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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 4 M7 s$ D' y! |# J8 F6 s
    # L& G7 J! g: X4 h
    解释的不错
    # T8 }2 w) |. z# k5 V4 ^: Y5 U) u& ~
    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。5 [4 ]0 i7 m2 E, S' W& T, @" t+ e

    ( ~8 r3 Q8 F; Y8 W 关键要素1 H, x: V2 L& n
    1. **基线条件(Base Case)**
    0 `! {/ I5 k2 D   - 递归终止的条件,防止无限循环
    / q  s7 g. ~: ?* G% S" q   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1* f& \& z" E4 O5 x
    0 K- ]4 ]0 z2 f' F( ^- c5 X
    2. **递归条件(Recursive Case)**
    ; K) j3 ~: M4 r8 L& V   - 将原问题分解为更小的子问题. M- u2 `: B- X
       - 例如:n! = n × (n-1)!
    , W3 q) e5 \) Y4 ~; n! U, ~# F7 P0 S+ H: P" K
    经典示例:计算阶乘
    * t+ z3 j; L  d+ |python( q7 i; @- P2 Q
    def factorial(n):
    - C' d1 q" X; I8 L6 c- R# X    if n == 0:        # 基线条件5 X* I& o) Q, }
            return 1
    ( L: |% t# M8 p    else:             # 递归条件
    4 q$ V; |' y5 m1 r7 y. u6 ~6 M        return n * factorial(n-1)6 q2 U, n4 F+ Y: E) r0 X4 ^
    执行过程(以计算 3! 为例):2 A: q/ `$ c* ^
    factorial(3), T$ f5 ~6 K  }% E/ z' V9 f$ ~
    3 * factorial(2)
    " I; N5 O7 ^9 F% A3 * (2 * factorial(1))# @0 T+ F9 p6 p! @
    3 * (2 * (1 * factorial(0)))$ f- z/ K; h/ G/ p3 m# o  i
    3 * (2 * (1 * 1)) = 6
    2 s" J4 t1 \8 Y3 ]& j3 t: Z/ [  K* n8 o; @/ v: S( S! N/ O+ u$ k; w( w
    递归思维要点, G  m1 w. N5 h
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑" s$ ]% L' i% ^6 k* r- o+ I* L' z7 `
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)$ q0 m3 s7 r* }' z" F) O
    3. **递推过程**:不断向下分解问题(递), ]; P7 l* c6 z
    4. **回溯过程**:组合子问题结果返回(归)
    + N7 ?, b- x' y0 l1 b8 i5 {* U4 D" E0 x; d& O% p
    注意事项: l7 D2 U# M! t5 t2 W" @+ H- b
    必须要有终止条件( O8 E$ f) I6 z$ D  Z2 O6 S' [% t/ q
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : @& d7 l4 J& W某些问题用递归更直观(如树遍历),但效率可能不如迭代
    8 w; h6 W; d. _- c* \尾递归优化可以提升效率(但Python不支持)/ B% G! P0 K/ K1 x7 d
      r# ?9 ]& L! B% ^5 ~) R) E" R
    递归 vs 迭代: B) k: Y9 H# [, S+ A8 h/ {
    |          | 递归                          | 迭代               |; d3 }. c# l9 W
    |----------|-----------------------------|------------------|
    ( r% f" Y3 S4 l% x. ?" u" D# b| 实现方式    | 函数自调用                        | 循环结构            |0 v( \: ?! S+ y
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    0 A: m' Q/ `; R* M- n| 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         |9 }1 e3 d  Z6 Y- g7 \! y
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |3 l0 r" [( X, b4 u3 ^8 W* I5 f8 f# U

    5 R- h7 @, Y6 U9 Y. f$ v; y 经典递归应用场景7 n9 O1 N$ W1 {$ y
    1. 文件系统遍历(目录树结构)
    3 E$ t8 C. a6 `0 H  J) T2. 快速排序/归并排序算法
    + k1 A2 w- X: B- L; Z3. 汉诺塔问题
    ; a" B4 A$ l- V) w4. 二叉树遍历(前序/中序/后序)
    , v/ g4 p. P( K& w( |0 S5. 生成所有可能的组合(回溯算法)! S# @( U5 w& S! d

    ' t& W* R; |) B' f/ t1 K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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    前天 01:20
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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,
    & y, u- z9 k2 t$ \# V我推理机的核心算法应该是二叉树遍历的变种。% T5 m2 i$ \( L! m
    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:
    ! y" Z6 C8 v0 m" N# tKey Idea of Recursion( G5 y& M; Y2 J

    + W- k' G$ T8 k& e) Q# WA recursive function solves a problem by:
    9 v; Y- |/ m7 U4 }/ E  |
    9 \% d" H* k  n6 T7 A. u9 {, d! k    Breaking the problem into smaller instances of the same problem.
    & U( S$ L, I9 `& ?* A& ?. s- J, E7 v+ Q9 r
        Solving the smallest instance directly (base case).6 ?2 o- f7 D1 [& S7 J) {
    ; |* u  A" t# h7 e1 [: A  X7 s1 k" o
        Combining the results of smaller instances to solve the larger problem.
    4 {9 R2 M9 A% I* Q0 M, w. j; ]4 n6 K" p
    Components of a Recursive Function
    1 o2 K9 c4 i, r" ]2 b* `' T* E% K2 A/ r( g" D, F5 L
        Base Case:: t: L' g; K/ N
    ' q. \/ r8 ^& t4 G4 R
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.# _$ J, j* s/ u; |6 ~8 {. W

    ) |; M: M3 Q5 ?% d  Z- d/ G! a: Y        It acts as the stopping condition to prevent infinite recursion.
    9 ~4 V* T% r5 j+ u: f* U
    * t0 v$ J- i7 x$ C        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% i9 R7 ]6 S( {7 X$ O0 d* B

    3 @0 O. o, a& V    Recursive Case:
    - \# n* {: v7 S- ^% @
    7 F8 p0 k  d$ D7 w$ j2 @        This is where the function calls itself with a smaller or simpler version of the problem.5 z2 y7 k8 ]( `# h  a6 l8 N! k

    1 F* E' U5 l0 U) @! P# P% U        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    6 |* b7 K. D$ h+ X. D
    5 g/ b. a2 \2 _Example: Factorial Calculation& M  A5 O* \+ Q# e1 x4 x7 B1 i

    / h1 \% n% N, D$ Y, P/ c7 J: L8 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:
    2 C* ?3 F: r# W6 e% L- B# K; B9 s, H# a/ C! E& V
        Base case: 0! = 1
    ! f4 d. l7 K) k+ [" l1 V8 Q2 b0 m8 G7 @
        Recursive case: n! = n * (n-1)!
    ( n+ c1 x' o  o$ E3 d( n0 C* [1 g' F0 O
    Here’s how it looks in code (Python):
    7 G; B  k7 _; c$ p3 xpython( t) h+ u* q" S0 [! H5 @" m

    , E7 _" D& J3 {5 p- L$ L, |: p; q
    ; t6 B' T! S5 ^' ~! H& u! {; K# Ydef factorial(n):, p9 E6 c5 m+ D
        # Base case
    4 G7 n5 S: X+ ^2 O    if n == 0:- ^% }$ s. u5 g5 n% m9 B4 X( @4 d" G2 L
            return 1
    - A2 V+ a' e2 B) I( m* R    # Recursive case5 F! E* s2 Q- [5 w" t$ F- B
        else:1 j$ e% s' k# i  B4 x- c' t
            return n * factorial(n - 1)7 F9 ~1 O8 d# u" b% s. U, Z

    5 o$ T/ ]8 V, l3 M# Example usage
    0 ]+ e  B3 m! N' i6 vprint(factorial(5))  # Output: 120
    ! o' V6 k7 h$ G- D  Q% t1 q- t" X0 a" ]# M' m( n# S, I- B
    How Recursion Works* p" v: S0 E* X5 U$ d9 |

    % q& g9 X1 ~) @+ Q  Z. Q    The function keeps calling itself with smaller inputs until it reaches the base case.7 u  L  D! x! R% e1 [7 z4 |
    / S2 x4 F1 x& B7 n
        Once the base case is reached, the function starts returning values back up the call stack.
    + W6 q4 V* ^/ I! l
    3 Y* J2 o7 g. ?+ y    These returned values are combined to produce the final result.
    ( k! Y1 b' n3 }% y: `* I0 k( E3 M, Z
    For factorial(5):
    6 W" _: A7 V, k9 {# D' l7 ^# Z
    1 T1 t) w0 X4 k( A5 G( m) o- U% ~+ Y6 q$ ]; B) ]- R6 M' h7 ~
    factorial(5) = 5 * factorial(4)0 t  _! X8 }: `" w% e: O7 F6 g1 I
    factorial(4) = 4 * factorial(3)
    ( Z& \; Q( _+ Z. t; L5 |factorial(3) = 3 * factorial(2)
    & W( ^1 p# \2 g' l8 o, Q% a4 t. t* [# zfactorial(2) = 2 * factorial(1)3 J6 P2 t1 P. @+ |3 T2 a
    factorial(1) = 1 * factorial(0)
    9 A( O$ _5 x% p2 ^: f8 {factorial(0) = 1  # Base case
    9 v% h' D7 Y& T4 R- C0 c+ [' s/ E+ _- |4 q# `
    Then, the results are combined:
    * s! V8 b& X6 ^1 R5 H  p* w' I3 n0 i' c5 p( m6 \) q6 F
    2 o# o& R4 ~, e: h$ u. k& H* i- B
    factorial(1) = 1 * 1 = 1) Y" b) Q7 F3 A0 N$ m
    factorial(2) = 2 * 1 = 2, K8 W9 d1 i* O$ s/ A+ }
    factorial(3) = 3 * 2 = 6& s# z" e! a1 G0 {
    factorial(4) = 4 * 6 = 24
    ; ]; x1 q- B" {/ A) U2 s) kfactorial(5) = 5 * 24 = 120
    1 b2 i& @" n9 @3 x
    9 N; u$ j' z: C0 s, e7 K9 u4 x8 K6 {1 o6 XAdvantages of Recursion
    5 m4 @- t- c# }6 X9 N, d9 f5 ~+ K6 n4 U8 ?' i5 P' [8 S
        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).9 o( O* r) Q6 v  u) K

    # K8 A# K  J9 Y- s. Y. a    Readability: Recursive code can be more readable and concise compared to iterative solutions.' P$ K/ E- H% q9 D0 ~- [! ?

    9 f' k, D* K" q: ADisadvantages of Recursion
    4 N2 W2 @) z( d) ?
    9 Q9 ?0 }. 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.0 e+ M- Y  f: d' R/ D
    9 t  ^5 v- h- s4 e# N- J
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).% M# s! N" R. ^0 ?: V: H

    : z) ]7 z/ O- i/ bWhen to Use Recursion
    9 e+ Y7 B5 u/ m  V1 E# M2 C, v+ W7 z- g% }% g6 @
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . g9 K# i9 r% c2 |; R0 x$ D
    * t" n( h! g8 E5 k5 }$ [( t    Problems with a clear base case and recursive case.7 b2 K: @% K- ^

    5 P; H8 z/ {4 i$ _7 V- M8 A7 l0 BExample: Fibonacci Sequence$ G3 u4 Z! O+ Q3 c1 ?

    - Z" z* P6 x( A; Y- B7 yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 ~. D1 F) G- E. M2 \; C1 C, \8 w& T7 f/ i, M' h
        Base case: fib(0) = 0, fib(1) = 1& N# a8 T8 \9 K3 n

    ! Z; \4 t/ A5 Z) A) Q0 i* b    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    8 N( O5 z( l  M% G- d; `% `" D5 I/ R1 x
    python
    & u4 T( W+ M: b: B+ k! o
    ( _8 d: |4 @- y4 r3 Q
    8 M4 y5 _& `  P1 S; ?' q, D6 W5 ddef fibonacci(n):
    : r$ H8 _3 ^+ m  J' |    # Base cases
    6 ]/ N6 r# k1 v) ~9 [7 k$ f: n    if n == 0:
    , ~1 y$ t$ v+ k. p  V        return 0# p: I: N" x- G8 X- Y
        elif n == 1:8 H6 r% @- k7 \+ U' j+ q& f
            return 18 Q$ Y' v; ]' L8 Y( G) a3 b2 C
        # Recursive case
    , P* k/ H! c: `% k' _0 R! ?    else:1 V- X! ]5 _' x
            return fibonacci(n - 1) + fibonacci(n - 2)
    ( G7 C" d9 N( @! T3 A5 e6 {
    5 \& I0 _# T: }& B. f; L* g: a# Example usage/ U9 o/ o; M9 M+ T
    print(fibonacci(6))  # Output: 8% O& O2 p6 m$ ^" f
    * t9 V2 d' Z* Y- V5 l2 ^# e
    Tail Recursion
      l8 q& h4 i4 G& ?
    1 e5 \6 [% T* 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).9 G$ g- Z5 G1 t2 u

    " l2 G- @; n3 E# k4 i# t  LIn 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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