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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
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    解释的不错  u/ h0 A5 ^+ h3 L
    4 }+ {6 b6 D$ T. m2 V: C
    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。
    8 g1 |( Q7 j8 V) J, Y4 S4 ^/ j1 {. Y9 P: R
    关键要素
    2 i0 M9 L1 @4 B. M1. **基线条件(Base Case)**0 Y8 u, J% P. }) C6 o$ _5 b
       - 递归终止的条件,防止无限循环- l9 }8 i3 s' ?0 ~2 r5 b
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    . @/ k3 T- M, M* I$ A" P
    9 E4 h( L* U  _, `) r  Q+ L& Q$ \2. **递归条件(Recursive Case)**
    6 X, n6 r2 R/ |6 y' K+ Z; @4 w# {   - 将原问题分解为更小的子问题8 P: w& ^; O$ b2 k1 V. d
       - 例如:n! = n × (n-1)!7 S7 r* B& O0 U

    5 o* |0 L4 S7 E6 i! v* R 经典示例:计算阶乘) a1 G) O, ~9 T) t
    python- D, {8 R% @: G5 c  O3 F1 W! H! P
    def factorial(n):
    , j$ x1 g, u; O    if n == 0:        # 基线条件% {& s2 ?( w  w
            return 1% S- @& v" ~9 J+ g7 r+ x
        else:             # 递归条件, O/ j: p( o1 z! _
            return n * factorial(n-1)
    6 [/ A( E# R, W" M2 r. s) g3 c执行过程(以计算 3! 为例):+ [7 r* s" ]' f+ [# Q
    factorial(3)
    % g- o4 k9 \  r' [3 * factorial(2)+ C% J9 Z, v0 P4 r, {
    3 * (2 * factorial(1))
    % d# _) }9 B* g" I3 * (2 * (1 * factorial(0)))
    ) _; L& a4 X8 i% T+ y3 * (2 * (1 * 1)) = 6: q6 C* ]2 h) V6 w; N& j1 m

    ) [) n, N2 }# ^8 J 递归思维要点
    8 u" Z" d2 |% j0 e! L9 j1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ( A; v6 Y  c& \- R% H  k# A& ^+ w2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' d2 r0 A. u; W' I& ]
    3. **递推过程**:不断向下分解问题(递)
    ; m8 q4 @4 I. \6 c4. **回溯过程**:组合子问题结果返回(归)0 G0 l$ t- H! M
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    注意事项
    ) H4 O: A! G5 G1 O0 O$ s9 b必须要有终止条件$ w4 J/ l6 ~' l- y& R7 r7 W
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)2 U! n! b+ H: Z. x5 |# H. U+ R
    某些问题用递归更直观(如树遍历),但效率可能不如迭代) {2 E; ^9 g, Q, s  q% J
    尾递归优化可以提升效率(但Python不支持)
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    递归 vs 迭代& b6 n/ q+ \& x6 b
    |          | 递归                          | 迭代               |, w3 b7 U: b9 R
    |----------|-----------------------------|------------------|
      R: s0 o# h' y4 K- }| 实现方式    | 函数自调用                        | 循环结构            |6 [: {% c" m/ Z, m
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 d; t9 Y" N8 R7 V7 {+ ~( j6 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         |6 b7 Y& h- I. o& v0 d0 |
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |# r# @* n, T( l: f/ v: c

    $ u( {( g7 [  p! ]1 ], w" p8 q8 E 经典递归应用场景) d; o0 }7 E- O0 _1 F/ `
    1. 文件系统遍历(目录树结构)+ _+ L6 U. m3 m4 Q! X2 C+ g
    2. 快速排序/归并排序算法5 E1 S% J! ~; z
    3. 汉诺塔问题% S% Q6 ]* Z7 M$ G4 m
    4. 二叉树遍历(前序/中序/后序)
    % k  [5 l7 ]; P5 Y! S' S2 t5. 生成所有可能的组合(回溯算法). ~+ ~1 ]' W  X6 O+ z$ C. b2 f, b/ l
    ( r2 N/ S7 r$ d- N3 d/ _
    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 01:20
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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,, ]; _, U0 W2 w8 {% \; Z7 ]( p
    我推理机的核心算法应该是二叉树遍历的变种。
    2 s1 }/ f* y! W$ v6 G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:
    & R: {1 b3 [0 Y8 F' f9 JKey Idea of Recursion
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    1 H6 ]8 S2 ]- i1 CA recursive function solves a problem by:( S) a* ]2 X: l/ K2 ~6 a( t4 X
    0 j' Z* f9 O/ V6 f/ k# j
        Breaking the problem into smaller instances of the same problem.
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    ( T4 N- o) z/ J0 I    Solving the smallest instance directly (base case).
    4 c8 r% D  k9 y4 \' a) r' R4 |
      x6 h8 t: Y0 t, l% f    Combining the results of smaller instances to solve the larger problem.
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      d* N" ~: A  XComponents of a Recursive Function
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        Base Case:
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    " }1 \; B1 d5 m4 w7 M        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            It acts as the stopping condition to prevent infinite recursion.
    : [1 C1 \# ]7 r) g1 z& J
    % s  B$ b" z+ W6 d3 }        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.# r, }* e' i3 A6 |6 n. W" m$ s( W
    6 B) G+ y* o: v
        Recursive Case:- e& e# |& U* V9 B
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            This is where the function calls itself with a smaller or simpler version of the problem.* i. u! s2 T' z3 N$ p) _

    0 g# ~2 X9 [, n6 Y0 H$ o2 n6 ~        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( h' [2 R4 U  d" S; T3 X4 L

    " w+ i, x0 }9 [9 {' oExample: Factorial Calculation
    # t" S8 k* |- L4 H- y2 J5 P1 C! ^$ ^* _! _6 Z+ _
    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:: E$ \4 O, D# o* ~4 y1 n

    . }. ~7 o- T2 N8 t* m    Base case: 0! = 1
    4 o$ e# h9 m/ t$ a4 G% ^1 d
    ; h6 j* G2 x8 h6 \& g2 l# S9 @9 R    Recursive case: n! = n * (n-1)!0 ]3 x- @- _8 F) i4 q. Y% Q2 K6 _$ o
    6 ?8 m3 U; h) M8 r! ^& h# r
    Here’s how it looks in code (Python):6 q4 |6 E1 `5 O' Y5 e$ r8 v9 D
    python
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    2 N  x1 b: Y- |- K) g6 E" u, {$ l. t# u
    def factorial(n):) c+ x. k7 l* J
        # Base case
    . Q  }4 m) a1 @" j/ ?( N" g" c, Y    if n == 0:
    * f7 K, z9 j' b- ]% }6 ^7 k( _+ E7 b, _        return 12 {( I  q! J  M$ C: o7 Q( o
        # Recursive case  l7 D8 n+ T% C# j% E8 L
        else:
    ( h: v( V& l2 |8 d        return n * factorial(n - 1)9 m* P: L8 t8 x/ a) A5 K
    / D2 t* G/ Z2 m6 @: v5 A
    # Example usage  \, L, [3 v: H8 a4 V2 A7 i
    print(factorial(5))  # Output: 1200 o2 R/ e, C' S/ g6 P9 E' R
    1 U" }1 @% L- {" @$ L* L
    How Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.
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    0 F, Z; f* \; m" T9 O' {# N    Once the base case is reached, the function starts returning values back up the call stack./ n% O  Y( p" L; @# ~2 J+ E1 Y

    - P$ O3 l$ g: h+ p5 Y    These returned values are combined to produce the final result.0 V# E9 U) L& K8 j4 R# K( p8 R
    / [' N$ E3 P/ N( G2 l
    For factorial(5):
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    ' P$ v% o3 }5 W1 c+ ^factorial(5) = 5 * factorial(4)
    & g& Y5 v6 H: r! i) ifactorial(4) = 4 * factorial(3)
    ! j! j' v6 Q# |0 Efactorial(3) = 3 * factorial(2)* [$ l" k) [$ V" B: h+ _) P
    factorial(2) = 2 * factorial(1)
    / G$ C- N- x9 H/ ]# @# H/ B! S. \: Wfactorial(1) = 1 * factorial(0)1 U; o5 o/ ~! M) F: h
    factorial(0) = 1  # Base case+ j: m7 H6 ]  X. w9 z, q/ q

    ' f) _! B% @% B4 g1 IThen, the results are combined:
    " I: G3 c' a, m" {- m- s# O& v5 C' f

    7 H: v8 b( ?% X* J+ C/ w) b6 }( lfactorial(1) = 1 * 1 = 1
    4 W) R6 B& b2 S( sfactorial(2) = 2 * 1 = 2
    , }6 T- ?, I+ q( _factorial(3) = 3 * 2 = 67 a% w; }& W; S; B6 c4 B
    factorial(4) = 4 * 6 = 24
    7 F5 C4 z- T- v2 P: ]factorial(5) = 5 * 24 = 120& ~* e9 ^1 i( ]; }$ }

    ' {: W5 ]9 C& l4 EAdvantages of Recursion0 @- |7 o3 F% q4 Z& y+ l* w% h+ O
    & x5 K8 F1 u! m3 c
        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).7 d0 n7 g, Y& v9 P* a0 ^  E
    8 {. s! H( B- [" H8 a; I! v7 i
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    : m$ L+ z# V- G, u" |) {* v2 u4 ?* d: N* c
    Disadvantages of Recursion
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        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.' f' S9 {1 l8 d; b$ W9 Q

    0 g& O/ L6 B: L" ]$ s2 Y, u; i9 a, S    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 X, Q( v, D) ~/ d( ~) W
    * k7 [, _3 L' |
    When to Use Recursion4 J4 V! @6 l6 T8 A* W! E- v+ e
    + n, U- }, H) T9 y9 h4 p
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    9 o+ h) _1 H& O- K! P' `: M( T4 w1 h4 D2 t' |1 `" w: Q, ^7 K# X9 W
        Problems with a clear base case and recursive case.
    # O$ `- x: ]' l% }; \1 w+ q; V. z- L$ i7 W% V
    Example: Fibonacci Sequence
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    9 ~# R8 |: T/ `& [1 eThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 W; m$ P/ w3 u) G5 P% ?
    3 q8 q( s4 I, B2 U4 W9 e- }1 n
        Base case: fib(0) = 0, fib(1) = 15 D7 P' j5 ?5 O1 b  b! i
    4 t, x- x2 S6 M9 N* P2 @3 Z5 t6 d/ m2 c
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    4 [/ _* o- ]/ ?/ V+ @5 B8 h5 k+ V& t! V2 b7 P9 W, n4 m# q
    python
    7 S) i, L0 Z5 u! d$ k
    $ u2 K8 ?% k/ `4 ~5 O8 y; g# [1 k- j4 r% ~/ t( D$ Z: R
    def fibonacci(n):0 s8 }1 I" }8 W0 Q
        # Base cases
    3 i6 ]' c, {* w; q    if n == 0:
    $ |; V( o) |- ~8 `        return 0
    . S5 x3 O: U3 d    elif n == 1:
    8 |) q9 T2 g! G1 e2 T4 k        return 1; }$ s/ Z3 I% a1 `; R- x! I: V
        # Recursive case
    ; G9 C+ |# k, x    else:
    4 `, I6 ]9 h/ U: O: Q        return fibonacci(n - 1) + fibonacci(n - 2)
    - h' m# l8 W& L3 r) r; m6 G
    ) G- {0 g! M  H2 c% D# Example usage
    8 `: D8 ~0 o; L/ D! [$ K% _3 mprint(fibonacci(6))  # Output: 85 o8 S2 L8 Y( @- V0 s& s4 a! c
    ; |  q; Z2 p0 m( w1 r2 p
    Tail Recursion
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    8 J# k; G, ?8 e7 ^5 sTail 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).
    ! T+ b+ r* X8 }4 o: C$ {3 x( C2 p' p4 c: Z& K7 @( }* o
    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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