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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
     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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    ! {. K  W1 |6 s: i' a* {解释的不错
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    5 G$ t, f8 ?9 r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。
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    5 q: [9 c8 ^. [2 }# k* w5 |1 x 关键要素! ~3 L1 w1 v, O4 ]: J
    1. **基线条件(Base Case)**
    ( q' {9 Y* _; i7 b  r   - 递归终止的条件,防止无限循环
    ( M8 F% @3 f( @- u1 e/ I0 w   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    ) D) q( B( ~/ h. |2. **递归条件(Recursive Case)**
    3 h+ v/ |8 p; A8 G6 \6 w6 b/ @   - 将原问题分解为更小的子问题
    4 _$ R7 h  Y1 ]* W   - 例如:n! = n × (n-1)!
    - t( \" d* [! Y8 K0 y* A! ?2 K/ F  O6 K4 d  C8 i( ^- N
    经典示例:计算阶乘
    , {. r! h9 n0 c9 a0 ^) j5 o, O. |python* b& C9 n+ a7 r
    def factorial(n):
    ' _  S: j( n) A! k    if n == 0:        # 基线条件
    0 f- Q6 v# ~( v8 \/ R% j; M        return 1# @+ ^5 n0 m" U2 r2 M4 N1 K; g
        else:             # 递归条件
    , h8 L! ^& n+ L8 _        return n * factorial(n-1)
    % \6 v( Y7 v; j- W( G3 m执行过程(以计算 3! 为例):
    6 R) T: ?" d. h+ p( y6 F' kfactorial(3): t" {" R1 M6 a* P
    3 * factorial(2)" ], l$ |) c8 ?$ i" [- u% \
    3 * (2 * factorial(1))$ k% o3 a: l4 V$ @$ B  L5 m2 E
    3 * (2 * (1 * factorial(0)))
    + e( R9 H3 f) a- o! L% `0 |/ ^3 * (2 * (1 * 1)) = 6: h8 X. n) U% Z( {; }* \4 m! L

    - u# C* y( @8 f7 V  j; D1 c 递归思维要点4 z- A* U2 G) Y
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ; |8 W) Y. y/ S" U8 I3 C: D/ v$ x2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- D( H- Q, f) C+ V9 O
    3. **递推过程**:不断向下分解问题(递)7 u' I& t, p* l
    4. **回溯过程**:组合子问题结果返回(归)
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    ) X5 J* S6 T2 R$ G. Y& g0 ] 注意事项: q; @/ \8 o% O0 h3 P/ L& w: g! \+ j
    必须要有终止条件
    0 w7 W! d1 C% A  \递归深度过大可能导致栈溢出(Python默认递归深度约1000层), T2 f  T: u" S* E
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ! ]. X0 u6 V  Y  F# I5 j尾递归优化可以提升效率(但Python不支持)
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    " p1 J: [. S# g# p2 ~. k5 m 递归 vs 迭代8 G, R9 E6 l# r6 N3 t, H8 {
    |          | 递归                          | 迭代               |
    " U9 L6 G  j& f- y4 U2 Z8 f|----------|-----------------------------|------------------|) y* N- {8 d/ m3 g2 F. U  U& U
    | 实现方式    | 函数自调用                        | 循环结构            |
    ; `! [$ M9 N6 ]+ `! Z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    # T; [0 Q: G: B, N1 {$ b| 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         |
    % O+ u5 Z8 J3 Y/ H- I7 m| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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    经典递归应用场景  c! n8 H7 W( a% b3 u0 \6 r
    1. 文件系统遍历(目录树结构)
    1 |9 q" x' u% r" z8 s% m2. 快速排序/归并排序算法
    2 Q% A7 ]- Y, c2 @  |  n( [3. 汉诺塔问题% d9 C9 ~3 V3 X' E2 m
    4. 二叉树遍历(前序/中序/后序)' k$ y0 t; d# }) S% ]/ |& ^; |0 E
    5. 生成所有可能的组合(回溯算法)3 V. Y7 q6 k: q! Y4 [

    " `) @, [6 q- Z) a3 d7 G% s" ]7 T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 06:20
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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,) ~. W  h  {- F9 b* ]  i1 u5 z+ K2 A. C
    我推理机的核心算法应该是二叉树遍历的变种。( v3 `1 t: M3 L! z
    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 l4 w( [# {& Q. \5 n- ]( _Key Idea of Recursion8 g: r' N+ ?( C1 d
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    A recursive function solves a problem by:* \* F' t3 Y- g' |2 I

    , U3 b1 Y* j' K: V& _+ B    Breaking the problem into smaller instances of the same problem.
    ; ~2 Y) ~# r! H/ w8 ]& u
    4 W  c' h5 Y2 \' e' j    Solving the smallest instance directly (base case).' i' D: {4 ]! Z( H/ h; w" \0 s8 q
    2 j/ a6 `5 _/ n9 Z$ T
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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    1 F2 [/ h: \: F3 O    Base Case:7 }/ G  W1 I' |! ?) H: x( ^
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    ' H: l# W/ O4 q" B+ W        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.! v7 v( O8 g4 B3 j4 z0 B! @
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        Recursive Case:
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            This is where the function calls itself with a smaller or simpler version of the problem.
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation
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    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:/ A  C* W1 C/ {1 s* E
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        Base case: 0! = 19 P$ h, Q. r* c) ^5 O
    1 d7 s5 `+ ^+ {. R6 _- m% g
        Recursive case: n! = n * (n-1)!# ^. N% W2 p+ f! I" a3 D

    2 d. Z9 N8 O/ y# R( y. L/ y9 OHere’s how it looks in code (Python):
    " n6 S( b, h; R9 @) i, m9 Q# Upython* C' J9 [5 g- ^6 x3 g
    6 Y" M7 y0 p- m) {4 h7 ?" Q

    % k1 F" G0 u: I# u* Zdef factorial(n):& h- k; L! k% ?$ R/ W  ]
        # Base case) J/ Z& p" l/ p6 h+ R
        if n == 0:  |) D; m, s! [6 @9 v
            return 1
    ' q, ?6 ^, x- m8 d' l    # Recursive case
    / @" r$ V# l- O" z/ h, ]! O0 Q    else:
    " a0 P2 C! m$ n% K) R1 z4 X8 G        return n * factorial(n - 1)7 u9 \6 d1 b# q' B1 ^# D

      V6 q9 d# z: m5 B# Example usage7 Z5 k3 S1 S# c0 u$ W$ _4 j
    print(factorial(5))  # Output: 120$ L( x0 Z4 w8 P) e2 C

      y! f7 @; O; J4 `1 AHow Recursion Works
    , _) I5 m# n% Y9 |: e( r2 v0 I1 a' N" K5 d8 ?% a. W! J1 Y
        The function keeps calling itself with smaller inputs until it reaches the base case.
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    1 T$ G1 c; }9 t* A, A' F    Once the base case is reached, the function starts returning values back up the call stack.9 @/ z: R; o$ }; w1 Q* D2 u

    2 A5 {! P& T9 E4 w) z    These returned values are combined to produce the final result.9 _  ?/ Y( ~. S
    . b/ @, A% c2 h4 l# K" Z/ |
    For factorial(5):
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    9 Q/ v0 L% F4 W0 m: A
    $ R# ]2 S8 V3 Nfactorial(5) = 5 * factorial(4)4 x4 x. G7 ^, g& g0 |9 C
    factorial(4) = 4 * factorial(3)* \( k. p6 T. M8 P0 s/ B: ]  N, a
    factorial(3) = 3 * factorial(2)# v: Y- K3 A& o7 {0 h. P) n
    factorial(2) = 2 * factorial(1); D% g1 V1 }! q+ w" {
    factorial(1) = 1 * factorial(0)4 ]9 x/ _9 {0 i& L9 I* V
    factorial(0) = 1  # Base case
    3 m: I2 K& I0 S
    " e  g, m; u; o: Q# [Then, the results are combined:
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    6 ?, }( D+ g1 @1 p" a+ ?3 u+ ufactorial(1) = 1 * 1 = 1
    1 s9 H# i$ r3 k6 p% Y6 O, Cfactorial(2) = 2 * 1 = 2
    / a; [" J" ?1 q. Dfactorial(3) = 3 * 2 = 6
    5 y8 _9 ~! \7 Ifactorial(4) = 4 * 6 = 241 k/ x! A& a* W$ G$ z6 B! e
    factorial(5) = 5 * 24 = 120" j6 u- s# }/ n: g

    5 N! n% @; z  q- {& IAdvantages of Recursion
    / N, X* c3 q4 b: A8 X; l. X6 Z0 v6 t$ P$ `! r7 s( ~
        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).
    ( _4 Y! G- U8 R/ `, K8 a- `) `7 N/ e4 k# o8 U9 r' o
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    , F, S, W3 g8 x- n7 u6 K. lDisadvantages of Recursion
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        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.
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        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    * U5 j% U4 u- }/ U3 Y! K; n) x7 b
    - X3 |+ E3 p- U& F) f0 AWhen to Use Recursion4 P% ^" m# B# i

    + Y' a' ^) }  T& u, H5 H; @( t6 F    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 [" T: u9 U# i8 ^

    : y$ v4 f% G0 |1 P    Problems with a clear base case and recursive case.: H; M+ E9 F- F9 _  B8 Y8 L
    7 ^/ ~' s) s% t+ r& q
    Example: Fibonacci Sequence
      ]; r7 H/ b: s2 z4 [4 L2 l8 q/ V5 y8 E& p: M( J
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 O; U. R1 W7 f+ H
    * T8 K% v  K* p0 m: V
        Base case: fib(0) = 0, fib(1) = 1
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    4 N  D7 [: t' A( i    Recursive case: fib(n) = fib(n-1) + fib(n-2). X( [3 F, m! p9 _( ^
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    python
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    9 c  F8 S# E% p. f/ a$ c5 V
    6 c  w0 f6 [. w9 b" Ddef fibonacci(n):3 U6 |' F! {$ ?, Z
        # Base cases
    & I% t2 R3 }& j  d    if n == 0:+ G; _7 M' f( ]: J
            return 0
    3 l$ N+ e2 z! \* x9 _    elif n == 1:. \8 L! w( v- Z' U
            return 1
    4 d& ], z6 w$ g    # Recursive case
    6 k$ O* B4 `2 F6 P    else:
    ( g2 q! @+ v# _; _, R        return fibonacci(n - 1) + fibonacci(n - 2); H) T( {; v( A' D

    ; j$ n; ?. x" U. h+ q- I# Example usage, s/ W' P, ?6 G% T  j6 P: X; j- c
    print(fibonacci(6))  # Output: 8! d3 t$ n8 m6 y0 S! ^! q, c: l
    $ S) ^& N1 J5 q" }8 m5 P# c
    Tail Recursion/ c' l" |: Y9 _" z

    # O7 e# f8 S9 l$ a3 rTail 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).
    # q5 Z+ q4 Z4 ?+ W: Y9 o( |$ Q" p
    4 p! B  s  g( H5 C  I0 |" G) KIn 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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