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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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  • TA的每日心情
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    2025-9-8 05:08
  • 签到天数: 3 天

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    楼主
     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 2 |- U$ T. ^+ _# R, ^* C

    9 o3 Z8 E7 @: R' [! e解释的不错$ V( B0 h2 ]/ S
    ' J4 P% _7 v! u# e& `( @
    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。/ J5 D7 a( I8 |% U# B8 ]7 t
    4 j. h/ e) Z! T2 N/ Z4 N$ Z% C* V
    关键要素
    3 \( x0 i$ E/ s1. **基线条件(Base Case)**# P: ?5 L; U6 d3 M0 G
       - 递归终止的条件,防止无限循环( Q+ F  D8 e' b- r. @4 E
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    # p7 E2 r( F, y% w2 G4 u6 Q" j; y8 b0 Z1 O) b  ?2 j
    2. **递归条件(Recursive Case)**
    8 T7 Y) m9 v3 v8 s2 |   - 将原问题分解为更小的子问题3 j" Y$ b7 @6 v0 e+ v! f  s
       - 例如:n! = n × (n-1)!
    ' e6 @$ Y1 b$ b/ }
    , P$ t) D- |; m' \/ z8 ]) n 经典示例:计算阶乘! J0 W& o9 C8 m5 X% p& h4 m
    python
    0 M1 E' A4 y+ A5 \8 M5 u. k! p1 J! Vdef factorial(n):/ f( v" I8 h8 _3 ~( X
        if n == 0:        # 基线条件
    ' E; Y( x  y( S$ {        return 16 q) c+ @: G3 j8 d0 X8 V
        else:             # 递归条件
    6 m$ F$ n/ a* F9 P9 b) p        return n * factorial(n-1)4 R: {3 r7 e& J- u3 k% x. V
    执行过程(以计算 3! 为例):
    8 S% y) N/ q/ y$ K! V% Bfactorial(3)
    $ F8 {! B* \0 s; i3 * factorial(2)
    9 J( e1 z$ m) H) F3 * (2 * factorial(1))
    + q$ Q" ~6 c2 \3 * (2 * (1 * factorial(0)))
    4 O; U+ j- \# Q. D, h& [% L0 z& U3 * (2 * (1 * 1)) = 6; I2 e5 X, E! W$ N6 Z+ N! K

    1 j) x# j7 V" Q6 a5 j' B  G( @ 递归思维要点% @1 N: L; {8 d3 T- p
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / z2 |7 d; Y* K- \& e* y+ S9 P2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' @2 a8 f, ^: ^) w& k% N6 N0 Z. \* ~# i
    3. **递推过程**:不断向下分解问题(递)
    # _; z# T3 Y' G$ M6 f  S6 U4. **回溯过程**:组合子问题结果返回(归)! I6 j- N6 _. X: Z% N

    , G# S( v- z2 u 注意事项
    2 m1 m# E$ i8 I必须要有终止条件
    9 ?# e1 Q9 Y5 n$ f- {3 M递归深度过大可能导致栈溢出(Python默认递归深度约1000层)" q8 ^! O( s9 H- d
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
      V  O* H& W% E# l4 p2 V: b尾递归优化可以提升效率(但Python不支持); q& a# o4 p7 o# }! A

    . ?( ?$ g" g; S 递归 vs 迭代
    ; y, H7 J! a: F! F! T|          | 递归                          | 迭代               |0 s6 K1 j2 m; o
    |----------|-----------------------------|------------------|
    - _2 F7 ^7 R+ ]| 实现方式    | 函数自调用                        | 循环结构            |
      ?$ y. J4 m, ^| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 h: ?$ Y8 X) N4 v
    | 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         |, f! n  X/ R0 z8 c$ i6 H8 |
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    $ r: Y# k1 |! `" r- I3 T' Z5 T- x9 i3 P4 e8 K( }4 u/ G$ _  a. w
    经典递归应用场景
    ; \' l8 o. `$ I( n1 J$ j1. 文件系统遍历(目录树结构)7 z: M" r1 G0 U/ p3 m0 Q1 Y, x
    2. 快速排序/归并排序算法
    , ]9 M6 c' J5 |0 r3. 汉诺塔问题2 i4 d/ U6 N; m: U6 g
    4. 二叉树遍历(前序/中序/后序)
    6 Z. b- [8 x& O# p; _7 I6 ?5. 生成所有可能的组合(回溯算法)
    7 i3 ?$ ~1 \( Y7 V# k# O4 d" B* I( F( H& {% ^0 O9 Y; e# w. \
    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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    郁闷
    3 小时前
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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,
    ) E: O% I% ^$ n$ b6 J我推理机的核心算法应该是二叉树遍历的变种。
    3 O8 {( n- v# U0 J" F7 {" M. ?6 ]3 W" y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:3 U+ E2 p, z6 N7 v* M6 h
    Key Idea of Recursion8 F% b% K9 d4 h: g" l
    . \9 e/ C3 l  o4 u8 |. t( k' c
    A recursive function solves a problem by:
    8 ^" ]# F7 j: c; k, p5 g4 l, s
    1 ]. }, W$ N2 L( \2 c+ Q) j    Breaking the problem into smaller instances of the same problem.
    % q8 w3 o- q( c5 |6 }) i% g0 ^+ V$ k8 a4 q3 ?( m7 ]1 ]
        Solving the smallest instance directly (base case).
    " ~- F. ]9 M+ x% `
    6 S5 u3 E+ E# y( Q/ D$ m! K% d    Combining the results of smaller instances to solve the larger problem.
    8 q( O/ H' A! n) X& g& \
    5 v1 S7 R5 f- m- X; j( T( ?Components of a Recursive Function# C" @: y0 R$ \8 p1 n4 R/ O

    . r' r8 f# T( j  v' j+ M4 V    Base Case:
    3 c" ^# p. [; G- g) S
    % M9 u8 z6 K8 L" @        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.+ v. g' H* Y2 g3 w: R
    ' [* q3 }) D8 ^4 y+ i. Z, \; {
            It acts as the stopping condition to prevent infinite recursion.0 p; [3 c$ f: j$ B, s. p
    $ |  E5 O7 X. `* F
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 q! t0 M5 j$ T. U5 \

    7 @- R( c4 R4 d4 I    Recursive Case:7 B+ ?3 j, j$ |6 Y2 i, }0 T  N
    : }% R" x5 P- h: m7 L
            This is where the function calls itself with a smaller or simpler version of the problem.3 R9 o* Q, n8 S- ^& w0 [1 u

    - s' C/ l8 c' D& d        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 `6 ~2 v! G: l" m, r2 v0 r
    % R$ _, _- x! u3 Y* MExample: Factorial Calculation
    / o$ x: l7 W* _$ w7 y8 \0 b# K& t8 t3 \5 V& V4 V- ~
    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:; b2 J- l' c) M3 t* E
    : U/ Y0 `8 W3 q7 X
        Base case: 0! = 1
    * c5 K" e. H  v: W2 E
      W* w: ]7 O: g    Recursive case: n! = n * (n-1)!4 \2 \; [, F: c  W! Y5 ~# U
    5 O; I% h1 j, b5 o# Z7 m
    Here’s how it looks in code (Python):
    7 [2 I2 R8 B% B& |1 W) spython
    . x+ G% z3 j' @6 w
    7 h: \3 Q& [' x" ]) z' u* ]
    + B( ?' ~4 X1 u$ j0 i% Q; Rdef factorial(n):
      k& f) @" \, M$ I* f4 z( |, Y    # Base case: A) p; T: P( q" L
        if n == 0:' n( w& A! B5 }4 A
            return 1
    7 ~, N; \% L. k+ ]    # Recursive case0 E9 C- _& n2 Z( K/ Q
        else:; V% L9 y; w% c+ P/ r8 C" @
            return n * factorial(n - 1)
    % t7 l% y! f9 [: W& A: y  m0 t
    % L! g6 P1 J$ K' e3 T. O# Example usage
    ! f9 ?, `$ B9 b1 h  I& [print(factorial(5))  # Output: 120. n3 b5 P7 |. ^5 R. K* L7 G6 N  ]

    & @0 G0 d# v  U' S0 _: C4 QHow Recursion Works2 u5 Z. h/ P( ^

    8 V' ^  v/ o8 A, }    The function keeps calling itself with smaller inputs until it reaches the base case.7 B# a# G0 w* @

    , e! Z. K, A! n    Once the base case is reached, the function starts returning values back up the call stack.. a: q9 z0 Y; e- m
    . _! W* }, C- U$ U* S: s5 k: E6 I
        These returned values are combined to produce the final result.
    1 L% A) B3 k( Z* e: _5 Z1 i$ \) e
    6 Q( o  l- A1 @: Z  v8 gFor factorial(5):+ O0 U' E# E" P, W7 R0 f
    ! k3 g1 M/ ~( V: N- u

      R: P% l+ r) D4 j" {factorial(5) = 5 * factorial(4)4 Y' R, C" h; C& [3 \
    factorial(4) = 4 * factorial(3). Z/ ~# M3 Z0 I3 v6 X5 l
    factorial(3) = 3 * factorial(2)& ~, r: T/ E1 q3 t" I5 k! G
    factorial(2) = 2 * factorial(1)
    2 X. s3 x9 _$ [0 e1 zfactorial(1) = 1 * factorial(0)
    5 \- {+ u+ Q7 k6 b+ G  Efactorial(0) = 1  # Base case
    / l3 I( w- Y' ?2 m$ g- s6 q3 b. W% P% y) B2 _4 `) B/ h  u6 N$ e/ p6 i* D. p2 q3 H( e
    Then, the results are combined:
    ! Q/ c. d9 \# W0 z3 o) ^5 E: k/ p! W4 t% a5 m7 R6 v
      ?2 j& |% t$ c$ l
    factorial(1) = 1 * 1 = 1: x' _% D3 \( o0 V# E( ]
    factorial(2) = 2 * 1 = 2' J3 t6 ^  o6 U& r
    factorial(3) = 3 * 2 = 6- p& l+ \, \4 X7 u2 L! e" |
    factorial(4) = 4 * 6 = 24( B. D& T  Z$ P, |
    factorial(5) = 5 * 24 = 120
    2 G/ {. u2 o( C2 Q* R* e. W9 M& H4 y+ Z. N0 C
    Advantages of Recursion
    : t: n  g7 M4 u9 ~2 M" G) L& C* f( p' M) q$ z: x
        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).5 }& {* C$ e& Q) R) V3 F; V3 b
    * I9 t0 T6 i4 e( V+ V/ E) {; R
        Readability: Recursive code can be more readable and concise compared to iterative solutions.' n1 Z. E) u+ N1 n. J5 _' d

    5 @7 b3 ^- y, ^" f& x* ?/ KDisadvantages of Recursion
    * M- G& c4 a6 B: Z5 S2 g. P! n8 V
    : p7 v1 o4 U/ y$ K  m    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.
    ) X+ X; n. i( K/ Y" i. k4 t& C" H! q3 \8 |- m- h" G1 ^6 ^
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    6 q- h& \4 e9 Z2 G( K9 f# m* j" B; I& R4 \
    When to Use Recursion% y! g4 ^, _1 ^" P  a

      h2 s+ P" Q6 X8 Z  I    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    $ `9 J: o% M0 o1 p; N/ F  k6 d0 e. [0 k) `& t# a2 z3 q
        Problems with a clear base case and recursive case.; p/ s6 g7 v# u3 G& {, F) q

    2 V! a) {+ s& P1 Y5 n0 dExample: Fibonacci Sequence
    . k1 H3 O1 G- p1 q% x7 z9 l* I& z) f
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ' e; t, }: A, X- X
    ; [  m7 b7 f" _9 i; g+ |; r9 Y    Base case: fib(0) = 0, fib(1) = 1
    / g2 @3 @$ o" \2 {0 _% Q2 n9 r, B* p2 i5 x0 Q  M: ^
        Recursive case: fib(n) = fib(n-1) + fib(n-2)3 q/ C0 T4 A5 ?2 w

    , V1 f! c7 a' D# ipython
    ' S: V" o) _0 b9 ?8 t: {+ c* n6 g9 E. }  J

    & l# Z+ D% J) i, D( z, @, Wdef fibonacci(n):
    7 r% Q/ H9 }5 h* T2 Y' n    # Base cases
    ! Y# k) }7 l0 o5 A    if n == 0:
    / `) T" G4 j+ [5 i0 z! x* C0 ]% l        return 0
    2 ^  z' h7 O" h" N. l. j/ t# y    elif n == 1:. h$ h9 ?1 G& u& k- ]. f
            return 1* \3 d& O7 T$ v' t  r% i7 V$ x. w0 s- w
        # Recursive case
    . [- k& o6 v/ A/ ?    else:1 h) Z7 Y' `. U1 R% n& s) W9 S  n% t
            return fibonacci(n - 1) + fibonacci(n - 2)
    $ ^/ o) Q/ x9 _/ h1 J2 w6 o( b4 t/ S+ j/ s, q8 }) E
    # Example usage/ U4 N" {+ y7 }! y& `
    print(fibonacci(6))  # Output: 8! w+ K! `6 r: }; c0 D
    ' d7 w" f7 T6 h- p$ ^7 W$ Z" t- J
    Tail Recursion7 ^; N# O: C5 ]1 H9 n# n0 h5 g  l$ e$ T

    9 D8 c, x' {: f( V5 C* GTail 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).
    % m  m& S9 ?, T( r/ T) D5 Q7 u1 q. q1 b' z4 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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