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 2 j$ Q; Z+ T4 y% f% f1 L% K! t8 h1 x: p4 B/ e
解释的不错9 a) r- B, X& G) W- U1 m) b; F
% [; D" v; L8 C9 D M. J% N: F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。6 m, @7 B, l- h: `0 k4 q- M/ K9 `
$ Q" S4 I6 g6 u2 `! h' P- a 关键要素 3 _9 o/ _' E7 p) z$ C; R1 V1. **基线条件(Base Case)** / Z$ `- P4 M& \' p' q/ K7 I% U - 递归终止的条件,防止无限循环 9 v z& ~% ?: S' p9 ^, B8 n - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 8 i' L/ [, j" u' z - D* I/ U: m! T" Z2. **递归条件(Recursive Case)** ; d( Y& ?8 e% B3 Q. \9 b3 \, D - 将原问题分解为更小的子问题5 I. f) c1 V* S0 D! M) |+ q
- 例如:n! = n × (n-1)!+ G, b3 T+ J# [" _
j4 G* w2 L! m7 D# I6 z 经典示例:计算阶乘+ `4 V4 S" t, [" b m) ^
python * S% L# P- M3 Bdef factorial(n):! I9 n. j m% n. d) o! Q( `$ O
if n == 0: # 基线条件. _. D) o- o+ i \) g7 S4 i
return 19 ?1 g# E- g2 l( u( y
else: # 递归条件 ; n. X' t$ o) c! `5 B2 g return n * factorial(n-1)( }) N8 n& n/ }& R" C8 V) O
执行过程(以计算 3! 为例):4 |: g( O* I4 q. H
factorial(3) ) w! ^& L: X3 c1 i5 S( q. i3 * factorial(2); m6 b! i) h0 y, j. \
3 * (2 * factorial(1)) w" M! m* j8 |1 b$ F3 * (2 * (1 * factorial(0)))& m. }; @8 K: _" \5 e1 w
3 * (2 * (1 * 1)) = 6 % @( F( {6 M; m9 p' s5 g* B# V, j1 f- ~2 Z
递归思维要点 # S" l: o7 R# T" s# ?1. **信任递归**:假设子问题已经解决,专注当前层逻辑 . E! F$ T- f- g2 C" d! `- a2. **栈结构**:每次调用都会创建新的栈帧(内存空间) / A4 T- p2 T9 X- }% F4 {2 Z3. **递推过程**:不断向下分解问题(递)$ E; C- {# @4 X9 R. e& y, C% z
4. **回溯过程**:组合子问题结果返回(归)/ n5 N8 p" S$ _1 g
# B- H- r0 y4 k3 Z 注意事项 + ~$ @3 Y8 H' \9 X9 _必须要有终止条件 $ X6 Q( t l0 I& q V( p! \递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 v, ]7 \) I s! D. u' ~
某些问题用递归更直观(如树遍历),但效率可能不如迭代, e$ ]2 R5 c) ~4 y
尾递归优化可以提升效率(但Python不支持)1 R6 ?* l2 J, A/ l& v- g$ p7 g9 T) }0 V) L
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递归 vs 迭代1 b! b5 C2 t2 S; I
| | 递归 | 迭代 |/ c/ N: N% Q4 c* P
|----------|-----------------------------|------------------|* A* C$ ~9 p( z" Q
| 实现方式 | 函数自调用 | 循环结构 |4 Q* K2 \. c- k5 E' U
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |! _* u6 F: S/ x
| 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 |& p4 `* [1 k5 n$ x5 | w* u' P
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | 5 K8 B+ c' C7 x 9 F/ r2 W* [$ ]) j, { 经典递归应用场景 ' b3 Z. O4 W V1. 文件系统遍历(目录树结构) 0 V" x) z8 m: P5 q2. 快速排序/归并排序算法 3 T+ I8 O; x1 B2 m% y' L3. 汉诺塔问题 3 ?$ h- d I/ Q, T; H% P4. 二叉树遍历(前序/中序/后序)% h+ _3 V6 N3 w0 x# I1 x/ ^2 {
5. 生成所有可能的组合(回溯算法) . n9 o+ E3 q0 W! m3 O- s; U4 s( v5 D; k2 N" o, ^ ^
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,2 ?2 y' `4 _6 X7 }' I1 E
我推理机的核心算法应该是二叉树遍历的变种。/ O+ [* j+ `. O2 ^* y0 c
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: 5 Y5 t) r4 N8 U, \2 O2 C x4 e v9 HKey Idea of Recursion! {8 x; d6 ~ _7 q2 G+ A
- Q1 K9 U. D* v: F8 Y9 k& ?3 KA recursive function solves a problem by:7 K, H( u" t% v, T* f# ^( O! @( r/ S
: k& I0 ~/ z4 u% {1 Q Breaking the problem into smaller instances of the same problem. 1 j# I% I0 W' N n& _) I7 |2 M, E) f& o0 z. k
Solving the smallest instance directly (base case). 7 f2 h% H' d) K9 p2 J* J- j2 |" [* R" n, H. [
Combining the results of smaller instances to solve the larger problem.9 u( p8 i4 Q7 o, i* @. O
, W& z/ f' j3 j/ J- |Components of a Recursive Function- s G9 n0 {' [
/ S2 ?6 u* N% w! W" @ Base Case:" j# B2 l, |5 t. w# Y" e3 K
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This is the simplest, smallest instance of the problem that can be solved directly without further recursion. . ~6 N2 A1 x9 p. M% N; f+ K; b; D6 _8 g1 b: D
It acts as the stopping condition to prevent infinite recursion. " v* G7 T$ k# S8 L0 ^9 T) h6 t: E / O; E: N* y& T/ G8 E7 }6 e N Example: In calculating the factorial of a number, the base case is factorial(0) = 1. : ]0 D9 B( c* F* K$ R3 M- Y+ u " Y8 r6 ?0 D: ?) k% M Recursive Case: 9 `0 T3 b b/ y! Y& w' `: Y. k/ S7 |/ _6 L5 k1 ]8 Z0 d
This is where the function calls itself with a smaller or simpler version of the problem./ f' T8 S' L7 r' ?2 F/ w2 `
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Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., h+ L3 v' J$ c( _ D7 j) v
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Example: Factorial Calculation# x$ g* h7 p( H$ I
9 E1 G/ T; U" M, n3 aThe 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: + Q; e, I; f% t; s; u5 l( G- `2 X8 Y9 }
Base case: 0! = 1; a0 Z& B' A9 t1 k/ J: r
0 f0 j0 c5 |' ^( s1 ` Recursive case: n! = n * (n-1)! ) O) y. l! ^. E: y' _& x J) q( V- `' Q2 u% U! \
Here’s how it looks in code (Python): ) o; v. w1 I$ xpython2 V. ]+ {& b6 L9 x
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def factorial(n):5 U# C+ b4 L6 W: q
# Base case ' c) p8 ]( M/ `- c) p( w2 v# R' p if n == 0:/ Z3 \& \/ z* Q6 K8 d7 r
return 13 y" \# e! c/ |* E% T" P- Z+ H
# Recursive case R. V& @. q7 d, u; P
else:% y4 _/ {( T0 z( G* Q8 i" N! @
return n * factorial(n - 1) / @/ a1 b8 }8 d; {/ r: G( Y" D! B7 s7 s; L0 e) ]
# Example usage $ k) R; u6 x. B, R% u, tprint(factorial(5)) # Output: 120) v/ h2 g! _* `3 l8 z: |
+ \8 V4 ~; x% r5 D& L
How Recursion Works 6 S/ ?8 [6 d6 Z$ X! }0 r) r 8 W* J4 l- @+ Q, w The function keeps calling itself with smaller inputs until it reaches the base case., r& I3 t/ I* V0 i. B4 H
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Once the base case is reached, the function starts returning values back up the call stack. 0 m, s9 h8 t- [, e * O; \5 Q1 f- U+ }; L; c These returned values are combined to produce the final result. ; A) M$ B. n& i! h3 l, o! n/ r o( ~
For factorial(5): 0 _; G! U$ Z+ f i- }# m' ^8 b" X) |5 W4 J% ]3 M9 i& g* J |
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factorial(5) = 5 * factorial(4) , f+ P: f" ?- ~7 I; x# E+ f" ^factorial(4) = 4 * factorial(3)6 \6 R# m' Z, _! p1 J
factorial(3) = 3 * factorial(2) 6 a, L5 T t% Z2 f3 Hfactorial(2) = 2 * factorial(1)& \4 I9 i2 ~/ k- ^/ a* s3 y
factorial(1) = 1 * factorial(0)2 [3 ^3 b( ~0 ~6 [; l. f
factorial(0) = 1 # Base case) d, H D, |. B! G5 P" j) z
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Then, the results are combined:! O$ V6 n# e5 J4 V) a
1 t1 G: j: ?/ U- l7 g" tAdvantages of Recursion2 ]" n/ {) b; o T7 t5 p1 w2 X, I
2 A6 P3 U- e1 V/ ~( o 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 b8 i# s# m1 Z" f* |2 \0 K ; Z3 x0 A( B: [- d- r Readability: Recursive code can be more readable and concise compared to iterative solutions. 6 c# D, O9 {9 z% @5 S5 w+ u A; \$ L' K( L: g
Disadvantages of Recursion $ b E' Y7 R) c$ v1 a! L9 K& j! E* _& K1 q8 o( ?
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. E0 m2 x5 ^+ P! R
5 z3 P& u J7 N1 w* w# ] Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).6 _: h5 U' F' F. E
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When to Use Recursion L- m. I9 F v) @4 Q1 ?& m6 x
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Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). 2 W9 \$ a4 K$ p; e: d/ y n D5 S3 H$ L: I6 X( | Problems with a clear base case and recursive case.5 O8 v, i# g- K0 J
3 I& A) [; @& IThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: ; g# e" @$ v E . k* W, Y( |! `& u Base case: fib(0) = 0, fib(1) = 1 # N, G* r+ {2 J% n% |1 S- X$ e6 P# B- N0 q! q) g2 q
Recursive case: fib(n) = fib(n-1) + fib(n-2)3 Q6 E/ d% e8 {( m% i6 d
- b7 a: k& |4 [5 q0 F6 cpython6 _0 A! x2 c b0 |
0 }% F2 m5 j! t2 a& B/ f! B5 h 7 |; P8 w ` e( y# Ddef fibonacci(n): ( |2 i. a( [/ g1 [8 c3 U G+ T # Base cases + L! {8 K# G6 Z" e( x* i if n == 0: + b# \! F/ }8 K return 0 8 L$ r J. p* i1 @ elif n == 1:1 o9 {+ P& ~. H3 N8 _+ D, u* O9 l" t
return 1# ?0 q. f7 c9 U1 X; k
# Recursive case ' D N; _7 i6 G z3 Z6 Y else:/ e5 G1 }0 C8 W& W
return fibonacci(n - 1) + fibonacci(n - 2) " E9 k) w% J2 N2 s& T9 M( p. y' ] r2 L8 A8 u
# Example usage3 g) ]( v& @5 Y6 a
print(fibonacci(6)) # Output: 8 , D1 D Y' u5 O; a) Z7 G$ D: g! w/ @! T6 W
Tail Recursion 1 H' P9 v1 W, M" V " U! |7 }: g; {3 G/ l: P9 iTail 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).+ X6 k6 F1 X- z$ f
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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。