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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 ]6 ^% a2 k0 R  I) R- b( K1 o+ o4 H2 X4 s" }# D" x
    解释的不错5 E5 M/ b  e: l6 f) R
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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。4 u  r7 N8 |3 _: e4 X  x
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    关键要素2 s* |5 Z! s$ m+ c
    1. **基线条件(Base Case)*** Y( _5 E+ j. `: s
       - 递归终止的条件,防止无限循环
    0 \8 u# f5 G" x. o6 P9 g& @5 T   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1( ?* Q3 R  y1 n, P

    ( d( w/ z4 \2 a7 A  h4 L+ a8 i2. **递归条件(Recursive Case)**
    1 }7 x) X4 ?! P: Z! V% E2 G   - 将原问题分解为更小的子问题
    - Z- W. S: r% R0 p9 y$ l   - 例如:n! = n × (n-1)!3 G$ q# @6 ~% i

    7 S0 S3 U( W& Q 经典示例:计算阶乘
    6 f9 X* b$ N" hpython7 U. c2 h  U& c/ F1 {  j
    def factorial(n):
    ! Q' I; Z& S5 H, ?' M+ i" t& p7 F    if n == 0:        # 基线条件" p4 P8 z1 A" I4 M  q
            return 1& b4 V# W3 ^6 u6 b# X6 S3 W
        else:             # 递归条件- `4 Z+ h+ ^( p7 z. F! u' v* Z
            return n * factorial(n-1)
    ) W& h$ w! b* ~2 u- d( B执行过程(以计算 3! 为例):
    9 x1 ?9 @) L3 P5 s% a3 g  Yfactorial(3): |" L' D, F1 B  \- B
    3 * factorial(2); m( @! I2 q5 a9 O! p3 f
    3 * (2 * factorial(1))
    7 h7 T1 b# J  d# u2 h3 * (2 * (1 * factorial(0)))1 B2 t, t* W# @/ c5 U% T
    3 * (2 * (1 * 1)) = 61 M' i. G7 B0 [+ V
    # z% h2 f: X) A
    递归思维要点6 T0 o, f" ~/ D" b
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ; V  Q- y  Q! C2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    - Q' c& d+ X- \/ Y$ d+ S7 W3. **递推过程**:不断向下分解问题(递)
    : m: S$ R% H7 Q' P, R4 Z6 z4. **回溯过程**:组合子问题结果返回(归)
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    / B+ o, K1 a: X. C9 [# H  V 注意事项
    & a2 }% M' _% E4 ^1 ?, a$ C9 b必须要有终止条件6 q0 G6 Q# S( d0 V* E
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)" W  A( y1 K- R- t2 X
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ! D1 g/ V6 Z( c) N8 d' x: P尾递归优化可以提升效率(但Python不支持)1 i2 G9 ?8 S$ \" p0 j$ ~
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    递归 vs 迭代/ n8 y2 X- ?9 a- ^, s
    |          | 递归                          | 迭代               |
    ' z$ t3 ^+ u- ]: M3 ?& C|----------|-----------------------------|------------------|
    0 @. p/ N$ g* I7 P; J| 实现方式    | 函数自调用                        | 循环结构            |" ~5 [/ a8 N, e: s  r* s& A
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ) q+ `. P8 X7 I( P' Q& u8 A| 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         |- C. i; N2 p: G
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 M/ N& j# `7 v0 O" h
    ( b. k- ^; G6 m5 K5 w, m 经典递归应用场景2 h3 ~' ?+ H* X5 f
    1. 文件系统遍历(目录树结构)
    % o% F5 V" ~5 c2 [! w2. 快速排序/归并排序算法' J7 }: {: `/ u: H2 E
    3. 汉诺塔问题! b# E* U6 v* ]7 `! o. I# M0 p
    4. 二叉树遍历(前序/中序/后序)* t6 a+ B2 z; ?* C( f* F# c
    5. 生成所有可能的组合(回溯算法)
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    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,, d0 H4 Y5 h. B+ l" S+ G$ [7 G1 H
    我推理机的核心算法应该是二叉树遍历的变种。
    ; O' G8 b( T$ o# ~- R9 P- ]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:( W5 b0 s& @9 {
    Key Idea of Recursion
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    A recursive function solves a problem by:
    # x5 E5 ^4 X6 o  P- W/ S9 {: B2 b7 g0 S% {/ H$ z
        Breaking the problem into smaller instances of the same problem.1 P0 z8 s' W% F7 ^, n' z

    6 O) [* S+ ?( |0 F    Solving the smallest instance directly (base case)., Z5 k4 y" v  U1 M5 C  q; ]- t
    % J* W, [- x' d
        Combining the results of smaller instances to solve the larger problem.
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    8 i6 P* p" N9 x5 D. c" J% aComponents of a Recursive Function
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    / ]# A$ S* l5 O9 f8 B# R    Base Case:0 E0 v; ~, ?# m3 C- H1 O
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 K; r: W, [  s
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            It acts as the stopping condition to prevent infinite recursion.
    " J1 A' M* F( u- T; {  n- r/ w# }5 A: w
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    0 h3 c+ O; Q, r% ^    Recursive Case:
    ) F, D4 {" i/ f  e, {( d
    * Z% u' [# f& W6 k7 g        This is where the function calls itself with a smaller or simpler version of the problem.
    # A; u6 G: B3 K9 ]) `2 L
    6 s5 {9 ]* e% g# p# X; E; t/ h        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    + {: M1 H# m: _. w' l+ g! J% v5 f# d( B6 U& m7 l$ d; x+ M1 X- Q! ~9 E
    Example: Factorial Calculation* M7 l( t# @) U. e  s3 {
    ' u( [, A, {0 O% o* ^, i- b
    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:
    4 I+ j" \- ~2 D$ H, K/ T
    % Q/ }7 G6 g! {1 }: m' U- r7 ~    Base case: 0! = 1
    , a, G$ e4 h( A, B" d  h- R& M  J. P3 j7 V# V( [
        Recursive case: n! = n * (n-1)!( f2 f2 l+ {8 z
    & c4 K* t5 q1 R% b) F" P% t+ {
    Here’s how it looks in code (Python):
    / c1 Y0 K  }' x: R. Kpython/ R: I3 W9 s. O8 A. y3 c# q5 a+ ^

    , a( b+ }+ u6 x/ S
    4 I9 d" L* _  K6 b" G- d6 vdef factorial(n):
    $ j: o  c# o( Q& z) ^    # Base case
    & y/ s* g& L2 @0 |+ C) L8 s# T. A    if n == 0:
    & l4 d0 M7 N9 @6 o2 Q; e: c        return 1* L1 Z! F' g8 J1 m
        # Recursive case
    " t8 F+ k. u; _2 F$ R1 n    else:
    3 ?1 k6 p# x) Y1 C        return n * factorial(n - 1)
    " d& w5 S1 e) k5 B8 {: N
    6 k) w  V- N. R# _, ?9 u1 w# Example usage! Y. @" o! e, J. O. ?' a' }
    print(factorial(5))  # Output: 120
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    How Recursion Works# c: N4 h* t5 i- c2 x
      ^1 u9 N! k: C% d6 H$ B1 ]6 j
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ' p! P: Q# n" {0 t( o. Q8 Q: L* ?. X9 [; \
        Once the base case is reached, the function starts returning values back up the call stack.
    $ W. Q: Z! y4 h7 X* w- d3 p3 w; i1 z
        These returned values are combined to produce the final result.
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    8 h) [4 z4 Z. J( L6 \  VFor factorial(5):# k9 S& i8 s8 @  A' `$ A7 p

    ) X" U) c) }  J+ p: Y
    7 u4 t4 J- H$ [, Ifactorial(5) = 5 * factorial(4)- I3 d+ }1 h! R' A0 i9 B  @: Z
    factorial(4) = 4 * factorial(3)
    5 X$ k& V) ]  c6 Q* J( Wfactorial(3) = 3 * factorial(2)
    9 S4 _1 ?3 H/ r" Z7 N1 a! [factorial(2) = 2 * factorial(1)
    + t0 a) d- S. J- i3 rfactorial(1) = 1 * factorial(0)
    ) @. \. @2 ]3 I* Q+ w5 b7 V0 a' lfactorial(0) = 1  # Base case
    * ]  H/ e: ~$ _3 v9 u* E, t
    % M3 }5 k& F4 }" i5 H7 R# bThen, the results are combined:+ }$ _* ]) c2 g3 k) H$ k$ I3 O

    : ~& B: }7 F6 E2 x$ q* W. S
    9 f* X8 r  j; n1 {. Wfactorial(1) = 1 * 1 = 19 I7 J: F/ b9 y+ u) x0 M+ i
    factorial(2) = 2 * 1 = 2
    3 {) M% A/ L6 x7 z9 o2 \; Ffactorial(3) = 3 * 2 = 6
    * T1 }4 r+ C% r$ efactorial(4) = 4 * 6 = 24
    ) O1 s0 p' `; L, V3 y; Nfactorial(5) = 5 * 24 = 120
    9 b4 |+ T5 _! p) Y
    $ g) X2 _- R( Q' A9 Q6 @& z$ zAdvantages of Recursion
      _9 s. v: h! u6 c9 A9 M3 B4 o& `+ s$ B; x! c9 ~9 }
        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' T" A/ ^% w& P6 ]  k
    4 u0 b4 _! y- g1 Z2 Y% `    Readability: Recursive code can be more readable and concise compared to iterative solutions.: ~+ b$ C2 A; U. ^8 ~) A
    , W  \: |) m- ^9 x0 s) ]3 e# k
    Disadvantages of Recursion
    $ k8 G- t3 F3 Z3 A* ]% o- ]
    / h) O9 c+ l" t    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.
    1 A! B$ [& c4 e8 ?0 P
    % v/ e) q% `0 u. |- i% ^  f& R    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    3 _7 K% l8 g+ J& q& U* b+ i$ N1 k1 G
    * ?& @2 z* b  h1 l& t2 VWhen to Use Recursion
    ) C1 J2 u( z' A1 a, x
    . J$ `/ c. N9 A9 P( L, G    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    2 ]" E3 j  R* x& s* n" F
    . w) r" K  i* T3 I1 s. n/ }    Problems with a clear base case and recursive case.
    ; \+ C6 }+ T3 _8 }! K8 J6 ?+ k" @% T) v
    Example: Fibonacci Sequence
    8 Y' Q. _. V% g% y0 p. R: f2 m4 H
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:. O) b+ [' T+ K$ N
    4 |4 g3 a9 j$ g; b  N( [
        Base case: fib(0) = 0, fib(1) = 18 v0 F' G7 ?/ V+ m0 x; t2 u

    & O5 n9 d6 m0 M% Q& N9 d4 I- b    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    * o# A2 L; _1 l7 ]# H3 |9 Q. ]' f5 q
    python
    " z$ ^, ^! D' c( `/ N0 o6 ~: W4 Y9 {2 v

    / {( u3 t3 d+ \( {% n% m: |5 Ddef fibonacci(n):: W  r+ s  d8 g( U
        # Base cases, g+ W$ z4 c3 r7 p9 }
        if n == 0:
    ) s1 _* a; a' [; \# j* k& F$ B8 K        return 0
    4 A* M0 E/ c5 ~- T# s    elif n == 1:3 \( m! c0 N' d2 O- c8 m
            return 1
    # ^; d, h6 y' [    # Recursive case4 g' U) U- Y: |( f* L. A. f& E9 M
        else:
    7 t2 C8 x. {; T. `  d( V% Q- X        return fibonacci(n - 1) + fibonacci(n - 2)
    ; L2 y3 ~0 m" o5 z, [) E
    0 G5 y4 Q( K4 x, d, Q4 B# Example usage
    & B' K4 {* R  }# Fprint(fibonacci(6))  # Output: 8, @: ?$ V0 y6 j; D0 i  R$ |

    ; R9 @8 L6 r, ]5 T; FTail Recursion' ]) v* J+ ]# ]: \) s1 o

    # ^! H$ [2 P& d* J" r2 z* A6 o1 ?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).
    3 j+ a5 {9 W: M8 @" o  x8 Q0 d" l2 ~' l# w
    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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