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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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) A3 @+ q% _' `6 ^; d5 K解释的不错
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6 \$ p5 i+ h) F( w; j2 Q/ }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。: c' C8 }; b$ x( V

4 U0 b3 S1 x9 L# R8 Y2 @ 关键要素
5 e9 d9 `% Q9 q3 s# T7 A4 U1. **基线条件(Base Case)**& G6 K* C4 [% N0 d9 f
   - 递归终止的条件,防止无限循环
( W5 q. I6 T- e7 c: _. w   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 14 H! M, H. Y; L
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2. **递归条件(Recursive Case)**2 L2 K4 o# q. k) x1 p
   - 将原问题分解为更小的子问题& O' I8 N9 l5 F8 L& O
   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘
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1 X. g) E: C8 X- |  d2 T3 \def factorial(n):
$ Z+ @; \2 U  Z! f  [" j, A    if n == 0:        # 基线条件6 t- B7 E2 d) w8 Z
        return 17 T1 a6 v' ~5 U& B
    else:             # 递归条件
8 B$ w/ C! G1 F  o* C+ ?, n        return n * factorial(n-1)+ D0 ?; A7 x* |
执行过程(以计算 3! 为例):" _# w0 s" o- b' J, p; ]" x
factorial(3)
& F8 I( R9 }% {9 S2 K3 * factorial(2)# ]8 t! g  E# ]4 V$ k8 F
3 * (2 * factorial(1))) E" F4 D( ]8 s2 W$ S
3 * (2 * (1 * factorial(0)))
& v: I( v5 ]% _0 p. X3 * (2 * (1 * 1)) = 6  w; x! M, J4 U

$ n0 b2 `1 M+ X4 a1 f/ U 递归思维要点
6 Y, A/ i' a3 }0 g- o/ E4 z9 K* v1. **信任递归**:假设子问题已经解决,专注当前层逻辑+ H& `1 R+ j+ S; u8 R( W; b' Q( J% V
2. **栈结构**:每次调用都会创建新的栈帧(内存空间). y* j. w5 ]* r7 f% a" {
3. **递推过程**:不断向下分解问题(递)3 s5 q- M7 ~6 c  P; L& K
4. **回溯过程**:组合子问题结果返回(归)- r  h! P8 j: [. }1 v$ D

3 Y! p3 V2 t0 V! b, i" d$ L8 Z 注意事项$ n9 Z+ U* t6 d6 X8 T$ \
必须要有终止条件) |- ~8 B; p5 I6 W0 K
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& z8 I& T( W) V1 _! K4 E
某些问题用递归更直观(如树遍历),但效率可能不如迭代6 q. R8 {; K% e$ J
尾递归优化可以提升效率(但Python不支持)
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4 B4 a" N0 g  T 递归 vs 迭代
$ n3 m+ S0 N4 E/ S|          | 递归                          | 迭代               |( P* F/ W; s  ?! M8 D8 d
|----------|-----------------------------|------------------|3 m, f4 P9 S+ O2 E, a0 B
| 实现方式    | 函数自调用                        | 循环结构            |
! I% g. h0 l+ B; ~% N: z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
1 }; J" d0 ?8 K4 I# F- h/ M- o| 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         |" U; X5 E- }6 i* J+ X$ e
| 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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经典递归应用场景2 F2 B% I2 F+ w8 ~0 n
1. 文件系统遍历(目录树结构)
2 M1 O: ^* @; Z3 L/ S) _2. 快速排序/归并排序算法
- ]0 Q. T0 W% X/ u6 R/ Q3. 汉诺塔问题
. m" ]7 B9 w8 G9 T9 }9 p% c4. 二叉树遍历(前序/中序/后序)9 `3 R1 R. z- m5 O+ c
5. 生成所有可能的组合(回溯算法)
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! \7 M7 J7 @9 e& N/ O: f% b1 g' N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,
* E' B& n" O. v( Z5 J  B( g% w, `我推理机的核心算法应该是二叉树遍历的变种。
2 p8 D; a" D. d5 o; T; R5 q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:
. V; ~# J4 t2 S, J/ {4 YKey Idea of Recursion
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A recursive function solves a problem by:9 R/ Y: B$ c4 z6 d* z3 e

  Y$ p: U8 r: o9 x    Breaking the problem into smaller instances of the same problem.
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    Solving the smallest instance directly (base case).9 m! j4 T, k$ \: F* n  M: H- y

0 ]6 [8 m' l. H, L3 I! C/ ?    Combining the results of smaller instances to solve the larger problem.
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Components of a Recursive Function
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+ f. m) \3 M  n1 }    Base Case:
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% L! U& ~0 V, `        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.: @- l* a* I* y$ p; d! }

( z0 f+ }6 N, |        It acts as the stopping condition to prevent infinite recursion.) e0 K, K4 t# z0 T, \% o$ E

3 b- s7 c# e: d5 U- C% B2 ?        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    Recursive Case:0 K8 y& P% H+ Z$ Y4 z0 U

' b) {* L3 H* \0 [: K1 c0 y" r        This is where the function calls itself with a smaller or simpler version of the problem.+ Y& o) b$ n. s1 a% G6 F

# X8 ^$ p5 s/ _/ C3 B5 y        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 o! s/ Q! |% I! ]0 l9 p# `& p
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Example: Factorial Calculation
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The 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:/ R( }4 v5 T- N: ?

! e/ f% ^. p! y: E) |' z/ j9 G    Base case: 0! = 1
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& B8 @6 a4 F3 |* [: p! C9 P4 y1 z    Recursive case: n! = n * (n-1)!
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' G+ \0 j+ d5 e/ I  EHere’s how it looks in code (Python):
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, u, C% H/ x4 B6 ^( ndef factorial(n):# l, b8 p* G* Z
    # Base case. y( }" n! o( l1 E& I9 o5 q5 y
    if n == 0:
7 Z* N0 c! l1 Z" z        return 1
3 m- N8 @/ R; H    # Recursive case0 t" q# r4 K- j7 p6 k% k
    else:
* V" @( k' Z6 U3 C: P5 C        return n * factorial(n - 1)5 S( E) B* N  ?; E
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# Example usage
) G' D+ m# D) R$ [print(factorial(5))  # Output: 120" U' s* m, ]! m; b

- Q) ]6 b8 {0 S) S8 t; u( L9 VHow Recursion Works
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    The function keeps calling itself with smaller inputs until it reaches the base case.. w! _5 X5 }3 q0 t& t

& y5 f1 ]7 O& g5 ]  M8 o" K* Q5 Z    Once the base case is reached, the function starts returning values back up the call stack.: d8 Q  j6 A% Q

  Y% g: ?) Y, d. w* \    These returned values are combined to produce the final result.$ E" j  m! V/ a% {

; V7 W2 ]9 t+ G4 qFor factorial(5):
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2 q) `5 q6 Q3 s- ufactorial(5) = 5 * factorial(4)
& d7 N% l. B8 J5 l+ b3 w3 Rfactorial(4) = 4 * factorial(3)
) N- d9 E, _6 l; K1 |2 L1 cfactorial(3) = 3 * factorial(2)
+ A$ A! N) b# U1 G  Ufactorial(2) = 2 * factorial(1)% g1 e4 n6 X3 u( ^
factorial(1) = 1 * factorial(0)
) |; F4 v! }) b& f$ \- lfactorial(0) = 1  # Base case
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Then, the results are combined:
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- m. C7 t* D+ U0 Kfactorial(1) = 1 * 1 = 1
9 s0 [+ T0 W8 W, ^2 a2 ]factorial(2) = 2 * 1 = 2
# e6 H- O% c$ }$ L! dfactorial(3) = 3 * 2 = 6/ s2 U7 f% n0 Z0 C- z
factorial(4) = 4 * 6 = 24
. ?  ]; j2 N9 p) d4 L% Dfactorial(5) = 5 * 24 = 120
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Advantages of Recursion
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8 S+ W, f- T- `9 S    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).& ~( Z- t8 a/ _8 E. T' f7 Z
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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8 K& j0 G3 I. H6 N2 ^Disadvantages of Recursion  @  S6 Q8 E4 d" h

1 }+ U0 Q! n- }1 o! I3 P    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.' d1 b8 {* K5 Q& N: ^! W' a! o9 J. p

. G8 |) A+ ]; e    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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' w$ V9 H; B/ @+ F1 w1 KWhen to Use Recursion
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).( n+ N# H+ I. J+ B. Q3 p7 z
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    Problems with a clear base case and recursive case.4 L) D- N3 R0 m6 `. y

* V2 ?% V7 z. n3 o2 _Example: Fibonacci Sequence4 B) D, g% F, `
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:1 G) _' N4 p$ C# t) p! r
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    Base case: fib(0) = 0, fib(1) = 10 Q0 Q& I. R4 K: j8 A

+ J$ y; A, w( w; J% @" T' T( D    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python! N4 Q6 G: _3 O
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def fibonacci(n):" J  r# ]9 y. ^- g0 m, k
    # Base cases% ~4 k* j: |7 \' Y! I! \. n+ Q& I( N
    if n == 0:
" K' M. d* K! D- Y. k        return 00 y" L5 a( ~: o% e6 ~( W/ `
    elif n == 1:8 P* t) h* Z4 z
        return 1$ Y; k- v7 h- W! T! V" Z1 Q
    # Recursive case" ?/ Q& r$ p# J7 H/ X
    else:
$ L) `& s) `7 Y6 l        return fibonacci(n - 1) + fibonacci(n - 2)+ z% ?( f9 M) J1 q& K  r0 K* `
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# Example usage
: R1 G  C; Y4 y# [* y; F' ]print(fibonacci(6))  # Output: 8
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$ s. M$ b* ?( J  y, YTail Recursion, r# m9 _9 P5 O) @: @! a, A# w$ ^

  d9 r- G# W; Q* KTail 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).* w! n4 q, X5 j

' {8 N% b7 w* mIn 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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