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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
    4 m% e0 r* n1 g, _: H7 F  M) v& V) H# |
    0 F: _; g! X1 g/ {. q* O. N& r解释的不错% c9 F! i2 }0 L- x% S' |
    $ {& c7 y* s, B
    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。! [- b( ^3 E9 _1 B
    5 r6 e% R7 I5 B9 @3 h' Y& Y
    关键要素* N: n: k4 C) x5 i; Y  f  U: y- e, n
    1. **基线条件(Base Case)**
    8 \+ d! `$ z0 Y  v   - 递归终止的条件,防止无限循环3 r) }1 u) r0 w; E+ K9 |
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 19 v* a) i4 h' S% |8 k9 D. d3 q, f- [

    2 \" L9 o2 |+ O7 T8 U2. **递归条件(Recursive Case)**
    . r; R- U! k5 K! i   - 将原问题分解为更小的子问题7 }) V6 M$ J7 ]& D2 u+ E$ {- _
       - 例如:n! = n × (n-1)!
    2 C: u, h+ s* e2 q( r+ b$ k  b
    + y. z0 t7 }; q( r0 ? 经典示例:计算阶乘" O/ E' |5 K6 I( @* C* n1 o
    python) |  C7 z7 `' {& z, F
    def factorial(n):3 A0 g6 M2 a$ n$ B/ G" p* a
        if n == 0:        # 基线条件+ a- j2 [% z2 U  t0 e, X; x
            return 1
    6 K) N0 l: n: u4 t; |8 p( x% W    else:             # 递归条件7 y- n8 H& q* m4 J# c$ P
            return n * factorial(n-1)
    ) [  D% |7 Y* s4 }, q* _执行过程(以计算 3! 为例):$ J# H! N) S$ L* S, s
    factorial(3)
    ; ^& q2 ~- ?/ U. |( K& k3 * factorial(2)
    ; u: f; @, ~' }+ m3 * (2 * factorial(1))3 z% m6 J9 l6 E0 J' C, Q
    3 * (2 * (1 * factorial(0)))# G/ v5 z& A% ^- ~1 J7 l
    3 * (2 * (1 * 1)) = 6
    , c- O% D$ i# Z! M3 l. A' j: R4 d/ m" W/ ?7 ?( I
    递归思维要点  T) C* ]! H6 b2 J# j' S
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑  \, Q; v, `$ v" ]9 _# ^
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间), U6 g2 u9 H+ j
    3. **递推过程**:不断向下分解问题(递)
    7 V# b  G( N. A. Q3 u4. **回溯过程**:组合子问题结果返回(归)
    & h/ T0 l: i' m' V; N- X: i% E  z0 I2 C* E% L
    注意事项
    - k0 V8 ]" g+ h2 A# f- O必须要有终止条件- P" r) Y+ D4 W9 E% R1 e! F. ^7 ~9 Q
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    5 I: V+ x/ A- Z$ e9 F某些问题用递归更直观(如树遍历),但效率可能不如迭代
    7 i3 h: ~# r7 W尾递归优化可以提升效率(但Python不支持)
      z& v* t3 @7 F/ E5 o
    ; h! S; O7 s( k$ p3 k 递归 vs 迭代! p' M( ~9 L+ K, V
    |          | 递归                          | 迭代               |/ z; O5 V, i  G4 Q: o
    |----------|-----------------------------|------------------|
    7 e  T7 v, f9 n9 A* j+ F| 实现方式    | 函数自调用                        | 循环结构            |  y# Q/ X" ^+ \7 r
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ' w! k$ i9 V$ M  p% O2 j) |7 N| 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         |4 ^, K) H/ |4 V+ S
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |% j) J" Z  y1 ?1 ]& {6 ^
    # d' z: I, M& v
    经典递归应用场景
    ) k8 p8 R5 ]: P1. 文件系统遍历(目录树结构)
      a( @# t9 @6 d: c& N& O8 L6 z2. 快速排序/归并排序算法6 p- I/ C+ ~2 ?! f' q
    3. 汉诺塔问题6 j: I. h% ~* M1 C/ R; d
    4. 二叉树遍历(前序/中序/后序)# |$ n8 x4 F0 z. N! d) G1 \  r6 y
    5. 生成所有可能的组合(回溯算法)% e' K1 u( C; K) j8 @; w4 x  R
    1 ]+ m1 P* Z" [4 k
    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 08:28
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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,
    ; z8 i& G9 `* P3 L我推理机的核心算法应该是二叉树遍历的变种。" u1 Z0 \. D1 j+ J# s% }5 D/ 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:
    7 j( M& D( ]+ b3 `- l- C& h0 `Key Idea of Recursion
    2 ]  O! j. P7 L/ K
    ) B( l0 Y% n. t" KA recursive function solves a problem by:
    ) D/ q% S) @: c
    ' y2 l2 [# w4 D: A$ h8 N    Breaking the problem into smaller instances of the same problem.
    # V( W. @, h5 X( ]3 S1 k& B8 e1 x% r
        Solving the smallest instance directly (base case).+ l0 N9 @, I: z6 I0 `
    $ t( i. o: L) F! t
        Combining the results of smaller instances to solve the larger problem.
    - k+ _& e6 G" P8 x
    7 m1 c6 ?; m0 \; I, H  |9 yComponents of a Recursive Function! M1 f  ~) G7 _! c9 f) X
    ; K% v% X/ R7 D; D
        Base Case:
    7 f( a. I/ G! @! p# L' q2 ?9 u$ B0 Q6 R
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.: G4 I5 V( T, {! W% D! y9 P* C
    9 ^+ [. T6 \' c! [/ @7 ]
            It acts as the stopping condition to prevent infinite recursion.) A& D& N% @* V
    - ?/ N7 M6 f* d# `+ i2 f- ^
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 J+ R* h3 Z  _  g2 ^. V. J

    6 h! S# a6 \( l$ h    Recursive Case:
    & l; S& m8 B  D5 f* j. V
    , v+ U1 R5 p+ n6 N* e        This is where the function calls itself with a smaller or simpler version of the problem.
    ! |, t. a8 Q6 g' I$ i: y  }- i/ k3 N- E
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 p- J9 B3 ]2 x, Y
    1 E9 P5 t. t: I3 f( E: Q! i8 C! Q3 @
    Example: Factorial Calculation' S9 O. A3 K' Z1 N

    ) ~3 Y: ~3 ~" g2 ]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:
    4 |/ O* Z/ J4 J  U% n& g5 g
    2 Y: H+ |, N- _1 D! _    Base case: 0! = 1
    # @. E  ~/ U" J+ v, y( ]7 V2 S: n2 c1 A9 X7 x  j* w
        Recursive case: n! = n * (n-1)!
    ! n  ~' {8 C9 N' G6 X& m6 C# L
      w8 E+ B, u6 _7 W) \+ IHere’s how it looks in code (Python):+ Q# [( _( S2 M% f% r3 {4 Y( P- u
    python
    + R+ {7 n  P$ L8 C* ?) q% H3 Z: b1 O# q% `3 @7 h

    - Z- q2 u, n9 ^! I, U: h" Jdef factorial(n):* ?0 l8 W& _* K
        # Base case
    ( d4 \: v1 l; D5 [( n8 {    if n == 0:
    " `4 f* Y8 m7 D        return 18 X) S* J! b! E: B: y
        # Recursive case6 {. `) F% d  V+ Z8 I1 F
        else:8 G  H: q+ y/ v& H0 `9 Q3 z
            return n * factorial(n - 1)
    % ?/ \9 ?( ~) s" ?- k
    7 H7 V# Y: \1 M( ^/ g# Example usage
    ( m$ o* _" n, {print(factorial(5))  # Output: 120# e! _! o  O/ N. N

    * K; O1 Y/ E% i9 T: E3 _  tHow Recursion Works4 i4 r' h( X, k% F. w: }# _
    9 y! Z1 f% V  U1 N
        The function keeps calling itself with smaller inputs until it reaches the base case.# r5 N. W8 g, k2 ?
    " m- M- m2 x1 z& }& z* u3 g
        Once the base case is reached, the function starts returning values back up the call stack.
    3 J6 {3 s9 g. g6 Z. b9 A
    8 S2 Z2 ?7 y: i$ U- \/ F" J' \    These returned values are combined to produce the final result.
    1 C5 e) w( ]& A: C4 e2 W7 R& ~  U+ C% z* A
    For factorial(5):
    0 d- [; X2 y. N  r" J1 ~- `" X) Y& x2 F9 a

    + l: i7 p; J8 ~+ zfactorial(5) = 5 * factorial(4)7 F- Y) ^! s4 b
    factorial(4) = 4 * factorial(3)7 @; H* T$ Y# m2 h! N9 n
    factorial(3) = 3 * factorial(2)
    ) ^" k4 t4 r9 Z  K: W8 ?factorial(2) = 2 * factorial(1)
    / |) P/ M3 S" h/ `' ifactorial(1) = 1 * factorial(0)
    ) D; o) |' \0 S/ Rfactorial(0) = 1  # Base case' A4 Y' X! M2 S9 _7 K* |) K

    , r6 I2 C) O$ @) r) Z4 hThen, the results are combined:
    * B% p0 O* U- x8 S1 g
    + F3 [1 P# c. o- Z+ E' p: g* F) R# N7 c5 }' ?1 r. Q
    factorial(1) = 1 * 1 = 1
    : Q* z( g" i6 n- r) z* mfactorial(2) = 2 * 1 = 2/ @( ?9 g' i2 K* s1 o
    factorial(3) = 3 * 2 = 6  h6 w$ t0 c4 c$ r
    factorial(4) = 4 * 6 = 24
    + Q) V: M) t0 M4 [) h& nfactorial(5) = 5 * 24 = 120
    7 U6 V: R$ w4 ]4 Z$ K) ]9 d
    & W5 f5 ^) Z& S* D: SAdvantages of Recursion
    # Y- B/ i( X3 Y1 J  C0 b: b! p* m( D; T, j& b4 \
        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).
    ) i5 T9 U4 q! H5 I
    6 C; R" k, \* K2 a    Readability: Recursive code can be more readable and concise compared to iterative solutions.- ~5 ^; T8 B, X  j
    8 p4 T  X% T7 a" R# e
    Disadvantages of Recursion0 E# n3 I' L/ |# n* M. t' e6 M
    # p0 T9 N1 A& S- B% h6 X
        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.
    " x5 H/ b$ c: O+ K* J6 `+ Q/ @* x" S. T: ^. x. q' `
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).: y( d7 o! _6 r+ {$ t. A

    7 W8 t9 C: A- t. G- }" ]1 m% ?% AWhen to Use Recursion% \9 a  B4 M8 F
    ' I# W, [9 x! s
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . K& q  @# ]+ M0 }9 T* c
    0 F# T+ F' c" s4 y( c    Problems with a clear base case and recursive case.) D5 N) E6 x3 L3 d
    ( ^% S7 ?; E' G5 h5 U
    Example: Fibonacci Sequence
    & E$ G: G  S/ J- n0 W4 y* g! _& a# ^' s/ P8 O$ O( P6 I& `
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 b, N) e4 ~8 K: D
    1 x1 ]$ N5 [' b7 d7 U+ A    Base case: fib(0) = 0, fib(1) = 1" K1 s  o% f$ n7 c8 p/ o. z

    $ l, m: Q; u4 q) Q- a# Q    Recursive case: fib(n) = fib(n-1) + fib(n-2)6 b1 `2 I. A! ~. g$ u& E

    . @. h" Q' ^% r7 x" Epython" X- K% w0 R4 {

    - ^- z3 \- z1 u' l* `3 ?3 b# T
    ' [# h7 b3 n1 W$ m0 @def fibonacci(n):
    - j% D& D: t# X  ^- @6 ^( a: h; S    # Base cases
    * n+ A" F7 n- w: f    if n == 0:
    : H0 U  V* K9 K8 G$ B2 u- H        return 0
    " [- G% }9 x5 ]4 e2 i; F    elif n == 1:$ ]8 l" U* Z2 X% t' M
            return 16 B' L4 c1 R7 C4 P, ]+ V& L. J! L
        # Recursive case
    / e4 D* U; a/ D0 z( L/ P    else:
    / n/ N# z4 g3 b5 r# r4 |* ?% T        return fibonacci(n - 1) + fibonacci(n - 2): k2 H. z1 R* b9 ^1 I

    ! l7 Q/ n7 f+ M" k# Example usage4 f( O/ R* \8 J4 a$ y  d* U
    print(fibonacci(6))  # Output: 8
    / _" G* I8 a/ ]( v9 i4 K
    2 m7 ^- R9 a9 K( J  zTail Recursion
      c* V2 L& ^4 X: T$ p4 @. C, ?) h  ~, j" U
    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).6 i$ a* i. _, `/ h$ |" k- T
    * [) A+ q' }# \& \. 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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