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[科技前沿] 突然想到让deepseek来解释一下递归

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    2025-9-8 05:08
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     楼主| 发表于 2025-1-29 14:16:55 | 只看该作者 回帖奖励 |倒序浏览 |阅读模式
    本帖最后由 密银 于 2025-1-29 14:19 编辑 6 g: w4 P' i' R4 f7 |6 D8 x

    - j# v( ~! Y+ c7 H; a% [, q0 C解释的不错& H, J" v$ B& [' i2 D! A5 F
    # t, h9 z5 D5 Q5 g8 o3 ?
    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。
    : j9 U; H4 A! F% O
    ) g& M3 l) ~& Z5 E: i3 @ 关键要素
    : T5 B5 e( m" p: _1. **基线条件(Base Case)**0 Z$ y. g2 l; o: S+ z9 m
       - 递归终止的条件,防止无限循环3 ~/ P1 c: B1 E8 o2 k
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ! B! d9 c* {  U) x2 c* h
    + }& P, ]4 C9 _4 Q2. **递归条件(Recursive Case)**
    % ~; u* C8 [: ^$ W3 P) @   - 将原问题分解为更小的子问题
    1 f: U8 Z5 z/ R   - 例如:n! = n × (n-1)!1 ^" ~. k3 L! U$ g& J

    ' ~) x- b' z; B+ N: p8 H' N  p% D& B 经典示例:计算阶乘
    - K4 v6 Y7 m9 Z! o1 V, ~; ?python
    / @8 p/ |: l- I4 a: R" kdef factorial(n):
    0 {$ A# f  f4 m. k2 h. E: g    if n == 0:        # 基线条件
      e6 @8 B* m/ ?& ~! \  t5 K        return 19 N. U/ D8 o2 ~4 o
        else:             # 递归条件
    0 ?4 U8 L# c% ]/ @        return n * factorial(n-1)0 @6 n* i! d' u' `2 h7 N* P1 j
    执行过程(以计算 3! 为例):0 g" j+ H/ s# D% ?8 D$ R9 a6 D
    factorial(3)
    # ~4 u, l  ~- g3 * factorial(2)& R" y4 q, n* a
    3 * (2 * factorial(1))( k4 T9 |& q6 H8 D. R5 O
    3 * (2 * (1 * factorial(0)))" W9 o3 Y2 k. q9 ^
    3 * (2 * (1 * 1)) = 6
    9 A) j6 q# C# k0 I4 Z9 G$ T
    5 S1 P# U. x3 u* o! k* c  Y9 c 递归思维要点
    ( v+ I8 J4 S  c. @, n5 e8 J: q1. **信任递归**:假设子问题已经解决,专注当前层逻辑/ F# d, j2 i2 i# U6 W; V6 ]
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    0 Q" a% @9 x! K+ Z; p3. **递推过程**:不断向下分解问题(递)& k1 G) g' b8 l, a  y& h$ o
    4. **回溯过程**:组合子问题结果返回(归)
    - ?5 b$ |6 c7 u4 G
    * h1 r/ C" q% A6 V0 a 注意事项
    ! e' d: F  T2 O* z8 z% f' B; G必须要有终止条件# Z5 ^4 M* B0 E5 B2 G" D
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)2 H, N! i3 X3 k9 q* O
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    * j# D; e+ r  z' r% N6 S尾递归优化可以提升效率(但Python不支持)
    6 s! `+ ~- j* S# b5 |1 T
    * O7 b+ I8 t) X( q* Q8 B8 f, C 递归 vs 迭代
    6 @  ?* v: e3 V' _5 Z$ `|          | 递归                          | 迭代               |
    ; b( W9 q. e0 [$ e- E  L3 G7 o# E|----------|-----------------------------|------------------|+ T& S: ~+ l8 `* Y
    | 实现方式    | 函数自调用                        | 循环结构            |, e2 z, u# }  M6 @2 E
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |( c0 S, J0 p4 t; ]
    | 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 k- W- h0 {9 P& p4 @| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |0 Y- }0 F2 {7 J  k8 }: }. }& |4 `
    & q! I- r& E+ t- e2 V
    经典递归应用场景# I2 |  p  r3 Y( r
    1. 文件系统遍历(目录树结构)- W/ Z1 ~1 v  j. N2 E
    2. 快速排序/归并排序算法7 @6 D2 q$ S: c# ^. k0 m
    3. 汉诺塔问题4 H! W' Q5 j0 ^! r
    4. 二叉树遍历(前序/中序/后序)  w( m, Q5 d; T4 S. F* N4 W  s8 ^
    5. 生成所有可能的组合(回溯算法)8 ?6 y8 Z5 m9 J+ `, j; [0 z9 @
    + K( T2 q9 F, u# k2 Z
    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。

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    沙发
    发表于 2025-1-30 00:07:50 | 只看该作者
    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,) ~' S7 Z: \! z6 f
    我推理机的核心算法应该是二叉树遍历的变种。
    9 ]0 t& z7 j' P5 T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。
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    板凳
    发表于 2025-2-2 00:45:59 | 只看该作者
    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:
    + L  b% O: s3 f, E7 B0 f/ wKey Idea of Recursion+ ]& [; o6 n0 F+ V) D
    9 e9 m+ i' O4 C/ b- ^
    A recursive function solves a problem by:
    0 R- C# Z. l/ k' l" g
    3 B# D) E: X# p7 j6 w9 o    Breaking the problem into smaller instances of the same problem.
    0 O7 F% T' q% A% l1 ]3 d% x) n# T4 o! b9 }, z% F
        Solving the smallest instance directly (base case).
    & X* H  {, ~& z9 t+ ?0 B
    8 _' F, [2 ]: X6 K! M4 I  R    Combining the results of smaller instances to solve the larger problem.
    5 j" p8 K8 u+ `4 u  a2 s) P
    9 `2 q8 `, r, P: B: I& [0 {Components of a Recursive Function
    9 _( [4 _0 {/ K( N
    6 v5 D5 V/ S9 S    Base Case:/ r/ c: o7 L* k1 g
    4 Q( c6 B3 |3 a5 X3 r
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ( @8 @9 d0 H- p3 ^5 H/ C0 U+ _' R- x# r+ V
            It acts as the stopping condition to prevent infinite recursion.
    : |: d% M& M/ U) B& d: g& d) F+ Q: c: G, t! Y
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' w: t" c8 N# [- [7 F
    - F2 I+ g9 ]  v' e
        Recursive Case:1 ~7 M" K; A! I9 O6 j
    + O. o) Q9 i/ G3 [1 [* Z% n
            This is where the function calls itself with a smaller or simpler version of the problem.
    4 ?' j6 l$ k  D6 A8 ~; T. g6 g& ?; J% G' I8 d
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    5 D, t% r5 l# L- }5 k" y, C5 \; F+ D* f: N+ u. Y
    Example: Factorial Calculation+ m! y% \) s- G" X, D2 ~5 A

    & _# U! m2 L- t" ^; RThe 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$ A) P" r3 t  O/ e5 w' ?0 o7 D) y: L  A. |' z6 Q  u
        Base case: 0! = 1
    4 b$ |% B6 {+ u
    - L: A5 j0 l2 w* Q2 U    Recursive case: n! = n * (n-1)!
    3 s5 r  |! B1 Z; R1 s3 E7 o1 s
    9 n6 o; l5 Z8 _5 \. RHere’s how it looks in code (Python):, C$ o1 S( t& ^3 Y) H
    python
    9 m$ d/ o) L: I: ?  a+ v0 L) @( i
    $ P6 ^8 m) G( H. E; a
    ! `- }+ X. e/ J% I; ddef factorial(n):, k& }* j) K+ F- T8 ]
        # Base case
    * l3 ^) \1 m( [    if n == 0:
      E/ V9 U3 N; m. r        return 1
    6 P( [& m' j. C1 Z    # Recursive case4 z' W" }/ e/ S" |# V
        else:
    3 ~3 _8 @; n- d! N        return n * factorial(n - 1)
    - B+ U+ u+ ?1 m8 t$ H
    , D0 p8 L# G! O+ _# Example usage
    ) o& e) v7 Z* L1 |% m# eprint(factorial(5))  # Output: 1208 B, Y  v/ k& w: Y- D& x  @0 w6 U/ N

    " s5 h8 j  [7 ^; h6 OHow Recursion Works
    . S7 C+ K$ }( w% x; p6 u5 l# W& x( F4 c: [
        The function keeps calling itself with smaller inputs until it reaches the base case.% {0 X# O; V& ~; R, g

      Q! K4 x9 w% S: Q4 q1 P6 N    Once the base case is reached, the function starts returning values back up the call stack.5 W2 `. R- S6 w! G6 ^
    ! ^$ J1 O1 z5 h0 [1 k' v- J
        These returned values are combined to produce the final result.
    . N! d' D8 q2 U# m: \1 j! W0 e5 A% b  ?) x# ^8 m5 z$ m
    For factorial(5):
    1 r: a4 z0 R) k: X  c% y$ a
    1 N7 ]4 a$ T  ~7 d* H1 t9 B: N/ v- i
    factorial(5) = 5 * factorial(4)) n  \* f: a* `. X/ E; E% t7 @
    factorial(4) = 4 * factorial(3)
    ! n- G& \1 A: w/ h0 \factorial(3) = 3 * factorial(2)
    & M/ w0 [5 l  ~  c- s- F2 ]factorial(2) = 2 * factorial(1)
    8 p& m- u+ [) R' u; Dfactorial(1) = 1 * factorial(0)
    ! i% N7 R' v' C0 z+ ~9 m% K) jfactorial(0) = 1  # Base case
    2 L1 I) Y' e/ w
    - Z3 T0 c% C, b2 s! \Then, the results are combined:/ L, J  Z& a4 h' s1 o; G

    0 [! M$ Y( {) }' @. O, ~- d: p. w7 m
    factorial(1) = 1 * 1 = 11 R) N9 ~& C% U# Y" b
    factorial(2) = 2 * 1 = 2
    # ~7 o. {8 S# s0 w2 H6 V; _factorial(3) = 3 * 2 = 6
    4 K6 {! k/ y# G  b- z) ^- Ufactorial(4) = 4 * 6 = 247 T- W6 g  d. ?6 |
    factorial(5) = 5 * 24 = 120
    * D) q/ u% W) n: @1 c3 O. [5 S7 s8 @# C; Z3 `. {
    Advantages of Recursion
    , \& P1 ~. x9 ]
    # |$ E$ P' g7 j: p4 m) k; l$ `    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).
    4 j9 Y7 w% r6 J
    " v, T- i' s4 {( _; c% d8 X    Readability: Recursive code can be more readable and concise compared to iterative solutions.1 ?8 O8 c3 ]# @* l$ Y" A/ t9 C& G- B
    1 [6 |$ g8 x0 H0 h$ \
    Disadvantages of Recursion: t9 r! ?. k, R: \: S. m

    8 ]4 u: e* V. p$ n    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.
    " G# ]! B% @6 U- ~7 Q" W* {) W9 |% z& ?2 E2 Q
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).! M$ N5 r6 f8 _( h$ _8 ]7 ~, ]

    - M* d! @: |5 B7 KWhen to Use Recursion
    " L% W5 e5 j  G7 w* G$ P/ K8 k! R6 E1 t
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . Z- g2 x  w. m8 l
    ! Q$ B: ?2 n; @  ~6 p5 ~1 t: _    Problems with a clear base case and recursive case.
    4 X/ h. x4 i5 l' j5 {% t
    + ~' A* B& l. }7 M: vExample: Fibonacci Sequence: r, w  i* u( u4 b9 @# S
    ) ^7 [& {! f$ _5 m7 r8 \5 c% z
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    3 h4 q( G4 K. \( w
    ! @8 a- b0 [8 O7 [8 T% y4 ]    Base case: fib(0) = 0, fib(1) = 1
    7 p2 r# K8 r) ^# C& M
    $ n. N0 b1 W, f* I( {4 N    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    : @1 S# h0 r; I+ n1 F3 a4 Q1 y8 @0 N
    python
    ) p0 ~- G2 \0 w- H8 J1 \9 v
    ) c& w0 J% ?" ~+ N4 E+ D3 c. R* i; T+ M( p6 \) }# Y0 ?, R
    def fibonacci(n):9 V  D: H. @0 a, |! D# A
        # Base cases
    2 e- P1 Y9 t! K$ ?/ P    if n == 0:/ n% f7 Y/ p1 i4 Z* |* S
            return 0
    9 W( k" {+ F( J0 ~" o% C* I: I    elif n == 1:9 d& |1 I& w! g6 w6 B
            return 1
    9 V9 t# D& X0 P, x6 q' Y/ y4 g$ `; c; q    # Recursive case
    - U4 f* r4 J6 @& T    else:
    " R1 b! j* D! d" ]        return fibonacci(n - 1) + fibonacci(n - 2)
    ; ~/ }, }" e3 t6 R+ D$ x
    $ t9 N0 I, _2 U- J5 B: i% g% h' y# Example usage
    . y; S) n" K6 Y3 _print(fibonacci(6))  # Output: 8
    3 U0 v; s2 e" m/ A* Q4 @5 j9 ]; B5 F) T/ R* N$ r" B9 M9 c
    Tail Recursion
    7 y; E) f5 q0 a) y: U
    % @# q9 s2 j! A. Q& Y* {9 ?. ~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).
    . l, _5 C" s) B9 L! Y% q% w" f
    6 e0 D) F$ M& `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.
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    地板
    发表于 2025-2-2 00:47:27 | 只看该作者
    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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