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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
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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 - W3 k" |6 Y4 z) ^! z
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    解释的不错5 K$ i0 E! r( l8 p

    : R+ H0 O6 s9 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。
    . ?1 q$ [( l  V0 X' b9 `7 Z. ~# `6 s/ D6 G) O9 L& K$ t/ ^
    关键要素8 C, J- R: Y8 W" k" O( x* q3 }
    1. **基线条件(Base Case)**7 _+ F- ^7 Y7 M* i
       - 递归终止的条件,防止无限循环
    8 i( ?$ ^- \" d   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1- f! |! J4 O) N) r5 H& e
    " ]8 r7 U( v$ f) r& y. x7 Z
    2. **递归条件(Recursive Case)**
    8 C. E* r7 S' S6 Q7 G   - 将原问题分解为更小的子问题
    2 E* A- `9 l" K8 ?   - 例如:n! = n × (n-1)!
    - M) M! l6 W! g0 p( W, c7 P) m; P+ ^9 c$ Q$ O1 l: H% S
    经典示例:计算阶乘! z7 I8 ]( @6 l  J: t
    python3 m. U$ B% |' W2 E! D! L
    def factorial(n):
    8 s2 a: Y6 f9 B) p& I8 [    if n == 0:        # 基线条件
    - T4 D3 g& k+ t        return 1
    ) i2 j1 K0 A, ^( d' O+ ?- B+ v1 G1 H) }5 E    else:             # 递归条件3 m$ R0 i, e, ~4 a- p& C' w9 F
            return n * factorial(n-1)
    - j8 G1 f6 S8 ~3 q& \9 L2 P执行过程(以计算 3! 为例):6 z/ }1 ]: h3 F9 B6 X
    factorial(3)
    2 h" R: q* _- G3 * factorial(2); h$ R& C; [: Y4 l& R
    3 * (2 * factorial(1)); X" P: L; _6 C3 E* F; d4 m
    3 * (2 * (1 * factorial(0)))( z7 e4 k. o8 g# e! C5 I
    3 * (2 * (1 * 1)) = 6
    . v  Q0 ~( O2 a8 a4 w3 p9 @
    ' _7 n. v. N, R* l 递归思维要点
    " o, Z0 y1 b. o# p2 |0 }1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    8 Z) Q$ J0 \' S( F% i2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' S; Y- p/ V6 `& Q2 W
    3. **递推过程**:不断向下分解问题(递)
    - v- Q; b3 a% z$ e/ p4. **回溯过程**:组合子问题结果返回(归)
    ; ^. f, \/ Q' A& t: L# F
    ' M9 ^! k8 X+ z# q2 D 注意事项
    ( r* e& ^( d6 j; g5 c必须要有终止条件1 N2 A) l5 k) p6 a  m" r
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ; ^+ P0 y2 z. v2 s/ t某些问题用递归更直观(如树遍历),但效率可能不如迭代+ }: o' p; s; e
    尾递归优化可以提升效率(但Python不支持)" z- q# T. n; V7 C( N

    % H1 l9 \+ b4 J7 D2 h+ R/ e  K' n9 v 递归 vs 迭代* p) H( Z% @) M0 \( Q
    |          | 递归                          | 迭代               |$ Q# D) g: L2 f+ M0 [: m# y5 ], c
    |----------|-----------------------------|------------------|
    8 S7 Z7 n- x* s- y" i| 实现方式    | 函数自调用                        | 循环结构            |' I: \! c. b; v/ b6 R
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 d! \! h4 X3 o  F3 o| 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         |' X4 e- v2 c7 i0 `( z
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |( W9 B$ X" V5 M# P- v
    ' g- m5 U1 @; L: _* }: A3 g& y
    经典递归应用场景: S* S7 E9 r- L+ P
    1. 文件系统遍历(目录树结构)  D5 }$ }( z( ^1 _
    2. 快速排序/归并排序算法
    * ~! K8 D4 ]9 C8 [3. 汉诺塔问题
    - F+ ~, ], n& S' Y  [/ b4. 二叉树遍历(前序/中序/后序)- \1 E( f) R3 P6 y- s8 f
    5. 生成所有可能的组合(回溯算法)
    2 G3 O1 W$ t9 T- _
    , h5 @. `/ T- v0 \" R2 ?4 L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,* z" P% W1 }) U1 a9 I' B
    我推理机的核心算法应该是二叉树遍历的变种。& ~, s5 S& w! 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:! ?2 [1 r  L, |3 j  {# K1 a4 u5 O
    Key Idea of Recursion
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    2 _5 K0 f" G  R, L4 jA recursive function solves a problem by:% E; u: b& t, r& k2 J( O) e

    ; e( b6 ]& h  J( q    Breaking the problem into smaller instances of the same problem.' k, ?1 ]. X8 {" t1 b

    % a5 ^  B7 ?$ {2 w( A    Solving the smallest instance directly (base case).1 _3 c4 E0 Q$ ]

    " P9 ?: n6 s. H- ~    Combining the results of smaller instances to solve the larger problem.) l; g* g7 _6 }7 C

    ! L: C1 i- }5 F2 A1 v+ i- ^" C  ]Components of a Recursive Function3 h  ~/ j+ q: v

    " N9 u4 G" N# B3 D    Base Case:; _5 R& }  ?, Y) E' `

    , s* c  B! ~4 m# F% f        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    * o2 B6 d( A; p" c/ G: |6 E! B- P4 X0 r, W" j2 B/ W
            It acts as the stopping condition to prevent infinite recursion.
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      b5 I  q! [( t% l' U        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.) F% r1 A# I5 l; S6 I

    ( z( ?' d  t" B$ R    Recursive Case:
    % U' h0 @0 n0 w) t' I! Z0 `0 o
    ; ]- A' H% [. m) ?# K        This is where the function calls itself with a smaller or simpler version of the problem.( |! F2 e4 \: o5 k6 p# ^" i
    * W( c& Y0 Q- h3 D
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( M+ G+ |0 k. N' O2 H( i3 ]. X3 O

    9 x+ }0 }$ _! tExample: Factorial Calculation
    8 P% [$ ?8 c. I$ k" V' T2 d6 Y4 `3 n3 F: x; u% V5 B
    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 E/ E9 O$ |6 h0 Y
    2 m7 r' Y( i2 i  {7 P    Base case: 0! = 1$ f6 i6 k% f; N5 k' s9 k

    + V$ h( e& R& W" p    Recursive case: n! = n * (n-1)!
    ; j1 X7 ?8 W+ Y, ?  E$ `+ e' t; P; B  {- a$ t1 H
    Here’s how it looks in code (Python):: U* w, q* P" u) j
    python
    % D+ U" J( D# z& q/ z: B# r8 h9 n, A5 E1 I

    ; H% }9 |+ q0 Z% }  J6 p5 tdef factorial(n):
    4 Q: Q& M* H. W2 s2 j+ o7 M    # Base case( s9 `* H/ Q+ d
        if n == 0:
    7 P2 A7 I0 u5 D9 B        return 1
    3 [3 c' f8 x3 j) ?4 r5 U( v$ U    # Recursive case* Z, l+ y- s. h' J3 z7 ~; t. N
        else:
    3 v1 e9 M- J" M* n1 J0 ]        return n * factorial(n - 1)
    6 H; n) P% C, l  U: A1 \3 U( ?( B" I: P
    # Example usage- w( ^" I& m1 ?. c+ ?5 p* @
    print(factorial(5))  # Output: 1209 M; B2 d% T. |$ D1 A
    1 x- c2 ?+ Q4 q$ M3 v
    How Recursion Works) Q# T) d5 c# D2 a" N9 h9 D

    7 W  o6 D' v( a4 W& z0 V1 a    The function keeps calling itself with smaller inputs until it reaches the base case.8 |) l5 ~' |8 v( W& |

    / I2 ?4 m% B) e( I) f    Once the base case is reached, the function starts returning values back up the call stack.
    8 }3 c# \& S% R. h, E9 K7 R: A" w* K1 {$ C0 t
        These returned values are combined to produce the final result.
    8 U0 A6 h1 e0 H
    . t& [! @+ V; F2 [; A; wFor factorial(5):' W# \4 D& O) r

    + T: D% M% J6 |/ `* E/ ]: J: ?3 n2 S1 E4 e( _
    factorial(5) = 5 * factorial(4)0 w# Q7 n" Y6 a& B* ?) D0 j
    factorial(4) = 4 * factorial(3)# Z8 f! x/ J% h. Q4 {
    factorial(3) = 3 * factorial(2)
    - O* ?6 j& r8 u' c* J4 Hfactorial(2) = 2 * factorial(1)
    # G! E' @8 V. j3 p- F4 ^# Dfactorial(1) = 1 * factorial(0)8 U0 V& g4 ^/ |  G4 t
    factorial(0) = 1  # Base case$ n9 I' `6 i  @0 C" `& ]( Q3 J
    6 B4 u/ [0 H* f3 L5 D: `8 O; q
    Then, the results are combined:
    ' K5 C( r5 q& j# c% b! _: E; |! U- F7 u' s" f4 B
    8 B. t2 i4 m: g' i$ e
    factorial(1) = 1 * 1 = 1
    * Y/ Z, H! N" q: Sfactorial(2) = 2 * 1 = 2* `: T5 g4 v% B) h  f# |
    factorial(3) = 3 * 2 = 6
    4 i; J" D- V4 d/ U* }0 F( @factorial(4) = 4 * 6 = 24! u6 Z, A0 ^' e
    factorial(5) = 5 * 24 = 120+ _4 f6 G( Y. W7 T. `

    9 F( m7 m5 b! ~0 AAdvantages of Recursion' }. b$ E7 t/ U, ^9 y

    , r/ n6 x- X0 I3 {4 S3 t: o    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).
    * V3 Q5 T( ^$ F: a+ Q
    4 h) X# }5 a# a5 s& j" F5 e# N; }    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    $ h3 ?  g8 V# F6 y
    ; P6 p% U2 a, P! P* IDisadvantages of Recursion
    1 {' o3 h5 A. y" L6 w7 G2 V4 I8 f8 q7 V+ n, ~
        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.: D, H4 e1 K5 S2 \7 _

      Y- Z" `2 k# m" h) g. V    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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    ( D$ W; S/ v9 _1 ^When to Use Recursion5 ^/ e8 y9 n; n' R- J+ l, J

    ) U) R/ G! b% C  A3 p5 k& e& F    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . ^5 x. Z: K$ ?, a. r$ A: A
    . F; p% Z7 q" M: m    Problems with a clear base case and recursive case.6 F" O- L5 I7 d6 G( B9 R2 N
    4 b( u2 h2 a4 c& B& x9 J& T4 q
    Example: Fibonacci Sequence$ q( t2 {: j0 ?6 Y+ q4 W

    ' Z6 X+ W" r. _The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" f8 q7 z  v, P

    ! n2 y/ {: }5 e& K% p0 I' ?5 n    Base case: fib(0) = 0, fib(1) = 13 u$ J9 ?( j+ X4 I  \

    , r  c4 V$ e  h0 J" P% l) X+ d    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    , @' x0 H7 u" X- D% b5 C& u5 z' }3 @( S  V+ v" N
    python- K6 {" h6 I7 _6 p1 B+ q
    ! ^7 Q( H1 L/ z+ t/ D/ U% ^1 a

    + ^" W5 ?6 l$ I8 h) g. i1 T8 Ldef fibonacci(n):) }3 k/ x$ ^0 e8 j8 d. M$ j
        # Base cases3 j* e* \* w8 o5 C0 `# p
        if n == 0:
    : }1 l  u2 k) D. O        return 0
    9 \5 v/ L: L& f$ o' X    elif n == 1:+ Z0 Y, \# n" v: K& a- F0 Y, Y
            return 1
    8 B8 K& W0 V9 l+ c* \7 ~    # Recursive case  w7 A+ j" U: V# {6 ~
        else:
    1 T$ P. t5 {. A3 X) v        return fibonacci(n - 1) + fibonacci(n - 2)- ]9 b+ Z/ W, T" ?

    6 c( n/ y( G& V2 ^9 i9 A0 m# Example usage" T) f& q. Y/ S
    print(fibonacci(6))  # Output: 8. G% S) Z! ^  S' D3 u* h

    . G. K" V! }$ ^) k0 t8 tTail Recursion& P& B% }0 X2 w) L
    # X1 |% R6 h  h1 f7 @% M
    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).
    $ B% `2 O; N7 N1 H3 U5 Y! M+ l2 ~2 J# Z5 ]
    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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