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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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    2015-11-30 11:11
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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
    & o( n8 N! u$ Y/ ^
    9 E0 i, e7 V8 t% C& H6 N0 N解释的不错
    5 ]  U+ K5 P! ?: J
    ; [% V' W8 P" s3 J# [# [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。0 d1 n3 x4 q. W( j" q( t
    ; o% X7 e  l6 ^1 h2 [! @4 G$ y& Q
    关键要素
    : I$ r; s3 z" t% V5 _/ M, @* K1. **基线条件(Base Case)**
    ( d: l8 E! F& L# L" R   - 递归终止的条件,防止无限循环
    / M6 v2 x6 |, a' g   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1- Z6 {/ e3 ~' g

    , C) w( ~* {% O# [2 F% M7 v  q! A1 y2. **递归条件(Recursive Case)**+ \7 Q3 l7 O" l3 d- P- ]- n
       - 将原问题分解为更小的子问题  w" e1 M7 `! F* E0 S
       - 例如:n! = n × (n-1)!1 [) _) X9 b" y, y8 w

    & Q( s7 p1 |3 ?" N$ n% M' n 经典示例:计算阶乘# R4 R6 i: `' c9 e, f
    python
    " P! m& d+ N- |# C- ?% Ldef factorial(n):( f. r0 l; U& V
        if n == 0:        # 基线条件
    ) S: v0 g3 z) |, ~: {" a+ h6 V        return 1
    ! Q( P& j1 W! Z2 Q. p' i1 z    else:             # 递归条件1 h  F: |  \$ }9 j# ]# |; T
            return n * factorial(n-1)
      ^# O1 ]3 W# o/ J+ m  u, x- z执行过程(以计算 3! 为例):
    8 Q0 y6 M& Z# bfactorial(3)
    ! x. p! t) P! L; M& k3 * factorial(2)
    " z9 e+ x0 a* u/ y) H2 j$ h! n2 \3 * (2 * factorial(1))
    ' \. L9 m0 [9 E/ S! O3 * (2 * (1 * factorial(0)))
    6 s. Z5 J, u: }2 Y3 * (2 * (1 * 1)) = 6
    9 h# R' k# K3 t0 e; ~+ J+ t/ L5 @; Z0 W
    递归思维要点1 v. y' A7 W: \, b) ?. ?6 M
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑" X, |' D: ], B/ v9 H
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)$ s$ g0 j/ w% a/ K9 u5 l, X* m
    3. **递推过程**:不断向下分解问题(递)
    3 J# g: y( f6 B5 l& D" ]% i4. **回溯过程**:组合子问题结果返回(归)
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    : c4 x7 e+ B) E 注意事项2 @# ~' ^) h  q5 `- J4 C
    必须要有终止条件) w/ h" ~' _8 j' W0 `
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)% L, ^; z( Z- X& c2 ]& |
    某些问题用递归更直观(如树遍历),但效率可能不如迭代: n/ Y- M5 G/ I' y" l4 i+ x% K
    尾递归优化可以提升效率(但Python不支持)
    ( [; H) ]8 |4 C' C, A4 C
    ' Q2 A' g7 `/ m" G 递归 vs 迭代
    ) e1 G( B- ?, p4 ^( e|          | 递归                          | 迭代               |
    # e6 B: `) Z5 Y7 q|----------|-----------------------------|------------------|6 B8 Z! d0 K! c: c
    | 实现方式    | 函数自调用                        | 循环结构            |
    ' I0 U9 ^  K' M0 O4 T' X| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    + ~# m% H% r; M8 Q& 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         |. t* O3 M& q4 _- f- g5 G: e
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    9 v7 q  }9 W8 f4 P0 ?, L5 ~+ O/ P3 Z0 w: c' s7 y2 J
    经典递归应用场景
    2 u1 P! G  T+ h" y1 {4 p. K9 ^1. 文件系统遍历(目录树结构)! i7 |- A. o5 B/ A0 U
    2. 快速排序/归并排序算法- Q& ]# S4 ]' Z; D$ ?
    3. 汉诺塔问题
    ' R) L* T  _2 ?/ z& c0 M* a: n  u4. 二叉树遍历(前序/中序/后序)
    & y. v9 H% K1 p% s  V, I5. 生成所有可能的组合(回溯算法)- g: J; b; ]/ W1 j0 P* }% \
    " _4 G# x  v) P) h* C6 c
    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,7 M1 n% R9 [: o  @/ e
    我推理机的核心算法应该是二叉树遍历的变种。
    - D, z$ l+ _+ F/ 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。
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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:/ W& c" `' \) h
    Key Idea of Recursion* E, t! e7 i! X- ^3 P8 A# i5 k% t  [

    6 }6 o& _/ _, o) @9 ]7 q0 NA recursive function solves a problem by:0 ?- T) _7 D% k7 n8 @+ d

    5 d: G3 O, f! N* j! [+ r    Breaking the problem into smaller instances of the same problem.$ Q1 \4 j$ |/ }4 X# i

    ) b* J! w/ i+ J, i5 t+ x% j    Solving the smallest instance directly (base case).
      `* _* D% Q3 W$ ^; t- x
    - U* ~4 k" M0 P( \4 f# b/ @5 F    Combining the results of smaller instances to solve the larger problem.! h* m( t6 O' k7 P# Y- r- D

    # g; z8 ^4 p7 M& \0 W' m$ Y7 o/ V. ]" gComponents of a Recursive Function
    % L# \7 m! ^- E' v) f" E) v7 j7 \4 I# {) `6 g% [$ a
        Base Case:4 c5 S3 H0 j& H2 S$ N; \2 _: h

    , S1 A& M2 X5 }  x7 m, V+ W: T" [        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.! e$ u, V! Y; I; g7 T
    7 n9 Y. u6 Y6 W
            It acts as the stopping condition to prevent infinite recursion.- _2 Z/ ?1 n/ q
    8 y9 D! F" V3 g" I3 [3 {! A) a. }0 R
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    % ^( D: m0 o7 W! l( j5 l( `& {; ^8 g3 F: Q/ n' N7 _' x
        Recursive Case:# O* z' R: s9 Y+ C) v/ M
    ; i2 X7 C# i; K. P
            This is where the function calls itself with a smaller or simpler version of the problem.
    2 t8 c) G! q* b4 r6 a8 o- x7 j$ b
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).# ^- Q0 f/ I3 D

    1 M3 R# o% W* q2 I9 X3 `/ m. e+ jExample: Factorial Calculation
    . ?; w" @9 U% F( M4 K7 `. a6 x! d$ F% ?3 V! p" G4 D
    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! Z# W6 ~+ B" ^& O  I* {3 v5 {; A( m
        Base case: 0! = 1
    * S3 F: [0 W# a9 [; t9 e3 B; V% h4 x8 V# K  s: V
        Recursive case: n! = n * (n-1)!
    % I1 d9 [1 N0 d! t) P  g
    ; k6 f; b$ `! R. jHere’s how it looks in code (Python):8 @" _) Q" Z: M: x9 }6 [+ r
    python
    8 q0 W2 n% y: S. s1 z/ I1 G. f5 Z: m% u6 ]; U/ O2 b- V
    & e/ l& W5 ^: |. Z6 a: S
    def factorial(n):
    : P2 D; \  X4 H' ?+ f* a    # Base case$ p! g( Q2 A* P; }
        if n == 0:
    ' _' `# ?) c0 [1 R1 H5 C        return 1
    : a1 g' A3 g9 |4 I5 x: P    # Recursive case. H0 [' N# i% l3 |; F$ A9 j
        else:
    ! ~* [& E: H! r$ C' I$ t! V# N        return n * factorial(n - 1)
    8 M6 `% a; S1 W+ ~% ~, S5 o' ?5 x6 e$ N) V, P6 P7 q
    # Example usage
    0 C9 n9 F8 r2 U, g5 q, ~  b/ r: vprint(factorial(5))  # Output: 120% J# Q4 {# t. y+ H% i4 e2 n- R+ o' t
    + y- e8 ~6 e* M  s
    How Recursion Works, K9 n  R' W, R. `- M! J) O

    3 I# Q6 [( F: c& W6 C+ f    The function keeps calling itself with smaller inputs until it reaches the base case.9 a' r: N: K/ J

    % L% X* P! u0 ^5 m( U/ C    Once the base case is reached, the function starts returning values back up the call stack.
      v% z4 H9 d- b' H1 _/ R1 J
    ( q9 @/ U* q% k9 y    These returned values are combined to produce the final result.
    9 s6 L- R! E4 V( e
    8 f$ x; s" q8 v- _- u0 b* ZFor factorial(5):
    ( f7 ^- j  b. x& Y0 z1 Q: T! v4 Y+ p4 H) k4 I) [; ?% x& P

    ( R  w9 v1 i' F0 Q6 z/ [' ?' Rfactorial(5) = 5 * factorial(4); |* m  r# I4 O8 f1 O* \" o
    factorial(4) = 4 * factorial(3)
    ! A- q8 j1 L9 C( Ifactorial(3) = 3 * factorial(2)4 `% Y2 a6 L. f" ?8 A
    factorial(2) = 2 * factorial(1)' f/ V! {: o( x* G
    factorial(1) = 1 * factorial(0)9 G$ Q% n% w9 U: f9 }4 P+ a$ q* z
    factorial(0) = 1  # Base case
    ( o# c" O" p: H3 y
      z( m" K4 H3 XThen, the results are combined:  [- z3 q5 j3 x9 r) W/ d1 C

    $ {' v- |/ o. k4 w. z9 g* c! M1 c' z3 u; c1 z8 N/ N- P" z) H" @
    factorial(1) = 1 * 1 = 1
    . j- f7 R/ h$ D9 S; _factorial(2) = 2 * 1 = 2' q  @, C! [* }! w( ^/ U: O4 V
    factorial(3) = 3 * 2 = 6" I+ u8 q. D& A
    factorial(4) = 4 * 6 = 244 z% s: }. g+ i7 Q6 O+ E! G0 y
    factorial(5) = 5 * 24 = 120# R  o% F+ j" A

    ' g1 |0 ?5 B* G2 Q" UAdvantages of Recursion
    4 t; G3 K( Y$ @* {7 i2 W8 H; W5 A, |+ O+ s, g! 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).1 w4 I& V8 z, V5 a1 H7 R
    4 c2 X: ^! _) n6 u" c
        Readability: Recursive code can be more readable and concise compared to iterative solutions.- h9 M, w0 a% R8 k- G, `' I  H# ~
    , N% R0 s; ~* x
    Disadvantages of Recursion
    # U2 a- B8 I  \. u- s$ \
    1 P  {8 K9 v% h    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.
    6 g; n# O3 t  ]+ A5 B4 x$ r
    0 ]# Q0 a7 @( b& c    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    6 ^9 M+ w! s* B/ Z( O" [0 [# P# v# d: ]" `4 E3 `
    When to Use Recursion
    5 u5 K7 }0 S& x) H! O) q. l5 G2 C& `0 `$ u/ L
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 y) R1 e5 i) j* y5 b. M
    + ]2 |& K/ I% T/ A" n1 o9 l
        Problems with a clear base case and recursive case.
    / f* P, k2 m3 ^9 G
    - h7 N, p6 O9 ~' {Example: Fibonacci Sequence
    0 o1 {. S+ }9 }0 h8 @7 A
    " u2 G" H% M/ c: tThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:0 A! L5 l7 C9 i6 _) p

    6 k0 h/ j1 h* `* z3 D- W    Base case: fib(0) = 0, fib(1) = 1
    . W4 A( P- L! Y' j2 \6 I
    - I+ Z0 t! `/ p5 @  M    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    * h( }5 [2 C# D' l, Q0 p7 |5 u; I; B0 {: ?% M' V0 o
    python
    : V6 V/ K, ]: X7 u3 c0 l& v
    # k, i" A, ?6 P9 T, c1 w7 N
    9 b9 K0 q7 m+ Z. |+ p2 v$ Zdef fibonacci(n):
    ! Y* i' [- g, M, n2 Y- F: x! |4 [    # Base cases2 O% r4 ]0 z/ e6 O
        if n == 0:/ X7 ^! a4 {8 m' P
            return 08 A. X4 Z+ \5 {
        elif n == 1:2 p3 o- B, w9 T7 b: u# P
            return 1
    % H+ ~& i' s6 q" f8 Z3 \    # Recursive case6 P) K2 S" f5 n3 {1 U+ t/ y  y8 x
        else:
    / }) j& i1 Y8 Y5 n# i' V) i& P        return fibonacci(n - 1) + fibonacci(n - 2)! [$ d' t2 ^" C% S0 ]" r4 W- Z8 e

    5 A3 W8 m5 h7 ^# Example usage2 A- C, F- i8 I6 T: W, G/ b
    print(fibonacci(6))  # Output: 85 P$ N, d3 P8 G  n
      n  ~0 k/ a7 B7 S- g, a# }
    Tail Recursion8 E" d# C' i/ J/ |$ r  @2 H4 F
    2 h, P) z1 Z9 B/ R
    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).
    ; C) x7 J* y( ^0 o& J' r9 m8 G- c- ^
    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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