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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 0 U) q# {1 d+ q, v9 c  |

      [8 K( v/ U# K3 D( H* l; o4 l解释的不错8 m, X- {  |/ ~# r7 j
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    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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    2 I; [; f% O; S2 O9 `2 k" F8 K 关键要素
    , B  q2 d0 R; U8 r; S0 p1. **基线条件(Base Case)**
    , N$ D0 u8 p/ _" x   - 递归终止的条件,防止无限循环
    3 \1 k4 {8 \9 z* d. @5 i   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; `. B" q; f! ~4 S' d- G0 w4 ~5 o! B( P" h# U0 W
    2. **递归条件(Recursive Case)*** x% D5 ]+ d. L* j& b
       - 将原问题分解为更小的子问题( W3 ~  Y$ u' o8 y8 i
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘
    " f6 g2 v8 N& q% j# L7 xpython! f1 q6 B* l) m4 Y) W3 f1 V# `
    def factorial(n):
    / g/ j+ g5 b8 A, q" X/ d    if n == 0:        # 基线条件0 H" d; W- Y4 ~6 ~$ b4 n+ z3 N
            return 16 A; F3 u( a! Y2 m* e5 F( W- I
        else:             # 递归条件; f' A# N" Q$ f+ t
            return n * factorial(n-1)
    , j2 d! y: ~( p# E3 U6 ]7 k% R执行过程(以计算 3! 为例):" H) i* a4 a" e) X+ `& a% `' ?
    factorial(3). ~, a- x. @) W, t" Y5 `" ^
    3 * factorial(2): {3 o2 ]$ x' m
    3 * (2 * factorial(1)): K: X2 |: B4 c& Q$ @
    3 * (2 * (1 * factorial(0)))
    ! @+ s- ?# U. D5 i2 c/ b5 I# c  w3 * (2 * (1 * 1)) = 6
    3 \' s1 B* ]/ L- C: A
    " h& c2 S9 `- O( c 递归思维要点
    : t( ~: u% ~3 M0 y+ R' L& U" j1. **信任递归**:假设子问题已经解决,专注当前层逻辑/ Q! \1 l: W- B/ V" J7 i1 k1 e2 a& Q
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    % V, H! U+ R$ W" a2 U3. **递推过程**:不断向下分解问题(递)# x- t: ~$ E* {+ \& z+ U. V
    4. **回溯过程**:组合子问题结果返回(归): |1 W/ K: v% w* r' P2 }: C

    * x' p3 X, W: V- P1 u' W 注意事项
    7 J/ p# a& N  O' X必须要有终止条件% X" q, g* R  i, }; d; |0 P
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层): @- h1 A; e/ [. s9 F1 I4 {
    某些问题用递归更直观(如树遍历),但效率可能不如迭代4 c" @5 i/ U1 l; `
    尾递归优化可以提升效率(但Python不支持)7 \* q9 X* {% A5 \& s5 I3 y' ?7 D

    ; B) }5 z2 u! v( k 递归 vs 迭代8 j  P) E( {9 C0 W
    |          | 递归                          | 迭代               |! b( n( Y7 y& |$ ?
    |----------|-----------------------------|------------------|% [2 l* F. _3 ^- t$ X" H
    | 实现方式    | 函数自调用                        | 循环结构            |
    . n' ?' N" q/ X( c5 L| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    * a5 `4 L& g" 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         |: E. d" H% P5 q, ?! @* B6 n  m3 l
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |4 ~+ l# q. d# o" @, H

    9 T3 k$ B. i4 Y 经典递归应用场景
    , o( C2 ~1 ~4 v- _/ V' c' u/ r1. 文件系统遍历(目录树结构)9 G. \# K; v6 G' t
    2. 快速排序/归并排序算法' B( k+ f0 A: ]! Q9 A7 ?
    3. 汉诺塔问题  F4 H# a) A/ n) g2 @
    4. 二叉树遍历(前序/中序/后序): R7 E" I! H8 {
    5. 生成所有可能的组合(回溯算法)
    & l7 r. y' V8 z; Z$ Y4 Z& V8 _8 n9 V1 N# w6 L4 Y: ]
    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

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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,
    & S) K( S) Z3 L& y5 ~5 k我推理机的核心算法应该是二叉树遍历的变种。
    & B7 d4 s$ w' F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:
    # g- l$ Y$ i. r; l2 CKey Idea of Recursion
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    5 `/ r0 \, J; g4 _A recursive function solves a problem by:; ?" P8 U2 V, L/ e  e

    % T7 P+ Q8 @0 R    Breaking the problem into smaller instances of the same problem.
    ( U6 s+ m, ]) v, x) ^$ K, Q8 }  z7 h6 g: d4 s4 R, l' U+ [
        Solving the smallest instance directly (base case).9 d1 m) I0 E* k2 k

    1 _( d8 E+ Q; U- J- T% N    Combining the results of smaller instances to solve the larger problem.0 S' |" e4 I8 Z3 _1 V& y# r* M

    * ?, w: x: _% a- j& e  N- _Components of a Recursive Function
    . x, e5 }& V* }/ I& H! N6 h3 Z% K
    0 B- b0 o2 W/ a; J    Base Case:
    4 N% F, A1 o" r6 V. Z0 }) T, v6 ^8 A# m% L1 K; p
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.7 b# C& ]& P5 }4 `

    ( k2 W+ ]0 d0 M. S3 w6 m& D% ~        It acts as the stopping condition to prevent infinite recursion.( F7 t6 K% j# a, \7 a/ V( d( `

    / W6 z' h7 g; K+ m5 B5 A        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # U, o: m' _6 A
    1 Z* K& V, S7 V* z    Recursive Case:
    ( @/ @6 d( G3 V+ V, J/ v
    ( L2 @+ Y  M* v' e# s        This is where the function calls itself with a smaller or simpler version of the problem.
    1 D" {& L6 f4 p! Q' _# b  [' u; E( \) L
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    4 A, k5 }# y& G3 K) \# qExample: Factorial Calculation2 o/ a. P! y# h" g5 X3 m! U4 s
    # j! b: _$ C. M4 @
    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:
    / s3 \$ f& q( F) M6 E
    " G# A  N7 x8 Q& k( T    Base case: 0! = 1! _' D! b7 _/ }

    2 f+ r, U7 {- \6 c    Recursive case: n! = n * (n-1)!
    , j) _. f& Q, z
    7 l" t( \5 ?8 {+ u+ v+ x0 wHere’s how it looks in code (Python):
    ; M8 p; C4 d7 xpython$ @1 a6 M: n& ~0 r* d# K

    3 _8 g$ d3 ]( q; x% y) f$ O  N( q3 y+ {
    def factorial(n):
    9 j) I( X+ S; J+ ?+ l$ I8 V" ]    # Base case
    + g5 }9 Y) G3 F/ g0 Y6 M    if n == 0:
    0 b1 W( r+ [% @1 }# Z5 R        return 1$ @: Y9 ~" w$ Q# B1 b
        # Recursive case
    + T' v& h% O% h  v7 |    else:3 G8 u% t$ _' g4 J0 v
            return n * factorial(n - 1)% a5 u: z" [& G" ^

    : q5 _+ x; j6 w& X8 P9 J: s# Example usage8 l+ m- c* x3 l8 j) M
    print(factorial(5))  # Output: 120* e7 P5 k: n- q9 q8 w: Q6 Z" `
    3 ?3 c: l+ O# ]/ l
    How Recursion Works* l4 J5 H8 L3 c" ^% z

    - Z3 r7 P' ^- t& ~& e5 Q$ P2 m    The function keeps calling itself with smaller inputs until it reaches the base case.
      Y1 m6 A0 i" b6 L6 A1 s8 l6 P
    0 v. p; G6 k0 E* T8 P    Once the base case is reached, the function starts returning values back up the call stack.
    4 M" M6 w- t0 h- z9 d: m+ R
    ! j5 h6 z( U: K% Z* e# p0 K/ I    These returned values are combined to produce the final result.
    1 U: V/ w& F3 l: ~8 f" `
    * _4 F- D, f# E) B1 @For factorial(5):
    , D* a" h. O% a! J/ p2 Y1 `% Z# z0 o) D" \6 a
    8 ]8 x0 d1 r  n: O" W/ g. @6 @0 g- k
    factorial(5) = 5 * factorial(4)
    " a: @- O  k  p- [' \/ Ufactorial(4) = 4 * factorial(3)
    3 R$ c/ U/ W, {: y2 Wfactorial(3) = 3 * factorial(2)
    $ `8 x7 ^0 Y& s! ^4 Kfactorial(2) = 2 * factorial(1)0 W6 H4 C. @4 u9 H
    factorial(1) = 1 * factorial(0)
    4 Z0 G8 ?0 B) I  O7 M( |factorial(0) = 1  # Base case
    3 v2 l* g- `. A; S& q+ b, e3 u( ~$ a5 @: `
    Then, the results are combined:1 }$ M8 Z* V' b9 o
    # F7 Y& h" x' s) g* f! a( t( G
    * Q7 `- h' v3 D+ k% p( K
    factorial(1) = 1 * 1 = 1. ^, @0 H1 V1 [5 ~. l+ E
    factorial(2) = 2 * 1 = 27 h: q7 b- X; }4 W
    factorial(3) = 3 * 2 = 69 h. d/ K2 T6 @. _' q' E
    factorial(4) = 4 * 6 = 24
    . Y2 J: W4 A3 e4 efactorial(5) = 5 * 24 = 120
    ! d  ^9 i; m- t* d: R
    / Q* b) X8 i+ |( Y4 Y  a" L" xAdvantages of Recursion
    . ?- a9 S5 f" l/ |
    , [. q9 f5 V0 H. M+ r' C    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, K1 w1 F/ _1 p; m

    0 X8 r+ t" S) X! M$ Z1 C    Readability: Recursive code can be more readable and concise compared to iterative solutions.  G! }9 |! ]8 T; T0 L4 Q5 V% M

    * w9 x  g4 A0 S! V+ dDisadvantages of Recursion9 h5 M2 Z8 X5 G1 W

    / U- S+ x5 w) ~- F( r1 w    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; i, P$ h5 |/ p  N
    6 a! W( e6 e9 b$ H# \8 |    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).) M; k' p+ ?4 ~6 }; ~

    + k" e% M  L; C/ y; eWhen to Use Recursion
    : u: s  V6 }8 g2 O* ]
    5 w. w4 _# G. L# `: R# C    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    3 h6 s" {, }7 m8 Z7 s% c: |3 F
    3 n2 W2 a; S  E  v; X; V    Problems with a clear base case and recursive case.
    ! R# y+ _6 v+ Q. D+ o5 I+ u8 _2 o) T4 _
    Example: Fibonacci Sequence
    1 p+ F. c- d5 q- y. `2 o% |% a: y- E7 F* b4 r* o
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 W- B  p1 I  R9 |
    $ I+ z3 d6 b" o, A; i; ?5 E
        Base case: fib(0) = 0, fib(1) = 1
    4 l$ V1 i6 P; _& C$ u& e6 v7 s. x; a# O9 e+ |9 {3 S. G
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    % t' h1 Y$ n4 e8 n" s* o, \
    : T. Y' _4 r% Gpython
    4 F: R" s: {( \
    & X  f& ^7 d" Y5 T6 P% N6 V7 }, H  ]% |7 z7 Z
    8 a/ O0 ?4 c4 \( S; hdef fibonacci(n):, F2 {. @* x& |- e6 t- O0 f
        # Base cases  j. z, Q% ?7 `
        if n == 0:
    / H# |2 o, N) {1 V        return 0
    5 w! R, G! W  O1 T5 e- O    elif n == 1:
    7 T6 s6 b2 h6 [' ^9 S        return 1
    7 B/ w8 j3 v9 ]: @2 ?) j( n    # Recursive case, v+ E7 z; K  d* c8 E; c3 r2 I
        else:
    % R' h. L% U* [! Z/ I* M1 z        return fibonacci(n - 1) + fibonacci(n - 2)& X4 e. m) p+ D) T& v; w

    ; L5 }% c1 B8 g+ M# Example usage+ E  T+ q) Q$ X9 q
    print(fibonacci(6))  # Output: 80 S! e1 N$ A1 @) V+ T* ?
    0 ~. C! H/ N$ |: J
    Tail Recursion
    ; q$ {* t) Z) d' ^  U3 D
    / q# [$ B# U8 p8 b1 V' CTail 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).
    " G# Q) d2 X" [' a5 z4 H3 u: }2 ?& Q
    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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