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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
    ' V( n9 \3 L( H3 S2 ~0 S! o; E4 x( E
    : E3 |$ E9 l7 G解释的不错. K7 p+ @/ c% _$ @
    / n: K2 ^. B4 c# h8 ?
    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。* w: C6 q3 f; a  s, ^
    : g$ Q: L0 g1 b
    关键要素
    / @7 U: O, g7 {1. **基线条件(Base Case)**
      V6 y: b# Z, h   - 递归终止的条件,防止无限循环
    6 }# u$ c& B( A. j4 d1 P. y. O   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& i* B5 L2 M8 N% u+ I
    9 j2 s( T) n. M1 Z8 s% f" N
    2. **递归条件(Recursive Case)**
    4 w) ^' p  y+ O   - 将原问题分解为更小的子问题
    0 h* \$ N9 O  q+ f+ g& C   - 例如:n! = n × (n-1)!$ [1 |* B5 x7 z9 n& D$ r- a

    ( [) D  c7 f" c 经典示例:计算阶乘7 b* k6 W# T, U4 ~' ]' K$ `
    python
    # @8 {* T9 C. a, i- x$ pdef factorial(n):
    1 H  k. W: j+ ~* O    if n == 0:        # 基线条件* p% A5 A' V" Y) Q' U, G# {
            return 1$ S5 g( S4 B5 A! I7 @
        else:             # 递归条件
    5 Q2 z) s2 ?& g  g        return n * factorial(n-1)) U' x  I0 M8 L! A
    执行过程(以计算 3! 为例):
    3 R. n+ k0 _  J, h, c1 b+ Ufactorial(3)
    $ C! o& c( L; }3 * factorial(2)9 R- R( {! F. v( a4 G
    3 * (2 * factorial(1))1 d( U7 ]: ?! l9 Z' L/ C7 k
    3 * (2 * (1 * factorial(0)))
    9 {! ~, r$ u, c8 F& ?3 * (2 * (1 * 1)) = 6
    + y  q! D2 h" W( g! h( n- N- N: P$ W/ x3 m& ?8 Y+ }5 T$ f
    递归思维要点
    1 D4 u6 Y. u! L) g+ l" o" P1. **信任递归**:假设子问题已经解决,专注当前层逻辑$ i* n9 d, N, r3 [1 U
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 d1 J% a6 r9 j$ H- g% F; V
    3. **递推过程**:不断向下分解问题(递)  b6 z: f: s4 Q# F, @" ?! q. Q
    4. **回溯过程**:组合子问题结果返回(归)5 p! e2 `' E  @, i1 K. z9 F. ^7 B% b

    & |  l8 d! G- {; B7 }6 ^1 O 注意事项
    " M+ `/ A! B3 w# p9 z必须要有终止条件4 i. E% M# _9 [: }* d$ N+ c2 O
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : o* ~' P* {% z( z* T  s) F' w( H某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # J) d+ ?1 i/ Q尾递归优化可以提升效率(但Python不支持)
    1 }0 t$ I, P( L% i% B) H$ S7 I! p" e- w/ i4 w# g) f* w% H; a5 @2 l
    递归 vs 迭代
    * s+ Z0 Y  P( y# Z) I9 q6 t; ~|          | 递归                          | 迭代               |! l' g: D8 `0 A2 b# \
    |----------|-----------------------------|------------------|
    ; B" U: A& R4 m# m' G) [| 实现方式    | 函数自调用                        | 循环结构            |% \* H. l' L/ d4 f. q2 Z' k, X
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 r4 R4 ?% c; i
    | 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         |3 Q+ a& G- e; k/ o9 ~6 w/ |
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |  p3 N0 B( X- n+ g0 i% V8 J8 P: ~
    % C5 w# W" _, A4 n/ _
    经典递归应用场景
    % F& [! i+ U' N4 P9 ?, a1. 文件系统遍历(目录树结构)3 ?: q8 C. R" {9 t* N$ y
    2. 快速排序/归并排序算法  {* b) f5 D1 {
    3. 汉诺塔问题6 P+ E7 ]+ I* J8 \% Z; z
    4. 二叉树遍历(前序/中序/后序)
    5 A8 B/ ?5 M: [5. 生成所有可能的组合(回溯算法)
    * s9 D+ `9 M9 m+ i4 X- e' L- ?9 |3 D/ X& Y1 }
    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,. C4 G, W3 C1 d# F+ e
    我推理机的核心算法应该是二叉树遍历的变种。
    , G4 {  b) Y9 _7 [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:( U: Q' G& _! e1 J) `
    Key Idea of Recursion
    8 o7 U( N8 Z9 o8 o& ?. U  N0 E1 {  ?$ r) \3 v/ x! H" y5 N% B
    A recursive function solves a problem by:
    + u: G4 |; [/ N$ f# a& `6 ~3 N: I, a) r0 E9 h3 v3 U& K9 Z- M3 ^
        Breaking the problem into smaller instances of the same problem.) I' y. ]0 Y/ w

    : v& k" F# u) c2 i    Solving the smallest instance directly (base case).; U" Q, k0 L' c! h

    ; U* A( w2 i9 V& H    Combining the results of smaller instances to solve the larger problem.
    ) J8 K2 E% q3 F( p5 D( i/ Q2 j& j  A% [: C* U& ^& i- c3 o/ c
    Components of a Recursive Function
    6 k! Q  Q, x. ~8 v# M8 k1 B- y/ k  M2 X- l$ @$ Z9 A
        Base Case:: w- U3 ~! d0 j4 V5 D: l, _
    + ~3 }+ b. K$ h, [" Z) J7 k! ]2 K
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.' p+ j9 ]6 [/ S- W' s8 P8 t

    ' M$ y% }* l' o& n0 c/ B        It acts as the stopping condition to prevent infinite recursion.3 m0 \0 V/ X& h$ p$ A& d: T

    " w- n3 b2 l2 J( x/ G        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ [2 y* i; m; J8 T- _0 ?

    1 h% J1 z1 `; [5 P    Recursive Case:$ Y( m9 D" N% \/ Q4 r
    + v; K% @8 Y, r0 t+ a
            This is where the function calls itself with a smaller or simpler version of the problem." q  a8 l3 P. ^3 e$ Q9 I. Z7 c

    ( h; G* l" t0 j        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).+ S3 Z+ X+ s# @- \1 `# G. g$ V

    % z# f6 ~$ k: d7 c1 iExample: Factorial Calculation. @) P% F! ?; `' k( a9 z+ c" n$ g

    : ]' |( F& ^# b' m5 ~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:
    5 F) o' q) ^, b' Z3 Y7 w& V# y% [* V/ [
        Base case: 0! = 1
    # Y. J4 y, s+ @  N! H4 w5 D. o- u9 }  t) b
        Recursive case: n! = n * (n-1)!6 c( K( H  Q/ P  r* |& d# D$ Q

    ; V/ ]  [6 _+ R. a9 F, aHere’s how it looks in code (Python):  }9 g( W! Y. `
    python
    # p' a& @, r3 t0 V& t. \( _0 z& b9 j! H# r6 w% c& [

    % {* B4 |8 L9 L6 }) m  tdef factorial(n):8 B' D2 a2 N# N& R6 t4 y6 w
        # Base case- N. Y5 ?7 _: |3 D
        if n == 0:
    ! R3 M8 L+ M/ @$ I8 o1 H        return 1& V# W( {( K1 }) F/ E1 d* p  b
        # Recursive case* h  \8 y' h" C  v
        else:
    2 U! A9 \- R" X7 p, x8 j        return n * factorial(n - 1)
    ) G' x( Z; b% H/ V# ~7 F! w1 {
    + Z  D5 g$ T7 d5 C7 e4 Z& D6 W# Example usage
    + ]0 d# h& v2 T% u3 Cprint(factorial(5))  # Output: 120
    % v7 _2 V- u: A0 Z( k! G3 ~. n7 d2 S- N7 u5 C0 e; F
    How Recursion Works
    # o7 Y2 ~( `1 f0 f/ [$ r  V! Q8 I. D$ S, v8 L
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ! @/ x! t! E' l# f# Q: |; ?
    + }* |, A" `! \; x% B( h    Once the base case is reached, the function starts returning values back up the call stack.
    ( `9 f! ^7 r' b1 i: L! B  c1 R. m, @4 Z, q: Y; q
        These returned values are combined to produce the final result.. ^+ o8 q; ^7 O& c+ m
    5 J/ ^9 C' k+ d4 {2 p) H3 m& N9 a
    For factorial(5):* b# ^) F. v: I) @5 I; B" t
    ( C% r* O( M; c
    / C) t2 K5 Q7 _$ E' T
    factorial(5) = 5 * factorial(4)* |9 I6 z$ Y  [' u& E. i* U" \2 D
    factorial(4) = 4 * factorial(3)+ g: h+ }& Y7 [+ e! j
    factorial(3) = 3 * factorial(2)
    % {0 Q" I( L) |% }( z: dfactorial(2) = 2 * factorial(1); ^2 S# O; _9 ~# j/ s
    factorial(1) = 1 * factorial(0)
    ! T+ C2 K3 B& ]. R) Vfactorial(0) = 1  # Base case
    0 A. F* l$ B, N4 k+ a4 F# ^
    ( I9 F( n2 K) l# m( a& RThen, the results are combined:
    / o* s* f. P( n! u) d# {; o( N1 C. v; S9 ]* p

    1 Q# W4 c. N) V( O& M0 dfactorial(1) = 1 * 1 = 1
    ) v7 J8 J* \/ p& |1 R9 i# F+ N& L+ xfactorial(2) = 2 * 1 = 24 u& D  t& W% W( F: x; ^2 F
    factorial(3) = 3 * 2 = 6
    % J( Z" x5 N7 h1 x; Ofactorial(4) = 4 * 6 = 24
    # x) B! T# p4 }, |- n; s7 h5 b/ Ffactorial(5) = 5 * 24 = 120
    5 D; }" \* M* L: `7 h+ k& q+ ], d# }
    " P+ h6 N9 k; I, H7 W" l' q7 C, qAdvantages of Recursion/ e3 W6 W3 E' W+ a" L) o2 O* y! k/ _4 }2 \
    3 J  b9 [. K; g0 I
        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).# B8 X4 i! S7 a2 B  `4 g; N+ t3 u  Y& F

    $ q  h# w* ]" ?    Readability: Recursive code can be more readable and concise compared to iterative solutions.; Y$ s9 x* s6 y

    % W" z; q3 u& |$ B5 }3 |Disadvantages of Recursion# e" D3 i7 q: c2 ^
    # C1 O2 p3 d; [- D7 v) L9 b7 Q) H) U2 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.
    # u- x; I. c! w  O- k, R( N  Q3 m- r6 H' D/ }9 @' u
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).+ |/ O% K6 `) z

    $ V  b) X- {' m* {, ~When to Use Recursion
    4 b! V3 S- u+ g! T6 P, v% v
    * v  r: @( M  x. B+ _+ z    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- K+ e8 k2 u/ \1 @

    8 l. V4 b* }/ Z: U4 ^    Problems with a clear base case and recursive case.% q; ]5 g' M7 R3 [6 z* x. ~

    1 E- ?1 \* t$ V5 e  i' [Example: Fibonacci Sequence
    % `# J- s+ j: w8 m; v
    + j% n& n1 w5 m& t; H# G" z, _6 aThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    / v0 g# ~& b. I% G3 y& @# v, ~; j/ R; s- Y# j
        Base case: fib(0) = 0, fib(1) = 16 ~- ?& h8 O+ ~$ Q
    2 D, `& F$ W) y8 A
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    9 V+ a5 J& J7 c! i3 f7 O5 l
    ' S( J8 N7 B" `" i: T* ?" k' bpython1 D( E, ^( P6 s$ W( L- ^

    0 t$ I& V# ^" h; E2 g# t
      A: N5 b" s" n& S- sdef fibonacci(n):! P# `7 c( B* G
        # Base cases
    6 H/ h: w7 u- Q: e% [* t: B1 ^    if n == 0:/ A* ], U. U, Z+ j4 E$ s- ~
            return 0
    % E# r% L$ g6 ?5 ]9 @) J4 l    elif n == 1:
    ! }. W  @; r  R* d7 Z7 Q" J( g        return 1
    ! Z/ \3 W- _9 x5 l    # Recursive case
    4 \% ^) v2 l% W# d    else:
    $ G' J# N3 [: J$ ~( e- T        return fibonacci(n - 1) + fibonacci(n - 2)
    - P8 d6 q0 \7 n  [2 |  L
      q, I2 C. q; e! C# Example usage! x: i  l1 Z6 j( J
    print(fibonacci(6))  # Output: 8
    - w3 Q( w* W7 E* T& B; O2 V
    5 s% F' I" z9 E) c. l3 M$ }. TTail Recursion" N" I2 W$ a  |& j$ k" ~& S

    3 t. w9 p9 W+ ?( TTail 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).
    : p' r% W5 V; E$ k- o7 K7 q$ M/ ^! \4 ~+ o! e
    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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