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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
    8 m" _% |* v' _! b$ x! d, u; @# [" O1 u; n: W  U! G1 M
    解释的不错
    3 D) F. p# T  Z# P/ ]& ?2 }5 S, d- K+ y. K6 n0 q
    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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    3 ]3 j5 Y! l7 w! @8 I! W 关键要素8 l2 c) ~9 o% J) Z2 F4 x
    1. **基线条件(Base Case)**/ A" ~, q+ c) {, D5 l% E; O& l
       - 递归终止的条件,防止无限循环
    . [8 x! e: r" X7 Z: f   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    $ p) {8 a6 Q6 z9 [% }: c5 V' y( `. J! h& ?, S7 w
    2. **递归条件(Recursive Case)**3 N0 E* Y9 M7 Z  }/ D5 @
       - 将原问题分解为更小的子问题
    6 t. v# E0 K, o5 ~$ i& P   - 例如:n! = n × (n-1)!. `) ?" ~- q2 n% U$ ?# ~5 }% \, N
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    经典示例:计算阶乘
    : P2 H% c9 m4 Npython; N3 R7 p: ~6 v  Q6 v( {# _
    def factorial(n):1 ~) l$ K5 j' r- n3 P
        if n == 0:        # 基线条件% c: \& u; F) y. P- [
            return 17 R" d" M2 n0 N# p
        else:             # 递归条件5 }2 k$ j: p- f- c) s
            return n * factorial(n-1)" ]# A; R: I  c9 g; z) g
    执行过程(以计算 3! 为例):
    , ]0 P1 T6 O+ c8 wfactorial(3)9 I9 e1 R( @! c5 H0 o8 q( l* v
    3 * factorial(2)# k) h* j' q! U5 a: K* X
    3 * (2 * factorial(1))& x% l# s; T: h/ [$ Q0 C
    3 * (2 * (1 * factorial(0))). {. K  b: E" o! B  x- Y/ C
    3 * (2 * (1 * 1)) = 6  s9 ?& {2 a& ~! K
    ! c* z5 E, s" D
    递归思维要点
    3 l' A: u% s1 R! G1. **信任递归**:假设子问题已经解决,专注当前层逻辑' C1 X6 r4 P/ W) h) V, ^/ e
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    " V9 d' c' }% G. z' O; a" y+ ^, u3. **递推过程**:不断向下分解问题(递)" D  D" E% f: T
    4. **回溯过程**:组合子问题结果返回(归)
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    % B5 ?4 ~  {9 A! s7 u8 \) } 注意事项! ~  m5 G0 l5 s1 Z8 e" ]2 B
    必须要有终止条件
    4 o, q& g0 i. F6 }1 F递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    0 B; z  U7 y$ S7 Q* s: q某些问题用递归更直观(如树遍历),但效率可能不如迭代
    . V& F2 B; z9 S% X$ \尾递归优化可以提升效率(但Python不支持)
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    0 `0 b3 {' ?, \  m# Z( D3 a 递归 vs 迭代# I1 f4 n2 R' V8 [$ y2 U4 J* l
    |          | 递归                          | 迭代               |9 ]3 y6 j. V  ~, V
    |----------|-----------------------------|------------------|
    9 Q$ \8 Q! C/ L/ k1 m( E# ?2 \+ z& _* U| 实现方式    | 函数自调用                        | 循环结构            |
    + S% V. Y5 n( C0 o# V8 T/ ?| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# t$ L% {: E, z2 a5 Y) V9 s
    | 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         |6 x0 i& b+ \9 q# `1 P  C
    | 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
    0 S3 q, j9 B  p1. 文件系统遍历(目录树结构)* t0 v2 t1 Y. v8 y* w2 v% w) o' V
    2. 快速排序/归并排序算法
    1 v, I- g4 ^5 d, q! k6 I$ k3. 汉诺塔问题9 H" U. \" B9 |3 o, q* j
    4. 二叉树遍历(前序/中序/后序)3 e  K. _* r( g! E) J) \' B
    5. 生成所有可能的组合(回溯算法)0 y6 X0 O0 u9 Y& Z- m

    6 i5 G. J, N+ u/ M7 J! D3 f) o( C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,
    " H& ?% _' }) V/ c) q" n我推理机的核心算法应该是二叉树遍历的变种。
    ; H: [) h9 f2 s, h! i" s; b' v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:6 w: I  f. N% c" D% H4 ?
    Key Idea of Recursion8 T$ i" ^& P& T8 [
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    A recursive function solves a problem by:
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    & k% s5 [+ s( p9 x! k9 a  L    Breaking the problem into smaller instances of the same problem.
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    ( I1 w5 @; N& u/ a  r& R    Solving the smallest instance directly (base case).; }2 E% k* E8 K/ s% Q/ y/ b

    + C2 ?# g3 L  w+ S- {7 o7 d    Combining the results of smaller instances to solve the larger problem.
    ( k+ [; n3 b0 ?3 B) o& F. y; n- |+ p/ P& p' x/ B: W9 B
    Components of a Recursive Function
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        Base Case:
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.0 V# F8 M0 S4 S9 e. O
    - ^1 n. j8 y$ g
            It acts as the stopping condition to prevent infinite recursion.: z1 }  x& f  Q

    / @4 G' p; p2 A8 f        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.& _+ t1 o. R1 Q2 o7 H
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        Recursive Case:
    0 `' ^. @2 X. |6 t5 Z( b- u& }+ J4 r$ k' t4 Z3 @' U+ c
            This is where the function calls itself with a smaller or simpler version of the problem.5 L9 Q8 [& P3 I+ X' X, ~
    - Z: ?) ^5 i- \" s# C; y
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    # M+ Q. \. o# G0 h7 T) L) p# YExample: Factorial Calculation2 P$ F2 D( f+ [! n' {

    8 r; |( V9 o- |1 \7 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:/ m* J2 t; i( u( P

    , T  }7 I& `9 x- @    Base case: 0! = 13 S! C$ H- ^6 b4 i% f6 M

    1 ?% c! @( i0 J& A" ?- [    Recursive case: n! = n * (n-1)!: J- ?7 r) F0 \- R
    + x7 J( W; ^4 q6 r5 W8 g
    Here’s how it looks in code (Python):$ k' f' D' X8 W* o5 Q9 j: `1 t% t" e
    python
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    ; S9 ^/ \( H1 Y- G. X
    . ?# F8 z9 \7 G2 C2 Z" b3 Ydef factorial(n):' o0 `1 H0 P2 o; s2 C+ C8 v" ^
        # Base case
    & h+ L+ Z% e: Y" [    if n == 0:
    1 s3 d2 |6 E! N5 C        return 1% J; }% L) G# m
        # Recursive case' h0 M5 |  Y0 w6 Y% R8 ]' V9 K: A
        else:9 Q7 ^9 V# S% q/ U
            return n * factorial(n - 1)
      ?9 j4 m' ]$ l" f! {: i& C7 y
    0 ~; n( b9 O- K) i' {" k" n; q# Example usage
    - x  o8 S) s6 {% L& \; s3 oprint(factorial(5))  # Output: 120
    " c& x9 s0 X5 d7 z
    1 Q. |" }3 C0 i6 F0 YHow Recursion Works
    ) \# E3 k# a9 v  c* n
    ' P; F) m/ W2 H2 i    The function keeps calling itself with smaller inputs until it reaches the base case.
    . ^0 T+ k4 _6 f- |+ a! T
    ; d) c; e; I1 s; }+ |$ R    Once the base case is reached, the function starts returning values back up the call stack.
    / T# ]* O: C% g5 O1 T2 P$ {9 P0 N) r* T3 V7 Z9 I5 l
        These returned values are combined to produce the final result.4 _: k, W0 D0 |3 x* a4 _5 G0 H( ^

    ' O9 S8 M6 I( r! y. S9 qFor factorial(5):
    6 ]5 ?3 ^) d& _4 L% ~
    . X3 E2 B2 I2 O9 \) C& l) q
    " E5 @6 T4 ?" z; V4 P+ ]factorial(5) = 5 * factorial(4)9 K: M5 g' p4 n
    factorial(4) = 4 * factorial(3)
      J! j; @, O& A* afactorial(3) = 3 * factorial(2)
    4 t6 ]# R' _* E$ {8 W9 sfactorial(2) = 2 * factorial(1)
    ; B. S" F0 A) C" l& Nfactorial(1) = 1 * factorial(0)
    & f% e% K( V) Lfactorial(0) = 1  # Base case
    # ?' ]; |4 o; K2 b3 F0 ]  k1 b  t% t, ^# W
    Then, the results are combined:
    " U) a! l* r" R+ ^. v6 `& q
    ; W0 S* X' S2 k& }2 N' f  E7 H( r5 Q' e3 k
    factorial(1) = 1 * 1 = 1
    0 x: d: O& C* K+ ?factorial(2) = 2 * 1 = 26 ]! A4 X! V/ C
    factorial(3) = 3 * 2 = 6
    % n$ q5 B* {% b) bfactorial(4) = 4 * 6 = 24
    ! O) \( A( V' ofactorial(5) = 5 * 24 = 120. E8 N" p' e5 D+ ^* n; N9 O* e/ \

    ' e" r, R5 O. Z1 y2 D9 NAdvantages of Recursion! J- ?6 @6 k1 l9 w0 J8 u% `/ t0 ?! C

    8 Z1 ?3 B5 _& o6 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).' u& B+ K  l7 c
    0 f5 x/ G1 e; v' ]. @
        Readability: Recursive code can be more readable and concise compared to iterative solutions.: R/ w$ ^6 v0 `/ q
    7 W0 U" g7 s2 i' \
    Disadvantages of Recursion
    * R. r$ J$ _3 q- d, i3 s
    : M+ l; X" K4 Y  k0 z/ |    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.
    & O  S7 U+ V- v3 R4 C9 }/ V# I0 P1 B6 y; p, Y1 Y4 \  N
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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    When to Use Recursion/ J8 k& Y3 S6 ~, R

    6 i* C, i0 W8 v/ C# j% {, c    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' M0 q) O2 x8 s3 Z$ m' l; e& i( `7 ?1 J6 X0 N) u* E7 |
        Problems with a clear base case and recursive case.3 d- \, U% b; ^$ u, Z
    ) P- L7 u% o/ [) Q5 K# m9 R/ `/ g
    Example: Fibonacci Sequence
    & X0 f/ C8 Q$ |  k( G3 n/ W: ]# p+ G; M4 v& w9 I- }0 i1 t6 s9 d' r
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:6 s2 k( Z4 p. ?& N. t9 y! c+ J+ }0 m

    % _" \" ^7 r4 ~5 i    Base case: fib(0) = 0, fib(1) = 1+ F, O' b0 X* p

    & |5 X/ ]& P/ G4 b- N' o    Recursive case: fib(n) = fib(n-1) + fib(n-2)
      b4 V5 T6 ]/ C8 O2 G0 \  h% |( \. y
    python
    4 t) X3 E0 l. ]+ V7 \; r4 H/ W* ]. p4 u, H9 w
    7 m; o7 X' J  k3 s2 y& K/ e0 v
    def fibonacci(n):
    ( u7 ?# E1 C# v1 N    # Base cases% d" O  Q: c) o9 j6 x
        if n == 0:
    * |) Y# P8 e* N( Q6 s3 @        return 0
    - `+ p5 l1 B0 e! l) L1 t  H    elif n == 1:
    8 j5 t5 m1 L" [- e$ f        return 1' S/ H) d) }0 ]. D( Z# }4 W9 [: C) ?
        # Recursive case  _; `9 A7 g3 x  R- z# P$ |& \
        else:
    3 {  X+ j* z9 d# Y" T. }        return fibonacci(n - 1) + fibonacci(n - 2)
    9 d% T6 ~* y2 j. Z7 y4 X+ F
    / {! u) W5 p0 o, T0 u) u7 e# Example usage' s# P* x$ Y- M
    print(fibonacci(6))  # Output: 8
    8 ]  A5 E0 x( C# v. e6 |% _4 n) j& Z+ \
    Tail Recursion
    ' }9 M' ~; l  s* M' t8 m$ E  i0 i# u; B7 {" @5 D3 c0 ]
    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).
    : ^' p: Q, U. b, u, ~3 O9 e- B& E9 A% @7 I5 `- X- d4 ~0 H
    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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