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标题: 突然想到让deepseek来解释一下递归 [打印本页]

作者: 密银    时间: 2025-1-29 14:16
标题: 突然想到让deepseek来解释一下递归
本帖最后由 密银 于 2025-1-29 14:19 编辑
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. ]1 ?. w4 D9 k- V# w+ j解释的不错6 E) g# B+ {1 o# X, p: v4 y& z- T
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递归是一种通过将问题分解为更小的同类子问题来解决问题的方法。它的核心思想是:**函数直接或间接地调用自身**,直到满足终止条件。
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7 G0 z7 T: G/ ], G, ^! J# r 关键要素
, U/ R. i- m2 S7 G7 ?3 E; }0 h1. **基线条件(Base Case)**
1 t, v* @) j6 X   - 递归终止的条件,防止无限循环# Q7 A! Q' c9 a& [+ ^. H
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1+ }3 ~$ Q, |! L$ t; P

! D# Y& |  O  W6 `' l3 y2. **递归条件(Recursive Case)**5 N& I- k+ e$ K. I0 x; a& z* l3 S" w% A
   - 将原问题分解为更小的子问题
1 @# U; C3 V" q( E  e   - 例如:n! = n × (n-1)!
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, n7 W9 @! y% i8 K 经典示例:计算阶乘/ a8 v$ Q6 a4 d0 p( T  N$ @: \
python
6 [  Z$ P/ G/ w5 e& Cdef factorial(n):, n: `: a1 u; G7 ]2 ~
    if n == 0:        # 基线条件
; q- |# P" c( E* b& h. h        return 1
+ H, ^0 }3 U% q1 O# s( W% L    else:             # 递归条件
" g0 V  }2 M: E  F        return n * factorial(n-1)5 r) F* |/ p8 W- R* V
执行过程(以计算 3! 为例):+ ~# i8 U7 y  Z; k4 z
factorial(3)* r- V0 w5 Z# L5 W
3 * factorial(2)
/ }$ j, N9 Z* B) \; o  F3 * (2 * factorial(1))
) S( z* ^' V, _$ y3 * (2 * (1 * factorial(0)))
( h2 s3 S# Y' o3 * (2 * (1 * 1)) = 6
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$ y; x( J% x0 s/ h/ x9 O8 f 递归思维要点
# j' m6 \9 ]# O8 |$ x! B1. **信任递归**:假设子问题已经解决,专注当前层逻辑
! f$ Z- ?8 u) P# l) X8 v4 D2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
% x  w. C3 l, o1 Z. k3. **递推过程**:不断向下分解问题(递)
  v8 ^" I' c7 G1 R) @% W* G; ]4. **回溯过程**:组合子问题结果返回(归)
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+ X' k% C' X# y- R 注意事项
% k) }& c" P1 N/ H" ^必须要有终止条件3 u. K' X9 \1 [$ |, }. s4 [/ Y. T
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
: y" C/ J0 w0 d: ^% F6 H, i8 {某些问题用递归更直观(如树遍历),但效率可能不如迭代
+ J" I+ u. v* P+ t9 a尾递归优化可以提升效率(但Python不支持)
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递归 vs 迭代
- r/ i: [  x! d. o0 _|          | 递归                          | 迭代               |
/ N$ e$ U* l# s" Q. \- v|----------|-----------------------------|------------------|  _  W& y% k  t; u) z/ o
| 实现方式    | 函数自调用                        | 循环结构            |
+ W% k# ^' p% p! A" P5 P0 p| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |7 h& G9 \6 {" L$ D5 `# K* }
| 代码可读性  | 对符合递归思维的问题更直观                | 线性流程更直接         |8 c1 w8 l9 [: s9 J, q) `, p
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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- e; }. t/ R2 V: V 经典递归应用场景7 e7 g  }1 n; m
1. 文件系统遍历(目录树结构)& Y/ H5 }9 w$ I1 t$ M, h
2. 快速排序/归并排序算法7 k1 R( P( R) B1 Y* ~& |
3. 汉诺塔问题  v9 q2 j3 W' C# Z
4. 二叉树遍历(前序/中序/后序)
- W5 e3 Y0 J  s  x5 o5. 生成所有可能的组合(回溯算法)
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试着用递归思维想象:你站在一面镜子前,镜子里有无数个逐渐变小的你在照镜子,这就是递归的直观体现。但记住每个"分身"最终都要有结束的时刻,这就是基线条件的重要性。
作者: testjhy    时间: 2025-1-30 00:07
挺好,递归思维要点与我能够回忆起来我当时写递归程序的思路很一致,,或者被它唤醒,8 ~- n0 v; {' ]
我推理机的核心算法应该是二叉树遍历的变种。
* H6 M7 i# P( n: v* N另外知识系统的推理机搜索深度(递归深度)并不长,没有超过10层的,如果输入变量多的话,搜索宽度很大,但对那时的286-386DOS系统,计算压力也不算大。
作者: nanimarcus    时间: 2025-2-2 00:45
Recursion in programming is a technique where a function calls itself in order to solve a problem. It is a powerful concept that allows you to break down complex problems into smaller, more manageable subproblems. Here's a detailed explanation:
0 S0 [6 k* }4 l: @# k& X: `Key Idea of Recursion& i1 v* ^2 z: f$ q, R0 B

$ c; x7 `  M# h: EA recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.
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3 n& I" f1 Z- E9 X1 \    Solving the smallest instance directly (base case).) |2 }) k# H+ `/ `" H5 E# [1 T

& a% F, [4 `) ~1 O% F    Combining the results of smaller instances to solve the larger problem.- R: {: S( g+ }" ?
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Components of a Recursive Function- o* L& g6 [- a
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    Base Case:
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        This is the simplest, smallest instance of the problem that can be solved directly without further recursion., B: E2 B0 W4 w; ^& J% Y/ k- H: u: A
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        It acts as the stopping condition to prevent infinite recursion.. ?$ {* H; ^9 H+ [

0 h$ |( }- o# R3 I        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; a* M4 i5 c) I, K: U
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    Recursive Case:
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        This is where the function calls itself with a smaller or simpler version of the problem.
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8 p$ a; {7 y" L3 ]+ }        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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- F- [+ m' [) v# L! mExample: Factorial Calculation3 U. ?8 G3 n" a9 n

; j7 u0 {2 W: F: {7 WThe factorial of a number n (denoted as n!) is the product of all positive integers less than or equal to n. It can be defined recursively as:
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    Base case: 0! = 13 S* C3 A; q+ r$ w% L4 N/ _# p* c' }8 {

+ e1 U% J; |: K- ]" Y    Recursive case: n! = n * (n-1)!
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- P7 l6 M) g' Q6 |) ~6 L) {. h9 OHere’s how it looks in code (Python):$ W  ~7 V* z7 X4 W2 ~1 g* Q
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def factorial(n):6 X) O* O! _: ^: c( `' ?
    # Base case
5 U# [  x* M/ w9 v7 m6 v# j    if n == 0:8 M" e. Q. U% H& l: s
        return 1
, j' d1 l% I9 y+ ?6 G) D    # Recursive case
7 d. w' [8 P4 C7 d    else:
4 b0 m3 o  A$ |: r  h        return n * factorial(n - 1)
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# Example usage
3 W, R9 v" |! J( X9 R- ]print(factorial(5))  # Output: 120
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How Recursion Works/ T. l. X7 y4 @/ T+ u$ }( \
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    The function keeps calling itself with smaller inputs until it reaches the base case.1 m" [) C. `9 f( E
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    Once the base case is reached, the function starts returning values back up the call stack.$ ~4 H  k' b; I0 B  y( q/ ?( x
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    These returned values are combined to produce the final result.
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For factorial(5):6 t$ E! O- G$ }+ e" g

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factorial(5) = 5 * factorial(4)
9 @$ H3 t4 @9 Y% s% `. B' Afactorial(4) = 4 * factorial(3)
! t7 I# t7 c0 S1 S/ e8 v- l, yfactorial(3) = 3 * factorial(2)* x4 g% I. r8 d9 ]: W
factorial(2) = 2 * factorial(1)* R; y1 y6 w! n( ~
factorial(1) = 1 * factorial(0)$ C3 b( E4 t& H
factorial(0) = 1  # Base case' ?" C) U* |! V: ~

5 v. F  y5 r! `3 ?: F# D/ n3 c- q( IThen, the results are combined:5 u9 D- M) s; ^8 w7 t( U/ |2 B
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factorial(1) = 1 * 1 = 1
; {" m: v) i7 ^- ^% dfactorial(2) = 2 * 1 = 23 Y- y) g/ d  C. A
factorial(3) = 3 * 2 = 6! Q* B3 }- @2 K: }) r
factorial(4) = 4 * 6 = 249 `# j1 a+ R& K: o5 e. s
factorial(5) = 5 * 24 = 120* q+ X+ m1 `2 v6 F' v

1 \5 R9 ^1 x7 j& r+ c( z% [$ SAdvantages of Recursion! q, z* d, e* q4 G

& |* E: v  n( h  ^1 x" k( [) i5 ~+ o! ^    Simplicity: Recursive solutions are often more intuitive and easier to write for problems that have a natural recursive structure (e.g., tree traversals, divide-and-conquer algorithms).
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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3 @- F+ c8 h/ c3 j3 D( g/ `Disadvantages of Recursion+ \. _4 r9 r; _- t( }7 J

! @* b- w* M; r    Performance Overhead: Each recursive call adds a new layer to the call stack, which can lead to high memory usage and potential stack overflow for deep recursion.3 s4 f7 `+ J9 r4 `7 w2 N

8 U! Y  F1 e  @- B8 z    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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4 e# Z# q  B) ]* QWhen to Use Recursion
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+ N6 N  H( d5 d- R4 W. S) f" r9 B" X; r    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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! ~+ f! E  s* D8 T    Problems with a clear base case and recursive case.
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% L0 j, q' L4 m) n) LExample: Fibonacci Sequence$ ~* B0 R- f  O0 ?/ s

# V- f! {1 p: d! H4 x2 H8 B0 LThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
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    Base case: fib(0) = 0, fib(1) = 1- ^( V0 a9 M* o5 H
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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; W8 v2 x& T& G, c$ j. b4 vpython
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& O6 w9 b' ?( K8 Edef fibonacci(n):/ k" P8 @( [7 x0 t9 ?3 R' Z
    # Base cases
# Y% G9 A9 n0 m( `/ Q' q    if n == 0:* f: I2 d+ z$ {* i$ U: E; m
        return 00 L4 {; t  B; i6 {5 u. s
    elif n == 1:
2 E$ Q" O0 X) k$ L        return 1
- K$ `0 {5 L' m' j    # Recursive case
8 M1 q9 ~# K( N( q" v    else:
1 O5 s# G) d& @4 F        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage
) X- |& I% w( j9 Jprint(fibonacci(6))  # Output: 80 B% Z! c% p6 k* X9 |

- R# d1 s  [4 oTail Recursion$ y1 v8 X  T% N8 Z  D) R

  @& E: R9 J6 D! `5 NTail recursion is a special case of recursion where the recursive call is the last operation in the function. Some programming languages optimize tail-recursive functions to avoid stack overflow, but not all languages (e.g., Python does not optimize tail recursion).: v1 ^! V$ [0 q# r% R

! f, [. f3 C' NIn summary, recursion is a fundamental concept in programming that allows you to solve problems by breaking them into smaller, self-similar subproblems. It’s important to define a base case to avoid infinite recursion and to understand the trade-offs between recursion and iteration.
作者: nanimarcus    时间: 2025-2-2 00:47
我还让Deepseek 给我讲讲Linux Kernel Driver 现在的开发流程,让一个老同志复习复习,快忘光了。




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