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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 ( @3 D% E- i5 M
    ( V4 P5 H! I( `4 I" q
    解释的不错/ |* D, j5 u% ~: O
    5 D: j& }  M5 @. [/ I
    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。9 ?* @: S- }% p0 L: y7 L
    & v: x6 b4 \) y& \3 X
    关键要素
    - @& s# I' q5 d$ k: D* e1. **基线条件(Base Case)**
    - I, b$ m1 X. J9 C: ]   - 递归终止的条件,防止无限循环
    # L6 t, H6 f3 z. B, e& w3 ~7 i   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    3 z) x/ T4 N0 w$ T. y# k) X2 k: |; d4 K/ C7 D$ ~
    2. **递归条件(Recursive Case)**3 @4 D; [7 f9 A7 F* }- M# o
       - 将原问题分解为更小的子问题
    & Y& r1 B4 y4 r- N; L1 n   - 例如:n! = n × (n-1)!
    & s* t( I- ^4 D6 [' @- W
    ' j* E2 R6 l/ ~4 d6 R% D 经典示例:计算阶乘1 ^& p! E# ^2 i
    python  c  m; K2 ^( j( \
    def factorial(n):/ `8 X1 V* T2 I0 |; ^2 m
        if n == 0:        # 基线条件
    ( a3 T1 F& f3 b- l8 w        return 1
    ) u7 S4 q" j- h& V    else:             # 递归条件
    8 K, r' w( o+ b3 c  c# Q2 |% Z        return n * factorial(n-1)- G# Y* f# q7 N/ A6 w
    执行过程(以计算 3! 为例):
    # f" ]2 m/ j: dfactorial(3)1 I- l1 t% e% Z9 ?
    3 * factorial(2)
    8 e) m& P$ s7 Z' e% k% \0 U) W! s3 * (2 * factorial(1)): F/ Y4 ~* ~2 v. `
    3 * (2 * (1 * factorial(0))); Q0 D7 Q) J, ?% ^+ E# X" W6 `/ S" b
    3 * (2 * (1 * 1)) = 6
    7 k( D0 a9 s; m8 B
    - D; h, D7 P% X3 W8 x 递归思维要点
    " P3 Q2 O, G0 ^1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 u" j% V) y: H) b3 _2. **栈结构**:每次调用都会创建新的栈帧(内存空间)$ K/ x. @+ e2 A6 q; N4 Q8 `, B# N6 o
    3. **递推过程**:不断向下分解问题(递)
    % T* R+ I1 f9 S4. **回溯过程**:组合子问题结果返回(归)
    3 D7 H% m! L  g2 V* ^8 Y
    ' Y, A2 e) I0 W7 B, Q 注意事项# j9 y' S' t7 c, v7 u6 f" k4 ^6 f: R
    必须要有终止条件
    4 Q7 U9 ?3 A5 z% _* J2 @递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    1 ~( h! P" C; S$ V0 B某些问题用递归更直观(如树遍历),但效率可能不如迭代2 m0 `- ~* ]. t2 I. ?$ l
    尾递归优化可以提升效率(但Python不支持)
    ) p: ]9 c6 E3 {; l9 u8 J0 v8 Y$ R; P( |& D' o% Z
    递归 vs 迭代' E' E4 n9 h  X1 e
    |          | 递归                          | 迭代               |
    6 U5 E  s: R6 k+ X' p/ Y|----------|-----------------------------|------------------|& D! f8 _$ X& [+ w
    | 实现方式    | 函数自调用                        | 循环结构            |$ l1 A. S5 b. J8 i- v0 F# p7 ]7 [8 P
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    : J* b5 E$ ~9 ^: V3 `9 O| 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         |
    5 H# K. d6 y& h3 @$ Z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    $ i4 D& |7 \  c7 N; C
    # d: J. e1 A( d2 H+ ] 经典递归应用场景8 F& S4 M1 F: j, W
    1. 文件系统遍历(目录树结构). @, s2 W9 V( l: Y: i9 y
    2. 快速排序/归并排序算法, C! v% s# s* v3 a/ H3 z
    3. 汉诺塔问题/ I! v4 T% I  k1 W+ ^$ G" U
    4. 二叉树遍历(前序/中序/后序)
    & {( y6 I' i5 u. Y9 G5. 生成所有可能的组合(回溯算法)
    + ]# R7 W$ V1 i: @! n
    3 U6 ~3 a$ ^1 e* C- Y* D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,
    ' R  z. V7 J% o$ ~我推理机的核心算法应该是二叉树遍历的变种。# \, N$ H* H2 E1 M3 q4 c" c
    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:
    # [3 Z+ k/ ~; p& {Key Idea of Recursion7 h: v" b0 j/ g9 c+ k3 W

    ) q6 n! @) a7 E% V. dA recursive function solves a problem by:! h% L9 F% g5 l8 N
    : K! @/ i% K2 B0 q, ]( X: b
        Breaking the problem into smaller instances of the same problem.
    & m0 P$ N; s7 g" p) ]; B5 b6 F
    , r% W- q. e0 |+ x' _    Solving the smallest instance directly (base case).
    0 l- l3 {9 _9 M9 b; n1 r7 |+ F/ f4 B$ ?+ ?" j  s
        Combining the results of smaller instances to solve the larger problem.
    ( x7 O1 C7 N7 _& B3 z% o8 d+ v$ j7 `
    Components of a Recursive Function
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        Base Case:0 p% {$ E0 Q, z  T: M# f& x

    / L5 N  m: _: T' f0 J3 f; a: g        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ( Z6 H$ y# ?# T7 c( _% ~8 U  l
    . Y! u  S7 g% X) ]. F        It acts as the stopping condition to prevent infinite recursion.
    6 A3 u4 m  F" T' J9 s/ Q( n+ ?* [5 B. ~' b+ S
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( I: L$ j. o$ V& K$ D

    8 d3 j9 N6 l; |' l# d% [* q    Recursive Case:
    3 j5 w7 Q! T6 S9 c1 y! g1 P5 M9 x3 j2 H; m2 p4 I# g$ u- ?
            This is where the function calls itself with a smaller or simpler version of the problem.
    2 [" V4 j/ P8 C+ a4 E# N' m) c4 S2 H' g7 N3 u# ~, @  V9 ?
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." M0 I6 y$ i7 J) G! U/ e# Y* H
    $ j4 ~8 O) J- o8 a
    Example: Factorial Calculation
    1 B/ m  \( C% }  \
    ) P! z6 ~3 l0 O3 j1 i9 B" yThe 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:
    1 I$ ]2 g) b4 U; q! k# R" E: l: X7 T' e/ q0 X# `
        Base case: 0! = 1
    ' A0 F" F8 u) s8 b
    , p' i$ |! o8 h- z# @    Recursive case: n! = n * (n-1)!% Q  n$ t- B8 O! r; k. Z* d! j7 T
    & p+ I# ?. _/ z: G5 f% H" t, ~
    Here’s how it looks in code (Python):5 g6 o; L% m' I" C
    python
    # x5 C4 t0 x; P" d% q: a# z' y2 I5 U$ i$ ~6 q
    . L, ?$ r6 X! ]0 A" T( o
    def factorial(n):
    & N- M! F& n, O5 _6 D$ m2 w4 v3 {    # Base case
    * ?1 |0 x0 v7 l7 Z; ^, y    if n == 0:: [& L0 t+ Q1 P2 B( p
            return 1
    $ Q, w/ D' O! F2 V# p( [    # Recursive case
    9 x7 B2 w6 Y1 t3 M    else:, C# V. j" ^4 Z+ z: W; ]/ o& Q
            return n * factorial(n - 1)
    4 C, t8 ^) _0 r7 a# v7 I0 h- a* Q% F& A8 D! j! v0 B
    # Example usage2 y0 x9 ], m1 L' E
    print(factorial(5))  # Output: 120
    / N% h, v0 E+ B+ u
    ; p# F+ ]( D' m5 Y' ?3 z/ BHow Recursion Works3 F' z2 c# @9 @9 [8 g: b
    $ h' E' K( {3 p! v/ M+ w
        The function keeps calling itself with smaller inputs until it reaches the base case./ |3 J3 Q; B, r
    / m0 L7 d3 m. d5 X8 H
        Once the base case is reached, the function starts returning values back up the call stack.$ s. n( c1 M* F. E. ]' f$ M0 i
    . G+ U9 b9 r2 F; K: |$ j2 G
        These returned values are combined to produce the final result.# R* I9 C0 q9 L* G, T% V+ q

    0 F$ Z1 r3 j6 u( H% p4 ]For factorial(5):4 K7 U' j1 v8 q5 t$ S' S
    ; O4 M1 Z, i& n' Q: }  G# R$ T
    % \. J% w6 |4 {8 m2 S
    factorial(5) = 5 * factorial(4)# U8 l7 H  s0 ?& k) H% B
    factorial(4) = 4 * factorial(3)
    . n9 q4 M/ X' ^factorial(3) = 3 * factorial(2)
    5 W% \* ^9 Z( Xfactorial(2) = 2 * factorial(1)
    0 X0 o8 B# j4 [5 P+ y1 `, j9 Qfactorial(1) = 1 * factorial(0)
    2 q" ?: Y/ f$ k' Jfactorial(0) = 1  # Base case
    " H/ h/ j' }6 f4 E
    , d. p: i$ o- ]2 q3 |/ ?4 LThen, the results are combined:
    * b8 z' i3 y( P& A6 F1 O$ J& k& E1 F
    , \& D0 H4 F! @! ]2 s
    ( c* n+ L# l) }  n3 Tfactorial(1) = 1 * 1 = 1' @2 B& A& H$ t$ Z8 k8 d0 U2 _
    factorial(2) = 2 * 1 = 29 u1 s6 g$ F9 @0 r" \9 G9 a
    factorial(3) = 3 * 2 = 6) s0 ]# W" Y8 b3 Y2 X& ~
    factorial(4) = 4 * 6 = 245 z6 y$ P+ f4 X9 q* C5 c
    factorial(5) = 5 * 24 = 120) |5 J0 G! |7 Z! X/ a. H4 T5 b: H5 l
    - G3 R, H# p# E3 S# }# C2 E) r
    Advantages of Recursion
    7 n2 c' F0 s9 V$ O- ?, X) Q! ^' O5 X1 `3 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).! j- j+ P: Y. I& `7 [0 o  g

    2 d+ @7 [  B( p0 L    Readability: Recursive code can be more readable and concise compared to iterative solutions.# E$ P: E5 U: ?& W* M( w4 |

    + u7 b( e9 \! C1 f' `3 z$ W3 C6 NDisadvantages of Recursion
    . t  l- A  r2 e- w2 v/ f$ D9 c) W& 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.8 h, U1 t8 ~7 d; W# n, d, d

    * q& A9 @; g8 S    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    0 F8 y* }4 X) j7 h4 O! S" j; _4 s- x0 v) f; i+ d
    When to Use Recursion
    5 ^* i7 _6 G$ F( }& R$ c4 x$ O: n/ f5 @' C; h+ e
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! H6 b; V" h5 t4 i/ ?. r! I' l
    : h5 r3 U+ n* x& W: p% T    Problems with a clear base case and recursive case.3 E5 z5 Q9 N2 d* T( Y1 _  S
    9 r7 C1 s& I8 q1 A, m: A; S- C
    Example: Fibonacci Sequence
    & p3 C  u- {3 t/ m) j: x: w9 u
    2 J) M. ^7 I/ p  z' _9 m6 wThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:' \, Q: O- N2 @9 M& B# P' z' C3 m

    ! A5 P6 ?  E3 A$ P7 X  ?    Base case: fib(0) = 0, fib(1) = 18 x& a9 G! l7 E
    - N/ u  S; O  X4 }
        Recursive case: fib(n) = fib(n-1) + fib(n-2)- u. P& Q3 R/ Q  l5 {5 h% q

    # ?/ v* O4 F9 r, M) gpython
    9 U) e5 H" M, X9 F- p2 ]. {- n/ G* O0 v" B! M/ m' b, U$ j- o
    - x2 {2 M" Z; o1 o% y* m
    def fibonacci(n):/ i" {; n- V" q. u! R. z( s
        # Base cases/ G( L* N; H8 K) c, t# W: J
        if n == 0:& o: [4 [& A3 o2 x5 N# l& r
            return 05 d8 |2 B- V( h  x/ j
        elif n == 1:4 {5 ?% v9 s8 d7 m
            return 1( q9 {" B" ]# t2 C- q" r/ q1 @
        # Recursive case4 @& M( b" J& B* G5 T
        else:4 n, q' A! ?6 ]" c7 S
            return fibonacci(n - 1) + fibonacci(n - 2)3 F5 Q3 D( [/ l& D
    % i2 D" H" {0 x
    # Example usage3 t: H5 f  }0 p( X6 W
    print(fibonacci(6))  # Output: 8# k) p8 x8 e7 q; T  k9 ?

    & L/ u0 P' a& H  E( S. N. dTail Recursion
    & c! A& r* q& l0 e' |5 ?2 ~0 y/ Z
    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).
    7 t$ a; a% d3 I9 s9 Z4 G! U( _5 V7 H; S
    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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