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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
    ( i: {5 @4 p* h5 o; B# o6 _7 `9 Q" Y8 _( v3 t0 p
    解释的不错% ~3 E2 F0 ~- c/ N3 F: u$ h
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    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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      j0 R# Q  u/ o3 G$ h- U' v0 u 关键要素. S/ A3 e. h+ t' o! a
    1. **基线条件(Base Case)**. O, F9 `& i. q9 g* G/ V- q/ ?
       - 递归终止的条件,防止无限循环4 `. k$ O& W7 k4 e% M6 h
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    % V4 T: s' h) B) z8 d! e# |1 H; p2 u1 ?4 F  L' A% v% q
    2. **递归条件(Recursive Case)**
    / C0 ~& t% n9 M9 _: W& [7 K) l   - 将原问题分解为更小的子问题
    ( |% _/ c& ~8 u9 [1 f. U. |   - 例如:n! = n × (n-1)!
    " Y3 U4 S" U0 Y$ v) e
    ( m) `4 g# N! A. G8 q 经典示例:计算阶乘
    ( V) m% a' q7 f% ?# Kpython& |, g+ T3 A/ v
    def factorial(n):2 Q" P7 B5 e- K3 N$ d$ V
        if n == 0:        # 基线条件
    0 d% e, Z* e" j# d        return 1
    / K3 w6 r. v8 O0 c; i    else:             # 递归条件2 V' e+ ?9 i1 `( f/ h% ]
            return n * factorial(n-1)5 t0 h9 [# x1 x$ Q
    执行过程(以计算 3! 为例):
    ! F3 C5 F) D2 Q/ Zfactorial(3)4 ]0 }/ _8 y  a0 n+ Z* [
    3 * factorial(2)9 a8 N4 n( f+ l
    3 * (2 * factorial(1))7 T* S* C5 m. G8 r+ C
    3 * (2 * (1 * factorial(0)))) f% r, x' F' }0 D$ \3 i
    3 * (2 * (1 * 1)) = 64 L8 L% e: u2 Y& Z% i

    ; F8 T  Z) V  b' P4 T! r 递归思维要点
    3 j- {# _6 `+ U4 _1. **信任递归**:假设子问题已经解决,专注当前层逻辑$ A7 V! Y1 ^5 C& U# |3 @! m
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)/ f, T+ c: P2 z4 E. I' R1 a
    3. **递推过程**:不断向下分解问题(递)& X; S0 a5 N% w) j' L
    4. **回溯过程**:组合子问题结果返回(归)
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    5 h& n$ m& k/ Z8 w, ? 注意事项3 s" ^: ]/ o3 M0 k% Z8 |$ ]( X, ~
    必须要有终止条件; ^' \" L2 A  A- `- j
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)1 v& L; w% U, m
    某些问题用递归更直观(如树遍历),但效率可能不如迭代" ^  Q4 [# \8 n2 N" T: O
    尾递归优化可以提升效率(但Python不支持)
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    - n: a! I3 @5 h 递归 vs 迭代" U; K4 \- \; k4 F& h
    |          | 递归                          | 迭代               |
    7 m4 f& y9 u% O2 G0 e% N! {& ]|----------|-----------------------------|------------------|$ Q$ A1 F9 x5 j; Q( t& k
    | 实现方式    | 函数自调用                        | 循环结构            |
    8 k! A: T. h/ F2 z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    5 V! A7 X: _+ L| 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         |1 R2 I  t1 t2 L1 b' ^- u- J
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |; U5 h1 V  D* K6 K5 ~& q
    # P' Z# o4 }0 t& i5 I
    经典递归应用场景
    * D8 _/ n% S% f0 E1. 文件系统遍历(目录树结构)
    7 Z5 Z1 f* J4 G% @3 _; t" `2. 快速排序/归并排序算法
    0 }" `' P! R1 J2 s. ]3. 汉诺塔问题  u( s& E) ]! h. Z+ o
    4. 二叉树遍历(前序/中序/后序)
    0 M, F: k, t. i. g4 k# t! i5. 生成所有可能的组合(回溯算法)
    5 i) r/ N% T3 i! t9 x  m9 n% R: ^' o, m' S+ C4 n' Z' y; 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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    昨天 11:16
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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,
    * F4 {, U7 F& d1 @6 [我推理机的核心算法应该是二叉树遍历的变种。6 ~  N! [& c* P0 @+ ?# M9 q* 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:
    2 ]3 \) ]7 G: H6 d* wKey Idea of Recursion
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    % c) U3 k& ^0 I$ D$ pA recursive function solves a problem by:5 P) ^2 n" T/ E7 a: M2 j% z9 q2 [" o

    0 U7 G7 V% ]. y" y& g    Breaking the problem into smaller instances of the same problem./ l9 R  ]9 j! E9 z0 B# `3 @

    7 b$ |" o7 ^1 q# j) j% i    Solving the smallest instance directly (base case).
    3 F" z5 l4 j3 h) p" `& W( K  e
    * b3 q* y8 F9 j  ]0 Y4 E% p, Y    Combining the results of smaller instances to solve the larger problem.
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    # q. E  @+ ]; w7 |4 C4 }Components of a Recursive Function4 w/ @4 F! `5 ?/ Q5 @

    8 J% a( D; x# J    Base Case:
    3 z" J9 l: b$ _
    - U% Z4 [: q; X        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.6 t+ S% w+ ?' H* @3 K( C+ S
    ; w, d( f) p( T
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    ! l4 {0 e- `( a0 @    Recursive Case:- j( j4 B& ]3 G2 V5 \
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            This is where the function calls itself with a smaller or simpler version of the problem.
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).- A0 D; n# J1 \. c8 M, U7 q
    7 y: A/ }3 S8 Z
    Example: Factorial Calculation
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    ' |2 {/ N' @' R/ [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:6 w6 C$ w. E' V; w# p

    ' b0 J; c1 j5 E0 k4 m7 O1 u( A    Base case: 0! = 1
    0 Y7 S: B1 d+ u9 T& R  k) p3 @/ ]6 I
        Recursive case: n! = n * (n-1)!
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    Here’s how it looks in code (Python):7 y, ~9 W+ a$ L
    python
    ; }, X% `" M7 \+ ?# z; X' o) Y; p6 F( Q9 B6 Z, q, A

    - O. [: U% F* Z9 A7 x7 Ldef factorial(n):
    " J% }# B! N6 A( l; B    # Base case; h+ g7 }4 d: n; ?7 L3 C
        if n == 0:
    7 q" V5 v% S; j& I0 Q7 J        return 1
    ( h. {# G2 u- Y. O" ]    # Recursive case: a3 P% T5 O" t
        else:" X; Q8 d& z; T/ K
            return n * factorial(n - 1)
    ; C$ @' {+ p, F" y( ?
    8 g# Y6 |& g6 o3 @# Example usage
    , _0 D$ E0 I: E) C2 rprint(factorial(5))  # Output: 120
    ( X) e0 L9 N: ]& T% Y( K$ q% n* f8 L2 J/ E# N5 h
    How Recursion Works. A! d* ~$ U$ r, H8 e9 \$ \# g4 L: I
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        The function keeps calling itself with smaller inputs until it reaches the base case.
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    * i/ Y9 G/ b) ^$ r    Once the base case is reached, the function starts returning values back up the call stack., T* e8 D0 o+ y/ o

    7 E- u$ L: [" I! A# L9 [  d    These returned values are combined to produce the final result.; y& e8 m! g" h

    . q' @' }& u3 g& N6 [6 QFor factorial(5):, J* `/ h8 I/ R; ?  p

    7 u5 a# j+ T6 c# a, p: G0 j1 X
    # n) ]" G! Z( `2 N9 S! zfactorial(5) = 5 * factorial(4)
    ( J/ E# ?( m2 g0 `factorial(4) = 4 * factorial(3)
    - R3 l3 n$ t* Yfactorial(3) = 3 * factorial(2)
    % j  B4 Z6 k/ E# ]% L9 }, p& }factorial(2) = 2 * factorial(1): p1 b: k2 l" i! J
    factorial(1) = 1 * factorial(0)
    ' B, T1 h3 `* \) p/ f3 ]$ I5 d' yfactorial(0) = 1  # Base case2 k" G* I+ ]5 X
    7 N: p6 q" V8 J/ D! x- Z  P" H9 q$ S
    Then, the results are combined:& N/ A6 f2 n2 Y9 o! @# [& X+ Z

    . i2 m+ r8 M8 _0 t  d/ m8 a% J0 _
    factorial(1) = 1 * 1 = 1
    # ]+ i) x% P% Y6 Ifactorial(2) = 2 * 1 = 2; M& B$ V; w! p- Y9 j/ P3 x
    factorial(3) = 3 * 2 = 6
    ' ^1 E: O+ i  @( u9 X* Gfactorial(4) = 4 * 6 = 24
    % n" E- G) F- Y  L& Y/ k  e* dfactorial(5) = 5 * 24 = 120! }7 K. h6 I* x: }6 A0 _6 `" \- ~
    ) r/ n4 v2 O& Q1 V9 a7 g. E/ }
    Advantages of Recursion
    . }5 n. E# {2 ~7 _$ I6 _' n, x) _! ]
        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 P1 o" N* n' Q5 [
    1 L) O, O( L& Q) q- g- I7 k  q9 q    Readability: Recursive code can be more readable and concise compared to iterative solutions.( |  S; l; h7 k3 \9 I% Q

    / h% r6 i$ u  Y7 o/ w" s5 VDisadvantages of Recursion6 n2 M/ f3 \, }! K/ a; D* S' `  i
    2 d0 Y, u: t+ a. I/ w" f4 W) T+ S
        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.
    + i9 \: I% _4 i0 e' j$ _* z7 [) M3 d. f: ^5 T7 ^1 f; v
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
      Y, V$ i% ~& `6 ^- J
    " G' g. }3 m1 cWhen to Use Recursion# d) @! ?0 H2 {' Y/ T- `7 u

    6 V" A' \  X. P    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 u: m  _' {$ u3 [, E; `. }
    + E8 ^3 V5 U9 N. O  m
        Problems with a clear base case and recursive case.
    ' [/ q/ Z2 m, ^# E6 Z, g* @& K+ V
    * W4 r: b2 m" _9 B" kExample: Fibonacci Sequence
    7 [  r5 l: m: i' W% X
    : X' G& T' b. jThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 d3 I4 ~3 ^8 E

    5 i+ [4 ~" S% m4 A2 M    Base case: fib(0) = 0, fib(1) = 1
    * K  ?8 ]+ R' U' v/ i* e2 r; }7 Y! M
        Recursive case: fib(n) = fib(n-1) + fib(n-2), f; z7 Z( e: i( q1 c0 V

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    & g- U) }3 i" y0 {7 T+ I  J2 h

    5 k6 Z- C; w9 t' f7 adef fibonacci(n):
    4 z; b  @% N! t" _& v) v2 x    # Base cases' F) H# i4 S- r* `, ]6 d
        if n == 0:. t- C: X( L+ Y; T% |0 {
            return 0- O$ D: I& e% V  V
        elif n == 1:2 v" j) D" U+ T( V" S' {/ d) N
            return 1; P9 {) H9 ^8 l# J* [
        # Recursive case
    ! p$ I3 @  v! {, W    else:
    9 K1 l+ h; Y& R7 N; B: ^' w        return fibonacci(n - 1) + fibonacci(n - 2)
    ; o0 f+ z3 N8 P" K, w* `1 d2 W1 q* S; D: m0 l+ \4 J# ]$ m, K0 s4 U
    # Example usage
    6 V' T* c0 R' T7 V- u0 d8 Dprint(fibonacci(6))  # Output: 8& B7 h( i9 c& `2 o0 ]# b

    - H2 u/ I* V0 _9 YTail Recursion; D% x- _* Q: p3 J8 _
    6 _8 v. [( x! y# d- A
    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).
    ! T* u/ |% P* S7 ?8 k4 W) R7 T* r6 a  O1 c& R
    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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