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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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    2015-11-30 11:11
  • 签到天数: 2 天

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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 |1 q8 O4 R% N2 n. a$ z' M

    ' o3 m8 K: C( N) e3 ~解释的不错
    + W% J+ s# o6 ~& u8 B# y3 l/ F  ?5 R. p
    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。* k& c" h' i! J  o$ K
    4 }3 C/ x% X' {: m
    关键要素
    8 X: S+ e; n4 r  a% c1. **基线条件(Base Case)**
    , j& }: O. N! x5 o: P6 C   - 递归终止的条件,防止无限循环* o  N% b! U( y3 Z+ p" ?9 e
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    # M9 E* a& g# B: {4 p3 R3 P6 b2 S3 l! B+ {
    2. **递归条件(Recursive Case)**
    ( u- K6 R( X5 k) v   - 将原问题分解为更小的子问题
    & ~( h) o) h' w/ [/ f   - 例如:n! = n × (n-1)!) d% H) u3 F- Z2 _

    $ |: q" i* _* H/ Y3 m" q$ x* Q 经典示例:计算阶乘
    " L% \) {: V+ z+ P2 Kpython# L  D" a( ?6 ~% H( y2 U
    def factorial(n):
    ' {/ Q" e1 I6 [% \  j    if n == 0:        # 基线条件
    ' W8 `; u! T* T2 x# O1 ~        return 19 E0 V! {" `8 c8 X
        else:             # 递归条件
    ) q8 v% l  H3 K4 Z6 x. C; o        return n * factorial(n-1)8 T0 f$ m" W! y0 f
    执行过程(以计算 3! 为例):# A5 p6 Q) j' Q5 ?+ n2 o
    factorial(3)
    ) V7 z' j* B! P3 * factorial(2)4 g% k$ V8 H! r4 ]5 f1 o
    3 * (2 * factorial(1))
    : R5 ~; w% l( P/ r2 s) m3 * (2 * (1 * factorial(0)))
    . S$ C  N7 L+ `, t5 |* A" `. U8 H0 ?3 * (2 * (1 * 1)) = 61 P. j* y% d2 o  J% o% U1 b  F
    ! D" P5 e# l; }
    递归思维要点" ?9 l, M4 k- m. W2 @) r9 @9 k
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    $ M: R! P3 _4 V* Q& R2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 X; L8 {, V  B3 f3. **递推过程**:不断向下分解问题(递)
    - f$ Z; Y' i) Y9 Q1 m4. **回溯过程**:组合子问题结果返回(归)" N3 {3 o7 R0 p! N" c" ?) g

    . g' E) H9 M6 P 注意事项. Z. J: x  ~; l- F/ w
    必须要有终止条件
    - g# U5 \% a9 d+ ?; i递归深度过大可能导致栈溢出(Python默认递归深度约1000层): p6 J* g) H  W/ R( P: c
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    % {1 w0 o& p9 e8 m/ x尾递归优化可以提升效率(但Python不支持)% v# [' ~% }8 j( N

    4 L) N' _/ [+ a 递归 vs 迭代) F$ v+ J+ j/ y7 p
    |          | 递归                          | 迭代               |% J) g% e5 i1 [3 ]
    |----------|-----------------------------|------------------|8 q& W+ \  E% `  T/ j1 m
    | 实现方式    | 函数自调用                        | 循环结构            |
    / V, q, c) P8 U/ f, }| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    * a0 e: W% B7 t- ]) y  H# e5 ]7 k) c| 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         |
    / q5 @: H. ^8 V' N+ \2 n5 o| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    2 V& F+ _6 ?* L$ f  p/ ?" ^# f" N1 f* {+ @5 \
    经典递归应用场景
    8 k  F, ]7 B* `3 a/ G" g1. 文件系统遍历(目录树结构)7 p: H' w0 \4 \+ T% j( K, y( V, Z
    2. 快速排序/归并排序算法
    , x( ?2 E4 s+ ~9 L3. 汉诺塔问题4 p- n; D* ~3 L; o* P
    4. 二叉树遍历(前序/中序/后序)
    0 _) ^, @. p1 q, Y5. 生成所有可能的组合(回溯算法)
    % G5 d. W% H6 ~, {" ^$ ?- R2 ?0 P2 @0 Q# L- q! {/ 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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    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,
    1 N) \- ^$ g8 r" s: B. X8 W# r我推理机的核心算法应该是二叉树遍历的变种。2 K5 o# A/ z7 {0 T# r
    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:
    # B/ O" O; ?) H4 M9 t" |- GKey Idea of Recursion
    % [7 I- P& L: s3 o- x  S- q, D; N6 J' i6 ?# {& M6 e! |
    A recursive function solves a problem by:
    3 l' c5 q% t3 N. C0 [- r
    9 ?/ X0 A" m* I. m& P& U! `4 h/ \    Breaking the problem into smaller instances of the same problem.
    3 @4 Y5 F  _3 H# ^) f3 `  k; y% _5 P1 Q( h- v+ T
        Solving the smallest instance directly (base case)." t, v9 ^$ L. s: r
    ! Z0 q6 A& Y. N0 B# R" L. \6 l
        Combining the results of smaller instances to solve the larger problem.
    1 U) G; |4 [, E! N( G/ A
    3 n& E6 w4 y! R: q/ xComponents of a Recursive Function) a% o3 ?! ?( m0 [7 {) @; v
    ( g; Z7 I' Z6 N1 L' o' m9 G. f
        Base Case:
    ( [% y8 j' i9 r: H% x9 L5 k
    5 G& B! R& i4 ?/ T  v+ s        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.' h5 k5 E4 D& ^
    # I( u2 r& M) Y0 z5 m" U0 d  }
            It acts as the stopping condition to prevent infinite recursion.7 A, t6 O# D$ o

    + G$ Z3 c1 E! g, ]; j! J8 Y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    + y. f* X  k: H; F$ K' A8 a1 m
    6 ]+ D/ ]  Q+ T    Recursive Case:6 H8 E) ~& \6 Z; h$ d& f1 T
    3 O7 ?$ n! h. v0 @, M" _
            This is where the function calls itself with a smaller or simpler version of the problem.( t/ K: y; ?; p( x" A3 c$ x
    / U$ y6 d9 d6 o2 m7 j
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* s* W! G6 {* O  O0 e

    $ w4 X& X9 Z( o- {4 ~Example: Factorial Calculation
    ' _# M2 L# G; V5 W- y
    & ~; y% ?/ k1 }' i$ }9 WThe 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:# t; _! b0 s; \7 C, g/ a0 [

    & {7 `3 X/ j" S    Base case: 0! = 1
    ! [0 i" o* j* D' K0 E  S6 {* B6 H+ q- x
        Recursive case: n! = n * (n-1)!
    , Z, s& X+ m, q/ ]- \5 V. b' U
    0 ~2 [- C9 L1 b) e" f/ mHere’s how it looks in code (Python):
    ' n9 r7 V4 V' Qpython( F& r6 ?9 Y1 J
    * G5 k# ]% U( f

    / a' ^; ^5 `$ }0 a! H. T& X5 ndef factorial(n):) Z# u; W9 s  ?; B$ T, p
        # Base case
    / G* \; T6 `& ]/ }& R  G8 {1 ]4 _    if n == 0:
    8 ?; O+ K. z1 e9 E$ ^/ }# U* E        return 1
    , |- Y7 o8 w; L$ l    # Recursive case6 o% n: J1 @; X% U9 Y
        else:
    1 q0 Q& X) Y$ \; r# {! s3 W+ G        return n * factorial(n - 1)
    . n& ]+ `5 [  ^8 u# R$ j6 X& u: S& p  b# x, @) ~/ _6 G2 k* ^
    # Example usage7 Y: B) u2 c5 C
    print(factorial(5))  # Output: 120
    9 X. I: x7 [- h/ D% H/ K% e+ w+ ~2 Q+ v, L3 Y% B  v; ^( b( e/ v; W
    How Recursion Works
    8 J, e2 i; H8 q& r. p2 Y
    / K) b: t  A9 N4 v1 ]" L    The function keeps calling itself with smaller inputs until it reaches the base case.
    . ^; g# Q8 I9 I* g7 j% g$ h. {
    2 d7 p( E& v) u- Z" H% N    Once the base case is reached, the function starts returning values back up the call stack.
    , i. u" |0 Q5 I: l0 J( |9 A+ |9 N% @9 A
        These returned values are combined to produce the final result.
    ( h0 ]: u3 |: B' |
    0 [* a/ t. Q6 qFor factorial(5):
    7 q0 W) h, F$ h
    / c" j. @' x3 }! h
      G0 a" m4 x' r1 u; K# Afactorial(5) = 5 * factorial(4). a' K+ e" ^6 O/ v. P! U( E$ t* X
    factorial(4) = 4 * factorial(3): l+ L2 p/ u2 T0 T1 L2 L
    factorial(3) = 3 * factorial(2)! W; F4 q' f5 u
    factorial(2) = 2 * factorial(1)) }+ V1 k0 I9 t4 _5 ?& F
    factorial(1) = 1 * factorial(0)$ j8 K1 u  s; L4 j: Q4 l
    factorial(0) = 1  # Base case
    " F2 h9 y1 ?& D7 v, v) v0 w% d2 c+ o8 A) Y/ g, E
    Then, the results are combined:- W  W; X$ Z$ I1 o

    1 u# k3 s2 H5 G& m( _9 G# R- S- D5 p6 }- }! z
    factorial(1) = 1 * 1 = 1  t" u. N- D/ v6 I
    factorial(2) = 2 * 1 = 2
    4 ?) [5 `# R/ ]+ R$ Ffactorial(3) = 3 * 2 = 6
    ! d9 w" j/ l* D  h5 F' b6 \factorial(4) = 4 * 6 = 24
      C. ]7 x! I; E+ n8 ]$ X& U# Y: ?factorial(5) = 5 * 24 = 120
    ) M) [# U3 y) r/ @/ V+ y! U/ e6 l! L3 B( w+ b$ b
    Advantages of Recursion
    : Y* }" ~# n/ P- {
    ( Y' Y+ c5 s- o% M/ b5 E    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).( N8 u+ I  r: U2 B+ B2 f4 l0 ^

    5 ~( B- D3 \, C0 L) O    Readability: Recursive code can be more readable and concise compared to iterative solutions.! U, O' C8 B0 L3 o
    - Y8 E6 X$ b1 X% s6 ], ~
    Disadvantages of Recursion
    2 w! d  m( B( u$ y5 f+ s* v" d4 M
    . S) P2 L% e' G6 ~$ p( p6 G+ h    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.
    ! N6 W8 M% |! `3 n9 I% N/ C6 }9 L- }5 U# b0 w+ `
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    $ b$ X6 G0 f5 M% s& [& Y, L! n2 W$ L6 ~6 o1 S
    When to Use Recursion2 Z7 J4 _$ q6 ]7 ]$ m

    ' r- N2 B% ~; d/ Z' U) N# e    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: `! X+ ^0 ]; Z2 Q$ L) b
    5 D1 I6 ?$ H2 Z& f' l9 d
        Problems with a clear base case and recursive case.
    ( D  M! ?  [% i" q' V8 C+ J
    / ~2 E1 U* ]- h+ a  Y" pExample: Fibonacci Sequence& O% ^% g7 z9 X+ h- X+ s
      ?! \( ?6 t$ y1 P
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    4 i0 O# V# c2 z* s( u3 p5 S  @
    7 ?+ t2 Q7 [1 P2 b# H    Base case: fib(0) = 0, fib(1) = 15 P! J! p9 j, e; i

    3 w- V% u" p' F1 V. B7 F    Recursive case: fib(n) = fib(n-1) + fib(n-2)" c2 b! u+ j3 ]- o5 |: X

    1 l! h1 _8 Z( ^( g" ^4 U9 E% Ipython
    & b/ I+ o+ |3 m3 g1 ?
    * S5 F2 ~! d- L. T9 C* t/ x' l) }& Y9 m& a; {8 s+ e# j
    def fibonacci(n):
    # U, ]0 Z, e3 w    # Base cases4 z6 c4 s. {" h7 ?: O6 H+ E
        if n == 0:
    # }- w( z1 q' {% U7 Y/ t. {( b        return 0/ }2 c! t) T2 c
        elif n == 1:
    ' i. }5 e$ u" N0 a        return 12 c9 K! m5 P" _+ T8 S- @, a
        # Recursive case
    , V. ~! e) o) J. M    else:/ ?6 w: y% ?9 I# a7 P: j# a- ]
            return fibonacci(n - 1) + fibonacci(n - 2)& s- I6 M( q# Z2 E& H5 ^. o9 b

    - ~  h8 u. E! m3 D& U, X# Example usage' i3 u' @- E: a5 ?# I
    print(fibonacci(6))  # Output: 8
    5 q$ T# ?2 a, H) Z5 [0 M) \0 {. m# l$ ]* F' v: @5 g& p
    Tail Recursion
    % E  Z& L6 _' r
    4 ?$ \8 @. j7 F! R. H; N: OTail 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).8 ~' c3 p+ C+ C9 [; @
    / P# S- K3 ~& n" D
    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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