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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 ; M2 ^" ]2 z( |: m! f4 T; `
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    解释的不错
    5 @- l$ a0 h; o! U/ l& g+ c: L% W( g
    . k# `8 r: H- F  W0 Q( d$ I& g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。6 J6 h6 z. W  e! F' H

    0 W1 I& ^2 L/ E) j0 _ 关键要素6 h! E- k* l1 M. I) |
    1. **基线条件(Base Case)**" e. P4 p3 Z3 T4 u0 \8 r% P
       - 递归终止的条件,防止无限循环2 Q/ I2 x% i7 N) T- G! b
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ! ]3 k' u4 A/ T; y
    ! r0 d6 l+ v1 L2. **递归条件(Recursive Case)**
    / ?3 |4 `4 y9 A( s: q   - 将原问题分解为更小的子问题. }9 N" s0 M# {( D  G9 B0 |9 N
       - 例如:n! = n × (n-1)!
    ( x# D; X% ^3 y, W& o! r5 u6 k
    8 P( c: F1 E1 ^9 b 经典示例:计算阶乘
    9 ?+ x! V+ u6 M* A' ipython
    3 |5 Y, e& s# u1 p4 Pdef factorial(n):  K6 s! S$ h  E5 K2 |
        if n == 0:        # 基线条件
    2 ?% a" N1 C; A1 q9 ]        return 1+ h3 d) R* q7 W8 `
        else:             # 递归条件
    8 d, W1 x/ W6 w# \- O" _        return n * factorial(n-1)
    9 J7 ]9 G3 w, `  O4 x8 G  G5 g执行过程(以计算 3! 为例):4 _/ M' Z3 d% m7 ^6 d
    factorial(3)" [3 j' [2 Z4 t' ]+ b% c2 {
    3 * factorial(2)
    & m5 Q8 [" l5 d+ ?% x0 {9 q3 * (2 * factorial(1)): G2 R  }6 [3 z9 B
    3 * (2 * (1 * factorial(0)))
    - o" n8 E& Z) ]) S+ E1 b3 * (2 * (1 * 1)) = 6
    3 r" K' l0 A3 `) G9 t; J$ K2 H
      L+ M. ?- Z% w: G% h; { 递归思维要点
    * b0 n, ], o  s6 E1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ; F9 C. _: p* B$ U1 b4 t0 v2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    8 Q4 @( d" D+ Z9 u4 f! c4 L' d3. **递推过程**:不断向下分解问题(递)$ o3 |1 ]  v; X$ ^
    4. **回溯过程**:组合子问题结果返回(归)
    ' d/ u& c0 M2 m# j0 E! j# F8 T( a  b
    注意事项
    3 E2 R3 @% r6 H( I必须要有终止条件
    ( Z4 @* M! W2 o% j6 K) b; @, k2 z) {递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    . s! R6 T: V7 G, ~" `2 @1 C1 v某些问题用递归更直观(如树遍历),但效率可能不如迭代1 f$ a, M9 W' i* x
    尾递归优化可以提升效率(但Python不支持)
    9 ^" |2 Z! l, r7 h$ F, {9 e; F
    $ j: M4 [6 Z" t, V+ v 递归 vs 迭代
    % ?# B# H% b4 u! Z|          | 递归                          | 迭代               |- ?$ ~" ]) N1 @# ~: w) X/ M
    |----------|-----------------------------|------------------|# Q. [! X+ |) O# u# ~/ D1 a
    | 实现方式    | 函数自调用                        | 循环结构            |& y  G8 L6 B0 z
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |" {7 I  q4 d; K2 m6 u
    | 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         |3 e. l9 O% U( n% L
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |" A/ c* r1 E, w6 p

    ! N2 `& J& x$ W0 b6 h0 Y$ Z 经典递归应用场景) I. N, Q0 `5 x1 L$ C! i
    1. 文件系统遍历(目录树结构)$ @& p; b, A+ @9 t, }
    2. 快速排序/归并排序算法/ c' |( V4 i% K7 F
    3. 汉诺塔问题% x0 e, s' y( ~4 r1 l" W
    4. 二叉树遍历(前序/中序/后序)
    1 q8 T2 S" \" _5. 生成所有可能的组合(回溯算法)7 k2 @4 h8 T# S8 j- L# g. A+ l. c
    ( P! T, X, h/ @, |0 \7 g1 r
    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,' V  \4 ?0 q1 X! F
    我推理机的核心算法应该是二叉树遍历的变种。8 X' _1 b7 O3 \
    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:$ F1 Q% p" i# e* q' B0 c
    Key Idea of Recursion7 c3 W# u* U! O# R7 Y- K

    6 C: k4 k0 n5 K' b# o8 |: GA recursive function solves a problem by:
    ( Q  t) b  X. t5 g9 Q
    & E2 `% q' @, b    Breaking the problem into smaller instances of the same problem.
    # A2 n- I: [4 J! V. D5 G0 j& A0 I. I/ S! T
        Solving the smallest instance directly (base case).
    ( g! \. d* [' o. j$ |, o4 x5 S) C! B$ b2 C1 D/ l  ~" M$ @5 k5 A
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function7 ^3 ~: ~9 u6 {) W9 _1 h. a5 _1 s

    & s) C+ D& D$ m8 a# I1 I# y: Q    Base Case:1 z4 B4 @( i$ U& I; ~

    9 P; u, W% B' b( b- g( S) t& ?' M        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    " p1 s9 L1 I* D
    0 Z/ w: j' v/ D  ~3 E        It acts as the stopping condition to prevent infinite recursion.% p8 ?' A% l3 q, p  Y. `0 O

    ) m; J+ M% z! H$ a        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    " G& |5 R- n) I+ `) I  Z) i    Recursive Case:: h- n3 y: {  C% \
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            This is where the function calls itself with a smaller or simpler version of the problem.# p7 R% m0 D6 H6 x" i: M

    4 [  V. i5 ^. ^5 L) G% `4 I' |% J7 \        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 K, C' @3 w3 z5 s0 @' M
    : F! I  o+ a% P& OExample: Factorial Calculation
    6 z3 z) @2 i( S; c; {) o
    ; ^& x: m6 k. `: r1 \, H  NThe 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:
    / _* Y4 v1 Y  _; g
    + u# I: ^- @  g7 v    Base case: 0! = 1* T+ T3 E: S; a
    6 X* Y  i3 s) v. M3 [2 L
        Recursive case: n! = n * (n-1)!( h& o7 c% Y8 @+ ]; {: [5 e

    ( n1 P# g) n( t( q9 q# GHere’s how it looks in code (Python):
    3 G( g" ~5 E8 E1 ?% ypython
    1 r% d# A9 @9 b" m' H4 h* N  h+ }' L$ l8 ~* D

    5 b1 C3 f# \" z& Z7 p$ F$ Edef factorial(n):+ [1 j# k( I' ?7 c8 [1 H. n" h2 m
        # Base case
    2 N$ @' @3 Z8 f3 f& ]    if n == 0:
    ! c8 n5 F6 ]! x& d! M: k# y/ Q8 Z        return 1* @5 q# a' T5 I& @0 S8 E! p
        # Recursive case
    ( i" e0 v9 g0 _: Z7 l5 A- y, F3 w; B    else:% o# C5 P# H1 B" R
            return n * factorial(n - 1)- v9 ^" C- X& N5 J" B

    4 g% B  V0 a2 ]; Y9 G, z# Example usage9 Y9 P3 K, I& i6 p( h6 E
    print(factorial(5))  # Output: 120" h+ f$ X1 e; N0 e) y- X
    8 E( j& P) g8 Q0 T( ]6 i" o
    How Recursion Works
    , ?- P# {8 n6 k: ?) S0 M. S+ e
    4 p; E' m$ C4 M0 J) ?2 W  C  A" }    The function keeps calling itself with smaller inputs until it reaches the base case.
    & n5 I2 P$ F' p$ e1 {& Q" P% q6 O9 {% s9 V2 S" t3 F( b# F
        Once the base case is reached, the function starts returning values back up the call stack.0 d7 M: ~% l' g' T) S  N* _  |
    0 o8 W% m! n* H3 `  l& i, t$ n$ |
        These returned values are combined to produce the final result.) N5 Y" V6 B8 m5 i
    2 x1 H" k. t' i9 h6 T( N% R
    For factorial(5):
      u0 D1 n  R, ]9 H2 g
    6 @! z, W' ]3 w& C( Q; R- z8 I  W" n7 a( W
    factorial(5) = 5 * factorial(4)) N- H0 H& i. I- _: f( L" E  M; e
    factorial(4) = 4 * factorial(3); n6 a% J+ |5 O' ~
    factorial(3) = 3 * factorial(2)
    3 C; Z3 F$ I! C* ]9 a, @  r* qfactorial(2) = 2 * factorial(1)
    9 H& p/ \# ~. Ufactorial(1) = 1 * factorial(0)
    8 L5 r4 ]0 B" t. z' Z% }factorial(0) = 1  # Base case2 q$ K3 ]! i& N& r( j

      W7 \# F6 s. E" Q7 Z/ [8 OThen, the results are combined:
    + V) ]2 H/ }4 K; h# l  k7 d3 ^+ m: P1 h0 e5 ?7 k- D
    * T& C6 y- T; h& p) N* h7 B9 o# W
    factorial(1) = 1 * 1 = 16 n7 p3 ?' l1 c# K- Z: u  f3 E
    factorial(2) = 2 * 1 = 2
    : {" i- D7 W* d. efactorial(3) = 3 * 2 = 63 o) _+ J6 E- o% g4 c" a& s/ `
    factorial(4) = 4 * 6 = 24
    $ F1 W, R( }! J2 O, P# hfactorial(5) = 5 * 24 = 120- z1 u/ w* a4 f4 g3 L! _6 }) l

    / Y& j$ h+ f5 M; {6 p8 @Advantages of Recursion
    . k& m) ]0 p1 }. z/ S( [: L$ h8 L9 r3 u
        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).5 Q1 K; E$ b2 B% U; i
    # }8 N2 e5 J6 J
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ) w5 F' i, |) o, h2 t  }- w6 v
    " {+ u/ Y6 N0 g8 q, vDisadvantages of Recursion
    % J4 Q* C6 u; ]  [! b' s4 ?& A( p! t2 [+ g2 j2 @
        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." g% [$ Z) b  i5 A# \6 q

    4 Z6 n1 [: q4 A3 w) Z    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' F2 V) \, {8 j3 f) n8 I; D3 `# A
    / {, s) m" t$ ^$ l& B& M+ }
    When to Use Recursion6 c# x. O! z8 A( s) [0 [" K- s' t
    / C' y- R9 y5 N; m  y6 d2 l) ?
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).5 B& ^1 C$ n" I  ]5 o3 C" y4 c+ g
    1 m9 N/ y/ C- R7 }& E, B
        Problems with a clear base case and recursive case.
    2 f; Y- v5 L# L2 R4 M* W3 R
    - G$ }$ K5 W8 @% d+ uExample: Fibonacci Sequence& F# @9 p  h4 q. t  g: D1 D3 k

    9 x5 M1 W2 y+ V; RThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:2 n; Y8 y. b; u1 Y) s
    , y* W8 S2 @1 |( M! \4 p% F& ]
        Base case: fib(0) = 0, fib(1) = 14 i" H4 @6 Z9 k+ P+ U
    7 o& \' H! A7 Q4 b# e- A( M9 X
        Recursive case: fib(n) = fib(n-1) + fib(n-2)( ~  C& M6 j8 D% o% R8 g

    . ]0 }/ D# S6 @+ q+ G; Apython. b+ ?% k. [% \) z" _' q5 ~
    , q$ f" a9 P  {  Z4 j! J$ E

    : n' S9 ]  t0 M/ [( ddef fibonacci(n):* h5 H; C3 S$ _4 E, W
        # Base cases
    1 `5 K& I5 x- {& V, p0 s$ Y    if n == 0:
    * {% ~$ `' ^0 ~0 x$ A        return 0
    1 K/ q: d/ l) V3 o* v6 {) e) m    elif n == 1:
    9 {+ {5 p; e$ F5 `# w( |5 S        return 1
    2 P! X2 G# \+ B& n    # Recursive case  s' {# {5 W# T. ~9 O" K( {
        else:2 _, L; }7 {8 Z& D& W6 \1 R$ Y
            return fibonacci(n - 1) + fibonacci(n - 2)( V9 u5 g+ {# c0 \4 U! W
    + f0 M# z/ A# j6 j3 A' `! V3 |
    # Example usage
    + x9 X3 m/ o7 L* M2 `' b7 j9 zprint(fibonacci(6))  # Output: 8
    . t0 d  h3 Y$ d( X% T. P" e1 Y( O( h) g: d5 X8 i' t0 D
    Tail Recursion& l4 Q) z2 f/ e  J0 q1 n; x

    . G% b2 D( P4 ]5 QTail 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).
    ! _; b" G1 }. @0 V! `+ H% ~6 b" }) |% L
    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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