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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 & q, q! e1 F) L

    & ~& Y% }7 P" j) M( _; ~4 r3 N解释的不错
    7 |* U, {( d/ F  Z# [3 F
    5 P7 q1 o+ }# q7 C4 Y+ M- s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。
    , H* y- V% M  r1 x7 [) b8 t( t" ]7 x2 Q& R8 ]$ r! \: ?+ W* ]5 j' e) w  o
    关键要素
    ) s* {' }8 P% f+ V2 r1. **基线条件(Base Case)**
    + i8 _. R- I, A- Y' F; ~% ~) A   - 递归终止的条件,防止无限循环
    3 p* B4 h, e7 N4 [8 ]* A: s) V   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ' ]' F2 P: q$ \% R1 I1 `
    & v) e, `4 g0 ~- ]! K$ `( B2. **递归条件(Recursive Case)**
    ) H0 K& ^4 ]/ ]' b7 r, t   - 将原问题分解为更小的子问题2 \( x) q4 W& _4 z+ p8 F
       - 例如:n! = n × (n-1)!8 u9 f0 O1 ~; u, s6 ~. z. Z; G
    6 x2 c' D+ M; y$ h' o  V  n
    经典示例:计算阶乘& \! c# Q- X$ j0 @
    python
    . |& V, [; f8 t# E2 J3 [- @def factorial(n):! f) i3 E2 [& P: G1 X6 ~
        if n == 0:        # 基线条件
    ' `% z) D9 Y/ P, R        return 1
    9 |( s' w' m: r- U1 a) M    else:             # 递归条件  X# x, x3 B" F% P
            return n * factorial(n-1)
      J6 q: }4 q" |7 X5 e7 i  x执行过程(以计算 3! 为例):; }8 C; h# e) L% {6 M/ m
    factorial(3)8 V7 C& g7 `* r0 O/ S; q. u
    3 * factorial(2)
    4 x6 G. u' B7 Q9 J! a8 C/ G3 * (2 * factorial(1))
    . j+ G/ u5 u$ ?, _& V& E+ h2 G3 * (2 * (1 * factorial(0)))
    6 E5 Z) b" Z  T# \! e6 `3 * (2 * (1 * 1)) = 6
    0 N  I. J3 n) h% H
    0 @) r3 X" d# n4 J' y* ?4 K 递归思维要点
    * p% B; Y' r: t  T  _6 g1. **信任递归**:假设子问题已经解决,专注当前层逻辑* G$ |1 L" `9 L) T8 K
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 O/ ]1 @  H6 o8 _* C- s/ r, Y
    3. **递推过程**:不断向下分解问题(递)% u, v' y  w: R3 n5 E& D
    4. **回溯过程**:组合子问题结果返回(归)1 R' K6 N. {$ \) g
    + y# p+ T  S% g4 W% u* Z
    注意事项) I7 `8 H% a2 o0 _2 C& c; n
    必须要有终止条件1 p! \! }. T9 O3 B, O3 ]4 [
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    6 I. m: C$ ~! v/ i) `; j' X; @4 }某些问题用递归更直观(如树遍历),但效率可能不如迭代
    $ I7 n& d* P9 O% O& |1 R" E尾递归优化可以提升效率(但Python不支持)
    1 G1 Y4 n3 [0 P+ a8 \" v" |- {' d
    递归 vs 迭代: I" b' ]. ]: A6 F. v. F' m; T! f1 B
    |          | 递归                          | 迭代               |# b9 Y& D8 {( k7 @" ?% F
    |----------|-----------------------------|------------------|
    $ l0 j: o$ G0 d6 M2 U| 实现方式    | 函数自调用                        | 循环结构            |( u6 G- Y& c" d9 g: l$ c0 Y
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- v4 }: k4 E" [! N( P) ]4 a
    | 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         |% Z8 v, R- e$ R8 f
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |# M6 Z8 P* [4 K2 W; h
    ( N& P2 I& p1 R- w6 ~$ o
    经典递归应用场景
    ; f# J0 Q2 p$ G1 ~4 C1. 文件系统遍历(目录树结构)
    ( h! z* h  C# A/ T- s* V6 J2. 快速排序/归并排序算法
    0 U0 \7 P: F) G5 D$ @3. 汉诺塔问题. j! e% e1 ~+ v2 t8 }4 @0 |  {+ O
    4. 二叉树遍历(前序/中序/后序)* N# l: @; P& i3 e2 g8 V0 U
    5. 生成所有可能的组合(回溯算法)& L6 r0 D8 Z4 l5 M" S" C3 r

    + N% C( o7 h% z" ^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,2 q7 j& M) ~# ?  C3 k  g4 g
    我推理机的核心算法应该是二叉树遍历的变种。: A( N; S0 [' X" W7 ^* W
    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:
    0 m  ?: s0 g7 r, a9 \/ B1 WKey Idea of Recursion
    0 {2 }) I  O3 B0 m4 i! O
    - D) n# }6 [- ]; y$ ZA recursive function solves a problem by:
    8 I8 ]; ^+ v2 y8 |+ J) W  n2 f$ C
    + \3 M6 H, R( [& ^$ N3 u    Breaking the problem into smaller instances of the same problem.
    ! }5 @1 Z- h0 t4 k+ o1 k8 ]" M9 e6 ]: h: h- _* [% f* E) {
        Solving the smallest instance directly (base case).
    6 _. l# C; h, Y3 N+ g' V3 [3 M6 g. {. F6 H  c7 @
        Combining the results of smaller instances to solve the larger problem.
    ; ]3 _  F" R4 G5 |# Y" V5 i" N; G5 s$ V0 A
    Components of a Recursive Function
    5 S8 z+ X- t. V3 c* C! ], Q4 X1 L1 n* R$ x# k( ?( Z5 L
        Base Case:$ c9 K* b* M6 Y- C6 V# o7 g: c

    * }1 }+ p$ ^1 n        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.$ R% p3 m6 x9 c& {9 p0 v
    * P, I  l1 N: h4 z! _7 |
            It acts as the stopping condition to prevent infinite recursion." E( e( S- p9 w0 l& _. G; N
    9 k; I/ [* Y% a9 ^0 Y/ f. f, Y- k
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    7 [' I! B- G& V0 N8 m( a
    6 U3 M' C, f3 W9 \8 M; R    Recursive Case:
    3 W2 V) ^! {9 g4 N
    * u! ^/ B0 F8 o, Q' B7 c& a        This is where the function calls itself with a smaller or simpler version of the problem.4 b$ A, Q) D! h0 ^
    , y- d5 D  D! ~
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." i1 B7 L, x! M9 n# D; r* o

    6 O. W5 B" T3 B9 cExample: Factorial Calculation
    + W; J" y, k5 [( a+ ^8 Y5 F( |0 Y# {0 q) y1 k( d
    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:- p! I. v* ?! A! ]0 f' S8 @

    1 s" ?2 z5 x' `5 S9 z+ {    Base case: 0! = 1
    : |  U# ?; }4 `9 r- `
    . ?. L1 t) ~0 _! @1 u    Recursive case: n! = n * (n-1)!
    ( m) b; Z  f$ n- \) I9 [! c  h; k7 u# {6 @. ]' `8 L: Y' R
    Here’s how it looks in code (Python):
    4 {9 M: O2 h7 a# `python2 E* H. `& b/ s) g/ b

    $ B7 M3 ^+ M! K& L& x5 J! O' o' C( ?2 C" k6 p: m. D
    def factorial(n):
    ) X) e9 ^) R9 N" w  ^    # Base case
    + Y9 O) E" Y9 K: Z3 ^    if n == 0:/ H6 z9 s- S; I# ?$ N: u5 F6 ]5 i
            return 1! @* h* l; j+ H2 A% v" [
        # Recursive case4 g; V$ d/ S) O. }$ _5 E
        else:3 P8 R* |  |# @8 w/ I6 p& n
            return n * factorial(n - 1)
    % M3 f8 Q/ d& l9 [$ |" I- r. T3 g- q% X. a( F! D5 H5 y
    # Example usage8 u; A; Y4 a& M. r& |- z6 a
    print(factorial(5))  # Output: 120
    5 o( T# a) ]& h" \9 b1 y
    3 ?8 Z" Q% R6 N- i. N3 zHow Recursion Works/ i/ Q( w- \( h! J

    , c( i" b1 `3 z9 E: m  K2 ^    The function keeps calling itself with smaller inputs until it reaches the base case.
    0 H/ f- [$ d- y8 N* l# w5 u- P/ l2 U4 g6 w
        Once the base case is reached, the function starts returning values back up the call stack.) `' i! n  @% ?! n5 s4 Y  B; V

    5 \) Z& ?6 K# X1 ^( G2 b  S& C    These returned values are combined to produce the final result." c2 d/ P- W' P9 f7 ^
    3 R6 j) X4 V" @: i2 s/ g% V) ?6 J
    For factorial(5):7 s3 S1 V. H8 A1 m6 g* C( N

    . \. `' \% J- M/ V  T% ]1 D# F+ t; F0 F8 @
    factorial(5) = 5 * factorial(4)
    9 d8 u5 D8 n+ nfactorial(4) = 4 * factorial(3)
    $ a7 e- x0 q1 x/ T$ c) gfactorial(3) = 3 * factorial(2)
    9 W. e- t& y; p6 |# u  L' _factorial(2) = 2 * factorial(1)
    " Z- ?0 g8 B: x! `7 C) Zfactorial(1) = 1 * factorial(0)
    ( ]4 Z: I1 K4 X: p. a7 \factorial(0) = 1  # Base case
    , d: k# E/ W: e2 ^. Z  R. x  H3 @) K4 e5 d  B
    Then, the results are combined:) \  M! j7 T4 [6 q! X5 F
    0 s6 x+ p; a5 u) V
    ( R. n5 W. J2 H/ ?7 Y4 ?
    factorial(1) = 1 * 1 = 1; |; ^) X% }4 N+ D2 M7 k" N
    factorial(2) = 2 * 1 = 2
    1 Y5 y4 v! ^1 S- M' Q& B# @: K: rfactorial(3) = 3 * 2 = 6- r# \$ b. x0 d! J" _
    factorial(4) = 4 * 6 = 24
    : U! j( B3 J  N; dfactorial(5) = 5 * 24 = 120
    / \( G6 I. V5 U3 |% X
    - X0 T0 X0 Q0 d4 E- n# V. UAdvantages of Recursion
    . e" E6 A  b; e6 J
    4 H: R7 R9 a/ A) X  n! W    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).- r: w! Y1 L% R& S& W

    ; F% n6 t4 M* G! v  K1 B5 [    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    & [, q6 r* N5 S  c) L2 A- r' v( g& }. C* j/ B. @" F8 c( K( R
    Disadvantages of Recursion% z, o& \6 h% O) {1 N
    5 B( e. f9 i/ |: V& v1 o
        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.
    ( m1 y5 ^% I; ]" G+ ^2 c( `1 h3 t1 {0 X, V
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).6 Q4 A9 G8 k8 V8 y( @  L# G# O
    ) e" s* U* @- u$ l! N9 G" W2 |
    When to Use Recursion
    - h# W/ M) v: P0 R0 k  r; d1 v% O
    $ k' Z0 J& H. I) i% U+ X  z    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# Y( S% p) o7 w9 c0 H' K1 c, [) G

    / `; a6 O. }: C1 Z- a    Problems with a clear base case and recursive case.# l9 ?. K( ]2 G  f/ m! L

    6 X8 f  k6 O6 {" a5 J" A! LExample: Fibonacci Sequence, h4 B2 _' D3 a6 [
    7 ~. R; }( S3 `/ H( A
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
      {- D, j" U& W
    % s$ v5 N4 s% X- e! q    Base case: fib(0) = 0, fib(1) = 1
    % d% D) T6 ^* H
    ) C0 w: S  f3 q& u4 }8 o4 @8 G    Recursive case: fib(n) = fib(n-1) + fib(n-2)" p" c& j8 O" b3 _6 \8 v

    " i- z) I. K3 i! i. C+ Qpython
    ' s+ `  w( O" a1 u8 z7 A% g! s+ w. s/ P/ Y! i
    # M6 `" f4 w  {, C) o: A4 G8 ~
    def fibonacci(n):
    9 T5 a- p1 _# J5 Y/ [    # Base cases
    # |# {# S6 B8 u% a6 E4 d    if n == 0:
    " n2 h2 `) X) e/ b- L9 X/ M7 e$ @9 Q4 o        return 0
    8 |3 i1 y; H8 [! i: {7 x! }5 Y' t  c    elif n == 1:/ J" B& j6 W$ E) V' o4 N
            return 1
    6 j- `/ w0 T* D; T- ^$ l    # Recursive case
    % |! t/ n9 k. [3 S  z; t    else:; k  E5 i) P1 r0 I5 ?( X. U2 q
            return fibonacci(n - 1) + fibonacci(n - 2). G0 U0 x  l7 d, I+ i- Z4 c

    . O  F1 `  H' r" {" s# Example usage* D0 {) L; E- j
    print(fibonacci(6))  # Output: 8
    ( c  w; S% l! {0 k: n/ @7 Y8 A% ^9 O
    Tail Recursion
    ) B, o' d! ?; d
    , Q8 V& H) F; i9 o- O2 yTail 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).
    ; m# y0 f- ?+ L$ d& L- p' P- Z: `) I! ?( ]8 D2 s! Q8 \7 Z3 B! M
    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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