【蓝桥杯】【啊哈！算法】深度优先搜索——坑爹的奥数
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目录
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- 【啊哈！算法】系列文章目录
-
- 需求介绍
- 解答
需求介绍
将三个空格分别填入两个三位数相加的形式中（如\alpha \beta \gamma + \delta \epsilon \zeta = \eta \theta \iota），并要求每个从1到9的数字仅能使用一次以满足等式要求。例如\alpha\beta\gamma + \delta\epsilon\zeta = \eta\theta\iota就是一个符合要求的例子实例。那么总共有多少种符合要求的组合方式呢？注意： \alpha\beta\gamma + \delta\epsilon\zeta = \eta\theta\iota与\delta\epsilon\zeta + \alpha\beta\gamma = \eta\theta\iota被视为同一种组合形式！
解答
#include <iostream>
#include <ctime>
using namespace std;
#define N 9
int map[N] = {1,2,3,4,5,6,7,8,9};
int a[N];
int book[N] = {0};
int count=0;
//暴力dfs查找
void dfs1(int step)
{
if(step==N)
{
if(a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5] == a[6]*100+a[7]*10+a[8])
{
count++;
}
return;
}
for(char i=0; i<N; i++)
{
if(book[i]==0)
{
a[step] = map[i];
book[i] = 1;
dfs1(step+1);
book[i] = 0;
}
}
}
//优化1：前两个数的和不可能大于1000
void dfs2(int step)
{
if(step==6)
{
if(a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5] >= 1000)
{
return;
}
}
if(step==N)
{
if(a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5] == a[6]*100+a[7]*10+a[8])
{
count++;
}
return;
}
for(char i=0; i<N; i++)
{
if(book[i]==0)
{
a[step] = map[i];
book[i] = 1;
dfs2(step+1);
book[i] = 0;
}
}
}
int max_element_f(void)
{
int max=0;
for(char i=0; i<N; i++)
{
if(book[i]==0 && map[i]>max)
{
max = map[i];
}
}
return max;
}
//优化2：排除 剩余的三个数字中的最大的数小于前两个数的和的百位
void dfs3(int step)
{
if(step==6)
{
int sum = a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5];
if(sum >= 1000)
{
return;
}
if(sum/100 > max_element_f())
{
return;
}
}
if(step==N)
{
if(a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5] == a[6]*100+a[7]*10+a[8])
{
count++;
}
return;
}
for(char i=0; i<N; i++)
{
if(book[i]==0)
{
a[step] = map[i];
book[i] = 1;
dfs3(step+1);
book[i] = 0;
}
}
}
//优化3：分别查看剩余的三个数字 是否与sum个十百位中的其中一个相等
void dfs4(int step)
{
if(step==6)
{
int sum = a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5];
if(sum >= 1000)
{
return;
}
int bai = sum/100;
int shi = sum/10%10;
int ge = sum%10;
for(char i=0; i<N; i++)
{
if(book[i]==0)
{
if(!(map[i]==ge||map[i]==shi||map[i]==bai)) return;
}
}
}
if(step==N)
{
if(a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5] == a[6]*100+a[7]*10+a[8])
{
count++;
}
return;
}
for(char i=0; i<N; i++)
{
if(book[i]==0)
{
a[step] = map[i];
book[i] = 1;
dfs4(step+1);
book[i] = 0;
}
}
}
//带输出的dfs
void dfs0(int step)
{
if(step==6)
{
int sum = a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5];
if(sum >= 1000)
{
return;
}
int bai = sum/100;
int shi = sum/10%10;
int ge = sum%10;
for(char i=0; i<N; i++)
{
if(book[i]==0)
{
if(!(map[i]==ge||map[i]==shi||map[i]==bai)) return;
}
}
}
if(step==N)
{
if(a[0]*100+a[1]*10+a[2] + a[3]*100+a[4]*10+a[5] == a[6]*100+a[7]*10+a[8])
{
cout << a[0] << a[1] << a[2] << " + " << a[3] << a[4] << a[5] << \
" = " << a[6] << a[7] << a[8] << endl;
}
return;
}
for(char i=0; i<N; i++)
{
if(book[i]==0)
{
a[step] = map[i];
book[i] = 1;
dfs0(step+1);
book[i] = 0;
}
}
}
int main()
{
clock_t start, end;
int clock_num=10;
count = 0;
start = clock();
for(char i=0; i<clock_num; i++)dfs1(0);
end = clock();
cout << "dfs1: ";
cout << (end-start) << endl;
//优化1：前两个数的和不可能大于1000
count = 0;
start = clock();
for(char i=0; i<clock_num; i++)dfs2(0);
end = clock();
cout << "dfs2: ";
cout << (end-start) << endl;
//优化2：排除 剩余的三个数字中的最大的数小于前两个数的和的百位
count = 0;
start = clock();
for(char i=0; i<clock_num; i++)dfs3(0);
end = clock();
cout << "dfs3: ";
cout << (end-start) << endl;
//优化3：分别查看剩余的三个数字 是否与sum个十百位中的其中一个相等
count = 0;
start = clock();
for(char i=0; i<clock_num; i++)dfs4(0);
end = clock();
cout << "dfs4: ";
cout << (end-start) << endl;
cout << "共有" << count/2/clock_num << "种组合，分别为";
cout << "（此处输出" << count/clock_num << "种，包括重复的）:" << endl;
dfs0(0);
return 0;
}输出为：
dfs1: 245
dfs2: 96
dfs3: 79
dfs4: 24
共有168种组合，分别为（此处输出336种，包括重复的）:
124 + 659 = 783
125 + 739 = 864
127 + 359 = 486
127 + 368 = 495
128 + 367 = 495
128 + 439 = 567
129 + 357 = 486
129 + 438 = 567
129 + 654 = 783
129 + 735 = 864
134 + 658 = 792
135 + 729 = 864
138 + 429 = 567
138 + 654 = 792
139 + 428 = 567
139 + 725 = 864
142 + 596 = 738
142 + 695 = 837
143 + 586 = 729
145 + 692 = 837
146 + 583 = 729
146 + 592 = 738
152 + 487 = 639
152 + 784 = 936
154 + 629 = 783
154 + 638 = 792
154 + 782 = 936
157 + 329 = 486
157 + 482 = 639
158 + 634 = 792
159 + 327 = 486
159 + 624 = 783
162 + 387 = 549
162 + 783 = 945
163 + 782 = 945
167 + 328 = 495
167 + 382 = 549
168 + 327 = 495
173 + 286 = 459
173 + 295 = 468
175 + 293 = 468
176 + 283 = 459
182 + 367 = 549
182 + 394 = 576
182 + 457 = 639
182 + 493 = 675
182 + 754 = 936
182 + 763 = 945
183 + 276 = 459
183 + 492 = 675
183 + 546 = 729
183 + 762 = 945
184 + 392 = 576
184 + 752 = 936
186 + 273 = 459
186 + 543 = 729
187 + 362 = 549
187 + 452 = 639
192 + 384 = 576
192 + 483 = 675
192 + 546 = 738
192 + 645 = 837
193 + 275 = 468
193 + 482 = 675
194 + 382 = 576
195 + 273 = 468
195 + 642 = 837
196 + 542 = 738
214 + 569 = 783
214 + 659 = 873
215 + 478 = 693
215 + 748 = 963
216 + 378 = 594
216 + 738 = 954
218 + 349 = 567
218 + 376 = 594
218 + 439 = 657
218 + 475 = 693
218 + 736 = 954
218 + 745 = 963
219 + 348 = 567
219 + 438 = 657
219 + 564 = 783
219 + 654 = 873
234 + 657 = 891
235 + 746 = 981
236 + 718 = 954
236 + 745 = 981
237 + 654 = 891
238 + 419 = 657
238 + 716 = 954
239 + 418 = 657
241 + 596 = 837
243 + 576 = 819
243 + 675 = 918
245 + 673 = 918
245 + 718 = 963
245 + 736 = 981
246 + 573 = 819
246 + 591 = 837
246 + 735 = 981
248 + 319 = 567
248 + 715 = 963
249 + 318 = 567
251 + 397 = 648
254 + 619 = 873
254 + 637 = 891
257 + 391 = 648
257 + 634 = 891
259 + 614 = 873
264 + 519 = 783
269 + 514 = 783
271 + 593 = 864
271 + 683 = 954
273 + 186 = 459
273 + 195 = 468
273 + 546 = 819
273 + 591 = 864
273 + 645 = 918
273 + 681 = 954
275 + 193 = 468
275 + 418 = 693
275 + 643 = 918
276 + 183 = 459
276 + 318 = 594
276 + 543 = 819
278 + 316 = 594
278 + 415 = 693
281 + 394 = 675
281 + 673 = 954
283 + 176 = 459
283 + 671 = 954
284 + 391 = 675
286 + 173 = 459
291 + 357 = 648
291 + 384 = 675
291 + 546 = 837
291 + 573 = 864
293 + 175 = 468
293 + 571 = 864
294 + 381 = 675
295 + 173 = 468
296 + 541 = 837
297 + 351 = 648
314 + 658 = 972
316 + 278 = 594
317 + 529 = 846
317 + 628 = 945
318 + 249 = 567
318 + 276 = 594
318 + 627 = 945
318 + 654 = 972
319 + 248 = 567
319 + 527 = 846
324 + 567 = 891
324 + 657 = 981
327 + 159 = 486
327 + 168 = 495
327 + 519 = 846
327 + 564 = 891
327 + 618 = 945
327 + 654 = 981
328 + 167 = 495
328 + 617 = 945
329 + 157 = 486
329 + 517 = 846
341 + 586 = 927
342 + 576 = 918
346 + 572 = 918
346 + 581 = 927
348 + 219 = 567
349 + 218 = 567
351 + 297 = 648
352 + 467 = 819
354 + 618 = 972
354 + 627 = 981
357 + 129 = 486
357 + 291 = 648
357 + 462 = 819
357 + 624 = 981
358 + 614 = 972
359 + 127 = 486
362 + 187 = 549
362 + 457 = 819
364 + 527 = 891
367 + 128 = 495
367 + 182 = 549
367 + 452 = 819
367 + 524 = 891
368 + 127 = 495
372 + 546 = 918
376 + 218 = 594
376 + 542 = 918
378 + 216 = 594
381 + 294 = 675
381 + 546 = 927
382 + 167 = 549
382 + 194 = 576
384 + 192 = 576
384 + 291 = 675
386 + 541 = 927
387 + 162 = 549
391 + 257 = 648
391 + 284 = 675
392 + 184 = 576
394 + 182 = 576
394 + 281 = 675
397 + 251 = 648
415 + 278 = 693
418 + 239 = 657
418 + 275 = 693
419 + 238 = 657
428 + 139 = 567
429 + 138 = 567
438 + 129 = 567
438 + 219 = 657
439 + 128 = 567
439 + 218 = 657
452 + 187 = 639
452 + 367 = 819
457 + 182 = 639
457 + 362 = 819
462 + 357 = 819
467 + 352 = 819
475 + 218 = 693
478 + 215 = 693
482 + 157 = 639
482 + 193 = 675
483 + 192 = 675
487 + 152 = 639
492 + 183 = 675
493 + 182 = 675
514 + 269 = 783
517 + 329 = 846
519 + 264 = 783
519 + 327 = 846
524 + 367 = 891
527 + 319 = 846
527 + 364 = 891
529 + 317 = 846
541 + 296 = 837
541 + 386 = 927
542 + 196 = 738
542 + 376 = 918
543 + 186 = 729
543 + 276 = 819
546 + 183 = 729
546 + 192 = 738
546 + 273 = 819
546 + 291 = 837
546 + 372 = 918
546 + 381 = 927
564 + 219 = 783
564 + 327 = 891
567 + 324 = 891
569 + 214 = 783
571 + 293 = 864
572 + 346 = 918
573 + 246 = 819
573 + 291 = 864
576 + 243 = 819
576 + 342 = 918
581 + 346 = 927
583 + 146 = 729
586 + 143 = 729
586 + 341 = 927
591 + 246 = 837
591 + 273 = 864
592 + 146 = 738
593 + 271 = 864
596 + 142 = 738
596 + 241 = 837
614 + 259 = 873
614 + 358 = 972
617 + 328 = 945
618 + 327 = 945
618 + 354 = 972
619 + 254 = 873
624 + 159 = 783
624 + 357 = 981
627 + 318 = 945
627 + 354 = 981
628 + 317 = 945
629 + 154 = 783
634 + 158 = 792
634 + 257 = 891
637 + 254 = 891
638 + 154 = 792
642 + 195 = 837
643 + 275 = 918
645 + 192 = 837
645 + 273 = 918
654 + 129 = 783
654 + 138 = 792
654 + 219 = 873
654 + 237 = 891
654 + 318 = 972
654 + 327 = 981
657 + 234 = 891
657 + 324 = 981
658 + 134 = 792
658 + 314 = 972
659 + 124 = 783
659 + 214 = 873
671 + 283 = 954
673 + 245 = 918
673 + 281 = 954
675 + 243 = 918
681 + 273 = 954
683 + 271 = 954
692 + 145 = 837
695 + 142 = 837
715 + 248 = 963
716 + 238 = 954
718 + 236 = 954
718 + 245 = 963
725 + 139 = 864
729 + 135 = 864
735 + 129 = 864
735 + 246 = 981
736 + 218 = 954
736 + 245 = 981
738 + 216 = 954
739 + 125 = 864
745 + 218 = 963
745 + 236 = 981
746 + 235 = 981
748 + 215 = 963
752 + 184 = 936
754 + 182 = 936
762 + 183 = 945
763 + 182 = 945
782 + 154 = 936
782 + 163 = 945
783 + 162 = 945
784 + 152 = 936全部评论 (0)
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