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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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    0 f/ t4 |' I. R解释的不错/ h. R7 a; R/ d+ L. ?& ^3 M" u

    7 h9 ]5 Y* f. N- w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。
    9 D3 R5 F4 X  c; X/ }$ \: g
    ; s/ @7 K- P2 f4 R* k6 m 关键要素5 ]1 U$ N. f5 k  V* g6 i& H
    1. **基线条件(Base Case)**
    5 l6 V, M' ]3 j3 D" a' G; a$ w, C   - 递归终止的条件,防止无限循环
    % i/ |) O# x( K3 Q   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1/ F+ _: V/ w3 R6 k- G: `/ z
    , c5 `* y5 H: N' D9 H
    2. **递归条件(Recursive Case)**1 U* j2 {# [$ Z! Z, c+ x, S% D6 P
       - 将原问题分解为更小的子问题% a% H) u5 y7 I* a2 q3 a
       - 例如:n! = n × (n-1)!
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    " H9 C* {; }) E2 r' w2 c3 Y 经典示例:计算阶乘% Y8 H8 w( M1 V. l5 g1 o" l
    python& c4 ]* V6 O# i2 S: T* C
    def factorial(n):* A0 C- X# D- a6 q& n$ r
        if n == 0:        # 基线条件
    . X2 C& F9 f' J* F: j! `" N        return 18 h. C8 T: H& i+ l" |
        else:             # 递归条件
    3 y7 S4 s* |6 Y$ q! w0 v        return n * factorial(n-1)
    9 S: Y, |; o7 l  P( z. v执行过程(以计算 3! 为例):
    8 g; d: K& b6 C+ o- pfactorial(3)6 @( X9 r! ]* k7 z1 Z
    3 * factorial(2)* U2 o  j" Q& [
    3 * (2 * factorial(1))
    9 z. J, S" I, n4 e3 * (2 * (1 * factorial(0)))2 u+ o4 u8 `& d) C, t9 g+ ~) `
    3 * (2 * (1 * 1)) = 61 |1 x4 X# L1 G% e

    : l3 C- \# y5 ` 递归思维要点
    , u* S4 o3 ^% R- R; a  H/ }1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 t+ c' ~6 X! }9 {5 ]
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)2 g4 Q- h* X" l" D% I+ `- ?6 @
    3. **递推过程**:不断向下分解问题(递)! x. N% z  S" t& x7 Y5 J/ Y
    4. **回溯过程**:组合子问题结果返回(归)
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    $ Q  d. R. V0 K& D! T! q 注意事项
    ! _2 H; P: D) f4 E  A必须要有终止条件8 ^5 Q( Z* f/ y$ [6 @, G
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    . f2 W; z, n& }5 N+ n  s4 ^5 H某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 P4 `7 M( M3 Y% B尾递归优化可以提升效率(但Python不支持)& W1 L  h+ h4 L
    $ V- k0 x4 U9 k6 {  Z9 t
    递归 vs 迭代* ^! h: |2 K4 s1 J. S
    |          | 递归                          | 迭代               |
    % S( w( ]2 @- U+ ]/ D# d# z|----------|-----------------------------|------------------|% @, j1 N/ N4 x& W2 j- X" D
    | 实现方式    | 函数自调用                        | 循环结构            |. |( c3 f* ~' B1 }
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |) a8 D$ R" `7 ?5 C% ~& p  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         |# m# n8 q+ W% `& J2 I* Y& e
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    & W2 D& X+ ~: l! N. T
    7 O1 J  [3 y8 A3 A% j! n 经典递归应用场景
    + R( v9 b' l' E% \1. 文件系统遍历(目录树结构)
    8 F# w5 @5 W" W* G' z* B2. 快速排序/归并排序算法
    ; j0 y+ E4 T- m- ?1 r3. 汉诺塔问题
    2 q$ t, e# j- c+ q+ ?2 z4. 二叉树遍历(前序/中序/后序)
    * T9 u$ D/ r+ Q% ]9 t5. 生成所有可能的组合(回溯算法)
    4 g( G2 r4 G  C  M; i% w
    0 f# m& p9 Q& X  z, R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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    郁闷
    7 小时前
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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,
    : e. j" t5 G) }, r我推理机的核心算法应该是二叉树遍历的变种。
    / M) e/ k- k6 w# K/ x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:
    8 D( f# X% R) t$ G7 O& t% v. eKey Idea of Recursion: v+ D. @! s" t2 v' j
    & b) u/ l: r2 b
    A recursive function solves a problem by:1 j8 K/ w' T! N. O, V* |, U/ p) D" I

    % H: A6 ?7 |5 f2 c6 X; u    Breaking the problem into smaller instances of the same problem.
    . b7 Z3 ]) o2 z8 S4 v* B; ^1 f4 |0 @
        Solving the smallest instance directly (base case).
    + g3 O6 F  x5 Q* q" n' a) N8 ~1 y& I, `
        Combining the results of smaller instances to solve the larger problem., z8 R. _# k; U/ m

    3 \/ p: }# T7 \4 ]7 HComponents of a Recursive Function0 k# B9 K1 L' s: @

      e/ `4 r. O  ^' i. f! a    Base Case:, q5 F- n7 _( g" V; z
    ( ^3 P; S* A4 ~# g4 e
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.5 d- C5 g" {2 r! m! Q- w

    : o3 r0 ?; ?) P  r        It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1." g) I) W( |$ F0 j  Z' ?
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        Recursive Case:: [' k' X& N" R

    : I9 {& A7 t% y3 ]- m0 }5 [  M" w        This is where the function calls itself with a smaller or simpler version of the problem.' B( |( T/ s) _) i& [
    * Y$ c7 G6 P0 {+ C5 X' V+ m5 m
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    ! `: ^+ R" Y- h7 VExample: Factorial Calculation6 h- {7 T# }+ A, I! W$ ^$ y4 y3 x" k0 I

    / G* S- B! ]7 I% h, Q0 QThe 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:0 i4 q! l! B; ~  u
    1 d- \4 ?" N% r1 d
        Base case: 0! = 1
    8 G% B% x  u( X2 t& x" a! S1 p9 K& y6 g* T: ~+ I+ l2 n. d
        Recursive case: n! = n * (n-1)!
    + f! ?' c* g8 s$ W/ t6 E, I8 w
    6 W& l( }: w5 h- }/ v( UHere’s how it looks in code (Python):
    3 E8 |# i3 s7 {. bpython, N! q* @  t8 r/ {8 ~

    3 h) t: X; T- Q" S( I5 X/ O$ j) e
    4 D1 u# s( [8 vdef factorial(n):& Z+ C% p, k8 A) j# V- u4 ]
        # Base case
    ! @, x9 j3 r- Y+ l" x- A; a    if n == 0:
    : i$ y/ w3 N0 v' e7 k0 q1 ^8 w1 z        return 1
    5 |# ^7 N/ F8 p, V0 s7 Q    # Recursive case' H. D) X( K, v# }. Q
        else:1 g- A, r' z( `+ i2 k
            return n * factorial(n - 1)6 c' m) {" `4 W7 s: D

    ! F: R4 `4 u* U! Y2 s- G# Example usage- `! j3 U' @: X3 [/ b! m/ E0 J1 W
    print(factorial(5))  # Output: 120, v5 z. b& s: Z9 `- @, I
    . _; u0 ]$ N: v& Y/ o! F' J
    How Recursion Works, E$ T8 f# i+ K! Y- s; R
    * a! ^! g5 B5 J4 ^" |6 g
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ! Z4 x+ T/ [  z; |9 B7 S3 Z/ |6 D$ J/ k( ^7 H! t* u8 z
        Once the base case is reached, the function starts returning values back up the call stack.) o+ W2 v* |( h' I( ?9 @0 G

    7 n! q# J7 Y# g( R. O  _2 b    These returned values are combined to produce the final result.- }# L' G! j: y

    ; z# d# K( R) c% }+ UFor factorial(5):
    $ B* M/ ?4 A0 _6 X; |) g
    ; ]' A0 f6 i  X9 ]# A- L, Z& L
    3 W# g$ [3 T4 v# p# h' Xfactorial(5) = 5 * factorial(4)
    : O. B3 b1 K; i# wfactorial(4) = 4 * factorial(3)
    * @8 t6 k" a. m% m. Y& D9 Afactorial(3) = 3 * factorial(2)
    # I2 ?: `: C3 @/ y! Sfactorial(2) = 2 * factorial(1)0 f- `+ w8 |/ [0 s
    factorial(1) = 1 * factorial(0)
    " ?; ]; q4 k0 C9 b3 z! Ofactorial(0) = 1  # Base case
    7 s6 h. S& F( K5 N( ?
    ( k' x2 }5 N8 s0 qThen, the results are combined:
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    + L1 u. N8 v; Z2 x- [+ b$ Y" t- n" a8 R
    factorial(1) = 1 * 1 = 1
    & |; r+ j2 g, h* p/ @. vfactorial(2) = 2 * 1 = 2
    " S+ F; `% t5 l0 E; Qfactorial(3) = 3 * 2 = 6
    8 W* Z3 t5 O+ ]- }$ j- }1 Lfactorial(4) = 4 * 6 = 24
    1 z) {8 [% p. qfactorial(5) = 5 * 24 = 120
    6 m9 u. j  R1 D8 u) j
    1 K+ N7 ^6 _, W* ?% i" mAdvantages of Recursion
    * C& A' s: d5 x9 q5 l( b8 U! 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).
    2 L& h0 G8 }0 ?  ^+ }$ I) m3 B5 P8 K& g( ^4 v; d( K+ |+ z5 r$ m
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    3 T7 M3 ]2 L8 B; K8 k7 Z: K& m% R! T* ?, n' t4 j3 p% V
    Disadvantages of Recursion. V/ L, r3 l) a& b0 o
    $ q) S' V2 s$ L  Z7 |# ?6 f# G
        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.
    2 R5 n8 {) S2 i- s: Q
    % X$ D" A+ N: t0 L) O" O    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    : B, ]' m% u  T  D5 f* A+ O- E8 b2 M0 J" g* \2 L1 Y
    When to Use Recursion
    * G$ m, d# z- |. X% F
    2 |2 O, D0 P* M# u& x  D    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& A- n8 `0 ]2 g. K* t( Q; H

    : n/ c; N! y  _+ y. r4 Q    Problems with a clear base case and recursive case.8 U# Z8 e6 O5 b6 ?# Z- n

    . N  o2 f% ]: D* n1 l6 V! Z' XExample: Fibonacci Sequence. V3 W: N& G! a$ R

      m- B4 `; f% l1 t3 ~5 G( H/ CThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ( T" r. P0 t4 W( y
    9 e" b1 t3 Q$ e0 ~; p5 }    Base case: fib(0) = 0, fib(1) = 1
    ; f6 Z- r! X; }
    - s9 z% M' f3 c, Q  y1 G1 q    Recursive case: fib(n) = fib(n-1) + fib(n-2): _) f; G* I$ c- _2 B0 t

    & D/ A% U6 D0 gpython3 \$ ^  V) Y) n2 Q1 _% Y/ |

    0 I8 D) J% H5 P- l3 A4 ]$ f) D- M0 R  |3 r. I: m# R( Q
    def fibonacci(n):) G/ }" p) e# m9 {% j2 J
        # Base cases+ R- n  r, F* K
        if n == 0:
    ' p  ^- t9 c8 L: O- z        return 04 h) H: X) \" B; P
        elif n == 1:6 J6 v4 j  u( \* J1 @$ }  g
            return 1% b: b( k* D6 u0 M3 B  Z" r
        # Recursive case% y3 v% a8 y; s. u7 K' k% P
        else:* n: _$ v+ ]) G+ C/ ?& n
            return fibonacci(n - 1) + fibonacci(n - 2)! n/ Q7 [, z( S( q+ L0 ?
    1 |! G/ t* k+ ~  r* \) {3 n7 V8 D
    # Example usage
    . r7 J+ l7 x; `print(fibonacci(6))  # Output: 8
    7 k+ D  _0 h" ^9 r: J0 c3 i- x: L4 q* x: w5 m6 z4 @; O
    Tail Recursion
    & ^* d: H) y+ n7 o' C: }
    # e' L# k* X7 m0 P5 bTail 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).
    4 Y9 l/ x/ ]! i* T+ Y
    4 z7 _+ r) B9 J% z' y4 jIn 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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