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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 5 A# C* E- G7 @4 D7 t

    4 R2 o6 M0 [, L. h解释的不错
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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。
    : t- U9 o% S4 X: ]$ S  }' I: [$ @% _# n+ H) u" s
    关键要素
    4 a3 P+ M4 p- e9 o9 V1. **基线条件(Base Case)*** f$ e9 b, x$ a5 _  O4 \% J
       - 递归终止的条件,防止无限循环5 K% `( i7 R' d* m0 C
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ' H7 W3 U) K' B# O- B
    ' `* K, _; g! Q+ |2 u2. **递归条件(Recursive Case)**
    9 p. I7 t" T% _" @   - 将原问题分解为更小的子问题0 L) i! R8 L+ a* x1 X
       - 例如:n! = n × (n-1)!7 U5 ?9 H0 f, p) U! t! R# E
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    经典示例:计算阶乘
    ; j! h4 s! n" y( z; [python# ^  e" ]' T4 z) f
    def factorial(n):: W5 b9 P" H0 O+ h& X/ f6 q) _
        if n == 0:        # 基线条件; z7 X+ L# c! R. M6 `( q
            return 1& y5 x( S5 ]1 K( l+ v" W4 g
        else:             # 递归条件& K7 t5 P: Z! e" L
            return n * factorial(n-1)
    4 e& D4 I& @9 g执行过程(以计算 3! 为例):* P# [3 e  n- u% V: T
    factorial(3)$ Z! F. M& O( Q. g9 ~3 I
    3 * factorial(2)9 s/ q. z6 j+ c9 \
    3 * (2 * factorial(1))
    4 r  G- s1 W* P  |8 j  a3 * (2 * (1 * factorial(0)))( t- L- `' Z4 a" M& Z
    3 * (2 * (1 * 1)) = 6" Z" P) [% r# k9 X% W
    ) K5 D. a2 |$ H
    递归思维要点
    / g; c* @4 P$ Q* _6 q1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , |7 t" P' E7 ]4 B( s% c2. **栈结构**:每次调用都会创建新的栈帧(内存空间)" n" S' q" E! p. U) d
    3. **递推过程**:不断向下分解问题(递)1 j6 g$ @! B6 ~2 \' y) p
    4. **回溯过程**:组合子问题结果返回(归)
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    8 k3 n9 H5 n0 j; }3 t# |, w 注意事项" D  F* X8 k) d9 M6 ?. i9 B
    必须要有终止条件* d) ]5 R. P4 `1 M, x) i
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    2 A, `7 ~, g, B4 P某些问题用递归更直观(如树遍历),但效率可能不如迭代
    : Q) G; l1 P; D% d: V. w尾递归优化可以提升效率(但Python不支持)0 [& L" g" i) D$ V" M
    , e  m0 @5 y# ]0 X
    递归 vs 迭代! i& `8 V- S) g' l- j6 A
    |          | 递归                          | 迭代               |" A9 W! I! ~' o1 k. N. O) x
    |----------|-----------------------------|------------------|3 t0 \6 }6 u; o8 ]
    | 实现方式    | 函数自调用                        | 循环结构            |
    - \% ^& b- G2 o% {) A9 t2 \6 G* s, k| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    % [: \& n" u& |/ O3 Q| 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 _, b; B+ C# ?! y| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    / M' c: U& ?" m4 I) v8 l" K& h/ D& S/ n* O
    经典递归应用场景
    * m+ y) k3 R4 Q3 Q9 |1. 文件系统遍历(目录树结构)9 A6 z- o9 V+ Q; o: N+ U( l
    2. 快速排序/归并排序算法1 m7 e: i0 p+ [1 H6 ~- S6 }6 }* o
    3. 汉诺塔问题) [2 m2 R! X: Z: t8 g" G
    4. 二叉树遍历(前序/中序/后序)
    ! I% @2 o; ]; Z( f; O6 h5. 生成所有可能的组合(回溯算法)
    0 q5 s9 q0 f: ]* H5 I4 q5 a/ c- q1 c$ b+ g8 |
    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,& O' ^) q4 ]* o
    我推理机的核心算法应该是二叉树遍历的变种。: b' o2 k9 v8 O( u6 b* ~
    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:" u' |% _1 z) p. C: J; {- Y
    Key Idea of Recursion
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    / Y2 M, c; D) S, _/ W1 Y# rA recursive function solves a problem by:
    9 L7 i! `+ w+ S) G2 \( ]/ R) b$ v2 \- P0 j, M8 h. h
        Breaking the problem into smaller instances of the same problem.
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    6 v  B. [/ o+ j  F4 c# A    Solving the smallest instance directly (base case).
    0 c& |( f$ e0 t8 k! l/ u) W9 u* r/ M5 d
        Combining the results of smaller instances to solve the larger problem., H6 h: u8 L3 B5 e
    2 }1 Y% J, c  R  g6 q
    Components of a Recursive Function$ l5 l5 D, b: b8 N4 j
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        Base Case:
    + H* \# u9 M, o  o( u+ ^5 _0 c- h7 ?  d  k
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    . h" g- k1 ^# {, g) J# f9 q! ]6 [5 w! h9 b4 B$ \& e! r
            It acts as the stopping condition to prevent infinite recursion.
    * Z8 e: G6 S- X- G8 F) s8 G! D  v  J. p- |" t" F5 ?5 U
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; ]$ I5 b( W; a2 r6 A' F$ p9 B9 L

    * b6 `' y) U4 y& z) M& L    Recursive Case:
    + T, E3 Q, G! E4 X9 l& J
    1 o1 }% x; g1 E        This is where the function calls itself with a smaller or simpler version of the problem.* W1 _" g9 f" S
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    / g% E: ^6 D) m1 S6 KExample: Factorial Calculation
    - G7 Z, Q4 `0 R- E: y# |! M4 F3 c6 T. f6 R; g6 }. S
    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:
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    % q$ @  e$ Q% Y0 |) W! V. c    Base case: 0! = 1
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    : r! r( }$ f) d0 u) F6 x  X5 \    Recursive case: n! = n * (n-1)!" C) W  T5 N0 X
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    Here’s how it looks in code (Python):
    ' z* |: \' P! X3 ]python
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    & [5 t8 t) o+ H
    " v$ g: v5 v" U/ ddef factorial(n):0 O, X" A, W2 f6 t/ F9 M! n
        # Base case
    # T+ X: ?% S* B4 J1 V, q    if n == 0:2 U/ ~% Q! V- H7 X7 O
            return 1$ P) F9 D# x7 @7 r' k! i
        # Recursive case9 ^8 n* A; _' n3 r. @# u( n
        else:7 I3 `0 _2 \8 A& P$ v$ k
            return n * factorial(n - 1)% v/ Q9 w( g' \+ }  O
    3 V  d# K: q& D) h5 H
    # Example usage" t. ]7 N- `- {" _( k- M& O1 B) N& h
    print(factorial(5))  # Output: 120
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    How Recursion Works1 `3 j% m3 A/ Z/ }, f. T
    ) J+ W/ `3 O1 E. [7 m% T
        The function keeps calling itself with smaller inputs until it reaches the base case.0 X  o! [5 [" B$ D( M  N
    , I0 }* J- V. [) I1 f' W
        Once the base case is reached, the function starts returning values back up the call stack.
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    : y8 P. \! m4 b6 t0 c# |2 l6 e    These returned values are combined to produce the final result.- k" h5 O7 Z& @9 r/ t

    1 H5 S, B2 z; y% PFor factorial(5):
    8 n, ~( x% E$ C& s( C2 ]  o, V' b4 }3 |$ j/ R
    * I# [2 f' E. G6 X- ]" Y
    factorial(5) = 5 * factorial(4)
    / p0 M) J9 T% X' Z4 kfactorial(4) = 4 * factorial(3)7 \5 H1 h8 T* m# p- @* @" M
    factorial(3) = 3 * factorial(2)% k$ Z* ^+ a" D
    factorial(2) = 2 * factorial(1)! R% c4 Z- x0 W3 T, H' l
    factorial(1) = 1 * factorial(0); F2 t$ r- n$ f$ c
    factorial(0) = 1  # Base case$ f4 [2 i& c' a0 z2 x+ M

    5 _# a# y9 A# AThen, the results are combined:: y& n5 n$ y1 d- O* ^5 k" l
    * [# X) R* D3 a8 X

    & r& k) L4 n* M3 tfactorial(1) = 1 * 1 = 1" w6 z% {  j7 m$ F% z, Z' `% w
    factorial(2) = 2 * 1 = 2( t/ o8 n* a/ p
    factorial(3) = 3 * 2 = 6: G" X/ a5 v$ m; Y+ T- W2 G6 ?
    factorial(4) = 4 * 6 = 24! T+ }/ x2 n, y! S1 f9 m
    factorial(5) = 5 * 24 = 120$ u* c1 U0 f. K- m9 f( M
    1 m5 _% H0 G! |1 J& Y  L0 ]
    Advantages of Recursion- O! R' o' Q5 \0 x7 o# Z
    # l) i2 N6 _( G4 }& c4 p5 _
        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).2 K$ L) V" e4 t- C  I

    2 k/ h% [' M( |    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    ' i2 X$ T+ `0 k! x6 {8 r- jDisadvantages of Recursion/ i. \/ v5 a/ U
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        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.
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        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)./ N; I  f) Y+ q0 ^; d# {, N
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    When to Use Recursion; m& M, E1 L; l5 H

    , H& E  }! L. W7 }    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).3 A: Y; s" J, B5 d5 \: i# D
    $ b& C4 o8 [1 A: r( b$ }- [
        Problems with a clear base case and recursive case.- z1 a4 ]( ]% b8 y1 O' `2 x1 }( o

    * p7 R. z" n+ G, Z7 f. v/ ]) J4 {Example: Fibonacci Sequence
    + B' |9 b2 z: T! T- X
    5 D, V  y# f9 p- H( u% jThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:: K" E7 Z& z7 D. d5 W/ W) M0 X
    7 d* ?- [  h3 N- K
        Base case: fib(0) = 0, fib(1) = 1
    ' z+ h' z& }. J# K9 T. f1 i7 H4 {* l) K. ^2 z9 p9 @3 g/ w2 n! |& Z3 t  u
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    9 Y% G! n" [. v& l  Z0 Mpython. I9 w4 a* @2 C$ ]1 Y( S' v0 k
    ! G( ?6 t4 a# u2 N: |

    . @" G3 m2 }' `  M8 a/ v6 b. fdef fibonacci(n):
    + N+ |8 v2 v, b' N2 G    # Base cases
    " u4 V3 K7 |7 y7 {5 v2 R    if n == 0:6 ?% o9 E* N& b* Z" A
            return 0$ h6 e" L7 q, O2 S
        elif n == 1:
    ! S6 R9 q# X2 ]  F/ U( x        return 1
      e; {2 k7 Y- {3 q' W( V1 V0 z    # Recursive case: x' ^+ m+ u& L- {
        else:
    + p( H0 M) d% a, `& Q        return fibonacci(n - 1) + fibonacci(n - 2)' _3 A0 W- K! [9 v- j' Y2 X- E
    ! S+ }2 t' ^7 A" N! }% f0 x* U
    # Example usage
    " g0 h) u- Q9 Nprint(fibonacci(6))  # Output: 8+ U* m3 T% y0 f/ S0 T1 o
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    Tail Recursion! t. Z+ d( _" s% A
    3 S- c! R; L5 E
    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).- Q7 j. y+ R6 b: E3 I+ @
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    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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