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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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解释的不错
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2 a5 f$ r* {/ x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。) F( K. n" ?2 g

5 `' @4 G0 D5 C7 M! r 关键要素2 }' c5 v  E! N8 I: g2 q' y
1. **基线条件(Base Case)**
1 L# M" T' p4 e  L$ f   - 递归终止的条件,防止无限循环
. v' ~# O, f$ G# o7 [0 \* \   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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# w1 I+ U7 ^' c& D2. **递归条件(Recursive Case)**6 J2 f4 ?  m7 L1 q, A: G- [
   - 将原问题分解为更小的子问题6 G0 f' k2 _+ B) f" Z3 S# @
   - 例如:n! = n × (n-1)!  `8 r! Z; f5 W6 r) _) n, \

# E5 n. ^: P. U/ W0 n, T: X 经典示例:计算阶乘
# x) w' q: R) O2 D8 M( T* ?python
6 b9 E0 C) k7 q% e0 `$ @! Odef factorial(n):
2 @1 y" A' c3 ~5 c! a3 @" @# P    if n == 0:        # 基线条件
4 I/ q  q/ b9 i* }        return 1
8 X# G! k7 H: D/ L: O  U    else:             # 递归条件8 Y( z, z$ v' z
        return n * factorial(n-1)
9 U2 f  U1 n/ \, F# O5 f2 |: t执行过程(以计算 3! 为例):# O8 L- E1 T: l5 \3 V0 U
factorial(3)
+ @6 p3 s9 X  w3 * factorial(2)
! h8 q% C# h. m3 * (2 * factorial(1)); ]6 `5 |1 C; |+ B; V3 y
3 * (2 * (1 * factorial(0))); V8 T6 I) s! e+ W+ ~$ I2 Z
3 * (2 * (1 * 1)) = 6
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) F" q5 a( p/ A7 z6 H1 h$ } 递归思维要点$ I, ?& C' J1 @0 X  t* v6 |
1. **信任递归**:假设子问题已经解决,专注当前层逻辑
! f2 l; a" |/ R2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 ~) L4 H7 |- _! R" O+ [7 W% B
3. **递推过程**:不断向下分解问题(递)" [/ y, u; o* T8 Q
4. **回溯过程**:组合子问题结果返回(归)/ \# m5 Y- m% @+ x3 y
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注意事项! y) T8 R" r% a
必须要有终止条件0 d4 @5 S' D( a5 C0 R9 I
递归深度过大可能导致栈溢出(Python默认递归深度约1000层); O; z- z+ I! u( ?! Q5 M5 m9 _
某些问题用递归更直观(如树遍历),但效率可能不如迭代
# p( N' `& q1 F# T, o尾递归优化可以提升效率(但Python不支持). s5 D8 L& {4 b2 F, s
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递归 vs 迭代
1 t7 w" e$ F$ ~5 A2 f+ w|          | 递归                          | 迭代               |5 S5 L2 o3 s3 S3 M( W5 U
|----------|-----------------------------|------------------|
, {0 _: o2 T5 w7 X3 f. F2 ]| 实现方式    | 函数自调用                        | 循环结构            |
- ]7 y( u; M* w( u| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- f8 X- \6 X% H# l5 w( k$ d
| 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         |
5 o+ ~6 I  m- z; c| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& x  |' w8 }/ b& C- I2 m# R% C' m

7 r- x/ l. x3 b( M4 L7 w2 y7 ? 经典递归应用场景1 @* L/ ?( a! R4 B
1. 文件系统遍历(目录树结构)0 S6 F' M- U1 j; }
2. 快速排序/归并排序算法; y$ X5 |! P) ~" `$ ^" C- w/ \
3. 汉诺塔问题: ^) G1 T* q8 d; K
4. 二叉树遍历(前序/中序/后序)
) o8 P  @) J4 C' [! N, s2 L5. 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,0 @2 f+ ~" F9 F$ ^* R% \: {
我推理机的核心算法应该是二叉树遍历的变种。
' o+ m( g, o) S# E2 K' N: b5 \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:
) b: V' Y- f6 }' n, uKey Idea of Recursion
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; [" ?$ U$ C: j0 `3 e8 ZA recursive function solves a problem by:: G1 `+ Z" I% N3 J& h
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    Breaking the problem into smaller instances of the same problem.( P( S0 U9 c1 i2 N& B

: C/ Q3 u' F5 H- y$ d* M    Solving the smallest instance directly (base case).; Z/ q! Q# H. F3 d6 ?1 S5 V! B

! N7 Y" Q5 n$ `6 [. j" J    Combining the results of smaller instances to solve the larger problem.
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Components of a Recursive Function* W+ f8 f; L% w
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    Base Case:
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7 y- U1 Y/ ~4 C5 w        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. W8 M* |* n+ Y* n+ k/ E' H9 y+ _

9 k1 O8 s1 }0 C, E: v        It acts as the stopping condition to prevent infinite recursion.
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        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.  T' D7 A, o3 f! r. A/ S$ Q- T
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    Recursive Case:
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- l$ g% h- c: X& `        This is where the function calls itself with a smaller or simpler version of the problem.
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3 r3 L" f  x* N+ y3 M        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).0 G& r/ B6 |0 X
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Example: Factorial Calculation) F% s  f( O% E4 k* ]- U

* f$ e+ n, b/ D: I" J6 B2 FThe 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:/ [  @2 W, S# x& d

6 W/ n# L: r2 I, Z. r8 i4 Y" `    Base case: 0! = 1
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1 {: Y1 V' X3 k! Y    Recursive case: n! = n * (n-1)!* T; P5 }2 \- |) m; m0 k- {. W* W

& U+ o" r  ]- I9 p7 M/ v7 E, u9 tHere’s how it looks in code (Python):: u1 J" z! ?6 A) D4 V& K
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, F9 K) X5 R0 fdef factorial(n):
" S5 ?- z& n( l) c$ d' v4 U    # Base case! B5 f# p; [; }9 B/ G7 I
    if n == 0:
) \$ B1 u0 ?8 d/ k) V1 F& N        return 1( }* U  w. m& m. G% {, s3 a* \8 o
    # Recursive case
/ q4 O9 n+ S0 W. k* z    else:
# V& h$ d1 K* m& M& n) V' w        return n * factorial(n - 1)
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# Example usage
% t0 K5 ~) s3 x2 z% Fprint(factorial(5))  # Output: 120
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: p' P) N9 N- r. m- I5 A  j( h1 ^) kHow Recursion Works" N6 i, r3 c2 b* ^
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    The function keeps calling itself with smaller inputs until it reaches the base case.
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  l) b4 x# W" Y3 X  ?4 e    Once the base case is reached, the function starts returning values back up the call stack.
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9 b9 m; }- z- Y. v" ]/ r. |, o    These returned values are combined to produce the final result.
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& @/ T3 j" p8 {; c$ _1 nFor factorial(5):6 @! J. |0 \$ Z* q% D7 l

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factorial(5) = 5 * factorial(4)7 s; ?3 Z0 q0 [. v9 v# W
factorial(4) = 4 * factorial(3)  l) [8 U. y* ~& J5 K8 N4 t
factorial(3) = 3 * factorial(2)+ I" }4 P4 g6 a% J
factorial(2) = 2 * factorial(1)
- B% G3 q! L% }1 kfactorial(1) = 1 * factorial(0). ~# D  M6 D* D4 l& V
factorial(0) = 1  # Base case2 I& M- g, W4 B# J7 |$ [# O/ O- o
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Then, the results are combined:' v# O, H2 g3 T4 ?8 d
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( y' ~: L# E; ]$ y' B) U; Jfactorial(1) = 1 * 1 = 1/ ?5 n6 s9 b$ g
factorial(2) = 2 * 1 = 2
4 z) K' P) v4 o# p4 {! @/ r# gfactorial(3) = 3 * 2 = 6
. z  X1 \! S# k: J, J7 tfactorial(4) = 4 * 6 = 24
0 z' d! k3 O. ?) g/ f8 T/ jfactorial(5) = 5 * 24 = 120
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  L; L# Y+ o) s! hAdvantages of Recursion  P3 E6 J: u6 H, w) ]; ?& _

. c' |6 {: J# z5 `    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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! n- T8 j" Y8 d; G" [    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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: F! r0 j1 W/ VDisadvantages of Recursion
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' r; R0 L# Y4 R4 b    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., v/ v/ W; Q% T( Q& F& |
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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7 q' _7 T3 `5 o0 Z5 |! |When 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).
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    Problems with a clear base case and recursive case.5 u( H. h" t) C; `& J0 Q

7 S1 C. u# m0 I8 [, NExample: Fibonacci Sequence
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
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' D! d  f2 q/ P, `. p# c; v    Base case: fib(0) = 0, fib(1) = 1' `5 C$ \; N$ ]% p6 g7 @' Z
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)* M& l) |- l" `8 g* r

$ {5 f* r7 d: s: H- {/ Bpython
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& k1 B& M3 p% q, h3 wdef fibonacci(n):
1 q; E3 f$ G, B/ c) G# E, n# O- s- E    # Base cases6 I3 Z- W' U, y+ d% U& a* i
    if n == 0:
; u7 ]% q# _5 m' N6 J) Z* K        return 0
0 u! t! z4 ?' Y1 K5 b5 t  [    elif n == 1:  Y% P9 I  H! e1 w  c& H) S/ h1 N4 C
        return 16 ]- V2 ~. i" b9 Y
    # Recursive case
! }9 U) J# p2 c+ T8 m3 }    else:, x7 U4 B7 _# j6 Q' N6 c6 y; j  k
        return fibonacci(n - 1) + fibonacci(n - 2)& C, t$ a- z% H, J, i: o& A

  J( v+ {: j! k3 ^: a9 V# Example usage
& W8 }3 c0 s$ x+ F# Bprint(fibonacci(6))  # Output: 82 a6 R9 I4 b+ K  e5 q

- f! h: U& C0 N& T0 E6 V6 o" uTail Recursion
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Tail 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).; u& d7 ~- x& o' K# \0 a
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In 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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