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 ' f6 W( U9 e8 G 0 c" B$ P* s( h解释的不错 0 Q" J# }& E# V/ t3 s7 n, O+ r7 F. N* A$ u0 l' Z
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。 # E; P3 x. _6 e0 |0 C# T: Y% R7 \" G+ I
关键要素 0 @ V5 e+ x2 R% }& y( s$ e1. **基线条件(Base Case)**" ~5 h; M$ J9 Z4 r" K" W
- 递归终止的条件,防止无限循环1 C' q* g4 i4 U: g+ I0 u' D& X% N
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 ( G( Y# r) _: Y# ^& P 3 C5 e0 n) @9 H2 N& P2. **递归条件(Recursive Case)**5 j# N7 Y- ^( d' c/ H3 z* @
- 将原问题分解为更小的子问题 1 O% Y0 k# N( H2 b+ L - 例如:n! = n × (n-1)! 3 u M& f! Z' j. c9 ? j/ h2 h 5 f1 i& U) K% { 经典示例:计算阶乘, q8 p% V7 i' D3 P+ S& |
python . [1 O9 E$ M; O1 k( W Z( l* a2 Odef factorial(n): k: Z3 ^& a- m. y. z9 u
if n == 0: # 基线条件. s1 @8 F0 t" g. D
return 1) q/ s0 z3 k3 t5 \% s7 p7 {
else: # 递归条件! c4 P1 |: g8 h$ s
return n * factorial(n-1) 5 h1 I+ N/ \; {' @3 |/ z* s执行过程(以计算 3! 为例): % y k; Z k( \& L8 U& _factorial(3) m9 W% k7 d, c8 r# _8 z& I
3 * factorial(2)0 [' y# P4 [) H+ G% U
3 * (2 * factorial(1)) ; E( `6 t; ~: X3 U3 * (2 * (1 * factorial(0)))5 U8 @4 X& T5 F0 Q5 S
3 * (2 * (1 * 1)) = 6 ! Y9 l0 \7 [# J; o+ ^4 b5 x- S% T5 b+ a
递归思维要点. }+ R* T0 Y9 e0 Z7 l' m; D: t
1. **信任递归**:假设子问题已经解决,专注当前层逻辑& b: x9 q: a+ R0 V J
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) % w9 ]" ?% c% K6 K5 C! E3. **递推过程**:不断向下分解问题(递) ; B8 r8 r: @0 c) d: n4. **回溯过程**:组合子问题结果返回(归)) w+ W2 K3 E# d
& K5 c5 b! S. w' w. b" A' o4 H3 B
注意事项 2 {- A% m9 `' ~' e! b: b/ W必须要有终止条件6 b# Q# B; j8 g$ I
递归深度过大可能导致栈溢出(Python默认递归深度约1000层) - d# k$ h0 r3 D& g( t+ g: Z: T某些问题用递归更直观(如树遍历),但效率可能不如迭代5 i* j; i0 U) S( N9 t( E4 |) b1 g
尾递归优化可以提升效率(但Python不支持) * G9 _: a3 w% S " f" X% w! T+ e9 j" v. L 递归 vs 迭代. b% r* l L) Z/ A$ I
| | 递归 | 迭代 | 0 a: k" i/ K6 M" W' r|----------|-----------------------------|------------------| . ?4 T8 Q- y0 g: h, S6 X! `& Q| 实现方式 | 函数自调用 | 循环结构 |" X6 M! Z" p7 `/ i6 t" C
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |% l& l$ F. o3 z% F" p3 b) e( V' D
| 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 |6 g* m; _* v. P4 s
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |* A4 P1 r$ p7 A2 w
* R: X- x8 I$ U
经典递归应用场景) ~; T7 x8 n0 {+ F- @5 ^5 b
1. 文件系统遍历(目录树结构) 2 w8 ^& _" Y4 D. g+ F2. 快速排序/归并排序算法 q" b. Q( A! E% c: p+ m7 d" t
3. 汉诺塔问题* z8 C, T2 B& F* |
4. 二叉树遍历(前序/中序/后序), ?8 Y# d W3 \
5. 生成所有可能的组合(回溯算法): k$ X) a7 \3 s4 p3 h! [4 N+ K
8 O; @+ C& K$ `8 _5 g6 B
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,/ ]* L- F- y( N+ e$ Q( j! k0 P
我推理机的核心算法应该是二叉树遍历的变种。 4 U# {( C- J8 D: `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:$ ]2 H$ x! R9 b9 m4 g
Key Idea of Recursion _1 m4 ^6 E7 d 7 _, d7 @) ^* c5 [# h5 d) vA recursive function solves a problem by: ) d0 A, E" n! Z" _3 P M. G3 W$ S5 l3 `& Y" A Breaking the problem into smaller instances of the same problem.' A6 T% `/ b" m2 ]) X: d
- O; L2 Q, b- w" ^8 @ Solving the smallest instance directly (base case). 0 D O$ l- h- W2 |2 u9 X, M1 b2 `* ?
Combining the results of smaller instances to solve the larger problem. 8 D0 y% v. ]( u, Y & P) C& j" y! O5 H; k; e5 C7 NComponents of a Recursive Function 9 {1 l$ G. E+ V; ~1 m8 {4 }: F # L& H! s M/ b0 h Base Case: ! k: J2 ?, t1 L7 X1 }' [1 K/ g1 ?4 `/ b _& B
This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* G2 T9 N) u. I" t4 u1 E. s
2 r. [- C" O. d N0 ?" k. j
It acts as the stopping condition to prevent infinite recursion./ H. C' Y# G7 d
* L' O+ |+ U0 \4 B7 D- Y1 Y, N
Example: In calculating the factorial of a number, the base case is factorial(0) = 1.. J# N: i& J+ a4 l" ` M
# {( f1 U& ]9 I+ Z* } Recursive Case:9 ] ^: U8 P$ w6 I! S; M$ H
* n1 k& Y7 P7 Z, @
This is where the function calls itself with a smaller or simpler version of the problem.8 v# H p6 W: C
0 ]% ?1 m2 _/ h9 b+ M. @ N
Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).$ c: a: v: a& x6 f0 h
1 b4 r" m, W& t. y. G5 O6 Z& hExample: Factorial Calculation # F% U! ~ k( k7 c) ]; f8 r6 H! ?! l% f" h
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: # G3 R" N" w3 s5 O6 N, w " W" Y+ V9 u8 m" _, J Base case: 0! = 1 ; b$ x4 ~6 {) z ~# u! t: m/ ]% _0 g/ x# ^& [8 E% V% A" }
Recursive case: n! = n * (n-1)!% x" d: t# N# Q$ z9 r* [; F
7 w- G. q8 a1 w5 ?
Here’s how it looks in code (Python): % e, M; B& {: Npython! I( c. W- t# C6 Y% p3 }2 y2 w+ i0 f
; E! l( i; x1 i& C- [ 0 W( O& V9 n9 v) J0 Pdef factorial(n):+ p7 u0 u6 u; b+ q+ d' _
# Base case+ _8 e/ Z8 X `* @
if n == 0:/ J7 s G3 ]7 R
return 1. J% q8 h: b6 q0 f& o7 c8 K
# Recursive case( T3 _( V0 q$ c
else: / [1 }! Z' D0 c) Q. P7 m return n * factorial(n - 1)' W- q2 E" @) K/ k
7 v8 O' _9 ?3 x: D
# Example usage - X/ r" t: p$ z1 j4 @7 y3 Uprint(factorial(5)) # Output: 120; d0 _! R7 h% ~8 P
4 s. O& q9 _( ]9 p
How Recursion Works' T" t8 q* i: D2 J
& e1 j$ Z9 m8 E: r
The function keeps calling itself with smaller inputs until it reaches the base case. 2 i. h+ y- R. M- c, @6 R7 A+ X* d" e. N9 m
Once the base case is reached, the function starts returning values back up the call stack.9 O" v, Y" v* O+ Y- y
8 @+ V" }5 t' ~ These returned values are combined to produce the final result. 4 J4 j' X. e" ]5 V7 T' o% l & T( R% e/ q" D4 p& n4 | ?For factorial(5):8 N- C4 \2 j% h! t
9 x J) @( _5 L* e# U
$ h" e. x% R! A3 S
factorial(5) = 5 * factorial(4) / s( y0 E& s( G8 afactorial(4) = 4 * factorial(3) 5 U P; u% r" U( b$ e/ _; bfactorial(3) = 3 * factorial(2) , j3 H! I! T3 |factorial(2) = 2 * factorial(1)7 m* L* J! w1 B8 M! X" ]. ~
factorial(1) = 1 * factorial(0)1 e& z' D* s4 W
factorial(0) = 1 # Base case : k8 s& T4 r8 B2 w$ z* ^$ q8 v9 p8 }9 I; K% b0 R+ I
Then, the results are combined: s" Z: J2 c% \' A- C% Y$ |3 J; z) Y- T% ?* P
' i& v9 g' I; l- S1 _7 d& o6 t
factorial(1) = 1 * 1 = 1: a% S* |) `# p# z: _6 C
factorial(2) = 2 * 1 = 2 e* H: l8 v6 Z3 ?# ]1 H: q$ d2 @$ Sfactorial(3) = 3 * 2 = 6; F) o* x# ^1 V* k, _
factorial(4) = 4 * 6 = 24 2 U. O( n' C* ?8 }3 s( Y9 tfactorial(5) = 5 * 24 = 120 ' Q% q0 P$ t5 H! E, W9 c4 h/ i$ n$ J X% @ U" L+ v3 X, p3 J0 lAdvantages of Recursion$ p6 d8 r" `- S- v8 a$ q' Q
G* N+ S- P5 P4 | 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). & B |0 R- I5 ^* B5 U. N5 K2 B2 X . D$ D a# d, s& I+ w4 s2 R Readability: Recursive code can be more readable and concise compared to iterative solutions.: M ^" ~7 T S
2 B! F( Y- m! B! x% a0 v3 HDisadvantages of Recursion- M; {) t) b" U! \1 H r, N
. r$ h6 A3 j2 `. i2 B0 f% B6 E
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.) V; W1 W: a2 q: j9 X! _
' O; E' p7 ^; z9 [& I& C9 p Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).% w* W! s7 F8 E. r8 \. {1 f x, y
1 D' X. M! I( o8 L4 f" x4 IWhen to Use Recursion 1 Y# p* d) F [# X! X+ z ' v& d6 N+ @/ r# O Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). . L3 w% ~8 B0 A; u* n / ^( g8 K# `4 i: _% Q Problems with a clear base case and recursive case.$ R% I6 w8 M0 T! ]3 R/ J2 K, a7 i) N
\+ Q# l' j- L) d9 d
Example: Fibonacci Sequence2 ^: t( x# J! f
' k) ?7 {7 M6 |3 w! q
The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: 4 X) E; Q4 P+ N$ e9 m& M ; w2 W* O# Q& l, B% B/ { Base case: fib(0) = 0, fib(1) = 1- C. ~8 T) E% `& n4 R
/ b0 F) i Y4 h. O( @
Recursive case: fib(n) = fib(n-1) + fib(n-2) & x1 \4 o0 ~1 ~0 n& B1 U7 q ! z9 M1 }! z1 r( xpython' q; @: v) N( {. f& k2 g
* m/ p0 a u! z' i) y f
! d9 i* F' @- V/ Y: i. Odef fibonacci(n): ) P2 G C! z" a1 V( [) Y # Base cases! _, f% N. ]# h, t/ g
if n == 0:0 J7 S9 ^: H0 P) ~
return 0 . ^! W3 j. _' X% G9 g- {; E elif n == 1: W6 F2 d9 a$ M5 c/ x5 S8 G3 R. L+ u return 1$ O" L* @" |8 A8 ]- }3 B
# Recursive case% q9 V4 P: y# L$ i( ]
else:0 `% N2 }4 ^/ L! z6 @" e* u+ F* J
return fibonacci(n - 1) + fibonacci(n - 2)0 J4 s3 l7 g* z$ S5 _
6 r2 c# D( s4 u) y# h; n) Q# Example usage # F# P. `5 w9 }' @2 }print(fibonacci(6)) # Output: 89 o( I( b& k8 b
0 M0 s# I# n- s9 A' ?$ r9 H( u
Tail Recursion / l. Y2 Q) o D& ^: E r9 L . W. C8 T1 `' s' i2 L) ]; O* d0 dTail 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).+ R" q$ T* h7 E% c8 o7 c
* o5 @1 f" u9 t8 oIn 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。