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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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    2015-11-30 11:11
  • 签到天数: 2 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
    3 H" X7 y) _. c
    ; `! D/ B; R& e3 @5 X- t, @解释的不错0 ~/ |$ N+ `! l) I
    3 z% d- V1 ~3 i) v# L& }5 Z1 g
    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。
    0 ]8 w+ }/ Y4 _1 v% f  r
    ; E8 A- q3 J: `/ @0 `- r 关键要素
    . d% [& I& S# G+ n1. **基线条件(Base Case)**
    : i8 A. \3 G2 J- k   - 递归终止的条件,防止无限循环
    8 n6 \9 Y7 v& S& U+ _% m( g   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# G, z2 ~( y5 k  A# |6 a% L

    / Z+ }+ f  j# c5 a0 L+ S2. **递归条件(Recursive Case)**/ L7 W) |! B7 p; p9 R
       - 将原问题分解为更小的子问题, v+ ]3 B' u7 C: C$ U# p, `/ ?2 z
       - 例如:n! = n × (n-1)!
    6 [9 h$ v3 D4 G* y! U. S; t2 }
    ( d% V3 {/ I& `1 \. ]) `! `" O 经典示例:计算阶乘
    ) g' o# z* z, q' Dpython1 W# A! R2 }6 F9 O
    def factorial(n):
    ) s5 C8 X- y  u1 w* ]    if n == 0:        # 基线条件
    % S" z( c8 W" G; p% d( ~        return 1
    - r! d+ R2 X0 k7 D2 J( Q/ _    else:             # 递归条件4 @8 }6 i8 f$ z: i4 w/ t8 X
            return n * factorial(n-1)2 n% D% T6 X4 |4 X% m
    执行过程(以计算 3! 为例):
    + |& f% U7 N0 _9 h$ C( [factorial(3)( b8 f" J- {; i2 S3 H
    3 * factorial(2)
    ' f# I# f& S! z8 Q: u" M2 J$ r3 * (2 * factorial(1))+ h2 _/ S1 S2 w
    3 * (2 * (1 * factorial(0)))5 B( a' S) c! r' S; T+ o
    3 * (2 * (1 * 1)) = 6
    1 G) J# I3 F" ~. U" x& ^- j1 N# L0 q) y. Z7 ^" m- ^
    递归思维要点/ X; ?+ Z) |9 u% H& v
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑# F+ W# T2 e6 K' }
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)5 I, }' @) q! G3 A8 C
    3. **递推过程**:不断向下分解问题(递)9 u& ]- r* C5 p1 F+ \
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项
    & x) [$ X) w+ d3 o8 Y+ w必须要有终止条件" g3 f: J9 M5 S4 ~: s# B3 ]
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    / \% F- y) q: E) t- R7 I- w( o8 ^某些问题用递归更直观(如树遍历),但效率可能不如迭代
    $ y4 M6 K& C2 v7 d' i8 K6 o尾递归优化可以提升效率(但Python不支持)3 u- A  |$ `3 A0 I
    0 ^1 ~) C# ?  j% S8 o. S1 e
    递归 vs 迭代
    , I6 b( `/ v# p9 s) m, `: l  y|          | 递归                          | 迭代               |! T) q1 @$ p; J# A! \. Z% V% _1 F2 W
    |----------|-----------------------------|------------------|
    4 [& n0 x2 C7 @& Z3 \* K| 实现方式    | 函数自调用                        | 循环结构            |  d0 W6 T4 m0 f" P7 B
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# l% y) ~( U5 S3 g& G, I# p# N7 }
    | 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         |
    4 m- E5 E( Z# w. L9 U; u* r* E1 ^% S| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |1 _$ P! i( L) r- b

    6 ~% ?1 O$ ]# f7 @0 l: X0 e0 a 经典递归应用场景- N/ h0 u8 t# S- h, m; _, q9 U
    1. 文件系统遍历(目录树结构)
    ) }# d' B. n8 d2. 快速排序/归并排序算法: R4 n# ^& ?$ s* S3 u
    3. 汉诺塔问题9 e! L, c2 q( C+ A3 Z
    4. 二叉树遍历(前序/中序/后序)
    ' T. `0 g- T$ X5. 生成所有可能的组合(回溯算法)3 h9 a+ U# S& X
    * z. j0 l4 J  C
    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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    昨天 05:55
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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,$ G- e- |0 p2 t. A
    我推理机的核心算法应该是二叉树遍历的变种。
    4 I6 f$ H' z4 q: I$ |, S; a8 V1 B) v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:
    5 F9 a/ t! I$ W1 Q- m1 o6 BKey Idea of Recursion8 }7 x2 ?; d! ?
    4 b2 n5 m% U8 A1 Z+ }: p
    A recursive function solves a problem by:
    ) i+ ]! K4 N/ r5 R$ H* M
    8 A% {, s6 y- Z, W- h- G    Breaking the problem into smaller instances of the same problem.
    % ~# X# F& M5 k- f: B$ e
    & f$ e8 m& i% C" V$ y6 c# |( E5 b    Solving the smallest instance directly (base case).5 B8 W& h+ R& a+ s
    3 {! ~4 ^, E- c  H. E3 o+ \  }2 ~
        Combining the results of smaller instances to solve the larger problem.+ B  ~1 \, g: i4 ~: L

    % Q' {, w! C! p$ x8 ZComponents of a Recursive Function! d5 W; W0 h0 ^( ]

    ' A5 ]8 m) \( H1 p( |    Base Case:& R; n. A: \7 @' x. W8 W

    : Z  B# ^: w, T# j- T, b        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.$ N5 z/ Y/ Y, s- ~! I8 S
    # w6 P6 Q) i  s1 N9 _. X9 V8 ?
            It acts as the stopping condition to prevent infinite recursion.7 ^" E- `4 Q# e, K# c

    4 u% R& F: c& s( [# R1 T/ n        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    1 V$ K# }: [8 ^  n+ ]/ m% ]. E( ?5 d! C& b
        Recursive Case:+ L  Y" I6 D) |; D5 ^# _+ I

    2 V, x  C* `6 I8 L6 ], o        This is where the function calls itself with a smaller or simpler version of the problem.
    8 I# {3 S: b: q$ y8 H
    8 n: a* ^0 @; x7 `+ k6 u$ C6 q        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    / o: `) u- |3 b( v
    & k. {3 C- x% _; d- Q: XExample: Factorial Calculation
    * J; h' b% X& A6 J+ J& ?8 z9 p4 G/ y# i  q9 G+ f. w1 c* 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:
    6 ]. D% U; N/ c1 C3 [0 Y: S; o  q0 H" R) A% @# K3 L( z+ p
        Base case: 0! = 1" G( T7 ~, {9 c

    $ C, i8 s, O8 |1 R    Recursive case: n! = n * (n-1)!* [8 o0 c  X. a5 M8 v" b' o

    ) P% Y( U' L5 q# [& @! d; _5 UHere’s how it looks in code (Python):
    ; I+ o1 M9 r% j' x8 c; N5 Cpython
    9 g* J9 g( G/ m. W3 }- U  l2 w, |+ Q% A/ K

    ) E' y$ H% b: t5 Y6 H$ W+ edef factorial(n):
    # k8 R2 L1 Z( {    # Base case% [! p- f5 ?& v+ L. i  ]5 G
        if n == 0:; K% z8 O$ z( G: X2 k6 R# c
            return 1; t4 J: g8 x5 E7 E* q
        # Recursive case
    2 F- |: H- I$ c/ F$ [1 m    else:' s" P. q. v7 F5 @8 [; G8 K
            return n * factorial(n - 1)% d& `. l% D4 z4 U: R

    7 ]) ]: i" A5 E+ b5 p; u( Z" q5 F# Example usage9 l: u6 w/ {! I0 G+ E0 E1 s
    print(factorial(5))  # Output: 120
    4 i( ~! P! |4 Q1 ?! U9 H8 p
    & K; b- L3 ]; f1 e  S3 JHow Recursion Works% L# @! u& {1 f2 T6 O) m- p+ A

      r6 G2 P0 X' W0 {3 g: B    The function keeps calling itself with smaller inputs until it reaches the base case.7 V1 Q( H5 R$ a4 |
    & U; ]# M# K! |# U* g/ R8 d$ `
        Once the base case is reached, the function starts returning values back up the call stack.
    * T3 z0 K  T0 a4 [( i( T" X/ n- G8 E0 s4 a
        These returned values are combined to produce the final result.: m% \# g9 I0 I2 c# F# R3 ?

    ) \# d& L, p/ H+ ]+ @% YFor factorial(5):  t) n% n* Y% ~1 T. Z& A8 G

    * e) q/ ], ~0 R6 T
    4 Y9 p* A8 p% H9 Y5 xfactorial(5) = 5 * factorial(4)
    - U* G& W9 e  ]& pfactorial(4) = 4 * factorial(3)5 M: D% |" y  D7 e6 O4 ]
    factorial(3) = 3 * factorial(2)
    5 d$ t2 d3 Y) `, j9 b# H( Lfactorial(2) = 2 * factorial(1)
    ) G  o$ f0 l/ i3 d0 _factorial(1) = 1 * factorial(0)
    ! b  C) Y* ^; e: @5 N+ g+ h. ?5 F) ^4 ifactorial(0) = 1  # Base case
      ^+ i! y& X$ {
    9 v: E) X1 G- c8 ?. A% @Then, the results are combined:
    " A( ~+ V0 \" a/ v& a
    % S! G; x2 S7 q2 o# Z4 k% [8 u0 h8 t; F; a0 Q/ ~5 A
    factorial(1) = 1 * 1 = 1
    5 `) l' Z6 R7 A# f; v9 E8 S- t7 nfactorial(2) = 2 * 1 = 2
    * }. d# K! k: z& U' Mfactorial(3) = 3 * 2 = 6
    0 n: h6 H( X- s* hfactorial(4) = 4 * 6 = 24
    9 \+ m- V# X$ L# D# K2 Lfactorial(5) = 5 * 24 = 120
    % q& j" X% G& c# l3 P5 z, s1 ]$ m8 C0 z
    Advantages of Recursion
    ) v, d% g7 u6 o6 A7 M" R. q* o- u9 x5 Z& F' 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 \2 |$ ^. C" K! c
    2 N, T9 S; ]/ E9 S$ c5 N1 d
        Readability: Recursive code can be more readable and concise compared to iterative solutions., C/ f" X5 T+ x& q2 t" B

    9 ^; D1 \1 E! l4 l2 q% nDisadvantages of Recursion
    ! i+ K0 H3 `7 M( K
    ) b8 [& C6 C; N$ D" N# Y1 x9 D    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.
    + p& ]" x) X( T2 v3 M* V+ i" I5 |1 H+ R$ s; \0 q5 Y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).; T( _6 l6 ?; r' s0 Q

    1 D1 M( ^6 R8 S4 f9 DWhen to Use Recursion- Y6 y' ?. X- u  k. v0 }  F
    % y: w4 R% I2 ~% ~
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).+ ~# u9 T- u: G! r8 Q' m4 ~
    0 H( Z- ]9 K  q- U, w6 ?
        Problems with a clear base case and recursive case.
      N5 l. y1 N6 v& L  D: w6 o
    , R8 ?2 f. e1 I" o$ z3 w4 e6 AExample: Fibonacci Sequence5 ~/ t* k  _, f4 W1 o+ w

    - _1 H$ i( P! U2 M$ U1 QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:0 b, q- ^0 s. V, Z6 j

    ' U% C- [; Q5 N6 c$ }! i. T    Base case: fib(0) = 0, fib(1) = 1" ~, [: D' H; R7 a
    - }; B- i# j9 l, S4 a
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ( h, f- y" [+ N- k% M$ c# L8 F9 K
    python7 [0 n! s% {% F8 i
    + F4 L+ U! ~2 d6 L  d

    ! x! U# k5 k) }) n. w5 P1 ?' w* Hdef fibonacci(n):- q7 S0 S% y" E# y2 D2 \
        # Base cases
    7 t% B6 K1 X& X5 W& u0 t7 y    if n == 0:2 Y: l" G  p% g/ x
            return 0; u4 v. O+ i0 ]; H+ j: t
        elif n == 1:2 m9 j# D/ l6 D" M9 P% L. Z
            return 16 d1 G1 Q" \8 i1 w5 f
        # Recursive case
    ' {( v; _0 r9 S4 e& X    else:
    * Z. r- u% G% N; V        return fibonacci(n - 1) + fibonacci(n - 2). N2 G! H5 c6 B! c( z% |+ l

    ( C, V! C" a9 T3 S& l0 z  t9 M# Example usage
    " s6 N# E* n" B! n* uprint(fibonacci(6))  # Output: 8
    * N8 A) ~2 w' X& s/ x# |
    , }8 F3 C3 Z, Z* m% z( sTail Recursion
    3 X8 P, Q0 `" k# t& R
    2 L- i) Z' |/ \6 J& |0 FTail 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).
    6 y9 S8 Y2 E: u% ~0 J6 K1 _( p3 o0 b% f% m0 h& o% Y* X4 U
    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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