标题: 突然想到让deepseek来解释一下递归 [打印本页] 作者: 密银 时间: 2025-1-29 14:16 标题: 突然想到让deepseek来解释一下递归 本帖最后由 密银 于 2025-1-29 14:19 编辑 5 F% i! E' t( M$ t
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解释的不错 0 }: W. U" \# w& V% E- I" F1 S# V* l
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。: N Y! n# b. y3 I& H) |
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关键要素% u: q* L" `" B E, z1 j
1. **基线条件(Base Case)**- r* Z1 q, |( O s- N0 K/ v; ?% p0 r
- 递归终止的条件,防止无限循环3 K: c+ U$ E6 I2 @! Y1 H
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 5 G* m1 \3 z4 F& b4 I ) P6 ]2 Y( F& i) U( X6 _2. **递归条件(Recursive Case)** 4 }0 u; C' r- ]2 K8 m6 t - 将原问题分解为更小的子问题2 `% f6 ~' X) R
- 例如:n! = n × (n-1)!2 Y, v. m: Q7 X: a4 C& h Z5 M
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经典示例:计算阶乘 7 G9 {$ z# z' t0 ], rpython 6 }' J$ K) \7 H( p: U: Odef factorial(n):1 p) w+ D' _# `9 U& u5 Q4 k2 E
if n == 0: # 基线条件 R9 w' j G D5 P4 ~( p
return 1 , o9 Q$ \. _; r9 d else: # 递归条件 4 k# E; m3 {1 ?& h return n * factorial(n-1)9 u2 ^( M( _+ N) E7 k
执行过程(以计算 3! 为例): + f( f3 E0 t& p! V4 |9 efactorial(3). c) D0 ?2 ~( Z5 W6 s
3 * factorial(2) * \/ q' q2 w; t8 e3 * (2 * factorial(1))' _; q/ i3 k& o& K# t$ g' N' R u( O
3 * (2 * (1 * factorial(0))) 1 e$ X( |4 t5 o% B1 v8 ~+ z- C3 * (2 * (1 * 1)) = 6 : B( T' Q5 D' L; L: S# l- h) i! H ; v) U& l( O* Q* k, Q 递归思维要点 ( ~ C- n, b6 P1 u/ z" e1 z; z1. **信任递归**:假设子问题已经解决,专注当前层逻辑 ; y+ z) G9 o( o! ]0 `2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 G3 X7 z1 t+ L
3. **递推过程**:不断向下分解问题(递) ; s, k2 p+ d8 C4 C4. **回溯过程**:组合子问题结果返回(归)/ p' y* i. ~ |6 H3 y
3 w. V- h5 v9 Y# p: x 注意事项2 X. K3 V' H# Q2 i: u
必须要有终止条件3 ~' z p# e5 {& H4 _6 @
递归深度过大可能导致栈溢出(Python默认递归深度约1000层) + N* ^3 A, r( J4 `" D7 m* Q) v某些问题用递归更直观(如树遍历),但效率可能不如迭代 - J4 n$ f* m2 d' g) G尾递归优化可以提升效率(但Python不支持)8 \- B1 X. _) A5 E. p
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递归 vs 迭代6 `, }9 A6 X/ ~" c
| | 递归 | 迭代 |1 Z, J& {0 C4 L& W3 A
|----------|-----------------------------|------------------|; u1 G* @! |1 m+ x! V: h
| 实现方式 | 函数自调用 | 循环结构 | $ N0 ~- }6 \( \6 r) R| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | ' ^5 n# K7 q, Y3 K5 i| 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 |& u$ t" h! n+ S% `# `& P, n
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | 2 b3 y& V# N' R- R! A( T5 q$ m% l% S
经典递归应用场景2 J3 c/ s# u8 j* [' J
1. 文件系统遍历(目录树结构)3 c+ K* C' |! d) Y
2. 快速排序/归并排序算法 3 t1 M0 @3 E6 s$ |4 u3. 汉诺塔问题 Q6 m" ^: z9 E1 ]8 x0 ]( A4. 二叉树遍历(前序/中序/后序) 5 j3 |5 |1 V4 q# f& d5. 生成所有可能的组合(回溯算法) 4 k! o4 {, H0 a2 ~& f. X" T: Y( |" E$ d" t) V, M: P
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。作者: testjhy 时间: 2025-1-30 00:07
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, 3 B. k T" d$ ^& h* E7 q- }, X我推理机的核心算法应该是二叉树遍历的变种。 - F' S5 t1 P" B- j4 [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。作者: nanimarcus 时间: 2025-2-2 00:45
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: * l5 {, ? @2 }+ s: i: m2 }Key Idea of Recursion2 d4 k+ l( K) I; e8 z
# t5 f9 |# D5 p! ?% hA recursive function solves a problem by:( _( f9 P( g' Q4 g/ y8 M
: X' ?3 i9 T4 w6 R Breaking the problem into smaller instances of the same problem. N3 C! ^! g: M* G \- W. R6 H
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Solving the smallest instance directly (base case).' L1 ~" T) _; c( D/ v- h
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Combining the results of smaller instances to solve the larger problem. : R$ Z! Q# c* j; @& m 1 Q2 d- `( R$ Y5 |5 w7 q; lComponents of a Recursive Function ; J) q: K* V+ X& X/ f* \ 5 v! I/ p V9 ^# a8 M- V. k Base Case: 0 T9 C: g- I% O# ]* h. u5 U+ V p0 U8 \
This is the simplest, smallest instance of the problem that can be solved directly without further recursion.) L$ P T G. m) D7 k
9 L0 [8 m7 z5 M; V2 { It acts as the stopping condition to prevent infinite recursion. 4 h7 X+ W: `$ G- q/ D. J. P( I# q! E1 K f1 Z1 W4 G
Example: In calculating the factorial of a number, the base case is factorial(0) = 1." q6 n7 A2 `* a
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Recursive Case:6 E% ]% M, |9 m: m
4 g# N" R- i+ |) R; }/ }) n2 g' J1 i This is where the function calls itself with a smaller or simpler version of the problem. ; v: S! \# r, l1 g4 d+ v* L" p6 S* I% }) U9 P/ K1 V( H' m& W% }2 m
Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).- G, L* W5 d+ ~: S) P/ v9 T
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Example: Factorial Calculation . b3 a: a. T& o2 B- R8 Q7 @$ @# Y/ p
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:) X% f) M8 v6 W- g1 _* ]
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Base case: 0! = 11 _8 Y7 e3 t# G8 ^, i* w6 @
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Recursive case: n! = n * (n-1)! 2 W: ?/ s4 R. q* d" s' E1 u: _/ d* {, w9 g2 R& ~
Here’s how it looks in code (Python): ~" C$ p) K% z
python ! I" k4 S y* S+ j! A1 V& p% G- w/ q7 g1 b4 h) H0 G% f5 v
0 \- G2 A2 _( G3 W) K+ x; Ldef factorial(n): 0 p8 {) a; P& B- c) f/ m! M # Base case$ L E! f, V* ^9 \& U! M8 \# U% j. b( I
if n == 0: 3 E/ @" p) G2 w" @6 e% s return 1 ( f: O5 B) S/ z. ? # Recursive case 1 i. n: w; ]0 U' T- v# X+ f else: ; O- Q( Q, _5 \ return n * factorial(n - 1) 9 h$ P% u6 ]6 k 7 b$ r1 F( v9 z/ r) Y% e( m# Example usage 0 U) X5 n- j0 S2 t0 ?- R* C+ q0 O! Lprint(factorial(5)) # Output: 120) N; `( R/ K# t' {
8 c" ]- n% z. P& W" W0 LHow Recursion Works , o) @4 Y1 D7 x- U+ Q9 Z" V! Z3 ~& U4 r, Z, g
The function keeps calling itself with smaller inputs until it reaches the base case. ' x" K! d# x$ D" c9 e6 ^ 2 ?6 i- i, O% x3 [+ @8 | Once the base case is reached, the function starts returning values back up the call stack.9 U s# J1 q$ t+ y% A
8 d# Y) V W$ E3 | These returned values are combined to produce the final result.! N5 D1 i7 Y ?# v% x# A# C1 D
" A- P. {" r" t% t$ j6 aFor factorial(5):4 k3 ~) h% s' [- A e
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factorial(5) = 5 * factorial(4) ! P! Z4 A7 |# k Q. {0 N9 o; P _factorial(4) = 4 * factorial(3) / R8 Y0 J+ K- u3 b+ O! g6 C ]factorial(3) = 3 * factorial(2) 2 ]2 B ~. j' N( Qfactorial(2) = 2 * factorial(1) * A' p0 ^! {2 |' C3 _5 ]0 M# lfactorial(1) = 1 * factorial(0)8 {' a# ?* K+ }+ G0 C
factorial(0) = 1 # Base case0 k% g4 e1 {% _' `. n1 Y
- v/ O& P0 K# m+ sThen, the results are combined:* V3 n3 \! S& ~1 V' K Y" L4 x
! W9 i- C9 g" d7 ] & ` A: ~& J$ h7 Z) O" u) v$ T) f' y: ]factorial(1) = 1 * 1 = 1+ }: D( G" Z$ Z( z
factorial(2) = 2 * 1 = 2/ `' z/ o8 _7 y- ^1 _& K
factorial(3) = 3 * 2 = 68 }/ P6 O2 o- s
factorial(4) = 4 * 6 = 242 `" e4 o- @9 X& W% p# \# E
factorial(5) = 5 * 24 = 120 2 [; h: n/ g! I0 G! w, m9 ]; h; S Y" B: O' i9 QAdvantages of Recursion 5 ^/ ]; X0 [& d9 g2 h q" g / L6 R. B t6 ?: x 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).+ K0 a8 I" A6 m4 L8 R# W
2 h/ \* X* ~# f, Q& S. ^' P Readability: Recursive code can be more readable and concise compared to iterative solutions. 2 c6 ~4 r( J D& ] - e7 V! |2 F0 o5 u( ?& KDisadvantages of Recursion* K' Q% V& s" d' ^9 G* c+ N1 f# {8 ~
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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." @# l4 g' V& V' h! S$ S
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Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). t6 `+ `$ `" B/ a0 Z5 a
: v: D. E+ |- H$ gWhen to Use Recursion" Z+ B# [6 V/ k+ G- x, L
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Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). 1 z5 @; S9 U% ~/ p , i8 i. r" Q3 Z# z. r2 d$ z Problems with a clear base case and recursive case. ' g, C, H! z& b. H & a7 ?1 k. H/ AExample: Fibonacci Sequence ) Z# n+ m9 [: X/ `. ] 0 o* A0 `- `; ZThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:( R8 [$ f* O5 m9 d) a o9 E; }' {, s
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Base case: fib(0) = 0, fib(1) = 1 ) p8 o( W) Y0 W0 t# j A: T$ W& }( v9 l n9 W4 q" h4 ]% ]
Recursive case: fib(n) = fib(n-1) + fib(n-2)+ S0 L N1 r& c# A8 K& x: Q: F& b# D
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def fibonacci(n): % S$ K1 W2 @9 y0 A& R: L6 D # Base cases # [8 T5 i! P/ r' v4 Y. c, H% { if n == 0: % F3 q- l5 p" x$ m# P% x: n' B return 0- t0 |* F# n/ [: K
elif n == 1: 8 y0 {2 c3 W. w' }* o4 G return 1% [# V0 `( w1 c: x( R
# Recursive case4 G8 k1 w/ i0 P1 c% [
else: , ~9 l P; L r/ t) k" a: D: L return fibonacci(n - 1) + fibonacci(n - 2) 2 O$ O5 k( ?% w) i6 {$ A6 ~2 E8 ^0 F, i# w- A7 H
# Example usage 5 y L( w* S1 X: }" i2 Jprint(fibonacci(6)) # Output: 8' d; c h. D) p6 Z9 i
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Tail Recursion * a4 |, R ?) V- g! L# i$ W/ s9 N4 l& k3 k: e! c5 x
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). K9 j, m" c( ?1 v
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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.作者: nanimarcus 时间: 2025-2-2 00:47
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。