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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
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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 2 H, r1 \- @1 [% C7 \4 K+ Q

    + }  p' U. l- ~5 K9 X$ a9 R解释的不错
      l) p% R7 K( O1 W
    & F  G4 Q0 E0 Q$ j5 f3 v2 f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。% `3 s* w4 ~# \( R& Y( i
    ; {' V2 ?! S: C" @$ s7 n. t! {0 ?
    关键要素& T" ?; q; j; S# O, o# @6 N
    1. **基线条件(Base Case)**: P! O  F  Z. m; k/ M
       - 递归终止的条件,防止无限循环0 n$ h/ |9 h+ l# l4 B) E
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      n- k/ ~9 E  r, T( z6 x4 ~0 ^
    ; S( B! i/ y# S) |! Y2. **递归条件(Recursive Case)**) [" I3 a7 s5 C# c' g. R
       - 将原问题分解为更小的子问题* x4 M' w0 m. C( J+ Q/ l$ d
       - 例如:n! = n × (n-1)!7 j: B% U/ H- u0 {& t! ^# Z( M

    ' X) p4 ?  Y. @* n 经典示例:计算阶乘7 A% Q* w3 [' U$ k4 y: d
    python3 G/ w4 K, I- Z& p
    def factorial(n):
    2 K  y4 A. t9 i* o* s    if n == 0:        # 基线条件
    / b4 ~8 i1 F' \0 G& V        return 1
    : H0 t$ n* W, _! d7 J    else:             # 递归条件
    ! m) h+ w1 n9 C, Y: `        return n * factorial(n-1)0 i; i9 c, E8 d& m# F/ W
    执行过程(以计算 3! 为例):* m" ?5 o- k0 Y% `+ q% t
    factorial(3)$ |3 h/ a# E5 r7 _! i- e
    3 * factorial(2)' k9 I- @. S( W- C
    3 * (2 * factorial(1))" Q* c, k/ X& _3 j' E4 ?" b
    3 * (2 * (1 * factorial(0)))5 f! |" M# T# r1 q3 B. @
    3 * (2 * (1 * 1)) = 6/ m' a3 t2 u8 p0 ]! k$ E
    & A; l2 f1 ]" @( u. m$ N4 u$ A
    递归思维要点  ]4 x" }+ u2 H- G8 I0 k/ P3 Q) q
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    8 P* q6 j& P+ b+ B+ y2 f2. **栈结构**:每次调用都会创建新的栈帧(内存空间)6 o! }7 X$ ^$ W& g+ v, V, M+ q
    3. **递推过程**:不断向下分解问题(递)
    ) C# X! h# d4 v3 {4 s$ o4. **回溯过程**:组合子问题结果返回(归)+ K3 y8 l! I5 }; C+ K( Y4 f
    + I/ f/ H6 J9 t- |) h
    注意事项
      O) ]; ]( Z# x) \必须要有终止条件
    $ R+ p  r6 s1 m' i# x递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    2 A% |  v( S; `; K  Z2 {7 X某些问题用递归更直观(如树遍历),但效率可能不如迭代
    . W# o* p7 L. `( w" u2 u# O尾递归优化可以提升效率(但Python不支持)
    0 d8 l. V9 Y* k0 Z
    ( V2 E. i) q% g8 _( ^ 递归 vs 迭代
    . `8 p% \) _; V3 W1 M|          | 递归                          | 迭代               |
      b1 ?2 w" `- m7 b& y|----------|-----------------------------|------------------|
    & V% Q3 \# P0 C! }; A0 E. A| 实现方式    | 函数自调用                        | 循环结构            |$ z* H( G( j" T$ P( a6 j
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - Z7 b+ m+ ]8 |) q0 x2 @% g2 L1 x| 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         |; H& z6 [9 T+ I. H. d- N) _
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& B4 k+ n* C- k) K4 L: }5 B6 Z

    # a$ }3 r3 [5 S, M1 x" n* O# A 经典递归应用场景
    ; S$ Y! m) a; L7 ]" m; T1. 文件系统遍历(目录树结构)
    5 P/ h  f8 a$ d: M2. 快速排序/归并排序算法" ^3 o( k$ Q( R
    3. 汉诺塔问题  E# N' v7 J- E# |" ]
    4. 二叉树遍历(前序/中序/后序)
    2 o: t4 S9 t3 ]( M5. 生成所有可能的组合(回溯算法)' H) s( D+ E8 i

    6 A4 \" y) Y. D2 P. R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,
    1 i8 C. k1 N! b+ Q) t* K我推理机的核心算法应该是二叉树遍历的变种。
    + C( ]0 |9 P* w9 E* w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:
    ( A$ e4 j7 `$ i/ \9 I' LKey Idea of Recursion
    & g/ r( r7 k6 q0 n5 p
    / J8 b& @8 ?; i, f$ c3 GA recursive function solves a problem by:
    4 B4 Y! I/ ]5 A/ D/ Z- \$ Z
    9 ]# W/ o! H2 x! t& p* g1 w. r    Breaking the problem into smaller instances of the same problem.. K# ]6 p8 a3 T7 d

    ; E" d) M2 l# ?3 d9 g/ l. q    Solving the smallest instance directly (base case).  P( d1 p, ^/ o* B% v1 [5 s. s* ^7 u

    3 ]7 h* N8 m. y5 c3 f) `. o' c5 Y    Combining the results of smaller instances to solve the larger problem.
    $ E2 N/ {3 W' M" X( U6 o  u7 X& ^$ ?0 T1 s9 n
    Components of a Recursive Function6 \8 y: x1 K' t
    ( m( d) r8 I+ N$ M( x
        Base Case:
    4 w) }1 _  u- B# I/ G) `$ N% m9 O* c4 Z/ \7 {9 G' q% H/ ?
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.8 X- u1 R" l, S/ V2 v

    5 F: k- b- ]* j" O+ Q3 G5 }% m" L        It acts as the stopping condition to prevent infinite recursion.
    7 l! Z  b7 O; G; l, `% [2 P$ u" I  f- U6 R, z* {( T
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.1 }8 |% e, P) E, m9 z
    7 X/ M: e. F6 ^
        Recursive Case:
    & L* j2 K, M4 ~% G/ }* _0 l
    * w2 Q; X! T# S        This is where the function calls itself with a smaller or simpler version of the problem.
    9 g. u( d, u1 J4 M' ]  v/ p& d' ^1 s3 t
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    - J+ I. I; d4 a- h" d5 v4 ^
    + w- u  F* f6 p" j( ]. YExample: Factorial Calculation
    : N7 {, P7 R9 u& |$ m" a8 C
    - W. R$ t9 s/ L7 g6 H4 E+ ZThe 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 X) a1 P7 [- c! S) |
    * F3 J, O" t# b    Base case: 0! = 1
    ! v! \# J0 P( w0 Q' i" L9 @
    . M. M, e$ w. q- [0 {* T- g6 L    Recursive case: n! = n * (n-1)!
    & D0 G0 P/ m) d8 {* N+ x. C: C- M+ m
    Here’s how it looks in code (Python):
    3 i3 m) V8 R- u. p6 G8 Q4 e- Zpython
    " @, ]) I/ Y" q: l* Z" Y" M- X2 p. }% p
    . x' _" u: m; A, f) ~' Q- B1 ^/ o5 Q9 ~; r
    def factorial(n):
    3 X* A; t0 e1 S  r& X    # Base case5 o" `  ]/ \! v1 R7 c0 Y
        if n == 0:
    / v& T4 Y* @" q( b        return 1$ n7 I6 i) r3 V7 E& C) v
        # Recursive case0 `: x$ X$ \+ q* Z" l: C
        else:/ i7 }/ V) m7 R0 g
            return n * factorial(n - 1)7 q& z4 P! s& u6 V, p, S

    $ |0 G" g; @  _0 q3 y& F2 l# Example usage% u6 o; v. B: I$ y
    print(factorial(5))  # Output: 120
    , h" R" ]- z- H* r. d9 ]
    2 m, z% e3 k  p. s" ~% I/ AHow Recursion Works2 F" q! o" }8 o- c
    7 ^( T# [- e& \/ J7 M: H7 `8 B/ L
        The function keeps calling itself with smaller inputs until it reaches the base case.
    . S* w9 D- s- L5 k. b
    . I% r" t: C+ r' p, j, g4 ?, m: [    Once the base case is reached, the function starts returning values back up the call stack.2 M$ E6 Z0 v7 N% a

    ; h8 d; e! p: t& S5 ^    These returned values are combined to produce the final result.
    : t) {# i/ m* e5 I) ~6 J& `6 S# m+ W! \% W, q4 e4 v# _9 z
    For factorial(5):
    % @9 n" w& r7 K$ F0 m. v3 K3 C9 f  a5 h5 u' _% h: r' M1 D/ w$ m

      s) i7 Z8 ]9 U8 f) Q7 W& }. S% cfactorial(5) = 5 * factorial(4)0 n' I8 ^' S, L. ^" l+ s2 @" q
    factorial(4) = 4 * factorial(3)
    ! R, w; {  m5 f2 M9 {# vfactorial(3) = 3 * factorial(2)+ V3 Y/ Y6 y% C
    factorial(2) = 2 * factorial(1); @  [* F. O. ]$ ?
    factorial(1) = 1 * factorial(0)
    9 M# g) ]4 B  }0 k* |factorial(0) = 1  # Base case
    5 n6 s/ s; I1 n% s) ]; C
    : L! u* y, _: x$ ^Then, the results are combined:
    1 ?5 `/ F8 I6 F: \; E6 Z  m* q/ a$ u* e6 g5 _- U  Q5 K* p- U
    , T: v" d, @5 A/ B; Q0 L% ]- Q
    factorial(1) = 1 * 1 = 1
    " z7 Q( K$ u  J* @( d! e6 [2 Nfactorial(2) = 2 * 1 = 2
    ! O$ q  Y. x* A! D7 @factorial(3) = 3 * 2 = 60 s5 n) h( Z2 P
    factorial(4) = 4 * 6 = 24- ?9 l. b. J' Y# Z/ K$ [
    factorial(5) = 5 * 24 = 120+ r( [5 w/ |- y) L9 g' |$ U* j
    ( {& e, D, J$ o( e/ g9 J7 }) r
    Advantages of Recursion
    7 D7 s& v4 y1 }5 [/ A" D4 T  q2 b$ q
        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).
    ( v; ?1 u6 E& Y# U
    5 U( Z1 J. C6 S+ n1 _' `    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ( Q2 E7 A1 X$ u# b0 u
    : X# [/ A6 C/ F* W! YDisadvantages of Recursion
    4 c9 M0 B+ j5 z9 b  y  P+ w2 t& ^1 I: @+ _
        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.; H. ^- v1 B" R, ~3 E

    ; a, P, {3 E% Q' g    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    0 \5 G: V- D' Z% Q, F( }" _7 h: v1 s% S  q- o0 Y
    When to Use Recursion
    + b- A8 Y+ r8 N
    1 y8 Z5 E/ p7 x5 h' n    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    5 e4 U$ M4 G% V, ?2 E9 F
    / O5 i( Y) e- p) |4 c" s    Problems with a clear base case and recursive case.
    ' ^8 a, ^4 g# o$ G+ v7 j9 |( O! H
    Example: Fibonacci Sequence
    ( J. b( m, d2 A8 H7 S7 C) U9 k. I7 t* Z; Y( H
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * F: o9 K" y- N8 t* b0 g: [- @9 ~/ e" B: \( ]5 u7 G
        Base case: fib(0) = 0, fib(1) = 1
    6 C% |5 G3 m" h6 A5 c0 O, p# s6 s
    " I% g9 s* I5 o& l, }$ p    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ' o( h. a2 {# n: J) G" M% m
    . F* M1 i# w7 ~! i+ {; a2 spython0 c/ ~1 X3 _5 f$ u$ q; C0 s8 |
    2 F) O, \& C9 l! ?" }' i
    / A: _% ~$ n2 P
    def fibonacci(n):
    . ^' y: B) k- O( T9 F" [1 w    # Base cases
    9 O' ~& j/ m  o0 @! i- s5 x    if n == 0:. ~* g7 t2 a4 ^( d( B
            return 01 U8 ]- R/ f5 N2 T3 k1 \* U
        elif n == 1:- S$ P- c+ d7 d0 w/ o, I
            return 1/ u& u1 L1 P9 \/ j) Q
        # Recursive case
    1 o: m0 G5 H% ^& R+ V! E6 A7 v    else:
    ; A; y; S3 V& {0 [, D0 _" l6 b& x        return fibonacci(n - 1) + fibonacci(n - 2)
    ! V# d3 N- W+ G* S
    " \5 k; M6 J: u% L& t" G" a7 s2 d% E# Example usage# H1 n( V. f! m' }+ M& w( q
    print(fibonacci(6))  # Output: 8; w# N; u/ D' h" S
    1 a! N; \, v3 v& b9 D
    Tail Recursion
    $ n. c* }% T$ Q5 H
    1 u5 k( P( `! [4 ITail 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).% ^1 ?6 o6 S  @9 [8 j7 j, T1 R

    - X9 }" p! Z; k; X8 PIn 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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