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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
  • 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
    3 r+ t3 ], B6 W% C- R4 n& W8 w9 B, ^0 u2 |0 e
    解释的不错
    $ r# J1 i! F; m6 @/ h) s2 V9 k7 C' I/ z/ V( O! n# s' L0 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。
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    关键要素$ J5 S/ V) q' U- F& p3 o
    1. **基线条件(Base Case)**
    $ [) x! V1 v; r8 I   - 递归终止的条件,防止无限循环: V: u' r+ s' E, q/ l
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 15 W7 h) ~! G' l) V) a! j
    " M) r1 R6 x# P/ g
    2. **递归条件(Recursive Case)**
    * v; n: H; y2 O2 T, }- H   - 将原问题分解为更小的子问题( O5 ]- o! U% q: a6 h( m
       - 例如:n! = n × (n-1)!
    & a" w4 \7 ]4 a6 U- S- V( X# u0 z- U- v2 p' d4 w
    经典示例:计算阶乘* j* v& j/ m) c
    python5 [" |0 C8 O7 E6 k; Y/ C4 e
    def factorial(n):* |  ^1 n9 y! `5 |" o' V* h
        if n == 0:        # 基线条件' M6 t. G9 x6 g2 i, s+ E2 l
            return 1) e  O) u$ L9 C/ h- \+ Z' C
        else:             # 递归条件  \0 w" t/ p5 `
            return n * factorial(n-1)+ u; s+ n) ]7 [" w8 X
    执行过程(以计算 3! 为例):# B4 h/ G6 }% x2 f! o. k3 W
    factorial(3)* ~2 Q, e6 [% ]# C3 X( d' S+ M
    3 * factorial(2)
    + H: K' {2 m- W0 q0 x( d3 * (2 * factorial(1))
    $ j! a4 G( X1 ~% Y  e/ ], T3 * (2 * (1 * factorial(0)))
    - f, L9 v5 G) X& o$ @9 H3 * (2 * (1 * 1)) = 6
    % N& C6 Y) q$ j4 Z7 b! S( v, d! r9 N3 X4 \& O
    递归思维要点+ v1 X. m1 r1 j, K9 c7 M3 `+ g
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑- J2 c: n/ t& {& b
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间): K2 B$ n3 q: B6 u/ u- |
    3. **递推过程**:不断向下分解问题(递)
    9 e( @! ]# g. b4 P  H% y) `4. **回溯过程**:组合子问题结果返回(归)
    , {' P/ w, m! P) c% ]$ v, h9 m& \9 f7 G1 ]; A% ^- E) d* J) m
    注意事项
    2 ]+ Z- X$ N3 b1 l) o必须要有终止条件3 X* D' P% L# H) A* b2 y
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)6 w) l- N" E5 P# s
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
      e" M! {- @- z4 N9 b尾递归优化可以提升效率(但Python不支持)8 h+ f. p# n% a1 s. \
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    递归 vs 迭代
    8 I, k* J7 ~4 v6 W4 C0 l( z|          | 递归                          | 迭代               |
    2 k( b4 @+ c% O. n0 e|----------|-----------------------------|------------------|* I; F" m+ M' Q
    | 实现方式    | 函数自调用                        | 循环结构            |3 c) }. F4 p  a2 D
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    / I4 N; Z1 N; 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         |
    # t4 d0 N+ r: [* m) ?; z" g| 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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    # @5 h0 P) \! u4 `$ q0 [( I2 e 经典递归应用场景8 ]5 Q; q5 u7 J8 U' ]! [" I
    1. 文件系统遍历(目录树结构)* s5 G6 Z" z, d$ z- V' o  N, K
    2. 快速排序/归并排序算法! \9 s$ [) e* s, O; @" V7 e
    3. 汉诺塔问题" c( F  J  |$ J1 y; B
    4. 二叉树遍历(前序/中序/后序)
    * Z) }# L# {( n) J9 x+ K' K5. 生成所有可能的组合(回溯算法); O8 c. |3 x; w. a( a" @8 o% G

    & T. y0 |: F, }0 C$ E$ q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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    2 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,
    ( |9 g, P3 w5 S2 @, F我推理机的核心算法应该是二叉树遍历的变种。
    " K+ B) P$ V9 [! Z: u5 K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:
    : P8 S5 H+ s1 bKey Idea of Recursion
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    . Z# W' z! Q  [" MA recursive function solves a problem by:% L& Y. M) r, v  {5 h# H- {, v
    ) ?) \6 {; d( a+ t+ C0 N& W& L
        Breaking the problem into smaller instances of the same problem.0 q, |/ G7 S1 |0 j: F1 l1 m- v
    1 c/ Z2 N" Q  H4 y2 X9 {8 {
        Solving the smallest instance directly (base case).
    ) |. Y" V: j- b6 ~1 }! q
    : R! Y: T, ?5 U" l    Combining the results of smaller instances to solve the larger problem.; q  k; N" X% H5 s% s

    0 u: C, D# G1 P7 }' T4 A0 u5 dComponents of a Recursive Function
    6 q1 P9 y( F$ v: s1 G9 \  B) Q- U; U" H
        Base Case:
    7 J# i7 c; o8 e- M
      c: t+ }6 a0 {& N9 U0 m        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    7 z# e. J9 G. \4 U& J7 u% U3 H* j+ p, x+ Q" T" `
            It acts as the stopping condition to prevent infinite recursion.; ?2 i; k# V" K; [
    # F4 F) e0 o$ {* H7 f  \' [
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    " ?9 j, _' n* N& Y4 @$ L% e
    6 @! R% h0 ~* D# J1 O7 L    Recursive Case:
    # q; |- V6 c, j% d* I5 o+ [% \4 V  }/ L) K! L) y- u- E
            This is where the function calls itself with a smaller or simpler version of the problem.$ I2 }. Z; ]: L, R+ w

    # B/ s& |: _" a* S+ ~        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    8 \" Q9 g. `; f% K$ M( S6 @Example: Factorial Calculation1 j+ ?8 C3 I& z
    * x" u7 S- x% j3 P; ~7 a, v) s4 \
    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:0 U- F  D( z0 I: {
    ' V, V- A# C3 x( h5 Z7 T
        Base case: 0! = 1
    / ?; p% Y: g+ ]$ U
    : d' ~4 x  h/ G3 T    Recursive case: n! = n * (n-1)!
    / t# M7 p8 L+ L7 F/ V) ?  B) r% P8 X0 X" w( m3 z) x) b
    Here’s how it looks in code (Python):
    ' t2 e' _- j! @# A7 k/ Spython5 x* A3 H  {" A. l) V2 d

    7 `" P# J! ^- d  y# c$ U. q4 b3 s; l# r! F' t2 w& n7 ~
    def factorial(n):
    5 _$ H' ~* h( U' `8 O    # Base case6 |$ v+ `7 v8 x: d1 Q% @, T  U
        if n == 0:$ S) [2 O% V* Z, p# \# o2 z: j
            return 1
    ! @* V2 s) h6 [- A  e    # Recursive case
    ! H7 k% M* V8 d5 m) v    else:# B1 N4 h4 [# s8 m, _* K8 ]! b3 d
            return n * factorial(n - 1)
    7 Z1 p) p+ k8 L4 Z6 S; ^  f
    # c( G; [  L$ _# Example usage
    6 G) g6 a: F' oprint(factorial(5))  # Output: 1207 M& h2 L  B* h7 j+ ]
    + q& }3 }% W1 L. ]6 I6 E
    How Recursion Works2 n" l. [: b% g+ V# l

    . p& P5 F( a7 L; T6 _+ E    The function keeps calling itself with smaller inputs until it reaches the base case.5 ]( q+ g" g+ x% `6 f# J
    ! C8 Q# F3 z6 ?. d6 v
        Once the base case is reached, the function starts returning values back up the call stack.' @) p: e" l4 ?# D2 Q- e

      ?! d: I7 n& M* `4 O. O    These returned values are combined to produce the final result.; `& E) d' z. r' `3 s0 q5 M
    ' N, ]* k: n3 C: q$ p. J$ ]* }
    For factorial(5):4 V( z* u  Y4 A- }, j  U+ `

    ! K* P- `8 C6 [1 X* p4 M' f' {* o. f& G0 T6 W# X8 |
    factorial(5) = 5 * factorial(4)
    6 H6 @, x/ Q" {" t" A6 ]1 g6 ~factorial(4) = 4 * factorial(3)) i+ |2 L# C. ~# U+ S+ j
    factorial(3) = 3 * factorial(2)% K6 g- T7 L9 Y% X' p
    factorial(2) = 2 * factorial(1)  K- S, F6 }4 E" R( l
    factorial(1) = 1 * factorial(0)
    ) C) ^( n1 \! s& R& O+ a9 Mfactorial(0) = 1  # Base case
    7 ^% n' Q: K) B3 Y! I
    $ c! ]; D, X/ B, x5 `( Y4 N1 w$ iThen, the results are combined:
    $ O& b9 }- A! t5 F8 b5 K  K0 V3 L1 ^" g4 j/ l

    - C( I6 N/ m& H) |  S- s  t$ D2 mfactorial(1) = 1 * 1 = 1
    " a. U1 |5 |/ r3 o$ Wfactorial(2) = 2 * 1 = 2: F; ~8 e" X, r' h0 b9 ]* g' X
    factorial(3) = 3 * 2 = 6
    # c* k3 q( k9 q1 y) Zfactorial(4) = 4 * 6 = 24( R+ ]4 d# R2 ]! A4 t9 t5 |
    factorial(5) = 5 * 24 = 120
    - @( o% ^$ Q8 K% S! @- p& B& r, V0 x3 h& s) W' a
    Advantages of Recursion
    ! ]0 i% l2 b: [& C2 b6 g/ `/ S- E2 w& t6 m% y# T. Y
        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).
    " D6 y8 _# i; y8 @. W3 S$ N
    7 B; y, h" j0 n! C    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ) {% \) B% S8 Y' p4 o8 l3 R6 {! d4 R3 w* z% h/ O" Y% Z
    Disadvantages of Recursion; M# ^) K! _; q# M6 O

    " O& X1 _* e) k9 x6 L- T, Y4 ^    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.
    2 i9 Z/ z# @( c5 |, J% d: f1 C1 B! v8 o$ M! Q0 b% a
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)." Y7 w' M# V' X2 ?

    : ?) |$ E8 D5 ?6 j6 C9 YWhen to Use Recursion% T2 M4 j" c) R! ?# ]8 I4 m

    , C: L! C/ }% e1 p, c+ g    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).4 C$ ^5 W- Z% z  Y
    ; C* g# {' {8 R5 h
        Problems with a clear base case and recursive case.( Z  \+ v& K/ E

    - W1 f2 l6 ^% W% i, L$ EExample: Fibonacci Sequence
    . {) U! @# c/ K* ~# j# a4 V$ J# a# |/ H! |1 `2 ~
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    " Z+ _* G  M, @, w4 Z' q
    ! v( W3 d8 E) {5 Y  a( C    Base case: fib(0) = 0, fib(1) = 18 T8 p, Z5 [# w* s0 K  N* B/ w& A
    5 p2 G. |( W3 V3 U2 ^5 v0 m
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ) \3 o/ C  Y' r2 T. \7 s& |
    : c8 ]3 G; w+ T; ipython
    7 }" I8 ^7 w& ]7 ~) h
    : p# Q( R3 ?- G
    7 X6 q# Z& X6 l( A  Zdef fibonacci(n):
      R8 g/ `) s$ u    # Base cases& R5 y4 L4 F- u8 ]! _$ K
        if n == 0:
    : I' C. E; r3 S  v2 W$ N* B! l3 d        return 0
    9 c& R; b  B4 r. z5 b    elif n == 1:4 E4 u1 N  C) d' s2 ^+ l/ a
            return 1
    / I* T  d# r1 o3 o5 y    # Recursive case: F+ v$ B& u' R9 y& E
        else:* p- U) q2 q/ s/ r
            return fibonacci(n - 1) + fibonacci(n - 2)3 ^& t. v1 }+ y( |5 d& K
    # t) Y6 b" i, \- r
    # Example usage
    # R( H$ t2 q" f; Tprint(fibonacci(6))  # Output: 8
    6 X1 ^2 Q, w+ J/ S- x1 E8 K% ^
    / x* q# Y; n) @& J; xTail Recursion. a$ ]+ U& ?. q; v& _7 O5 P

    8 X& S3 ?3 m3 _% M& B3 q' A; x3 c5 ^; YTail 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).
    . d' g, x% B3 Z4 i& I$ v* A! _5 D: V
    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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