Сборник задач и упражнений по теории алгоритмов. Учебно-методическое пособие
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Федеральное государственное автономное образовательное учреждение высшего образования «РОССИЙСКИЙ УНИВЕРСИТЕТ ДРУЖБЫ НАРОДОВ»
Л.И. Кулькова, С.И. Салпагаров
СБОРНИК ЗАДАЧ И УПРАЖНЕНИЙ ПО ТЕОРИИ АЛГОРИТМОВ
Учебно-методическое пособие
Москва Российский университет дружбы народов
2017
УДК 510.5:004.43(072.8) |
У т в е р ж д е н о |
ББК 22.12+32.973.26-018.1 |
РИС Ученого совета |
К90 |
Российского университета |
|
дружбы народов |
Кулькова, Л. И.
К90 Сборник задач и упражнений по теории алгоритмов : учебно-методическое пособие / Л. И. Кулькова, С. И. Салпагаров. – Москва : РУДН, 2017. – 48 с. : ил.
В работе изложены как базовые, так и наиболее распространенные и используемые разделы теории алгоритмов. Пособие составлено в виде сборника практических заданий и упражнений, при этом содержит, необходимые для ознакомления, пояснения по рассматриваемым разделам.
Предназначено для студентов средних и старших курсов математического профиля.
ISBN 978-5-209-08067-1 |
© Кулькова Л.И., Салпагаров С.И., 2017 |
|
© Российский университет |
|
дружбы народов, 2017 |
, ɇɟɮɨɪɦɚɥɶɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɚɥɝɨɪɢɬɦɚ
ɉɨɧɹɬɢɟ ɚɥɝɨɪɢɬɦɚ ɧɟɮɨɪɦɚɥɶɧɨ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɤɚɤ ɩɨɫɥɟɞɨɜɚɬɟɥɶ
ɧɨɫɬɶ ɷɬɚɩɨɜ ɨɩɢɫɵɜɚɸɳɭɸ ɫɩɨɫɨɛ ɪɟɲɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ
Ⱦɚɧɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɧɟ ɹɜɥɹɟɬɫɹ ɩɨɥɧɵɦ Ⱦɥɹ ɩɨɥɧɨɬɵ ɨɩɪɟɞɟɥɟɧɢɹ ɩɟ
ɪɟɱɢɫɥɢɦ ɫɜɨɣɫɬɜɚ ɚɥɝɨɪɢɬɦɚȾɢɫɤɪɟɬɧɨɫɬɶ± ɨɞɧɨ ɢɡ ɜɚɠɧɟɣɲɢɯ ɬɪɟɛɨɜɚɧɢɣ ɤɨɬɨɪɨɟ ɡɚɤɥɸɱɚɟɬɫɹ
ɜ ɬɨɦ ɱɬɨ ɚɥɝɨɪɢɬɦ ɞɨɥɠɟɧ ɛɵɬɶ ɪɚɡɛɢɬ ɧɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɨɬɞɟɥɶɧɵɯ ɲɚ
ɝɨɜ ȼɨɡɧɢɤɚɸɳɚɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɬɚɤɨɝɨ ɪɚɡɛɢɟɧɢɹ ɡɚɩɢɫɶ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ
ɭɩɨɪɹɞɨɱɟɧɧɭɸ ɫɨɜɨɤɭɩɧɨɫɬɶ ɱɟɬɤɨ ɪɚɡɞɟɥɟɧɧɵɯ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɩɪɟɞɩɢɫɚɧɢɣ
ɲɚɝɨɜ ɤɨɦɚɧɞ ɨɩɟɪɚɬɨɪɨɜ ɨɛɪɚɡɭɸɳɢɯ ɞɢɫɤɪɟɬɧɭɸ ɫɬɪɭɤɬɭɪɭ ɚɥɝɨɪɢɬɦɚ
Ɍɨɥɶɤɨ ɜɵɩɨɥɧɢɜ ɬɪɟɛɨɜɚɧɢɹ ɨɞɧɨɝɨ ɲɚɝɚ ɦɨɠɧɨ ɩɟɪɟɣɬɢ ɤ ɜɵɩɨɥɧɟɧɢɸ ɫɥɟ
ɞɭɸɳɟɝɨȾɟɬɟɪɦɢɧɢɪɨɜɚɧɧɨɫɬɶɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ ɱɬɨ ɥɸɛɨɣ ɲɚɝ ɚɥɝɨɪɢɬɦɚ
ɞɨɥɠɟɧ ɛɵɬɶ ɱɟɬɤɢɦ ɢ ɧɟ ɨɫɬɚɜɥɹɬɶ ɦɟɫɬɚ ɞɥɹ ɩɪɨɢɡɜɨɥɚ ɢɫɩɨɥɧɢɬɟɥɹ Ⱦɨɥɠɧɨ
ɛɵɬɶ ɞɨɫɬɚɬɨɱɧɨ ɢɧɮɨɪɦɚɰɢɢ ɞɥɹ ɨɞɧɨɡɧɚɱɧɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɞɟɣɫɬɜɢɣ Ⱦɪɭ
ɝɢɦɢ ɫɥɨɜɚɦɢ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɥɸɛɨɝɨ ɲɚɝɚ ɞɨɥɠɧɨ ɬɪɟɛɨɜɚɬɶɫɹ ɧɟ ɧɚɥɢɱɢɟ
ɬɜɨɪɱɟɫɤɢɯ ɫɩɨɫɨɛɧɨɫɬɟɣ ɚ ɦɟɯɚɧɢɱɟɫɤɨɟ ɫɥɟɞɨɜɚɧɢɟ ɭɤɚɡɚɧɢɹɦ ɚɥɝɨɪɢɬɦɚ Ɍɪɟɛɨɜɚɧɢɟɭɩɨɪɹɞɨɱɟɧɧɨɫɬɢɨɡɧɚɱɚɟɬ ɱɟɬɤɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɜɵɩɨɥɧɟɧɢɹ ɲɚɝɨɜ ɚɥɝɨɪɢɬɦɚ Ɉɞɧɚɤɨ ɷɬɨ ɧɟ ɨɡɧɚɱɚɟɬ ɱɬɨ ɷɬɚɩɵ ɞɨɥɠɧɵ ɜɵ ɩɨɥɧɹɬɶɫɹ ɜ ɫɬɪɨɝɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɨɞɢɧ ɡɚ ɞɪɭɝɢɦ ɇɚɩɪɢɦɟɪ ɜ ɦɧɨɝɨ ɩɨɬɨɱɧɵɯ ɩɪɨɝɪɚɦɦɚɯ ɫɭɳɟɫɬɜɭɟɬ ɪɹɞ ɲɚɝɨɜ ɤɨɬɨɪɵɟ ɜɵɩɨɥɧɹɸɬɫɹ ɨɞɧɨɜɪɟ ɦɟɧɧɨ ɬɨ ɥɢ ɧɚ ɪɚɡɧɵɯ ɩɪɨɰɟɫɫɨɪɚɯ ɦɧɨɝɨɩɪɨɰɟɫɫɨɪɧɨɣ ɦɚɲɢɧɵ ɬɨ ɥɢ ɜ ɪɟ ɠɢɦɟ ɪɚɡɞɟɥɟɧɢɹ ɜɪɟɦɟɧɢ ɧɚ ɨɞɧɨɩɪɨɰɟɫɫɨɪɧɨɦ ɤɨɦɩɶɸɬɟɪɟ ȼ ɬɚɤɢɯ ɫɥɭɱɚɹɯ ɚɥɝɨɪɢɬɦɫɨɞɟɪɠɢɬɧɟɤɨɬɨɪɨɟɤɨɥɢɱɟɫɬɜɨɲɚɝɨɜɤɨɬɨɪɵɟɪɚɡɜɟɬɜɥɹɸɬɫɹɢ
ɜɧɨɜɶ ɨɛɴɟɞɢɧɹɸɬɫɹ ɩɨ ɦɟɪɟ ɢɯ ɜɵɩɨɥɧɟɧɢɹ Ⱥɥɝɨɪɢɬɦ ɞɨɥɠɟɧ ɫɨɫɬɨɹɬɶ ɢɡɜɵɩɨɥɧɢɦɵɯɷɬɚɩɨɜ ɑɬɨɛɵ ɨɰɟɧɢɬɶ
ɜɚɠɧɨɫɬɶ ɷɬɨɝɨ ɭɫɥɨɜɢɹ ɪɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɲɚɝɨɜ ɗɬɚɩ ɋɨɫɬɚɜɢɬɶ ɫɩɢɫɨɤ ɜɫɟɯ ɰɟɥɵɯ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɱɢɫɟɥ ɗɬɚɩ ɍɩɨɪɹɞɨɱɢɬɶ ɷɬɨɬ ɫɩɢɫɨɤ ɜ ɭɛɵɜɚɸɳɟɦ ɩɨɪɹɞɤɟ ɨɬ ɛɨɥɶɲɟɝɨ
ɡɧɚɱɟɧɢɹ ɤ ɦɟɧɶɲɟɦɭ
ϯ
ɗɬɚɩ ȼɵɛɪɚɬɶ ɩɟɪɜɨɟ ɱɢɫɥɨ ɢɡ ɨɬɫɨɪɬɢɪɨɜɚɧɧɨɝɨ ɫɩɢɫɤɚ ɉɟɪɜɵɟ ɞɜɚ ɷɬɚɩɚ ɜɵɩɨɥɧɢɬɶ ɧɟɜɨɡɦɨɠɧɨ ɩɨɷɬɨɦɭ ɷɬɨɬ ɧɚɛɨɪ ɢɧɫɬɪɭɤ
ɰɢɣ ɧɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɚɥɝɨɪɢɬɦ Ɉɛɹɡɚɬɟɥɶɧɨɟ ɬɪɟɛɨɜɚɧɢɟ ɤ ɚɥɝɨɪɢɬɦɚɦ ±ɪɟɡɭɥɶɬɚɬɢɜɧɨɫɬɶɋɦɵɫɥ
ɷɬɨɝɨ ɬɪɟɛɨɜɚɧɢɹ ɫɨɫɬɨɢɬ ɜ ɬɨɦ ɱɬɨ ɩɪɢ ɬɨɱɧɨɦ ɢɫɩɨɥɧɟɧɢɢ ɜɫɟɯ ɩɪɟɞɩɢɫɚɧɢɣ
ɚɥɝɨɪɢɬɦɚ ɩɪɨɰɟɫɫ ɞɨɥɠɟɧ ɩɪɟɤɪɚɬɢɬɶɫɹ ɡɚ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɲɚɝɨɜ ɢ ɩɪɢ ɷɬɨɦ
ɞɨɥɠɟɧ ɩɨɥɭɱɢɬɶɫɹ ɨɩɪɟɞɟɥɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ ȼɵɜɨɞ ɨ ɬɨɦ ɱɬɨ ɪɟɲɟɧɢɹ ɧɟ ɫɭ
ɳɟɫɬɜɭɟɬ ± ɬɨɠɟ ɪɟɡɭɥɶɬɚɬ
ɇɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɵ ɚɥɝɨɪɢɬɦɵ ɨɛɟɫɩɟɱɢɜɚɸɳɢɟ ɪɟɲɟɧɢɟ ɧɟ
ɨɞɧɨɣ ɤɨɧɤɪɟɬɧɨɣ ɡɚɞɚɱɢ ɚ ɧɟɤɨɬɨɪɨɝɨ ɤɥɚɫɫɚ ɡɚɞɚɱ ɞɚɧɧɨɝɨ ɬɢɩɚ ɗɬɨ ɫɜɨɣ ɫɬɜɨ ɚɥɝɨɪɢɬɦɚ ɧɚɡɵɜɚɸɬ ɦɚɫɫɨɜɨɫɬɶɸ ȼ ɩɪɨɫɬɟɣɲɟɦ ɫɥɭɱɚɟ ɦɚɫɫɨɜɨɫɬɶ ɨɛɟɫɩɟɱɢɜɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɪɚɡɥɢɱɧɵɯ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ
Ɍɨɝɞɚ ɩɨɥɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɚɥɝɨɪɢɬɦɚ ɫɜɨɞɢɬɫɹ ɤ ɫɥɟɞɭɸɳɟɦɭ
Ⱥɥɝɨɪɢɬɦ ± ɷɬɨ ɭɩɨɪɹɞɨɱɟɧɧɵɣ ɤɨɧɟɱɧɵɣ ɧɚɛɨɪ ɞɢɫɤɪɟɬɧɵɯ ɞɟɬɟɪɦɢɧɢɪɨɜɚɧ
ɧɵɯ ɜɵɩɨɥɧɢɦɵɯ ɲɚɝɨɜ ɩɪɢɜɨɞɹɳɢɣ ɤ ɪɟɲɟɧɢɸ ɧɟɤɨɬɨɪɨɝɨ ɤɥɚɫɫɚ ɡɚɞɚɱ
Ɂɚɞɚɧɢɹ
ȼɨ ɜɫɟɯ ɡɚɞɚɧɢɹɯ ɧɟɨɛɯɨɞɢɦɨ ɪɚɡɪɚɛɨɬɚɬɶ ɫɯɟɦɵ ɚɥɝɨɪɢɬɦɨɜ ɢ ɩɪɨɚɧɚ
ɥɢɡɢɪɨɜɚɬɶ ɩɪɨɰɟɫɫ ɪɟɚɥɢɡɚɰɢɢ ɚɥɝɨɪɢɬɦɚ ɬ ɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɲɚɝɨɜ ɤɨ
ɬɨɪɚɹ ɛɭɞɟɬ ɩɨɪɨɠɞɟɧɚ ɩɪɢ ɩɪɢɦɟɧɟɧɢɢ ɚɥɝɨɪɢɬɦɚ ɤ ɤɨɧɤɪɟɬɧɵɦ ɢɫɯɨɞɧɵɦ
ɞɚɧɧɵɦ Ɋɚɡɪɚɛɨɬɚɬɶ ɫɥɨɜɟɫɧɵɟ ɚɥɝɨɪɢɬɦɵ ɜɵɱɢɬɚɧɢɹ ɢɡ ɧɟɤɨɬɨɪɨɝɨ ɱɢɫɥɚ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɱɢɫɟɥ ǡ ǡǥǡ ɚ ɚɥɝɨɪɢɬɦ ɜɵɱɢɫɥɟɧɢɹ ɩɨ ɮɨɪɦɭɥɟ
ǥ ڮ ɛ ɚɥɝɨɪɢɬɦ ɜɵɱɢɫɥɟɧɢɹ ɩɨ ɮɨɪɦɭɥɟ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɜɵɱɢɫɥɟɧɢɹ ɩɨ ɮɨɪɦɭɥɟ
σ ǡ
ϰ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɜɵɱɢɫɥɟɧɢɹ ɩɨ ɮɨɪɦɭɥɟ
ς ή ǡ Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɩɨɢɫɤɚ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɨɬɪɢɰɚɬɟɥɶɧɨɝɨ ɢ ɦɢɧɢ
ɦɚɥɶɧɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɷɥɟɦɟɧɬɨɜ ɢɡ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɱɢɫɟɥ ɡɚɞɚɧɧɨɣ ɜ ɜɢɞɟ ɨɞɧɨɦɟɪɧɨɝɨ ɦɚɫɫɢɜɚ ǡ ǡǥǡ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɨɞɢɧɚɤɨɜɵɯ ɱɢɫɟɥ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɡɚɞɚɧɧɨɣ ɜ ɜɢɞɟ ɨɞɧɨɦɟɪɧɨɝɨ ɦɚɫɫɢɜɚǡ ǡǥǡ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɩɨɞɫɱɟɬɚ ɤɨɥɢɱɟɫɬɜɚ ɨɞɢɧɚɤɨɜɵɯ ɷɥɟɦɟɧɬɨɜ ɜ ɦɚɬɪɢɰɟBɪɚɡɦɟɪɧɨɫɬɶɸn×mɊɚɡɦɟɪɧɨɫɬɶ ɦɚɬɪɢɰɵ ɜɜɨɞɢɬɫɹ ɫ ɤɥɚɜɢɚɬɭɪɵ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɭɦɧɨɠɟɧɢɹ ɦɚɬɪɢɰɵ ɧɚ ɜɟɤɬɨɪ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɭɦɧɨɠɟɧɢɹ ɦɚɬɪɢɰɵ ɧɚ ɦɚɬɪɢɰɭ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɬɪɚɧɫɩɨɧɢɪɨɜɚɧɢɹ ɦɚɬɪɢɰɵ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɪɟɚɥɢɡɭɸɳɢɣ ɨɩɟɪɚɰɢɸ ɨɛɴɟɞɢɧɟɧɢɹ ɫɥɟɞɭ ɸɳɢɯ ɞɜɭɯ ɦɧɨɠɟɫɬɜ ǡ ǡǥǡ ɢ ǡ ǡǥǡ
Ɂɚɩɢɫɚɬɶ ɚɥɝɨɪɢɬɦ ɫɨɪɬɢɪɨɜɤɢ ɦɚɫɫɢɜɚ ɪɚɡɦɟɪɧɨɫɬɢNɫ ɩɨɦɨɳɶɸ ɩɪɹɦɨɝɨ ɜɵɛɨɪɚ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɫɨɪɬɢɪɨɜɤɢ ɦɚɫɫɢɜɚ ɪɚɡɦɟɪɧɨɫɬɢNɫ ɩɨɦɨ ɳɶɸ ɨɛɦɟɧɨɜ ɢ ɡɚɩɢɫɚɬɶ ɟɝɨ ɜ ɩɫɟɜɞɨɤɨɞɟ
Ɋɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɫɨɪɬɢɪɨɜɤɢ ɦɚɫɫɢɜɚ ɪɚɡɦɟɪɧɨɫɬɢNɫ ɩɨɦɨɳɶɸ ɜɤɥɸɱɟɧɢɹ
Ɂɚɩɢɫɚɬɶ ɫ ɩɨɦɨɳɶɸ ɩɫɟɜɞɨɤɨɞɚ ɚɥɝɨɪɢɬɦ ɥɢɧɟɣɧɨɝɨ ɩɨɢɫɤɚ ɢ ɥɢ ɧɟɣɧɨɝɨ ɩɨɢɫɤɚ ɫ ɛɚɪɶɟɪɨɦ ɜ ɦɚɫɫɢɜɟ ɪɚɡɦɟɪɧɨɫɬɢN.
Ɂɚɩɢɫɚɬɶ ɫ ɩɨɦɨɳɶɸ ɩɫɟɜɞɨɤɨɞɚ ɚɥɝɨɪɢɬɦ ɩɨɢɫɤɚ ɦɟɬɨɞɨɦ ɞɟɥɟɧɢɹ ɩɨɩɨɥɚɦ ɜ ɦɚɫɫɢɜɟ ɪɚɡɦɟɪɧɨɫɬɢN
ϱ
,, Ɏɨɪɦɚɥɶɧɵɟ ɨɩɪɟɞɟɥɟɧɢɹ ɚɥɝɨɪɢɬɦɚ
Ɇɚɲɢɧɚ Ɍɶɸɪɢɧɝɚ ɆɌɆɚɲɢɧɚ Ɍɶɸɪɢɧɝɚ ± ɷɬɨ ɫɩɨɫɨɛ ɨɩɢɫɚɧɢɹ ɚɥɝɨɪɢɬɦɚ ɩɪɢ ɩɨɦɨɳɢ ɤɨɬɨ
ɪɨɝɨ ɮɨɪɦɚɥɢɡɭɟɬɫɹ ɷɬɨ ɩɨɧɹɬɢɟ
ɋɨɫɬɚɜ ɆɌ
ɚȻɟɫɤɨɧɟɱɧɚɹ ɥɟɧɬɚ ɢɫɩɨɥɶɡɭɟɦɚɹ ɞɥɹ ɯɪɚɧɟɧɢɹ ɢɧɮɨɪɦɚɰɢɢ ɪɚɡɛɢɬɚ ɧɚ ɹɱɟɣɤɢ ɜ ɤɚɠɞɭɸ ɢɡ ɤɨɬɨɪɵɯ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧ ɨɞɢɧ ɢɡ ɫɢɦɜɨɥɨɜ ɤɨɧɟɱ ɧɨɝɨ ɚɥɮɚɜɢɬɚ ǡ ǡǥǡ – ɜɧɟɲɧɢɣ ɚɥɮɚɜɢɬ ȁ ɩɭɫɬɨɣ ɫɢɦɜɨɥ ɂɫɯɨɞɧɵɟ ɞɚɧɧɵɟPɢ ɨɤɨɧɱɚɬɟɥɶɧɵɣ ɪɟɡɭɥɶɬɚɬ ± ɷɬɨ ɫɥɨɜɨ ɧɚɛɨɪ ɫɢɦɜɨɥɨɜ ɜ ɚɥɮɚɜɢɬɟ ɪɚɫɩɨɥɨɠɟɧɧɨɟ ɧɚ ɥɟɧɬɟ
b. ɍɫɬɪɨɣɫɬɜɨ ɭɩɪɚɜɥɟɧɢɹ ɍɍ ɧɚɯɨɞɢɬɫɹ ɜ ɨɞɧɨɦ ɢɡ ɫɨɫɬɨɹɧɢɣ ɦɧɨɠɟ ɫɬɜɚ ǡ ǡǥǡ ± ɜɧɭɬɪɟɧɧɢɣ ɚɥɮɚɜɢɬ ȼ ɷɬɨɦ ɦɧɨɠɟɫɬɜɟ ɜɵɞɟɥɹɸɬɫɹ ɞɜɚ ɫɨɫɬɨɹɧɢɹ ± ɧɚɱɚɥɶɧɨɟ ɤɨɝɞɚ ɫɱɢɬɵɜɚɸɳɚɹ ɝɨɥɨɜɤɚ ɧɚɯɨɞɢɬɫɹ ɧɚ ɨɦ ɫɢɦɜɨɥɟ ɜɯɨɞɧɨɝɨ ɫɥɨɜɚ ± ɡɚɤɥɸɱɢɬɟɥɶɧɨɟǤɉɨɩɚɜ ɜ ɡɚɤɥɸɱɢɬɟɥɶɧɨɟ ɫɨɫɬɨ ɹɧɢɟ ɆɌ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ
cɋɱɢɬɵɜɚɸɳɚɹ ɝɨɥɨɜɤɚ ɤɨɬɨɪɚɹ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɱɢɬɵɜɚɟɬ ɫɢɦɜɨɥ ɜ ɹɱɟɣɤɟ ɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɧɟɝɨ ɢ ɫɨɫɬɨɹɧɢɹ ɍɍ ɡɚɩɢɫɵɜɚɟɬ ɜ ɷɬɭ ɹɱɟɣɤɭ ɫɢɦɜɨɥ ɢɡ ɚɥɮɚɜɢɬɚ ɢ ɫɨɜɟɪɲɚɟɬ ɫɞɜɢɝ ɜɞɨɥɶ ɥɟɧɬɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɷɥɟɦɟɧɬɭ ɢɡ ɦɧɨɠɟɫɬɜɚ ǡ ǡɝɞɟ ± ɫɞɜɢɝ ɧɚ ɨɞɧɭ ɹɱɟɣɤɭ ɜɥɟɜɨ ± ɫɞɜɢɝ ɜɩɪɚɜɨ ± ɫɱɢɬɵɜɚɸɳɚɹ ɝɨɥɨɜɤɚ ɨɫɬɚɟɬɫɹ ɧɚ ɦɟɫɬɟ
ɉɪɨɝɪɚɦɦɚ ɆɌ ɋɚɦɚ ɩɨ ɫɟɛɟ ɆɌ ɧɢɱɟɝɨ ɧɟ ɞɟɥɚɟɬ ɑɬɨɛɵ ɡɚɫɬɚɜɢɬɶ ɟɟ ɪɚɛɨɬɚɬɶ ɧɚɞɨ
ɧɚɩɢɫɚɬɶ ɞɥɹ ɧɟɟ ɩɪɨɝɪɚɦɦɭ ɉɪɨɝɪɚɦɦɚ ɡɚɞɚɟɬɫɹ ɬɪɟɦɹ ɫɩɨɫɨɛɚɦɢ ɫɨɜɨɤɭɩ
ɧɨɫɬɶɸ ɤɨɦɚɧɞ ɜ ɜɢɞɟ ɝɪɚɮɚ ɬɚɛɥɢɰɟɣ ɫɨɨɬɜɟɬɫɬɜɢɹ
ɚ ɉɪɟɞɫɬɚɜɥɟɧɢɟ ɩɪɨɝɪɚɦɦɵ ɆɌ ɫɨɜɨɤɭɩɧɨɫɬɶɸ ɤɨɦɚɧɞ
ɉɨɥɧɨɟ ɫɨɫɬɨɹɧɢɟ ɆɌ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɫɬɨɹɧɢɟɦ ɥɟɧɬɵ ɜɧɭɬ
ɪɟɧɧɢɦ ɫɨɫɬɨɹɧɢɟɦ ɍɍ ɢ ɩɨɥɨɠɟɧɢɟɦ ɝɨɥɨɜɤɢ Ȼɭɞɟɦ ɧɚɡɵɜɚɬɶ ɟɝɨ ɤɨɧɮɢɝɭ ɪɚɰɢɟɣ ɢ ɨɛɨɡɧɚɱɚɬɶ ן ן ɝɞɟ ± ɫɨɫɬɨɹɧɢɟ ɍɍן ± ɫɥɨɜɨ ɫɥɟɜɚ ɨɬ ɝɨɥɨɜɤɢן ± ɤɨɧɤɚɬɟɧɚɰɢɹ ɫɢɦɜɨɥɚ ɧɚ ɤɨɬɨɪɨɦ ɫɬɨɢɬ ɝɨɥɨɜɤɚ ɢ ɫɥɨɜɚ ɫɩɪɚɜɚ
ϲ
ɨɬ ɧɟɝɨ ɋɥɟɜɚ ɨɬן ɢ ɫɩɪɚɜɚ ɨɬן ± ɩɭɫɬɵɟ ɫɢɦɜɨɥɵ ɋɬɚɧɞɚɪɬɧɵɟ ɧɚɱɚɥɶɧɚɹ ɤɨɧɮɢɝɭɪɚɰɢɹ ןɢ ɤɨɧɟɱɧɚɹ ȕ ɢɥɢȾ Ⱦɚɧɧɵɣ ɫɩɨɫɨɛ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɆɌ ɩɨɞɪɚɡɭɦɟɜɚɟɬ ɦɧɨɠɟɫɬɜɨ ɤɨɦɚɧɞ Ʉɚɠɞɚɹ ɤɨɦɚɧɞɚ ɡɚɩɢɫɵɜɚɟɬɫɹ ɫɥɟɞɭ ɸɳɢɦ ɨɛɪɚɡɨɦ ɝɞɟ ɧɚ ɥɸɛɨɦ ɲɚɝɟ ɫɨɫɬɨɹɧɢɟ˃ ɩɟɪɟɯɨɞɢɬ ɜ ɫɨɫɬɨɹɧɢɟ ɢ ɝɨɥɨɜɤɚ ɫɞɜɢɝɚɟɬɫɹ ɩɨ ɥɟɧɬɟ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ א
b ɉɪɟɞɫɬɚɜɥɟɧɢɟ ɆɌ ɝɪɚɮɨɦ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɤɚɠɞɨɦɭ ɫɨɫɬɨɹɧɢɸ ɍɍ ɫɬɚɜɢɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɜɟɪɲɢɧɚ
ɝɪɚɮɚ ɚ ɤɚɠɞɨɣ ɤɨɦɚɧɞɟ ± ɩɨɦɟɱɟɧɧɚɹ ɞɭɝɚ Ⱦɭɝɚ ɩɨɦɟɱɚɟɬɫɹ ɤɨɦɚɧɞɨɣ ɜ ɤɨ
ɬɨɪɨɣ ɢɫɤɥɸɱɚɸɬɫɹ ɫɨɫɬɨɹɧɢɹ ɍɍ ȿɫɥɢ ɧɚ ɨɱɟɪɟɞɧɨɦ ɬɚɤɬɟ ɪɚɛɨɬɵ ɫɢɦɜɨɥ ɧɚ
ɥɟɧɬɟ ɧɟ ɦɟɧɹɟɬɫɹ ɬɨ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɤɨɦɚɧɞɵ ɟɝɨ ɦɨɠɧɨ ɧɟ ɩɢɫɚɬɶ
ɫ ɉɪɟɞɫɬɚɜɥɟɧɢɟ ɆɌ ɬɚɛɥɢɰɟɣ ɫɨɨɬɜɟɬɫɬɜɢɹ ɉɪɢ ɞɚɧɧɨɦ ɫɩɨɫɨɛɟ ɫɨɫɬɚɜɥɹɟɬɫɹ ɬɚɛɥɢɰɚ ɜ ɤɨɬɨɪɨɣ ɤɚɠɞɨɦɭ ɫɨɫɬɨɹ
ɧɢɸ ɍɍ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɬɪɨɤɚ ɚ ɤɚɠɞɨɦɭ ɫɢɦɜɨɥɭ ɢɡ ɜɧɟɲɧɟɝɨ ɚɥɮɚɜɢɬɚ ±
ɫɬɨɥɛɟɰ ȼ ɤɥɟɬɤɚɯ ɬɚɛɥɢɰɵ ɧɚ ɩɟɪɟɫɟɱɟɧɢɢ ɫɬɪɨɤɢ ɢ ɫɬɨɥɛɰɚ ɪɚɫɩɨɥɚɝɚɟɬɫɹ
ɩɪɚɜɚɹ ɱɚɫɬɶ ɤɨɦɚɧɞɵ ɉɪɢɦɟɪɉɨɫɬɪɨɢɬɶ ɩɪɨɝɪɚɦɦɭ ɞɥɹ ɆɌ ɤɨɬɨɪɚɹ ɢɧɜɟɪɬɢɪɭɟɬ ɜɯɨɞɧɭɸ
ɰɟɩɨɱɤɭ ɡɚɩɢɫɚɧɧɭɸ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɧɭɥɟɣ ɢ ɟɞɢɧɢɰ Ⱥɥɮɚɜɢɬ ɡɚɞɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨɦ ǡ ǡɚ ɦɧɨɠɟɫɬɜɨ ɫɨɫɬɨɹɧɢɣ
ɍɍ ɢɦɟɟɬ ɜɢɞ ǡ ǡ Ɍɨɝɞɚ ɟɫɥɢ ɧɚɱɚɥɶɧɚɹ ɤɨɧɮɢɝɭɪɚɰɢɹ ɢɦɟɟɬ ɜɢɞ ɬɨ ɩɭɫɬɶ ɤɨ
ɧɟɱɧɚɹ ɢɦɟɟɬ ɜɢɞ Ⱦɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɜɵɩɢɲɟɦ ɫɨɜɨɤɭɩɧɨɫɬɶ ɤɨɦɚɧɞ
ȁ ȦȦ Ȧ
ȼ ɜɢɞɟ ɝɪɚɮɚ ɩɪɨɝɪɚɦɦɚ ɞɥɹ ɞɚɧɧɨɣ ɆɌ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
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Ɍɚɛɥɢɰɚ ɫɨɨɬɜɟɬɫɬɜɢɹ ɞɥɹ ɞɚɧɧɨɣ ɆɌ ɢɦɟɟɬ ɜɢɞ
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ɍɩɪɚɠɧɟɧɢɹ
ɉɨ ɡɚɞɚɧɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɤɨɦɚɧɞ ɆɌ ɢ ɧɚɱɚɥɶɧɨɣ ɤɨɧɮɢɝɭɪɚɰɢɢ ɧɚɣɬɢ ɡɚɤɥɸɱɢɬɟɥɶɧɭɸ ɤɨɧɮɢɝɭɪɚɰɢɸ
ɚ
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Ʉɚɤɢɟ ɞɟɣɫɬɜɢɹ ɜɵɩɨɥɧɹɟɬ ɆɌ ɫ ɚɥɮɚɜɢɬɨɦ ǡ, ɫɨɫɬɨɹɧɢɹɦɢ
^ ǡ ɢ ɫɢɫɬɟɦɨɣ ɤɨɦɚɧɞ R Ȧ R ɞɥɹ ɥɸɛɨɣ ɧɚɱɚɥɶ ɧɨɣ ɤɨɧɮɢɝɭɪɚɰɢɢ ȼ ɭɩɪɚɠɧɟɧɢɹɯ ± ɩɪɢɧɹɬɵ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ
ȼɵɩɢɫɚɬɶ ɫɢɫɬɟɦɭ ɤɨɦɚɧɞ ɞɥɹ ɆɌ ɡɚɞɚɧɧɨɣ ɝɪɚɮɨɦ ɧɚ ɪɢɫɭɧɤɟ
Ɉɩɪɟɞɟɥɢɬɶ ɞɟɣɫɬɜɢɹ ɆɌ ɡɚɞɚɧɧɨɣ ɩɪɨɝɪɚɦɦɨɣ ȼɵɩɢɫɚɬɶ ɩɨɫɥɟɞɨɜɚ
ɬɟɥɶɧɨɫɬɶ ɤɨɧɮɢɝɭɪɚɰɢɣ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ ɫɥɨɜɚ.ȼɵɩɢɫɚɬɶ ɫɢɫɬɟɦɭ ɤɨɦɚɧɞ ɞɥɹ ɆɌ ɡɚɞɚɧɧɨɣ ɝɪɚɮɨɦ ɧɚ ɪɢɫɭɧɤɟ ɫɦ
ɧɢɠɟ ɫ ɚɥɮɚɜɢɬɨɦ ǡ ǡ. Ɂɞɟɫɶ ɟɫɥɢ ɢɡ ɜ ɜɟɞɭɬ ɞɜɚ ɪɟɛɪɚ ɫ ɨɞɢ ɧɚɤɨɜɨɣ ɩɪɚɜɨɣ ɱɚɫɬɶɸ ɬɨ ɨɧɢ ɨɛɴɟɞɢɧɹɸɬɫɹ ɜ ɨɞɧɨ ɪɟɛɪɨ ɧɚ ɤɨɬɨɪɨɦ ɥɟɜɵɟ ɱɚɫɬɢ ɡɚɩɢɫɚɧɵ ɱɟɪɟɡ ɡɚɩɹɬɭɸ ɢ ɟɫɥɢ ɫɢɦɜɨɥ ɧɚ ɥɟɧɬɟ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɬɨ ɜ ɩɪɚ ɜɨɣ ɱɚɫɬɢ ɤɨɦɚɧɞɵ ɨɧ ɧɟ ɩɢɲɟɬɫɹ
ϴ
ɈɩɪɟɞɟɥɢɬɶɞɟɣɫɬɜɢɹɆɌɡɚɞɚɧɧɨɣɩɪɨɝɪɚɦɦɨɣȼɵɩɢɫɚɬɶɩɨɫɥɟɞɨɜɚɬɟɥɶ ɧɨɫɬɶ ɤɨɧɮɢɝɭɪɚɰɢɣ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ ɫɥɨɜɚ
.ȼɵɩɢɫɚɬɶ ɫɢɫɬɟɦɭ ɤɨɦɚɧɞ ɞɥɹ ɆɌ ɡɚɞɚɧɧɨɣ ɝɪɚɮɨɦ ɧɚ ɪɢɫɭɧɤɟ ɫɦ
ɧɢɠɟ ɫ ɚɥɮɚɜɢɬɨɦ ǡ ǡɁɞɟɫɶ ɟɫɥɢ ɢɡ ɜ ɜɟɞɭɬ ɞɜɚ ɪɟɛɪɚ ɫ ɨɞɢ ɧɚɤɨɜɨɣ ɩɪɚɜɨɣ ɱɚɫɬɶɸ ɬɨ ɨɧɢ ɨɛɴɟɞɢɧɹɸɬɫɹ ɜ ɨɞɧɨ ɪɟɛɪɨ ɧɚ ɤɨɬɨɪɨɦ ɥɟɜɵɟ ɱɚɫɬɢ ɡɚɩɢɫɚɧɵ ɱɟɪɟɡ ɡɚɩɹɬɭɸ ɢ ɟɫɥɢ ɫɢɦɜɨɥ ɧɚ ɥɟɧɬɟ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɬɨ ɜ ɩɪɚ ɜɨɣ ɱɚɫɬɢ ɤɨɦɚɧɞɵ ɨɧ ɧɟ ɩɢɲɟɬɫɹ
Ɉɩɪɟɞɟɥɢɬɶ ɞɟɣɫɬɜɢɹ ɆɌ ɡɚɞɚɧɧɨɣ ɩɪɨɝɪɚɦɦɨɣ ȼɵɩɢɫɚɬɶ ɩɨɫɥɟɞɨɜɚ
ɬɟɥɶɧɨɫɬɶ ɤɨɧɮɢɝɭɪɚɰɢɣ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ ɫɥɨɜɚ
ϵ
Ɂɚɞɚɱɢ ɧɚɞ ɱɢɫɥɚɦɢ ɜ ɟɞɢɧɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ
ɉɨɞ ©ɟɞɢɧɢɱɧɨɣª ɫɢɫɬɟɦɨɣ ɫɱɢɫɥɟɧɢɹ ɩɨɧɢɦɚɟɬɫɹ ɡɚɩɢɫɶ ɧɟɨɬɪɢɰɚɬɟɥɶ
ɧɨɝɨ ɰɟɥɨɝɨ ɱɢɫɥɚ ɫ ɩɨɦɨɳɶɸ ɩɚɥɨɱɟɤ ± ɞɨɥɠɧɨ ɛɵɬɶ ɜɵɩɢɫɚɧɨ ɫɬɨɥɶɤɨ ɩɚɥɨ ɱɟɤ ɤɚɤɨɜɚ ɜɟɥɢɱɢɧɚ ɱɢɫɥɚ ɧɚɩɪɢɦɟɪ _ _ _ _ _ _ _ ɩɭɫɬɨɟ ɫɥɨɜɨ!
ɉɭɫɬɶ ɫɥɨɜɨ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ __ _ٔ __ _
ɝɞɟٔ Ȃɨɞɢɧ ɢɡ ɡɧɚɤɨɜ ±î · Ĺ Ļɫɥɟɜɚ ɨɬ ɤɨɬɨɪɨɝɨ ɭɤɚɡɚɧɨ ɩɚɥɨɱɟɤ
ɚ ɫɩɪɚɜɚ ± |
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Ɋɟɚɥɢɡɨɜɚɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɩɟɪɚɰɢɢ ɜ ɟɞɢɧɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟ |
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ɧɢɹ ɜ ɤɚɱɟɫɬɜɟ ɨɬɜɟɬɚ ɜɵɞɚɬɶ ɫɥɨɜɨ ɭɤɚɡɚɧɧɨɟ ɫɩɪɚɜɚ ɨɬ |
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ɫɬɪɟɥɤɢ |
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ɚ ɫɥɨɠɟɧɢɟ__ _ __ _ |
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E ɜɵɱɢɬɚɧɢɟ__ |
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F ɭɦɧɨɠɟɧɢɟ__ _î__ |
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G ɞɟɥɟɧɢɟ ɧɚɰɟɥɨ__ _ __ _ __ _ |
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H ɜɡɹɬɢɟ ɨɫɬɚɬɤɚ__ |
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I ɦɚɤɫɢɦɭɦ__ _ Ĺ _ |
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J ɦɢɧɢɦɭɦ__ _ Ļ __ |
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Ⱦɥɹ ɤɚɠɞɨɝɨ ɢɡ ɜɵɲɟɩɪɢɜɟɞɟɧɧɵɯ ɩɪɢɦɟɪɨɜ ɜɵɩɢɫɚɬɶ ɫɢɫɬɟɦɭ ɤɨɦɚɧɞ
ɢ ɩɨɫɬɪɨɢɬɶ ɬɚɛɥɢɰɭ ɫɨɨɬɜɟɬɫɬɜɢɹ ɉɭɫɬɶ ɜɧɟɲɧɢɣ ɚɥɮɚɜɢɬ ȁǡ.ɋɱɢɬɚɹ ɫɥɨɜɨ ɡɚɩɢɫɶɸ ɱɢɫɥɚ ɜ
ɟɞɢɧɢɱɧɨɣ ɫɢɫɬɟɦɟ ɨɩɪɟɞɟɥɢɬɶ ɹɜɥɹɟɬɫɹ ɥɢ ɷɬɨ ɱɢɫɥɨ ɫɬɟɩɟɧɶɸ ͵ǡ͵ǡͻǡʹ ǡǤǤǤȿɫɥɢ ɹɜɥɹɟɬɫɹ ɜɵɯɨɞɧɨɟ ɫɥɨɜɨ ɞɨɥɠɧɨ ɛɵɬɶ ɢɧɚɱɟ ±
ɉɭɫɬɶ ɜɧɟɲɧɢɣ ɚɥɮɚɜɢɬ ȁǡ.ɋɱɢɬɚɹ ɫɥɨɜɨ ɡɚɩɢɫɶɸ ɱɢɫɥɚ ɜ ɟɞɢɧɢɱɧɨɣ ɫɢɫɬɟɦɟ ɩɨɥɭɱɢɬɶ ɱɢɫɥɨʹ ɜ ɷɬɨɣ ɠɟ ɫɢɫɬɟɦɟ
ɉɭɫɬɶ ɜɧɟɲɧɢɣ ɚɥɮɚɜɢɬ ȁǡ.ɋɱɢɬɚɹ ɫɥɨɜɨ ɡɚɩɢɫɶɸ ɱɢɫɥɚʹ ɜ ɟɞɢɧɢɱɧɨɣ ɫɢɫɬɟɦɟ ɩɨɥɭɱɢɬɶ ɱɢɫɥɨ ɜ ɷɬɨɣ ɠɟ ɫɢɫɬɟɦɟ
ϭϬ