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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
     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
    9 V$ H( R5 I5 T5 V* W/ ]) v
    % d* n3 U! ?$ r. `解释的不错
    . P8 W6 K( l; O; g* j( ~2 j0 E4 {/ s: p/ _9 c: B
    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。1 T+ ?. [7 ?: g1 z

    ; P% `1 g' ]3 k 关键要素% m; G2 x  k0 W: {; I
    1. **基线条件(Base Case)**
    . {$ k6 e* z0 T' s   - 递归终止的条件,防止无限循环7 R! `6 X  n$ |; H
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% d# A* D- i' |
    ) B) k6 U" J$ @4 m) X* X
    2. **递归条件(Recursive Case)**! Z* b! X. Q4 c8 O% l' Y
       - 将原问题分解为更小的子问题
    * u) V: u. {9 P8 F& E" O   - 例如:n! = n × (n-1)!
    % |1 |) H9 ?' A  H3 K; z
    0 U1 Q7 u2 ^; R( o 经典示例:计算阶乘
    % Z5 f5 s8 i! A3 O. ipython8 A, i7 ~' _2 R( M
    def factorial(n):2 @+ U: @: d4 S* t9 u
        if n == 0:        # 基线条件
    . k& v9 X8 \4 A+ r# x        return 1) F7 f, r. N! i4 I6 W5 ?& l
        else:             # 递归条件
    0 d) R+ g) n3 w! Q9 s7 e9 Q        return n * factorial(n-1)
    & T' p8 ]- l) @- D) U& Q执行过程(以计算 3! 为例):
    - a4 w3 M  G& g# a7 Y6 Cfactorial(3)
    & Q- d8 V% P6 Z2 e3 * factorial(2)5 |6 r- Y6 ~6 L+ ~1 y  D
    3 * (2 * factorial(1))* Q! F4 u  W6 e. ~  U6 A9 S: l$ @
    3 * (2 * (1 * factorial(0)))
    ! D! J6 U2 R- X! z0 [( k8 T3 k  {3 * (2 * (1 * 1)) = 6% X) }' W: a* x4 [
    ) d6 {4 \% p/ U2 i
    递归思维要点
    3 o  ]' B) I" x1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 c7 r/ ?& g* i/ X" A2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    9 p+ @# W2 g+ {) r) c9 T3. **递推过程**:不断向下分解问题(递): G( a3 t& q  R/ }  }2 p7 y
    4. **回溯过程**:组合子问题结果返回(归)
    8 m) q4 M. y7 _" f" A; ]. i* h# `6 ~9 J+ X8 B
    注意事项6 k! \" F7 ^# w. U( V
    必须要有终止条件4 B. V( Y. F, K& I; L
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    3 e- }2 B! @5 {, |某些问题用递归更直观(如树遍历),但效率可能不如迭代
    2 _* j3 F# V6 W0 t: G尾递归优化可以提升效率(但Python不支持)4 G9 Q* S! S/ p0 U# N1 \1 u
    6 Y- ~( G" l# S( a4 }
    递归 vs 迭代+ J- P9 Z0 K0 e* U2 Y
    |          | 递归                          | 迭代               |
    ; I  ^- Z/ U( |( E+ x4 B" ~|----------|-----------------------------|------------------|
    3 K( e' s1 A4 c6 Q% L/ Z1 Q5 y% Q| 实现方式    | 函数自调用                        | 循环结构            |9 s1 K- h4 c* N, e* J$ E, E3 ^
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    . a( p' W/ @- c9 ~- G" ~; @3 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         |% E9 j9 ?" Z' f  a
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    9 H/ ?9 P3 d* V7 c; B) L# X
    : d9 j7 D4 A1 r% f8 j" u9 `! T% P( J 经典递归应用场景
    % d+ z) _. n! ]0 i1. 文件系统遍历(目录树结构)
      ]) t0 n5 l0 u/ ^5 h2. 快速排序/归并排序算法
    4 P* [4 [) t& P; A" j3. 汉诺塔问题- a' p! v0 B: V8 y  k
    4. 二叉树遍历(前序/中序/后序)  H1 Q+ i1 [1 n1 g8 m5 w
    5. 生成所有可能的组合(回溯算法)- {% c8 c0 ^2 l( n9 ]* F/ c
    8 D, v/ M6 @5 w$ A
    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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    昨天 07:29
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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,
    8 G9 w: B9 D* t. o* t- R3 A/ g我推理机的核心算法应该是二叉树遍历的变种。- [5 c) A& ~+ K" 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:* D8 m" B& K! A0 n$ J
    Key Idea of Recursion
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    A recursive function solves a problem by:
    : G# p# \/ E7 `
    8 F8 |! G6 E& t% g( a; |    Breaking the problem into smaller instances of the same problem.( r  L- p+ y0 I$ L# N! a. F$ M. K
    , z% F( E# w5 U5 ]0 r
        Solving the smallest instance directly (base case).( o6 B0 \& z% U3 U8 i* a* g% H
    & D8 F0 T6 N9 h' r- x& X; w6 ?. B; @
        Combining the results of smaller instances to solve the larger problem./ \) t1 o. T' R

    ( P( K5 d+ L9 \+ \Components of a Recursive Function/ Z9 c! E: w( U$ t! q+ @5 q

    * y* o2 c5 m& C6 b$ }( e& z4 q    Base Case:
    6 j  P  X7 J6 `5 l! q* r( i5 a
    & P$ A& v6 m6 @1 _" s        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    7 y. O! s9 X" K% E5 \( p/ V8 C; t- s/ b
            It acts as the stopping condition to prevent infinite recursion.; [" P* G# A$ a- h# x0 {

    # Y. F' s$ n4 _. P$ \, t        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    0 h+ T3 r5 K8 G- X$ `1 B) P/ k# h( |! Z7 r
        Recursive Case:) v; L" F" c- I1 |

    6 @) c8 a, J# \' L4 X  L        This is where the function calls itself with a smaller or simpler version of the problem.
    2 W- |  c. @$ A* C, W6 D& U0 u; P8 w
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." n0 y% z4 p4 E5 T& W

    " f- |" e5 R3 K  z- KExample: Factorial Calculation
    # j) Q5 y3 z4 L3 j7 d( t7 m6 x$ r1 E9 x8 N
    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:; g7 }0 ^3 @# a5 ~

    % A7 e# x5 w  Q( B7 O3 ~; V, U    Base case: 0! = 1
    6 y0 ~% ?3 X& C# b  X' u
    5 a& J: S* k9 H% L: l9 i" V    Recursive case: n! = n * (n-1)!7 C9 m7 o% Q( `4 y3 j" T4 X

    6 F- ?  r5 _8 `Here’s how it looks in code (Python):& K6 x9 v+ ?& X6 K4 U
    python
    : w6 x; q2 Z! m8 A6 `
    ; [: H6 w8 @) h* X9 E) v# L' @' o) p4 _% ^; [
    def factorial(n):
    ' t/ }8 t- F; d! b    # Base case
    : B+ x* I3 ?! n. \# f    if n == 0:5 d7 M( |/ U2 Y( B7 H, `6 w- u! I  S
            return 1  M& F/ d& G+ n7 n" r% H7 l2 j) e1 q
        # Recursive case
    0 h' ]1 T( g! x7 D2 Y    else:$ j4 N3 P7 F8 X# E* y
            return n * factorial(n - 1)) J$ Q1 h, r5 ^

    * L& q( a, I+ x+ K' W4 U. B# Example usage' d0 D) C  i) F3 ^6 _
    print(factorial(5))  # Output: 120$ w$ M  v; r6 v; V) a

    4 @  Y; c$ \# X, o5 Q  vHow Recursion Works
    0 F3 \5 I' W$ }* v
    + t. q6 I0 f; z# J) r  g! S    The function keeps calling itself with smaller inputs until it reaches the base case.
    1 B% b' y1 O. J. B$ V; n% B% r
    5 ?9 A4 ~% S3 Q! P: z! _    Once the base case is reached, the function starts returning values back up the call stack.
    # @1 `2 A7 t5 r5 {/ Z
    : Z" k; J  J% w3 m3 z& v! w    These returned values are combined to produce the final result.% i( R* Y7 V; h. M- C, Z4 V" p* h/ O6 k

    9 m( x' m8 Y9 H$ j: d7 D6 zFor factorial(5):' E) T# Y1 X( ?. A
    2 h% Z4 G8 ]: a/ \! I5 e( v

    7 Z7 r9 x( A7 f. G. xfactorial(5) = 5 * factorial(4)( Q5 a; @: ]5 d
    factorial(4) = 4 * factorial(3)& H1 C& L6 ~  O" u6 L* y
    factorial(3) = 3 * factorial(2)
    7 N: |! R; a# ~5 _4 x' O' r; z: i1 tfactorial(2) = 2 * factorial(1)
    7 m6 N/ `9 \& X2 k$ Ifactorial(1) = 1 * factorial(0)6 P3 D9 b& o4 @# d$ u
    factorial(0) = 1  # Base case
    6 o) r" {' ^, [0 L( v8 j7 }6 l
    % C9 V- c( i8 Q, `8 ^Then, the results are combined:4 I4 R. K. I4 B5 R
    % ?' Z: |$ d9 E& x/ ]* _- A
    * }" k) N# S% ~, H/ @
    factorial(1) = 1 * 1 = 1: i' a8 a# ?( Y$ G, X. c( K5 `
    factorial(2) = 2 * 1 = 2% N3 n6 s, p. r7 s
    factorial(3) = 3 * 2 = 6
    2 _3 `: A& n6 T- h- H4 Tfactorial(4) = 4 * 6 = 24$ Z* j- a" r0 N; x% F
    factorial(5) = 5 * 24 = 120
    " e# U3 w" T& F
    8 @, W7 s9 i/ F+ n+ G1 }Advantages of Recursion! l; V) [) `* ~! z

    ' C# l. [, b0 @0 d( @0 V    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 q+ x. z: J! J1 y4 A- y5 R0 H# U; K# D
        Readability: Recursive code can be more readable and concise compared to iterative solutions.1 i( ^- ?, o) L; T, o$ j7 `

    1 {' {( ~4 `. v; gDisadvantages of Recursion
    + a+ T& l# Z2 Q: b. u4 \: D
    ; f5 q/ u! Y1 f    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.
    : m, J; V* D7 B& s. \+ }# x5 |- V
    : j) [. @4 Z0 E& Y* D    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 B( z: T  d$ C3 T7 F
    $ w4 g6 {% T+ X  p
    When to Use Recursion
    6 D: n$ |. J3 ^: l
    1 i+ U. [4 j. N4 `! ]0 a    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    % J: w8 X# u+ e" z. {$ C! N3 d+ g
      A9 I2 K0 N% Y3 a' \    Problems with a clear base case and recursive case.
    ( R/ u+ Z3 u# I; @4 q" M
    # `. ?! X$ y8 X! T5 |/ BExample: Fibonacci Sequence8 g; K+ p4 z, J5 j. m
    ( L3 [9 A* E" ~0 h) C2 a
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    8 l- L8 k! z3 Z, @7 S
    : f# Z2 M6 ^& g/ h! v- `% K    Base case: fib(0) = 0, fib(1) = 1
    1 t- Y$ N' D) F4 c: Y! p  a- s8 ~7 C+ M& X- ]6 e
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    , |# r' ^7 e$ j& ^; j2 I. C
    # n, h6 o8 @$ l, Hpython* I; b7 t0 @7 r! B% y0 S; g
    % S0 \# U: j( w
    9 P7 V6 B6 q$ b% ]
    def fibonacci(n):, O4 w( ]4 O6 {6 L7 a6 N
        # Base cases
    5 F2 Z" |$ d' J7 E+ b    if n == 0:
    ) U/ J7 l: I* d: }) C* O        return 0  T. c$ w' @8 k4 F+ Q+ J4 G- V- ^
        elif n == 1:! v3 Y) S9 _. b
            return 1
    ; C. ~( d3 W; A. Z    # Recursive case, K# y! }/ O5 d9 v7 Z( n$ B
        else:' \0 ]6 w2 b' J. l* H# }0 ]+ u
            return fibonacci(n - 1) + fibonacci(n - 2)
    * w( w% c$ z$ @1 I& j
    4 x' ?3 `5 L+ y# E1 c# Example usage
    , R( Q! V, L1 t- Q6 rprint(fibonacci(6))  # Output: 8
    ! f6 E9 Y0 `5 }4 k2 M% G2 ^# ^* R/ e' R& x/ L! x4 h
    Tail Recursion! g4 K$ o0 k% o4 Z7 o' d
    ; t) z, ?' \5 P2 A! ]# M, d9 x# ]9 {
    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).' [0 I; u; z% n8 n1 V; P; \; K

    9 Q$ a+ p" z) E) }; _, GIn 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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