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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| 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 0 u& o, u4 [# u$ G
    1 r) I1 O; D* u& w. q: H
    解释的不错
    3 Y0 s0 G0 l; D1 q& U" O% N- c" `  a; q# `. p
    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。
      Z8 ~0 O6 Z0 l5 _- l/ S
    + ]. `4 m4 ^- h4 Y 关键要素( g% j0 ~0 b1 U0 @
    1. **基线条件(Base Case)**
    7 J$ h& G% M3 |) a8 b9 X) l3 U   - 递归终止的条件,防止无限循环7 i' @0 f. X2 c# x  h. _7 |# q
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1! A$ `! P. X- U4 C% v! n+ X

    ; H* |% J( p" ^6 H7 h2. **递归条件(Recursive Case)**
    9 K1 ?' _3 E  l2 g! d8 E! E   - 将原问题分解为更小的子问题
    ( h* E+ P# E7 e9 B# [! F$ J# C0 ?- f   - 例如:n! = n × (n-1)!6 e6 E# L# s4 X# h8 E

    2 b4 _0 b! ]$ | 经典示例:计算阶乘
    9 @' R5 x. m( r5 h, ~python8 x4 X% n! V0 {7 l& c3 y
    def factorial(n):% g9 s+ _4 u9 z2 M
        if n == 0:        # 基线条件
    & C4 A1 k3 N0 e: E        return 1
    ! n% f7 {. p; X$ @9 c    else:             # 递归条件3 q' w9 ]$ x: [
            return n * factorial(n-1)
    1 A/ R6 E0 t% M8 A! v4 {! \执行过程(以计算 3! 为例):
    ) t9 n) W: a6 F2 H; H5 I7 s3 }factorial(3)0 `5 `4 b. x$ Y# h# d8 p, d, a" f
    3 * factorial(2)5 k6 W( G9 X' G4 w7 q
    3 * (2 * factorial(1))
      M5 g6 X: n  K. t3 * (2 * (1 * factorial(0)))
    , ]  H- c; t, e# u7 T8 ]0 I3 * (2 * (1 * 1)) = 6
    & q6 z8 m' h  F: _& ?8 `% b3 _& z
    ! s, X" ]0 E$ Y( q* ^" S 递归思维要点' z8 G' o4 T7 ]6 d- J
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    # q% `1 i* s+ F4 w+ }4 T% N) O2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- G: k* Y# x( t! C  M$ w5 O4 E
    3. **递推过程**:不断向下分解问题(递)5 L% r* C+ M! E( P+ e$ J" ?
    4. **回溯过程**:组合子问题结果返回(归)3 d4 J2 C4 r1 p4 h

    2 `% p  k6 d, g4 U. }; V! x  w9 g 注意事项( k4 @% o+ R" S0 N6 H% R
    必须要有终止条件2 L6 V* O( l: I& B  b( i3 h
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ( y4 C& U5 {" N5 p) |1 @某些问题用递归更直观(如树遍历),但效率可能不如迭代* S# c( y% i3 D+ \1 Q
    尾递归优化可以提升效率(但Python不支持)
    ' R0 _) b3 ?  H2 ?
    : `* ]- p: Y1 \4 y- S 递归 vs 迭代
    % T* l$ b# o6 F% Q3 L|          | 递归                          | 迭代               |* P. \& K$ v* V4 D$ X
    |----------|-----------------------------|------------------|
    ! n: [: |& m; m3 v| 实现方式    | 函数自调用                        | 循环结构            |" y+ B6 M+ g6 C7 B9 g6 X/ D
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 x8 E8 X8 r% l0 y+ g8 E
    | 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         |
    2 _1 r: ~/ Q+ X3 r# N8 z8 |$ j3 j6 z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |) r2 P3 m8 k) n8 z

    " M. Q; X4 l" k" v3 T' N4 ` 经典递归应用场景
    0 m1 p2 h4 S( Q" h* O1. 文件系统遍历(目录树结构)
    1 D6 i' x5 x5 |3 @2. 快速排序/归并排序算法
    3 F0 A; ]8 z  \/ o) C, }3. 汉诺塔问题
    2 T/ ?, G$ U, I2 N, Q5 m! l- Q/ t4. 二叉树遍历(前序/中序/后序), x* N/ ~0 s% u9 b7 M4 p$ J4 M
    5. 生成所有可能的组合(回溯算法)
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    . z% ~( D+ L' I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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  • TA的每日心情
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    昨天 07:40
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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,
    ; x' l% S3 u. \+ E我推理机的核心算法应该是二叉树遍历的变种。+ r) I0 n0 M4 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:
    + M0 a6 v5 H3 a$ k( q$ X6 e) b7 YKey Idea of Recursion
    1 G2 ?& |& X) F# @, _5 |' O2 a! P% M; h* n2 j
    A recursive function solves a problem by:
    9 I9 b4 }: D' p! D3 }( X! S$ c* J0 L5 z6 d8 o
        Breaking the problem into smaller instances of the same problem.$ Q$ @3 ?' n; `* P
    5 g1 v1 M2 ?" I" \6 i
        Solving the smallest instance directly (base case).
    ) e0 ?3 \8 U! j. J) s
    $ ?  ]/ I+ ^; J+ Q6 w' _    Combining the results of smaller instances to solve the larger problem.
    3 ?6 m' Z; ?1 {! a6 P" ]0 b. M5 k7 N* I5 c
    ; @7 g" h, y+ v9 d, oComponents of a Recursive Function
    2 E- V# P0 M/ F- |6 B6 j
    7 V% ~- v% c3 ]6 d    Base Case:
    ) k) _  r! J7 k# e) k0 Y9 c) C
    * J; p0 [, i# d; P5 a        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    , q% C; g' l! }
    , X" U- Q+ P" G) |. \2 e( x        It acts as the stopping condition to prevent infinite recursion.
    " R  u. D* u/ n6 |0 k$ W9 l% P( E% B* _5 d3 w
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* A5 G& P8 z# `

    ) c. W# f. u2 U2 P    Recursive Case:, J+ j2 @; B5 O& e5 ~

    $ X# l' W8 ^6 H+ g4 j+ j6 t        This is where the function calls itself with a smaller or simpler version of the problem.
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    ; u# n$ J9 t* a. A2 o' ?        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    4 n* ]3 w7 N& O& F2 n* |$ J1 Q5 @0 |# r- Y6 i- t
    Example: Factorial Calculation
    ; r8 l& l3 h. h- P: r; i+ l# m
    ' Q) n! }. ?; g" yThe 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:
    ' X5 `/ |, u) w$ [+ ]( N. p
    . I/ z& Y  d. w. y! S# w    Base case: 0! = 1" v( n4 i) Z6 i5 P: W

    . q% R% T3 O9 V# O4 O0 h    Recursive case: n! = n * (n-1)!) V# h, v. A, @" f

    # a, r/ h% h0 i! _) ~% ?Here’s how it looks in code (Python):
    ; o! C1 @/ t3 V7 h4 M; E5 f$ ]9 }python7 m3 n, R7 j" `

    6 X) p6 P; H& j+ B# _
    / L% b& k1 S9 U8 f4 A% J2 odef factorial(n):; x/ H3 ]. @4 l' B
        # Base case
    " |/ ]* V1 v( R. Q$ N8 P9 H! V    if n == 0:
    ' r9 |9 Z/ }9 y        return 1
    $ b* V- C- d9 g/ w- e  s8 u0 E    # Recursive case
    : y6 p  T) {0 U' d  Y$ r  N    else:, e4 R" X, {' Y) m' |  J
            return n * factorial(n - 1)
    4 H4 ?4 V# D0 O* p, Q& P  y& p) X5 u7 T7 [
    # Example usage
    ' R2 b5 X8 Q+ J! Z+ C/ p- xprint(factorial(5))  # Output: 120
    # n2 {& }# E3 D% @7 P
    9 B: Z" g9 O; jHow Recursion Works6 m# A) n' g, q; m5 S) V
    # c  u4 _- n0 M4 o. b0 u% M" I, [/ o
        The function keeps calling itself with smaller inputs until it reaches the base case.
    , z4 `5 q9 u2 s$ H/ y/ ~6 c1 S
    , w$ g. ~+ Y# V- z" x' U% J+ c# ^    Once the base case is reached, the function starts returning values back up the call stack.3 w  H& w8 [8 M4 e: P

    & J$ H# }# O: }% L" l    These returned values are combined to produce the final result.! o: f. @0 a5 m1 {1 a1 L
    + t& Q' x6 [) v  B; V% z3 a# t
    For factorial(5):+ v6 {- t+ L. q2 i' i! _
    2 V5 c4 r& E9 v: Z2 d
    4 A3 ]: n2 v& L; Y+ d
    factorial(5) = 5 * factorial(4)/ l( z3 {9 N( c5 u) R" B
    factorial(4) = 4 * factorial(3)( d  @- x; D; v: N7 _' O$ h
    factorial(3) = 3 * factorial(2)
    7 A& S( j# z) t7 @3 B3 ?factorial(2) = 2 * factorial(1)  Z  `. U0 t' z9 T  v
    factorial(1) = 1 * factorial(0)( Z, t! M: @( H  s8 z# D- K
    factorial(0) = 1  # Base case
    " C% J/ \# B( h! M9 w9 f% j
    6 [9 o8 b; e; o. U; d3 FThen, the results are combined:
      l4 A8 _7 w! _, N" r# j' I* r0 \. \1 _. k. N. X2 T

    ) ~/ r! s% C2 ~" R8 i0 a+ I- xfactorial(1) = 1 * 1 = 1
    0 k* f" F% G! c4 D  yfactorial(2) = 2 * 1 = 2
    , P' {4 ]6 x9 `. h5 Y7 R; u; ]& lfactorial(3) = 3 * 2 = 63 F/ Y3 W2 S3 a9 i+ f( ~% K
    factorial(4) = 4 * 6 = 240 x& V$ ~( m& M. G) X" m( D6 d9 s! n
    factorial(5) = 5 * 24 = 120  j& o9 k2 @; g
    1 ]+ ]4 |) ]: ?# b$ _1 \4 ?8 r
    Advantages of Recursion
    " P4 d9 |' e7 ?% `$ C7 m6 S$ w3 \" g1 F% s6 i' K& p
        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).
    9 W3 R4 O1 a+ U1 H" V6 ~5 e! P" b* E+ b) W, c9 f
        Readability: Recursive code can be more readable and concise compared to iterative solutions.( C6 n9 B* q5 R
    ) H% D# B8 r5 B. X% X& z
    Disadvantages of Recursion
    : y* M% L$ I- Y  \0 z
    . K- P$ q  q2 B, Z' V    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.; |! l* F/ y" T. w. ~! k
    4 [9 l4 x# R$ u. Y% w/ I
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)., I/ D5 h! a1 e9 v
    4 {2 W- y; O7 C8 y+ {9 W& A$ @( E
    When to Use Recursion
    ' B$ \8 O8 F& ~% c2 ~, L3 y/ F1 O2 h& O! J3 i/ b( ~% ^
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).6 n2 |  V* @  l6 y$ C

    4 @) |  d6 v9 ^, [8 @' p9 I: J    Problems with a clear base case and recursive case.
    + x5 j! w0 }) a0 _# u9 r/ w# D; o9 k( j' Y2 f" ]* \- A# C9 N: T
    Example: Fibonacci Sequence* R$ Q, f! E2 N* E0 c2 A4 S
    1 X2 A: v3 ?" t3 }: S, h
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:6 S: b3 u: @; @8 |% C

    / {2 d! n. H! c  ^0 g' H/ h    Base case: fib(0) = 0, fib(1) = 1
    ) L# E* T  a! n$ K8 ?) ]  t, \3 U! F! Y" X' P" j5 T# o; x8 i
        Recursive case: fib(n) = fib(n-1) + fib(n-2), F) m7 j3 {' l/ ]1 [! g$ P
    / W( L3 `) J$ O. ?
    python
    ( d1 }, j: |, Y; Z. X3 \. p
    , C; ]6 d; Z+ ^
    3 w6 b) z3 N  c, @4 j' Odef fibonacci(n):3 @' E" M: M5 F/ }8 }
        # Base cases  ]2 v; l1 j! S4 U
        if n == 0:
      V1 i& ]0 h0 K6 `$ u; |        return 0" ^. h0 x" C+ }
        elif n == 1:; o4 A9 {$ B+ W  F* W
            return 1
    * i& @$ \& h/ k6 r    # Recursive case6 J. O* K* F$ W% _9 G; L9 l
        else:( N, |7 V  Y6 P% F) v( A  c
            return fibonacci(n - 1) + fibonacci(n - 2)2 B( y: ^- P& ]7 [
    # q; h' j- s/ w' P0 f5 g( {
    # Example usage
    3 ?" g0 R# v. g+ V- x* Cprint(fibonacci(6))  # Output: 8
    ) M) P  R3 W' m; w1 h
    . o9 x1 m" G9 \( Q& a( wTail Recursion4 I* @8 ?0 y6 ?( c- e5 y" _1 P
    * z5 P1 b1 a  N; G2 }5 S
    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).
    0 D4 w$ n3 b, l( p4 m4 D0 W3 p! Z) s0 {: l: A
    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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