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 + n- q- ?0 t1 N2 e+ Q7 X3 d; Z; }! e Y) Q" Z3 B2 M
解释的不错* q7 ~/ A( q* X' ]) R2 v0 [
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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。 ! g9 d6 v8 ]. j2 s9 ]" n, a* y/ }8 `& P; ] m1 a' _7 S
关键要素* ^9 z: t' p9 N3 P
1. **基线条件(Base Case)**: {* k9 s: U3 Q" Y9 }- v& t1 a- |
- 递归终止的条件,防止无限循环7 I* |& m9 N9 X0 t8 k6 X
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 17 A' t6 J' d/ h$ I' P" c
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2. **递归条件(Recursive Case)**# [2 ~8 ?+ _* I
- 将原问题分解为更小的子问题 ' L4 [! O1 k* P5 b# v3 v+ T+ R: [ - 例如:n! = n × (n-1)!5 G* j, C8 G6 m/ \, p6 s: H3 ?0 i' C
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经典示例:计算阶乘( P! O1 I1 V7 n8 _# n
python : V% V2 b/ U. U5 K( Q" k' |def factorial(n):' n+ i- _; v* N L
if n == 0: # 基线条件 Z7 `2 I, u' w" T& O
return 1 + j" D' r9 U/ h0 B+ ~! P8 u. C else: # 递归条件* P+ C6 B3 S" ?- P. R
return n * factorial(n-1) ) G% q2 d3 u: X8 F( m# g执行过程(以计算 3! 为例): $ _0 D: O3 [+ u( R& F! xfactorial(3) : P% E" d) U! `3 E4 F$ J: n% J3 * factorial(2)( W ~4 R9 s; k' I
3 * (2 * factorial(1))* F, S9 C/ f0 m3 u5 x6 h& `
3 * (2 * (1 * factorial(0))) 5 b8 u% ~3 z& E) g3 * (2 * (1 * 1)) = 62 O6 i: Y* s- \0 p1 ~0 R9 h% s4 K
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递归思维要点 ; ~# s1 A/ `* d0 S) P0 [8 i0 w1. **信任递归**:假设子问题已经解决,专注当前层逻辑' p4 q, \5 c. R3 x& i
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) ( F3 s& T3 ~0 ~. u H, ^( E8 F# j! g3. **递推过程**:不断向下分解问题(递) 6 G) [) ?. X- r4 ~4. **回溯过程**:组合子问题结果返回(归)" |2 A9 G& L- m- V2 U
% M6 s. \8 }+ U/ ~8 {. K 注意事项7 ]' N! u1 r, I; H6 O7 v
必须要有终止条件$ R2 p; M& E9 I7 c2 R& v1 ?. |
递归深度过大可能导致栈溢出(Python默认递归深度约1000层) + e$ A8 N+ M9 ^某些问题用递归更直观(如树遍历),但效率可能不如迭代 ! l0 ~0 X9 l0 l, _, K5 `尾递归优化可以提升效率(但Python不支持) % m( U2 X5 N: }. ?* A9 A 9 y0 z7 N8 j( \' p% \" q1 Y 递归 vs 迭代+ o. a, J. L" ]0 `9 v- ^
| | 递归 | 迭代 | 9 \8 m, B; h9 B4 H a8 l|----------|-----------------------------|------------------| + e, P( w0 E$ \% P @0 r, X| 实现方式 | 函数自调用 | 循环结构 | 0 c$ R1 |8 \0 h& n! ^+ Q2 r8 e| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |7 _* }( R( N" |9 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 |1 a6 a) y$ u7 O. v0 ], {; ~
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | : X) f8 z- X/ y- N6 ~: H) d! y) a1 w6 L: X5 D$ S3 v
经典递归应用场景 8 C4 m- B: J9 Y! C7 d- {5 Y1. 文件系统遍历(目录树结构)' W1 V' X$ c* g5 s/ |2 H( q, m
2. 快速排序/归并排序算法; S4 T- l6 J% t# ]) g" n9 C! j
3. 汉诺塔问题% Z8 q# Y2 O( }# {( l" M
4. 二叉树遍历(前序/中序/后序): N0 N/ W1 E, E0 m4 c
5. 生成所有可能的组合(回溯算法)7 t- p. A5 |6 ^9 u* T
0 e3 F; M, Z- n, ]4 F+ x" R5 Z' G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, 6 J# G( R* s( c6 {" `我推理机的核心算法应该是二叉树遍历的变种。! z B7 @" H& ?8 |+ `( Z/ n. Z
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:. W: T. C+ _' L. n0 G9 f4 r7 E2 ^: b
Key Idea of Recursion : U" \" G* L$ F( j/ Q; J- V 1 `9 L' U! k; O7 ?2 d9 `7 YA recursive function solves a problem by:5 q# @$ p/ V3 r' w5 K5 s) q+ W
/ w! Q- ~5 V. f1 n* o9 i Breaking the problem into smaller instances of the same problem.# \ g+ \& l5 W& q' v
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Solving the smallest instance directly (base case). + `* h) u8 p5 `$ B/ f ! [4 |& u- J& m( T: A Combining the results of smaller instances to solve the larger problem.$ q/ _5 P3 O# N
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Components of a Recursive Function( ~* ?" a! a$ ?: C- [; q; S G
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Base Case: " Q! c) Z. ^7 U' r1 g, g4 F: y% A% x4 S% r; B+ G0 y
This is the simplest, smallest instance of the problem that can be solved directly without further recursion.+ e+ d7 P- g7 z/ |
: T7 _1 W+ j& Q It acts as the stopping condition to prevent infinite recursion.8 o X7 Y6 r9 B6 h5 K+ U0 D
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Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* D& r5 R# H# t7 v* ^+ d
7 _/ Y2 t3 \) G& b& r) V! v Recursive Case: . P9 Y5 B, @* r a6 n+ j 6 s; D$ g2 T% X8 L This is where the function calls itself with a smaller or simpler version of the problem. # S: r$ c& G$ W5 J8 w % X, V j3 j- c/ T8 N* v I Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). 1 Q4 o( a* v4 |+ @: m1 c5 ?# P / m8 @1 P0 ^1 Y! a, |# |5 E! ^ eExample: Factorial Calculation 4 O: F7 \7 n$ A) H: a) r 6 U) f" {0 |, t; |- K) b& {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:0 B# {4 W- D( w% M! A, o; Y5 M
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Base case: 0! = 1 ; T3 Y' s+ f8 u( P0 I+ X' ]4 A9 b, C. u, x6 ^4 M
Recursive case: n! = n * (n-1)! * Z( I0 D3 v' u/ \ & W' N6 s; V$ v ZHere’s how it looks in code (Python):* ^, c4 A1 h9 \# `
python% c6 |, V% k u+ u
0 Y0 Q7 q: H7 J; c5 [) w" Y3 h N# G! e. [7 ]2 c# idef factorial(n): ( F, a) D9 i+ A" Q1 V: H # Base case* \( m4 y# d1 K! |* {0 ]9 E' ]! E1 t
if n == 0: 0 g9 v! N$ J! U3 N return 1 : N8 b5 \) E# T( a5 F) U/ Y, ` # Recursive case ! g: l5 S' |0 a, ?- t* m else: ' p" ^3 J* {4 V7 [5 n* E6 \ return n * factorial(n - 1)5 x" t4 _2 x9 R( s
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# Example usage + ~5 o3 l2 Y0 s# `print(factorial(5)) # Output: 120 9 `$ ?9 ?' ~8 G0 i' k% U$ ]$ c, S- p% c8 l
How Recursion Works+ u) [/ n/ V( x, m8 `/ p
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The function keeps calling itself with smaller inputs until it reaches the base case." n) m. Y, I' ~
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Once the base case is reached, the function starts returning values back up the call stack.5 _/ C# B4 M2 [% p
, ^6 K* ]5 s8 }- b% Y1 c These returned values are combined to produce the final result. : k, M6 c% ?; M& d2 ?% R4 x; x4 O1 I' U # l9 N; h$ y; l) s& P5 R; _0 oFor factorial(5):+ {, N9 k8 T% K7 _% w
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factorial(5) = 5 * factorial(4)4 f: c* P2 @6 ?; M: e
factorial(4) = 4 * factorial(3) 5 o! n% a$ R! pfactorial(3) = 3 * factorial(2)6 A4 T' w3 n1 J3 V6 M/ z/ o
factorial(2) = 2 * factorial(1) 3 _' X1 f9 ? E3 Pfactorial(1) = 1 * factorial(0) ) ]6 ~ l- K$ z6 V5 [+ T" ]( yfactorial(0) = 1 # Base case " R3 |& R# ?7 t. G' N1 q 0 o! @+ `" @5 {* g+ M3 y. k3 ^Then, the results are combined:0 s C6 }5 n+ K) M% Y% f( G' a$ x Z
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factorial(1) = 1 * 1 = 1 / r- c! M& b4 @4 A" Y9 y! gfactorial(2) = 2 * 1 = 2" @0 c: R4 E0 h3 Z3 @
factorial(3) = 3 * 2 = 6. {/ G& [( L, L# p3 H
factorial(4) = 4 * 6 = 24/ {1 C2 o3 a+ ]4 h# t! @+ l
factorial(5) = 5 * 24 = 120& G! o2 n" R+ K8 r) d( U
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Advantages of Recursion- K( p, O% h" ~7 n! @
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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). . E1 p, v% l) [' U+ J& [. V- R: _$ Q$ |7 E, Z) h
Readability: Recursive code can be more readable and concise compared to iterative solutions. 5 J2 m1 c; K$ y- n- P! F: h- b! c3 d
Disadvantages of Recursion4 ^$ v9 {$ [. o$ l' Q4 ^4 v
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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. 4 k; F; e: h; z# A" ` 8 N% M3 x+ p/ _4 J: b Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# ]+ n$ W% O5 f/ D# b, `
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When to Use Recursion- j& a3 ~/ X+ Z" Z: T. S6 b$ q# x. e! s
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Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).4 I" a! S- ]/ i$ w' p$ a+ g
6 [! m/ W% I. ^ f8 j, E: a Problems with a clear base case and recursive case.0 O( @1 p0 p( @- J
; W4 e- P$ E- v; Q& x" eExample: Fibonacci Sequence* h" }) V' o( D/ u: P: ^
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:5 k6 o+ M8 y+ `$ K ?1 e7 ]
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Base case: fib(0) = 0, fib(1) = 1 g; G) R; y6 `) M, E3 h; m/ ~+ g" @/ l2 C0 e0 u1 C* g8 ^2 Y
Recursive case: fib(n) = fib(n-1) + fib(n-2)6 f) u/ s5 D6 e" J7 R! D# g1 m! L3 n
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python 2 b: Q7 H. Q/ N3 w4 c# \0 y) u" V* j* K' c+ ?1 g Q
1 l; U! o# V, Y8 Y( L+ x: P5 @def fibonacci(n): 3 R$ q P1 g) ?$ r # Base cases/ I3 \- u7 j" i' _
if n == 0: ; S5 |5 _* Q$ m: t' B# l return 0/ Q$ m; N A6 e- |/ ~
elif n == 1: - @4 M X, f/ O) \# n) f return 11 U/ D3 ^2 z5 H5 m. i
# Recursive case 0 }, I& d' N+ d( K# m else: 5 |+ M1 ~: \. X7 |3 l# j5 @ return fibonacci(n - 1) + fibonacci(n - 2): p# V5 U$ ~- Z$ V8 t+ J! s1 w& `5 `
# C) {. |# \8 J% @Tail Recursion : R% B3 \7 N5 ~" S/ @* \' ]$ P 3 V/ y0 ~- T3 W% R. E$ RTail 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). 1 J x7 ~+ Z; L# b6 E8 B. G7 V1 @8 j8 t$ L1 x
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。