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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 ( _2 ]4 V: I! ]
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    解释的不错
    7 ?# N% E$ e- d* T( K3 z1 x7 T! I1 S  ?: ?  T2 V; K( q8 v
    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( v$ u9 l5 b8 `0 y" T
    ; {9 }/ H; N* l( y9 J. @ 关键要素
    0 i0 i  c) r" m1. **基线条件(Base Case)**9 G5 {) H% f" R1 s+ ~
       - 递归终止的条件,防止无限循环
    9 X( p% i# Y! z1 }  M. R; f   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    1 C+ X7 V7 F9 y3 L, W& \7 L) @2 t; E. ^& Y( Q. Y0 Z$ ~
    2. **递归条件(Recursive Case)**
    0 K9 T# E" b2 W* u, b3 k   - 将原问题分解为更小的子问题
    2 [+ J. [7 L2 g) K   - 例如:n! = n × (n-1)!( A9 o  r1 Q( [; f

    - }9 i, s! Y$ F# Y5 O8 x 经典示例:计算阶乘
    * |2 _7 w) `* y/ z( F" B% b& kpython. w, {8 o, q) _! b% c3 A7 B
    def factorial(n):5 ?7 A7 K/ ~6 h4 d/ b5 E7 J
        if n == 0:        # 基线条件
    + m# C* M% z5 O        return 1  L6 ?  X: Q1 I* V! ]
        else:             # 递归条件$ Q9 _! n, F* r- {+ t
            return n * factorial(n-1)0 U/ I; Z; P6 Z/ c
    执行过程(以计算 3! 为例):& C2 F  z" B- K! L
    factorial(3)
    & y  O) G6 x8 y' d4 H3 k+ p7 v/ s3 * factorial(2)7 o: |" z- K# f9 b! f! G0 [
    3 * (2 * factorial(1)); O8 B! s) G% g3 q0 u. z- u, \
    3 * (2 * (1 * factorial(0)))
    % G- G: v7 N4 l+ G7 y3 * (2 * (1 * 1)) = 6
    & T4 _3 z! g+ e( J. i3 T7 D$ p% E8 h  F" [+ y
    递归思维要点6 G2 h+ c4 Z3 f7 w+ e3 Z
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ) D5 l  p* D6 V6 f/ s) T2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    " B) V6 P  ?8 P% M0 V3. **递推过程**:不断向下分解问题(递)- ^3 L1 U1 H$ X( J, r. m( `0 Q
    4. **回溯过程**:组合子问题结果返回(归)
    3 L7 y, M3 A- x4 X/ g7 N- p' d1 w' P1 M7 q- |1 G
    注意事项
    $ o- B/ U0 y8 M- l" j, l2 T必须要有终止条件
    3 K9 s% S0 g6 f0 b7 g4 R递归深度过大可能导致栈溢出(Python默认递归深度约1000层)7 O( P, z$ u( V- ]6 X8 Z3 A4 {
    某些问题用递归更直观(如树遍历),但效率可能不如迭代3 K3 r0 a/ W* \  L3 v/ g' q
    尾递归优化可以提升效率(但Python不支持)
    " X; k& U/ h( J! \; i6 c- i. k, Q, A, e& u
    递归 vs 迭代5 `4 T# J  a/ H
    |          | 递归                          | 迭代               |; D+ [9 T* J$ d' W, m; e
    |----------|-----------------------------|------------------|
    * }9 N+ C: o9 ~: C5 E  || 实现方式    | 函数自调用                        | 循环结构            |
    2 d4 N8 Y& o0 r# i| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    0 e( J; ]1 }* m6 `$ I4 N* `2 f| 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         |2 N2 w; u  L7 M) }- ~7 @( c
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    6 V& g3 ~# k3 b- l
    ( i( }& ^' \- x8 Y7 P" d- W, q 经典递归应用场景
    " m) s! f  k# `" D' f6 G0 V' w1. 文件系统遍历(目录树结构). L  J. j0 {7 G! D4 y0 o
    2. 快速排序/归并排序算法! g9 T5 U+ ^9 Y+ v. m$ X/ Y7 q
    3. 汉诺塔问题+ J. V; B* {/ z
    4. 二叉树遍历(前序/中序/后序)$ ^* I( d% `! N- {" |( ~
    5. 生成所有可能的组合(回溯算法)
    " {/ S3 _) j- W  z; ~
    & ?" b/ m. G' a+ c( m' M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,
    4 g' Z+ K8 i7 n" n我推理机的核心算法应该是二叉树遍历的变种。
    - c/ w- g6 m: O( A* ?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 @( h2 P* Y) d  M2 w4 GKey Idea of Recursion
    % l6 l( V% H; Z) P6 w! H* d! U1 \, x1 E2 h* K% Y+ [: t1 E
    A recursive function solves a problem by:- o  r: i3 O* @2 P
    7 _3 t4 f) v; U. P5 w1 |* g
        Breaking the problem into smaller instances of the same problem.
    7 r: @6 D+ |* T& N8 l; p
    % h9 t' n) F# K3 B7 x( d# B  D$ |# _    Solving the smallest instance directly (base case).! g3 h% d; S7 v' F7 [+ _
    3 s: w$ V4 v1 o: N5 V
        Combining the results of smaller instances to solve the larger problem.2 F- ^; x) d% t! a7 l& s
    / ]: F8 z7 {2 J. J$ Y
    Components of a Recursive Function
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    + I) c4 R5 ^% u4 L/ L" g    Base Case:2 F5 M3 D' v2 q. K) j" W6 t4 c

    / p+ W) W; C* N( t        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.+ s; R) m' O  e( M3 M
    % M3 U9 q; }/ }1 J- Y
            It acts as the stopping condition to prevent infinite recursion." {% M' p! N8 s8 ]

    6 P# q* E; K+ d& p& m! K        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.1 h' q8 c; A7 U" }# g! s
    6 e# s; r4 J1 a2 w5 ]
        Recursive Case:8 c0 K) y5 x; f: V

    : i7 ]1 Y  k$ K: |1 x0 ?/ l& c        This is where the function calls itself with a smaller or simpler version of the problem.
    1 u1 C9 H) b9 v; j* j+ ^- |) Z# h4 k5 K
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 a+ b8 K0 D0 ^8 s  _  w' y

    ! L; V$ U- K) [  s2 f$ RExample: Factorial Calculation
    7 D2 m! v  ~+ X4 d6 g
    # f, Z. K* C0 B& S( u$ r; AThe 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:7 G7 r7 z6 i7 S; X( B: z

      D& I; l. X9 K4 }: E    Base case: 0! = 1
    0 @, ~4 o2 f- r4 w+ W4 B# @! n# t* N5 u& C% o
        Recursive case: n! = n * (n-1)!
    3 {/ R2 C: a- T1 M0 I6 ?- \% ]6 M) o
    # N9 F# p, f3 j8 ^, U- D) _Here’s how it looks in code (Python):
    $ [7 J% h  [* C3 _5 V$ F& ^2 t/ G. Mpython. Q: P1 t/ v8 K' E; s7 Z

    ! ]6 Z/ k& b6 E0 u) h" n! t7 y  ?7 x5 f/ E/ Z3 @6 V
    def factorial(n):
    4 `( f1 i- B4 ?! `- B    # Base case  v+ g7 Y3 ^0 g: N( H( B
        if n == 0:" _# h+ ^+ {! ]( K1 M& Z( q! |) |
            return 1- V6 L! M% T3 Z& c
        # Recursive case, Q( G  Z+ j4 O& z: ]9 k/ E
        else:# x' {4 ^3 b5 x* w: t
            return n * factorial(n - 1)
    ' F* i. }2 ~6 m: l" k
    * n1 }) D% [2 m% |! v# Example usage2 y8 {! s( c3 }
    print(factorial(5))  # Output: 120
    ' @, ~; n  i8 l/ {* I
      C+ S  t/ d$ A) GHow Recursion Works, L. d$ A- y2 |( n4 \2 X! Q

    - h6 F$ A0 y: p# r# o4 y4 y    The function keeps calling itself with smaller inputs until it reaches the base case.
    5 G8 S* U7 c! Y! i7 `6 R# P: e, c- T3 u! _3 b' F
        Once the base case is reached, the function starts returning values back up the call stack.6 R8 X, G) j0 {" L; L

    + N3 Y; f# V0 M9 _1 Z! {    These returned values are combined to produce the final result." ?7 f! @/ j7 G9 G
    5 _* V9 F  {% S0 U
    For factorial(5):
    & v& o% K2 U7 P, |+ Y+ j5 ?1 O& Y! u. y. j
    2 i) Z$ R5 t- ?, ~- c& E: I: p
    factorial(5) = 5 * factorial(4)! m/ M; l4 c0 [
    factorial(4) = 4 * factorial(3)
    2 ^: x( G$ R& q' gfactorial(3) = 3 * factorial(2)7 ]" W- _  @5 [6 |
    factorial(2) = 2 * factorial(1)& A3 v. G) H9 r' U  J4 w" B
    factorial(1) = 1 * factorial(0)* D. e6 l/ B5 X; I* R- A+ s  I
    factorial(0) = 1  # Base case
    ) H2 \- D: k2 @: I/ \  o) d2 z0 d2 `% w, ^, x( g; V: I
    Then, the results are combined:
    9 C, {8 P* Q' V7 m
    2 N3 V8 C$ A4 @, l" U1 Y' E/ i: N
    factorial(1) = 1 * 1 = 1: d% T$ \$ N( U7 |9 [
    factorial(2) = 2 * 1 = 2- I, X1 N7 K% O5 p9 X
    factorial(3) = 3 * 2 = 64 B7 `2 ~( p- a4 T* O' A
    factorial(4) = 4 * 6 = 24
    8 C6 A$ n2 L/ {) ~; ]1 tfactorial(5) = 5 * 24 = 1203 z) I% M4 x. g+ n2 }

    0 Q' R: j5 E4 R8 I( ~" HAdvantages of Recursion- e0 w( f, u8 f/ a
    ) i- _  n; ~- A2 b. S9 g  a
        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).8 [' s' ~, [! [9 j  z

    0 O- K" W) X: s7 S  U  v( q    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ; |, }7 [  c. J  M' x! f4 p$ w& Z9 {0 V+ j3 Z  ~4 A9 Y
    Disadvantages of Recursion
    - r1 h" O2 I% S- X6 H- M/ C0 r. K, i, h6 D3 F
        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.
    , E4 ~8 Q+ D' T# S5 d1 y
    ) @% ^4 s2 l4 [* t0 e; ]8 b    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
      X8 ~) k3 i' i, m5 ~' }% k" Q7 m( M
    When to Use Recursion
    / F3 s+ D" B$ c" `% b* U" f; P
    + ~2 U7 C+ m* \    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).( M5 K/ L7 Z3 w/ ?
    ; P1 z1 W+ H* ~3 P/ G/ K
        Problems with a clear base case and recursive case.2 P6 S" N9 y( P1 ]" k. e/ F2 A

    5 l" L, k6 r* R* R. G0 w$ JExample: Fibonacci Sequence3 z* ~9 M$ G( {7 Q. |

    # u2 h; c  J7 @; i: _The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    0 C9 _0 c7 J" u3 w. L) s6 |* U# a5 y( G, Q. ]% ~
        Base case: fib(0) = 0, fib(1) = 1
    - w  y# R' \- }. N6 P. a. ^
    " E# T% v! D$ D+ @3 ]; }' R    Recursive case: fib(n) = fib(n-1) + fib(n-2)+ ]+ C; q/ Y4 w. i/ k% ~

    , }# |0 u* u. B2 M3 {python
    % M5 k3 ^' @1 ?8 R+ h7 V" f; ~* e8 k# |

    + p, V5 C7 w- _2 e1 V  qdef fibonacci(n):
    2 r2 [* P& i3 {2 d; l    # Base cases' n" O( n% d8 g7 [, N6 ?8 L
        if n == 0:
    : o9 G5 ]/ [; z* z4 A# N        return 0  C' e* V; g+ s+ F- w9 [, Z) Q4 Y, V
        elif n == 1:
    , b" q! C. m# Z5 h5 Y        return 1
      q+ z3 C) S0 d, w8 }9 v; _    # Recursive case
    ( s- M& ^0 u3 y) Z2 B, D    else:- c4 K. O+ X3 K8 ]
            return fibonacci(n - 1) + fibonacci(n - 2)3 w1 c5 o# G5 N: H% g# A2 L* B. I9 K: k
    ; B6 k; G' x! }+ p+ G# [1 G( I
    # Example usage6 H+ N/ `5 }7 Z4 w+ t5 j) d2 U
    print(fibonacci(6))  # Output: 8
    & _  p" L. Y  u1 i  |, `
    8 x. u8 X0 g! TTail Recursion
    ( j( o" L, \* {2 I9 ^$ Y6 m+ Q" e/ g' H) I2 q  ?9 o! ?. R/ n
    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).
    . f" y" b$ r" |7 V9 y8 B3 }& n6 a
    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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