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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| 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 . N3 e7 y, |* c
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    解释的不错+ y! {  ^# m2 s  ^, p; a" A

      _% n% o- c" M) N- y7 z/ t$ i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。  Z+ c, ^6 q( f! R% h7 Y
    ' C  C. N9 U1 I& G0 G6 S6 g
    关键要素
    , y. y: A3 f. ~- F+ y  O1. **基线条件(Base Case)**
    ' D# }2 F5 X& o8 c   - 递归终止的条件,防止无限循环% ?. q3 Q$ a" E% S" \
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    $ W' N1 n: N2 N, r3 V9 A$ }# Z1 P
    2. **递归条件(Recursive Case)**3 u9 f- O$ _0 q7 v
       - 将原问题分解为更小的子问题1 N! v: o/ E- U+ x2 t4 X5 F
       - 例如:n! = n × (n-1)!
    5 _! B2 @7 j( k  L
    * l5 t! l. c2 y1 Z- i 经典示例:计算阶乘7 P6 ]' I& `$ O+ |
    python
    ! j$ j( I( h1 C3 u6 d1 sdef factorial(n):
    4 F  e' c, j0 N" [1 F    if n == 0:        # 基线条件
    " p. h" j/ `& |2 ~        return 1
    ' K. ]: Z% ?4 E4 Z1 R( X  h    else:             # 递归条件
    ; X* d( y3 n8 w3 u0 P5 p        return n * factorial(n-1)
    7 N0 Z; Y5 X9 b( Q执行过程(以计算 3! 为例):' [+ v7 F( N% K
    factorial(3)
    8 @$ J" y7 l- a3 * factorial(2)+ P5 B0 X" d. ], y) P- x0 o
    3 * (2 * factorial(1))* m+ i6 c+ T6 N# s+ k& d
    3 * (2 * (1 * factorial(0)))4 {9 I# T) r/ N5 U7 S
    3 * (2 * (1 * 1)) = 6* Z+ R8 {. J; g/ W+ ~% Z

    7 i# n8 b& }( g 递归思维要点
    4 A8 d- V2 v5 l7 M& ^1 u8 U1. **信任递归**:假设子问题已经解决,专注当前层逻辑& ~4 X# c4 h; \7 @& h9 S/ w0 w
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)% W$ `0 Z# U3 D+ m4 v
    3. **递推过程**:不断向下分解问题(递)
    : Q* ^/ G6 q$ F7 C4. **回溯过程**:组合子问题结果返回(归)
    5 w7 V- ~+ x4 q; r. G- B$ f& W0 H; ~5 Z' T6 v& M2 E
    注意事项
    7 ]/ {$ ]. ]2 L+ c$ J0 R必须要有终止条件  [9 j& U9 `) T5 B# q
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)0 [2 x: a& D! O: o1 S9 n9 a0 u% ?+ S
    某些问题用递归更直观(如树遍历),但效率可能不如迭代  X4 O6 Y3 E: ]4 r1 J% s( W
    尾递归优化可以提升效率(但Python不支持)
    7 H5 W# R( f6 B1 X4 Z- K; J3 E5 X$ W  L3 ?3 }
    递归 vs 迭代
    4 m( y* W3 k* W' |1 K/ V|          | 递归                          | 迭代               |4 V* w9 E( c3 Q3 b
    |----------|-----------------------------|------------------|2 e3 Q) n! a0 T2 T- F
    | 实现方式    | 函数自调用                        | 循环结构            |6 q4 h* p0 p  R) E! ]
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ) ]4 |, L, y, ~# E3 O| 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         |% C" N8 Q9 q% q/ i  L
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    5 _, x* [: D/ i) @2 F- w9 D1 j- y- Y$ p9 x5 I$ A
    经典递归应用场景" Z  e# [. |+ S
    1. 文件系统遍历(目录树结构)0 G" k  p3 x- H/ L. I. P$ ^+ Z
    2. 快速排序/归并排序算法
    , q/ S2 Z4 O: a" I% V* g) s! I: p3. 汉诺塔问题+ E! r! R1 I( C0 V6 m  l
    4. 二叉树遍历(前序/中序/后序)  p% B8 [  `3 W7 Y
    5. 生成所有可能的组合(回溯算法)
    1 F2 R" m0 [' N6 E' c8 u$ u- H* P* @  j) U/ C0 d
    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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  • TA的每日心情

    2026-4-9 07:45
  • 签到天数: 3217 天

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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,
    1 Y6 l0 s  m3 _2 l我推理机的核心算法应该是二叉树遍历的变种。
    # |3 U- _# J, O( n2 S: ^7 m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:, a  e; I, ]( \- R0 b  j0 K
    Key Idea of Recursion* V1 ]( f* @+ g

    2 s* Q8 I+ i' V5 o1 i& RA recursive function solves a problem by:
    # m  }* ^1 O2 C! m7 @, X4 L; f8 @. Y- \5 I" K
        Breaking the problem into smaller instances of the same problem.
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    ; j* t; z& V  v" Y5 P    Solving the smallest instance directly (base case).* D. p" M4 Q; C( c: x6 \. z4 ?

    " ?. w; D& ~& a  b" E2 b0 m    Combining the results of smaller instances to solve the larger problem.- w: c# @( d1 \

    1 M5 e) W4 [" G1 N/ UComponents of a Recursive Function
    " A1 H. p8 c0 s- X  R# ]3 `- S# c/ J7 q" D2 W$ \
        Base Case:
    1 d2 L# C& z8 d# U0 V3 A) i: Y5 e2 p2 @  u- ~# I
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    1 ^+ O. |2 K! A( R
    7 o5 K+ K7 N" @1 q2 D4 F# `        It acts as the stopping condition to prevent infinite recursion.$ S6 h: O  @9 V
    7 p7 f- [* \8 M8 [" F
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ; O$ s& R8 ^2 L: _5 a5 z
    % t2 V  N3 |/ V( d% [% |    Recursive Case:! J6 {1 J- D) {# v% C$ T

    6 o( ?" U- s. {! ~3 l        This is where the function calls itself with a smaller or simpler version of the problem.- o1 r5 n5 Q+ b, s5 ]; h/ E3 t; K
    6 w8 O- |$ N  i* \8 n6 p
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( t2 H! _# e1 M

    ; G8 Z; `# ^8 _Example: Factorial Calculation
    ; w4 ]% _. z! G
    ; W7 L1 P) x# O3 ^- gThe 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:
    4 u' G1 d9 V( B9 Q1 A2 W6 W% p1 h% U. o& v
        Base case: 0! = 1/ X! o) h' Y- y5 o. p8 G

    5 n/ l! D4 k, _: l+ p    Recursive case: n! = n * (n-1)!7 E. M9 c2 [* e+ ?' l( I" |/ E
    5 s3 v" a, ]: Y6 n
    Here’s how it looks in code (Python):6 N! V5 W) }* Q( {: k/ z
    python
    * P+ y* S$ `% i% _' n7 ~; ?5 J3 w$ b
    5 T$ |& w, _% m8 O" g" L. m$ [
    def factorial(n):3 s6 O2 J, W$ n, g$ C
        # Base case- q2 C' m) y# U. N
        if n == 0:5 W. |" F; D3 E1 A8 ^5 a4 k6 F
            return 15 V. u9 ]$ s9 R- x0 O0 a
        # Recursive case
    ) ~# [1 a; J: r  w+ ]    else:
    + j7 K1 C$ n+ A        return n * factorial(n - 1)
    1 w  ?1 ~. u/ m6 R
    ! Y7 Y0 G6 }. D% k# Example usage
    / }6 W$ Q2 l. ~# Q' s8 [# Z9 E9 Pprint(factorial(5))  # Output: 120
    7 `. b3 ^$ A! ?/ Q$ g* G+ H
    7 H& U0 E7 K5 E  ?How Recursion Works- k- P5 [- {! {% P( D/ `
    ( V4 q3 ~+ L0 n0 S
        The function keeps calling itself with smaller inputs until it reaches the base case.. U! N, r9 q) t1 k9 C( m$ \- Z

    % H: r# O" y/ f, {    Once the base case is reached, the function starts returning values back up the call stack.
    - i$ E8 s/ f% ?; H$ O" y' A& V, d% r  X5 G
        These returned values are combined to produce the final result.
    4 @' o/ F$ N; Y9 P8 k
    3 o$ P7 Y4 L9 B$ }7 Q4 |For factorial(5):
    , t6 H( }, ?' G: N+ q- M4 Z& v" {% _4 e+ l  ?
    5 s% L+ j5 y5 [& }" p' m! f* a5 b
    factorial(5) = 5 * factorial(4)3 j8 ?1 I! x4 W4 g
    factorial(4) = 4 * factorial(3)
    6 q5 X/ c, B# g1 Y6 j  z2 _factorial(3) = 3 * factorial(2)
      z0 e4 A7 D1 sfactorial(2) = 2 * factorial(1)
    + ]) R, y( z- G" c5 M- h5 X0 sfactorial(1) = 1 * factorial(0), s" J# q9 Z6 i3 |5 @
    factorial(0) = 1  # Base case
    3 S7 A' t! n* O/ o
    $ i9 k" Y) F. U+ n( p* G1 _3 N4 DThen, the results are combined:2 L/ Y6 D$ r  q

    6 b  h  R) |+ u" E% M& F6 a- M) W& z' H& J
    factorial(1) = 1 * 1 = 13 L1 b7 R6 Q7 V) c4 e
    factorial(2) = 2 * 1 = 2
    " c" p7 f, @3 K5 [( wfactorial(3) = 3 * 2 = 6# S$ P) b9 x+ v& S' I2 C
    factorial(4) = 4 * 6 = 24  \; k  ?* T; @3 Q6 ^1 ?  r: a  C
    factorial(5) = 5 * 24 = 120
    ; M- a  F6 A8 z! q( ]
    " n& A; ?, G9 [1 h. A, jAdvantages of Recursion
    $ h* b; [- U: P" Q: f7 W: A" i2 M! k# _# }
        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 i" G6 p( G: y/ h6 \9 F
    ) j/ T/ Y% f" B5 }1 u, r  ], ?    Readability: Recursive code can be more readable and concise compared to iterative solutions.+ j2 V+ r1 j/ |; \5 [3 u2 a
    ! N) l8 b6 d: c3 b% t
    Disadvantages of Recursion
    0 _3 v7 i! \9 `( q. `* J
    , J. s& |! ^/ |1 h    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.
    5 W, R5 ?9 Y# i% _+ }/ U7 \' Z3 l2 _( n! j2 i, q3 ?
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# t0 g% T# p  k& F1 a
    , p% S: t, G7 c9 T+ f+ T
    When to Use Recursion3 Y" r2 t! N  o7 Z* r, v
    . m- E; X' U$ o, }) T5 d
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' V- X2 L8 d- n9 C) R1 @$ U

    2 A6 j8 O, w1 B- j% t8 j: w/ o( C; d    Problems with a clear base case and recursive case.8 B8 [- K3 L& w7 i, y
    " z' N) L; A8 p% W
    Example: Fibonacci Sequence; ?4 O; L' O, ^/ y8 N

    1 s% P: O1 ?! v3 H! f5 {The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    / u/ b! u9 m. x9 Z& ?; I0 V; P$ d6 l$ e- C5 j9 u0 U5 |! W
        Base case: fib(0) = 0, fib(1) = 1) ?5 i. Y4 J3 x
    # M0 O8 r' N, p' c0 B  [1 i- T
        Recursive case: fib(n) = fib(n-1) + fib(n-2)4 A' V" q' j2 J

    . }' a' x0 u7 H( l" Apython
    # d7 [9 [' l3 g, }( E' j. ?, [8 g/ A5 L* u( S9 Q2 T) d% a2 S
    4 e, `! g) ~! L: [( }2 @" i# J5 ~
    def fibonacci(n):
    ( g" U( l, z- m2 E    # Base cases
    ( s6 [& s* N6 }1 P; f    if n == 0:
    , T* x$ R3 R& u. ^. L2 N/ U        return 0# d  o7 J2 Y/ _* l) ]
        elif n == 1:  _0 v" g/ ~3 |: ^1 p# G; _0 {: V
            return 14 [$ ~, L" ?3 t1 {5 b4 o1 T
        # Recursive case; u0 ?5 `3 o. k, G8 |" x
        else:) ~3 a7 V8 C& h' c' p8 N3 |
            return fibonacci(n - 1) + fibonacci(n - 2)$ A8 |- [+ z1 ?/ _9 F
    0 v7 m( q* V6 o
    # Example usage) U' W. S1 X9 _. Z" g
    print(fibonacci(6))  # Output: 8
    4 o5 A) s0 O/ M: P+ a$ n. D5 E! ~; K
    Tail Recursion$ J: F& \; o1 Y& P2 g# c; j

    + R8 Y2 }) b3 |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).
    ( F6 E! h6 w2 E' J2 _- }0 n9 M0 I6 s% C, d9 f- C
    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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