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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
  • 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 ( b1 F" a0 y6 i; g6 Q

    # U& b9 D. z& K5 E, i2 ^解释的不错
    : C9 p* a  A. }4 P! `0 \  }
    $ N- a# a# t3 p( i9 R3 F8 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。; G+ M+ d3 F. w$ B  m
    8 k# M. T& i- c/ L- J6 y
    关键要素
    9 G1 s4 N  W9 ~% h7 ]1. **基线条件(Base Case)**  R( o6 ]1 f6 I
       - 递归终止的条件,防止无限循环
    7 H/ |- V+ o8 V) _% L   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1/ P4 Z6 ~- L2 b3 A

    $ B0 t) O8 X  W. ~1 q6 Q- ?2. **递归条件(Recursive Case)**
    - l: f, v. Z9 f9 H* T   - 将原问题分解为更小的子问题" W& `1 p! s" e7 C4 \# l  a& P5 ]
       - 例如:n! = n × (n-1)!
      N: I' ]: O; t% U7 E3 o. _- M! F# G3 Y! `  K: `) U  O& u
    经典示例:计算阶乘, `4 F0 q) R' ~/ h' A- [
    python/ W% `5 n/ y) H  K! e
    def factorial(n):
    2 w* D1 h( e8 v+ a7 M    if n == 0:        # 基线条件+ m. A3 }$ ]6 O& B2 S6 b  n
            return 1
    9 L2 J9 Z* e9 q6 j. E: Q% A    else:             # 递归条件
    5 y2 C  u0 z( l4 K# M6 Y        return n * factorial(n-1)! ~8 |7 T; Q, p: I9 y1 M
    执行过程(以计算 3! 为例):
      _" x4 P5 X: r; H+ ufactorial(3)9 F6 E& s  d5 j
    3 * factorial(2), _6 z' q( R- @- F' O3 @& {
    3 * (2 * factorial(1))
    6 c* n; m8 W6 d3 n8 o, e( d' w3 * (2 * (1 * factorial(0)))
    % x/ j$ d4 G- d+ h: o9 M3 * (2 * (1 * 1)) = 6
    0 E3 X" V, i. e' Y7 T+ _8 M
    0 X) P1 J- _4 v1 z 递归思维要点
    7 @3 F; ^7 j0 t! E1. **信任递归**:假设子问题已经解决,专注当前层逻辑/ v" I5 v$ q( v: d6 U) i6 |7 O
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    % ?: U& |- B  c& k3. **递推过程**:不断向下分解问题(递)
    * p2 H- l8 n, [5 n4. **回溯过程**:组合子问题结果返回(归)
    ( g" \# ?: V. w2 L' b/ I
    & M5 Y+ d* D" {. T: q 注意事项+ l; T% e+ F- @9 g- p( H9 g+ I) m) g
    必须要有终止条件" m" c# j" A8 R% W- l; y* \1 B; m
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ `& [, K+ o3 Z2 q/ ?) V1 Q/ j* I
    某些问题用递归更直观(如树遍历),但效率可能不如迭代& w$ _" s2 {7 f( q
    尾递归优化可以提升效率(但Python不支持); J3 ]$ J* _5 R1 I& ]
    : B! L( a$ D$ q2 c: E
    递归 vs 迭代1 ~  T" k5 U7 f9 D" R# g
    |          | 递归                          | 迭代               |! e) k+ N" V# H- _+ c2 s
    |----------|-----------------------------|------------------|
    9 D0 w  F, C" B, o0 C  P| 实现方式    | 函数自调用                        | 循环结构            |
    + |$ p! x" k6 X; E& R2 U| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |! M) _1 e4 w+ s3 K, Y; M% z
    | 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         |
    ; l2 x0 Q, p9 R. B| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    * U8 z; R; X$ o
    2 Z4 b/ R: c! \$ W  J! e 经典递归应用场景! S# s- v7 w5 h- U& D" X8 f
    1. 文件系统遍历(目录树结构)
    2 ~* |7 `/ D5 ~2. 快速排序/归并排序算法6 H6 A9 C/ ?" j# b
    3. 汉诺塔问题! Q+ m3 n" {" Q. c( p; j* z. R& O
    4. 二叉树遍历(前序/中序/后序)
    % Q$ ~$ \# `% s( K5. 生成所有可能的组合(回溯算法)- b+ L2 j! @! C; `7 t" d0 [

    8 w% U, U" ^- U  l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,7 O, k; T, |1 H. F& R) `' a8 i8 ~
    我推理机的核心算法应该是二叉树遍历的变种。
    ' c* b8 n! z6 f0 W- a4 U5 \7 e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:
      H8 {7 H% f) V' T: E! k' m1 FKey Idea of Recursion( c; {+ \- Z+ h' v' a' U7 B

    1 t: e6 v6 }" r9 q5 ?/ F' A+ [A recursive function solves a problem by:+ r7 ?! [. B, r

    8 K4 I; I( h( @% _    Breaking the problem into smaller instances of the same problem.
    - m0 \2 R* m2 X/ j. D3 {! Y0 n  I6 Z, R; z% J1 B/ c
        Solving the smallest instance directly (base case).
    # r$ `2 i- @/ U+ o1 ?+ y8 P7 U1 k/ ^6 A9 ]. @+ O6 B, |$ i' l) {
        Combining the results of smaller instances to solve the larger problem.
    4 ~5 U! T3 f' O" @  T2 {7 ~) k9 h% r) |& q
    Components of a Recursive Function9 o: K4 {% @# F) N) b/ x1 f
    3 a  O. l2 n4 Z" p4 @8 |
        Base Case:8 ^9 i- P$ ?* u
    9 U' x. g+ d4 C
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 P1 r+ n8 x+ A7 c1 U& j# [7 [
    ! c' G: Y: c5 I
            It acts as the stopping condition to prevent infinite recursion.5 |; J6 m! |5 K, y* t2 o. Q* J

    8 J2 V, d) C, g        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.. F% A0 z* z6 ]4 w+ l9 M
    - k8 k  M) i; k2 H( j
        Recursive Case:
    6 h4 }8 K; z  o( \" k& E3 g7 r9 s9 H! x6 E) ]* k+ [% _' A
            This is where the function calls itself with a smaller or simpler version of the problem.
    + t5 Z; t% o7 _' H' n, F* T
    . G4 ^0 |& Q* n5 X        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( j- `6 G$ I6 S& v  N9 E- \3 A

    ( x2 S& A* ?$ N# m3 `Example: Factorial Calculation$ X. m) }$ I- |8 ]" \9 i2 a
    # `4 @; H6 c- Q: R2 E8 U6 b
    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:  T5 a  w/ X# S# {; `" K
    7 {) ?; D- G' a- o- i3 b/ X
        Base case: 0! = 1
    # W# V* T1 K6 X& U/ u; X5 q) Q- \2 k6 E0 p% z6 M
        Recursive case: n! = n * (n-1)!
    3 _/ X. A6 g! ]4 D' j5 Y0 q: |* a3 m: ?0 E3 @" M
    Here’s how it looks in code (Python):
    9 v  _0 t% O: \: k# r' @python' @5 S+ y' Q$ ]3 B0 R: f2 o3 c/ P

    + K. T- g4 B3 i! Q3 Q/ K! F& ~  s4 B/ u# p. g) f* j  D
    def factorial(n):! |. I7 x8 u' X1 c* x$ ^4 x
        # Base case
    ) w+ \. h: _, `    if n == 0:
    1 ?- t$ U; }8 P- ~& w& V        return 1
    ' G6 u4 P8 j9 w    # Recursive case
    7 q" W" G: ?/ r) `# c2 I    else:
    : M1 A- a9 `5 p$ p1 D* G/ k& h2 ~% O        return n * factorial(n - 1)
      e( B1 v0 c3 [* H3 a" Y5 h' B8 u7 i7 l' b* |" G  w0 y3 G2 U# C
    # Example usage; z. f2 }! q. p+ O( K
    print(factorial(5))  # Output: 120
    5 g" h) v$ ^6 N9 N5 ?; D% Z, u/ j  h5 s6 T
    How Recursion Works
    - p5 N8 z& p. h/ {  w+ {5 J- b" J* A9 F, [* X$ @. R2 }, D
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ; M3 n8 n! @$ F2 k# c3 l: \/ I0 K, Z- H& t" A+ R( w' W
        Once the base case is reached, the function starts returning values back up the call stack.
    2 k0 R; N5 j% Y) \8 g# x
    0 X( \7 h% c6 j. Z; r: o    These returned values are combined to produce the final result.- Y: v( v; T# n& V
    $ B+ ^0 O* I2 K7 U
    For factorial(5):
    : y; ^" _" w5 Z9 ~! A+ g" [7 T
    - C/ ~5 \0 N7 ]: G& {  g- P
    ( q* N( d7 O  k! G( m! cfactorial(5) = 5 * factorial(4)
    3 V! u/ f3 d1 U! @; ]$ o7 {factorial(4) = 4 * factorial(3)
    ! v1 q! T6 ]: `# ]factorial(3) = 3 * factorial(2)
    ) _2 z0 m* P7 w4 wfactorial(2) = 2 * factorial(1)
    # W& j( L9 `7 ~' B. C( N+ Efactorial(1) = 1 * factorial(0): |/ R. S, u  s3 D' D5 X/ a
    factorial(0) = 1  # Base case0 P2 O3 Q7 t/ {, U, {5 l

    # G" X' O: [# H& BThen, the results are combined:
    , n; [1 `/ f, x9 R/ t* D5 z% T# \% E( }) [# y/ s4 X% J
    # H/ k6 c* r9 m8 T
    factorial(1) = 1 * 1 = 1' O2 ?4 L% P+ J
    factorial(2) = 2 * 1 = 24 O3 p4 V  ~9 q7 T0 @
    factorial(3) = 3 * 2 = 6
    7 I8 u1 a4 k' Mfactorial(4) = 4 * 6 = 24% r' S6 |3 S9 }, f+ O7 M
    factorial(5) = 5 * 24 = 120
    0 i0 C  W# m4 O
    ' D! `2 M) E: j. xAdvantages of Recursion
    / _1 v( }) I( I" V* W  A6 k) ~1 u1 |1 z. X$ M
        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 W, s0 T, ]/ K1 ~' P
    % k5 E6 R, w0 o) |. Y
        Readability: Recursive code can be more readable and concise compared to iterative solutions.3 W3 Z) V" h% f2 `

    3 [" w7 J0 |+ u8 [  u2 F' u7 QDisadvantages of Recursion8 @) P& o' V* Y7 B
    ! x0 e* y' @3 V9 G$ v& S
        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.; r% ^% I) }; ], T" s- C

    ( k" u/ u. ~7 T9 ]- i    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ; ^0 Y2 U- T9 i" i3 D6 t/ k$ }8 x  l- o
    ) c& F- I# |4 \. {' ~4 K* eWhen to Use Recursion
    . A. K6 p; Z- w( W8 v! A" N: _/ l0 h& e+ d8 A
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    4 F" ?8 }" L3 v+ j, |2 _, J, W/ [' s% y9 A
        Problems with a clear base case and recursive case.
    & `' R, a& B  T1 r7 U- E% D. l( k5 P6 _: b
    Example: Fibonacci Sequence& I, i0 W' R+ S. u! e6 N3 k5 @" O
    2 b) |# [" s4 u3 y5 D3 V
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . a3 o0 o2 [" A+ e9 E
    0 u2 Y1 T8 k: n+ b    Base case: fib(0) = 0, fib(1) = 1' o9 K! {6 ?/ R" b. d
    + W' X) F8 R) E: o0 J9 @
        Recursive case: fib(n) = fib(n-1) + fib(n-2)) x/ Z* Z# p" ?6 y

    / r4 a; K6 R' W5 b; Wpython
    / {! f' Q  D! H$ i3 i9 z( {# i5 {% d/ q- P% B3 J0 f. G, j- L+ |

    . Y5 ~; V& |  U! C% w# Fdef fibonacci(n):' ]. r- ~4 f% I6 D0 T7 }# g
        # Base cases0 N( X9 j3 V/ }  `- W$ C  p
        if n == 0:4 N7 }$ h5 D. i1 F0 `8 w3 N
            return 0
    ( p$ q$ ^+ B! C5 D: `2 k$ V    elif n == 1:$ a" f; |% b. D3 e& m
            return 13 A" H+ G* `. M# g
        # Recursive case
    5 o# c) O) v; ^" ?    else:: ~) p& K+ }3 _* G7 b; t
            return fibonacci(n - 1) + fibonacci(n - 2)
    # r6 X# ?; q( q8 R, E, G& \8 X8 A5 T- v/ |/ ~
    # Example usage
    * [2 O) Q& h" F( V/ l; y- P8 p3 g" iprint(fibonacci(6))  # Output: 8
    . e% h5 H9 I8 o# d6 l% b5 K/ Q& G; i% K$ ]
    Tail Recursion2 h0 I/ G7 c9 ~4 a) C: a' a; G

    ! c8 F/ M, `3 }2 pTail 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).
    ( `9 |+ u# }- [  g+ v9 d$ _: F& {( t) G9 b2 C, H4 s4 K2 z
    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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