MathJax_Main-Web;">(</span><span class="mi" id="MathJax-Span-48" style="font-family: MathJax_Math-italic-Web;">x</span><span class="mo" id="MathJax-Span-49" style="font-family: MathJax_Main-Web;">)</span><span class="mo" id="MathJax-Span-50" style="padding-left: 0.27em; font-family: MathJax_Main-Web;">=</span><span class="mo" id="MathJax-Span-51" style="padding-left: 0.27em; font-family: MathJax_Main-Web;">−</span><span class="mn" id="MathJax-Span-52" style="font-family: MathJax_Main-Web;">3.4</span><span class="msubsup" id="MathJax-Span-53"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 1.34em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-54" style="font-family: MathJax_Math-italic-Web;">x</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.57em; top: -2.95em; position: absolute;"><span class="texatom" id="MathJax-Span-55"><span class="mrow" id="MathJax-Span-56"><span class="mn" id="MathJax-Span-57" style="font-family: MathJax_Main-Web; font-size: 70.7%;"><span style="font-size:14px;">10</span></span></span></span> <span style="width: 0px; height: 2.59em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-58" style="padding-left: 0.22em; font-family: MathJax_Main-Web;">−</span><span class="mn" id="MathJax-Span-59" style="padding-left: 0.22em; font-family: MathJax_Main-Web;">9.6</span><span class="msubsup" id="MathJax-Span-60"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 0.99em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-61" style="font-family: MathJax_Math-italic-Web;">x</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.57em; top: -2.55em; position: absolute;"><span class="mn" id="MathJax-Span-62" style="font-family: MathJax_Main-Web; font-size: 70.7%;"><span style="font-size:14px;">8</span></span> <span style="width: 0px; height: 2.18em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-63" style="padding-left: 0.22em; font-family: MathJax_Main-Web;">+</span><span class="mn" id="MathJax-Span-64" style="padding-left: 0.22em; font-family: MathJax_Main-Web;">7.2</span><span class="msubsup" id="MathJax-Span-65"><span style="width: 1px; height: 0px; overflow: hidden; margin-right: -1px; display: inline-block;"></span><span style="width: 0.99em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -2.47em; position: absolute; clip: rect(1.86em, 1000em, 2.66em, -0.48em);"><span class="mi" id="MathJax-Span-66" style="font-family: MathJax_Math-italic-Web;">x</span> <span style="width: 0px; height: 2.47em; display: inline-block;"></span></span><span style="left: 0.57em; top: -2.55em; position: absolute;"><span class="mn" id="MathJax-Span-67" style="font-family: MathJax_Main-Web; font-size: 70.7%;"><span style="font-size:14px;">2</span></span> <span style="width: 0px; height: 2.18em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-68" style="padding-left: 0.22em; font-family: MathJax_Main-Web;">+</span><span class="mi" id="MathJax-Span-69" style="padding-left: 0.22em; font-family: MathJax_Math-italic-Web;">x</span></span> <span style="width: 0px; height: 3.97em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.48em; overflow: hidden; vertical-align: -0.38em; border-left-color: currentColor; border-left-width: 0em; border-left-style: solid; display: inline-block;"></span></span></nobr></span>

为例，存储这个多项式的数组如下图：
<img title="" alt="这里写图片描述" src="https://img-blog.csdn.net/20150912111913706" />
　　可以看出，这种方案适合对某些多项式的处理。但是，在处理一些次数高但项数少的多项式时，存在浪费空间的现象，会有很多闲置的0。
　　可以使用如下定义的单链表结构存储多项式：链表中的每一个结点是多项式中的一项，结点的数据域包括指数和系数两部分，由指针域连接起多项式中的各项。
<code>
typedef struct pnode  //定义单链表结点类型，保存多项式中的一项，链表构成多项式 {
double coef;      //系数，浮点数
int exp;          //指数，正整数*
struct pnode *next; //指向下一项的指针
} PolyNode;
</code>
　　用于表示多项式的链表将如下图所示，在建立多项式的链表时，已经令结点按指数由大到小的顺序排列。
<img title="" alt="这里写图片描述" src="https://img-blog.csdn.net/20150912112053759" /></p><p><strong>2、多项式加法在链表存储结构下的实现</strong>
　　链表存储结构下，多项式加法的实现 在如上定义的单链表存储结构基础上，讨论实现多项式加法的算法。
　　两个多项式相加，其规则是对具有相同指数的项，令其系数相加。设两个待相加的多项式的链表的头指针分别为head1（第一个多项式）和head2（第二个多项式），两者的和保存到链表head1中。只需要先将head1和head2链表的首结点作为当前结点（分别用p1和p2指向）开始检测，在遍历链表的过程中，分情况作如下处理：
　　（1）若两个多项式中当前结点的指数值相同，则它们的系数相加，结果保存到p1结点，并将p2结点删除。如果相加后的系数不为0，p1指向第一个多项式的下一个结点，准备随后的工作，否则，不保存系数为0的项，将当前p1结点删除。
　　（2）当两个多项式中对应结点的指数值不相等时，若p1指向的结点的指数大，则p1简单地指向下一结点即可；而p2指向的结点大时，需要将p2结点插入到p1前，然后p2再重新指回到第二个多项式中的下一结点，继续进行处理。
　　（3）检测过程直到其中的任一个链表结束。若p1不为空，第一个多项式中的剩余项已经在链表中，不作处理，如果p2不为空，只需要将p2链接到相加后的第一个多项式末尾。
　　上面的讨论假设多项式链表中，已经按指数由大到小排序，在加法之前，采取多种手段保证这一前提成立。</p><p>*/</p></blockquote>

#include <stdio.h>
#include <malloc.h>
#define MAX 20          //多项式最多项数
typedef struct      //定义存放多项式的数组类型
{
double coef;        //系数
int exp;            //指数
} PolyArray;

typedef struct pnode    //定义单链表结点类型，保存多项式中的一项，链表构成多项式
{
double coef;        //系数
int exp;            //指数
struct pnode *next;
} PolyNode;

void DispPoly(PolyNode *L)  //输出多项式
{
bool first=true;        //first为true表示是第一项
PolyNode *p=L->next;
while (p!=NULL)
{
if (first)
first=false;
else if (p->coef>0)
printf("+");
if (p->exp==0)
printf("%g",p->coef);
else if (p->exp==1)
printf("%gx",p->coef);
else
printf("%gx^%d",p->coef,p->exp);
p=p->next;
}
printf("\n");
}

void DestroyList(PolyNode *&L)  //销毁单链表
{
PolyNode *p=L,*q=p->next;
while (q!=NULL)
{
free(p);
p=q;
q=p->next;
}
free(p);
}

void CreateListR(PolyNode *&L, PolyArray a[], int n) //尾插法建表
{
PolyNode *s,*r;
int i;
L=(PolyNode *)malloc(sizeof(PolyNode)); //创建头结点
L->next=NULL;
r=L;                        //r始终指向终端结点,开始时指向头结点
for (i=0; i<n; i++)
{
s=(PolyNode *)malloc(sizeof(PolyNode));//创建新结点
s->coef=a[i].coef;
s->exp=a[i].exp;
r->next=s;              //将*s插入*r之后
r=s;
}
r->next=NULL;               //终端结点next域置为NULL
}

void Sort(PolyNode *&head)      //按exp域递减排序
{
PolyNode *p=head->next,*q,*r;
if (p!=NULL)                //若原单链表中有一个或以上的数据结点
{
r=p->next;              //r保存*p结点后继结点的指针
p->next=NULL;           //构造只含一个数据结点的有序表
p=r;
while (p!=NULL)
{
r=p->next;          //r保存*p结点后继结点的指针
q=head;
while (q->next!=NULL && q->next->exp>p->exp)
q=q->next;      //在有序表中找插入*p的前驱结点*q
p->next=q->next;    //将*p插入到*q之后
q->next=p;
p=r;
}
}
}

void Add(PolyNode *ha,PolyNode *hb,PolyNode *&hc)  //求两有序集合的并，完成加法
{
PolyNode *pa=ha->next,*pb=hb->next,*s,*tc;
double c;
hc=(PolyNode *)malloc(sizeof(PolyNode));        //创建头结点
tc=hc;
while (pa!=NULL && pb!=NULL)
{
if (pa->exp>pb->exp)
{
s=(PolyNode *)malloc(sizeof(PolyNode)); //复制结点
s->exp=pa->exp;
s->coef=pa->coef;
tc->next=s;
tc=s;
pa=pa->next;
}
else if (pa->exp<pb->exp)
{
s=(PolyNode *)malloc(sizeof(PolyNode)); //复制结点
s->exp=pb->exp;
s->coef=pb->coef;
tc->next=s;
tc=s;
pb=pb->next;
}
else                //pa->exp=pb->exp
{
c=pa->coef+pb->coef;
if (c!=0)       //系数之和不为0时创建新结点
{
s=(PolyNode *)malloc(sizeof(PolyNode)); //复制结点
s->exp=pa->exp;
s->coef=c;
tc->next=s;
tc=s;
}
pa=pa->next;
pb=pb->next;
}
}
if (pb!=NULL) pa=pb;    //复制余下的结点
while (pa!=NULL)
{
s=(PolyNode *)malloc(sizeof(PolyNode)); //复制结点
s->exp=pa->exp;
s->coef=pa->coef;
tc->next=s;
tc=s;
pa=pa->next;
}
tc->next=NULL;
}

int main()
{
PolyNode *ha,*hb,*hc;
PolyArray a[]= {{1.2,0},{2.5,1},{3.2,3},{-2.5,5}};
PolyArray b[]= {{-1.2,0},{2.5,1},{3.2,3},{2.5,5},{5.4,10}};
CreateListR(ha,a,4);
CreateListR(hb,b,5);
printf("原多项式A:   ");
DispPoly(ha);
printf("原多项式B:   ");
DispPoly(hb);
Sort(ha);
Sort(hb);
printf("有序多项式A: ");
DispPoly(ha);
printf("有序多项式B: ");
DispPoly(hb);
Add(ha,hb,hc);
printf("多项式相加:  ");
DispPoly(hc);
DestroyList(ha);
DestroyList(hb);
DestroyList(hc);
return 0;
}